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Lemmas/ImoSteps.lean

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+-- This module serves as the root of the `ImoSteps` library.
+-- Import modules here that should be built as part of the library.
+import ImoSteps.Basic

Lemmas/imo_1959_p1_lemmas.lean

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+import Mathlib
+set_option linter.unusedVariables.analyzeTactics true
+
+open Nat
+
+lemma imo_1959_p1_1
+  (n : ℕ) :
+  Nat.gcd (21 * n + 4) (14 * n + 3) = Nat.gcd (7 * n + 1) (14 * n + 3) := by
+  have g₀: (21 * n + 4) = (7*n + 1) + 1 * (14 * n + 3) := by linarith
+  rw [g₀]
+  exact gcd_add_mul_right_left (7 * n + 1) (14 * n + 3) 1
+
+
+lemma imo_1959_p1_2
+  (n : ℕ) :
+  Nat.gcd (7 * n + 1) (14 * n + 3) = Nat.gcd (7 * n + 1) 1 := by
+  have g₁: 14 * n + 3 = (7 * n + 1) * 2 + 1 := by linarith
+  rw [g₁]
+  exact gcd_mul_left_add_right (7 * n + 1) 1 2
+
+
+lemma imo_1959_p1_3
+  (n : ℕ) :
+  Nat.gcd (7 * n + 1) 1 = 1 := by
+  exact Nat.gcd_one_right (7 * n + 1)
+
+
+lemma imo_1959_p1_4
+  (n : ℕ)
+  (h₁ : Nat.gcd (21 * n + 4) (14 * n + 3) = Nat.gcd (7 * n + 1) (14 * n + 3))
+  (h₂ : Nat.gcd (7 * n + 1) (14 * n + 3) = Nat.gcd (7 * n + 1) 1)
+  (h₃ : Nat.gcd (7 * n + 1) 1 = 1) :
+  Nat.gcd (21 * n + 4) (14 * n + 3) = 1 := by
+  rw [← h₃, ← h₂, ← h₁]

Lemmas/imo_1960_p2_lemmas.lean

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+import Mathlib
+set_option linter.unusedVariables.analyzeTactics true
+
+open Real
+
+theorem imo_1960_p2_1
+  (x : ℝ)
+  (h₀ : 0 ≤ 1 + 2 * x)
+  -- (h₁ : (1 - Real.sqrt (1 + 2 * x))^2 ≠ 0)
+  -- (h₂ : (4 * x^2) / (1 - Real.sqrt (1 + 2*x))^2 < 2*x + 9)
+  (h₃ : 7 * x ≤ -(7/4)) :
+  x ^ 3 + x ^ 2 * (2 / 5) ≤ (15/400) ∧ x / 16 + 3 / 160 ≤ (5/100) * x ^ 2 := by
+  have h₄: -(1/2) ≤ x := by linarith
+  have h₅: x ≤ -(1/4) := by linarith
+  have h₆: x ^ 2 ≤ (-(1 / 2)) ^ 2 := by
+    refine sq_le_sq.mpr ?_
+    norm_num
+    have h₆₁: x < 0 := by linarith
+    rw [abs_of_neg h₆₁]
+    rw [abs_of_pos (by norm_num)]
+    exact neg_le.mp h₄
+  have h₇: (-(1 / 4)) ^ 2 ≤ x ^ 2 := by
+    refine sq_le_sq.mpr ?_
+    have h₆₁: x < 0 := by linarith
+    rw [abs_of_neg h₆₁]
+    rw [abs_of_neg (by norm_num)]
+    norm_num
+    exact le_neg_of_le_neg h₅
+  norm_num at h₆ h₇
+  constructor
+  . have h₈: x + (4/10) ≤ (15/100) := by linarith
+    have h₉: (x + (4/10)) * x ^ 2 ≤ (15/100) * (1 / 4) := by
+      refine mul_le_mul h₈ h₆ ?_ ?_
+      . exact sq_nonneg x
+      . norm_num
+    linarith
+  . linarith
+
+
+theorem imo_1960_p2_2
+  (x : ℝ)
+  -- (h₀ : 0 ≤ 1 + 2 * x)
+  (h₁ : (1 - Real.sqrt (1 + 2 * x))^2 ≠ 0)
+  (h₂ : (4 * x^2) / (1 - Real.sqrt (1 + 2*x))^2 < 2*x + 9) :
+  4 * x ^ 2 < (2 * x + 9) * (1 - √(1 + 2 * x)) ^ 2 := by
+  refine' (div_lt_iff₀ ?_).mp h₂
+  refine Ne.lt_of_le (id (Ne.symm h₁)) ?_
+  exact sq_nonneg (1 - sqrt (1 + 2 * x))
+
+theorem imo_1960_p2_3
+  (x : ℝ)
+  (h₀ : 0 ≤ 1 + 2 * x) :
+  -- (h₁ : (1 - Real.sqrt (1 + 2 * x))^2 ≠ 0)
+  -- (h₂ : (4 * x^2) / (1 - Real.sqrt (1 + 2*x))^2 < 2*x + 9) :
+  (1 - sqrt (1 + 2 * x)) ^ 2 = (2 + 2 * x) - 2 * sqrt (1 + 2 * x) := by
+  ring_nf
+  ring_nf at h₀
+  rw [Real.sq_sqrt h₀]
+  ring_nf
+
+theorem imo_1960_p2_4
+  (x : ℝ)
+  (h₀ : 0 ≤ 1 + 2 * x)
+  -- (h₁ : (1 - Real.sqrt (1 + 2 * x))^2 ≠ 0)
+  -- (h₂ : (4 * x^2) / (1 - Real.sqrt (1 + 2*x))^2 < 2*x + 9)
+  (h₃: 4 * x ^ 2 < (2 * x + 9) * (1 - sqrt (1 + 2 * x) ) ^ 2)
+  (h₄: (1 - sqrt (1 + 2 * x)) ^ 2 = (2 + 2 * x) - 2 * sqrt (1 + 2 * x)) :
+  (2 * x + 9) ^ 2 * (sqrt (1 + 2 * x)) ^ 2 < (11 * x + 9) ^ 2 := by
+  rw [← mul_pow]
+  refine' pow_lt_pow_left₀ ?_  ?_ (by norm_num)
+  . rw [h₄] at h₃
+    linarith
+  . refine' mul_nonneg ?_ ?_
+    . linarith
+    . exact sqrt_nonneg (1 + 2 * x)
+
+
+theorem imo_1960_p2_5
+  (x : ℝ)
+  (h₀ : 0 ≤ 1 + 2 * x)
+  -- (h₁ : (1 - Real.sqrt (1 + 2 * x))^2 ≠ 0)
+  -- (h₂ : (4 * x^2) / (1 - Real.sqrt (1 + 2*x))^2 < 2*x + 9)
+  (h₃: (2 * x + 9) ^ 2 * (sqrt (1 + 2 * x)) ^ 2 < (11 * x + 9) ^ 2) :
+  8 * x^3 < 45 * x^2 := by
+  rw [Real.sq_sqrt h₀] at h₃
+  ring_nf at h₃
+  linarith
+
+
+theorem imo_1960_p2_6
+  (x : ℝ)
+  -- (h₀ : 0 ≤ 1 + 2 * x)
+  -- (h₁ : (1 - Real.sqrt (1 + 2 * x))^2 ≠ 0)
+  -- (h₂ : (4 * x^2) / (1 - Real.sqrt (1 + 2*x))^2 < 2*x + 9)
+  (h₃: x^3 * 8 < x^2 * 45) :
+  x < 45/8 := by
+  have h₇₁: 0 ≤ x^2 := by exact sq_nonneg x
+  refine (lt_div_iff₀ (by norm_num)).mpr ?_
+  refine' lt_of_mul_lt_mul_right ?_ h₇₁
+  ring_nf
+  exact h₃
+
+
+theorem imo_1960_p2_7
+  (x : ℝ)
+  (h₀ : 0 ≤ 1 + 2 * x)
+  (h₁ : (1 - Real.sqrt (1 + 2 * x))^2 ≠ 0)
+  (h₂ : (4 * x^2) / (1 - Real.sqrt (1 + 2*x))^2 < 2*x + 9) :
+  0 < x ^ 2 ∨ x ^ 2 = 0 := by
+  have h₄: 0 ≤ x ^ 2 := by
+    exact sq_nonneg x
+  exact LE.le.gt_or_eq h₄
+
+
+theorem imo_1960_p2_8
+  (x : ℝ)
+  -- (h₀ : 0 ≤ 1 + 2 * x)
+  -- (h₁ : (1 - Real.sqrt (1 + 2 * x))^2 ≠ 0)
+  -- (h₂ : (4 * x^2) / (1 - Real.sqrt (1 + 2*x))^2 < 2*x + 9)
+  (h₃: 4 * x ^ 2 < (2 * x + 9) * (1 - sqrt (1 + 2 * x) ) ^ 2)
+  (h₄: (1 - sqrt (1 + 2 * x)) ^ 2 = (2 + 2 * x) - 2 * sqrt (1 + 2 * x)) :
+  (2 * x + 9) * √(1 + 2 * x) < 11 * x + 9 := by
+  rw [h₄] at h₃
+  linarith
+
+
+theorem imo_1960_p2_9
+  (x : ℝ)
+  (h₀ : 0 ≤ 1 + 2 * x) :
+  -- (h₁ : (1 - Real.sqrt (1 + 2 * x)) ^ 2 ≠ 0)
+  -- (h₂ : (4 * x^2) / (1 - Real.sqrt (1 + 2*x))^2 < 2*x + 9) :
+  0 ≤ (2 * x + 9) * √(1 + 2 * x) := by
+  refine' mul_nonneg ?_ ?_
+  . linarith
+  . exact sqrt_nonneg (1 + 2 * x)

Lemmas/imo_1962_p2_lemmas.lean

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+import Mathlib
+set_option linter.unusedVariables.analyzeTactics true
+
+
+open Real
+
+
+
+theorem imo_1962_p2_1
+  (x : ℝ)
+  -- (h₀ : 0 ≤ 3 - x)
+  -- (h₁ : 0 ≤ x + 1)
+  (h₂ : 1 / 2 < Real.sqrt (x - 3) - Real.sqrt (x + 1)) :
+  -1 ≤ x := by
+  refine neg_le_iff_add_nonneg.mpr ?_
+  contrapose! h₂
+  have h₃: x - 3 < 0 := by linarith [h₂]
+  have h₄: Real.sqrt (x + 1) = 0 := by
+    refine Real.sqrt_eq_zero'.mpr ?_
+    exact le_of_lt h₂
+  have h₅: Real.sqrt (x -3) = 0 := by
+    refine Real.sqrt_eq_zero'.mpr ?_
+    exact le_of_lt h₃
+  rw [h₄, h₅, sub_zero]
+  refine div_nonneg ?_ ?_
+  all_goals try linarith
+
+
+theorem imo_1962_p2_2
+  (x : ℝ)
+  (h₀ : 0 ≤ 3 - x)
+  (h₁ : 0 ≤ x + 1)
+  (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) :
+  (2 * √(3 - x) * √(x + 1)) ^ 2 < (4 - 1 / 4) ^ 2 := by
+  refine' pow_lt_pow_left₀ _ _ (by norm_num)
+  . refine lt_tsub_iff_left.mpr ?_
+    refine lt_tsub_iff_right.mp ?_
+    suffices g₀: 4 - 2 * sqrt (3 - x) * sqrt (x + 1) = (sqrt (3 - x) - sqrt (x + 1)) ^ 2
+    . rw [g₀]
+      have g₁:  (1:ℝ) / 4 = (1/2)^2 := by norm_num
+      rw [g₁]
+      exact pow_lt_pow_left₀ h₂ (by norm_num) (by norm_num)
+    rw [sub_sq]
+    rw [sq_sqrt h₀, sq_sqrt h₁]
+    ring_nf
+  . refine' mul_nonneg _ _
+    . refine mul_nonneg (by linarith) ?_
+      exact sqrt_nonneg (3 - x)
+    . exact sqrt_nonneg (x + 1)
+
+
+theorem imo_1962_p2_3
+  (x : ℝ)
+  (h₀ : 0 ≤ 3 - x)
+  (h₁ : 0 ≤ x + 1)
+  (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) :
+  2 * √(3 - x) * √(x + 1) < 4 - 1 / 4 := by
+  refine lt_tsub_iff_left.mpr ?refine'_1.a
+  refine lt_tsub_iff_right.mp ?refine'_1.a.a
+  suffices g₀: 4 - 2 * sqrt (3 - x) * sqrt (x + 1) = (sqrt (3 - x) - sqrt (x + 1)) ^ 2
+  . rw [g₀]
+    have g₁:  (1:ℝ) / 4 = (1/2)^2 := by norm_num
+    rw [g₁]
+    exact pow_lt_pow_left₀ h₂ (by norm_num) (by norm_num)
+  rw [sub_sq]
+  rw [sq_sqrt h₀, sq_sqrt h₁]
+  ring_nf
+
+
+theorem imo_1962_p2_4
+  (x : ℝ) :
+  -- (h₀ : 0 ≤ 3 - x)
+  -- (h₁ : 0 ≤ x + 1)
+  -- (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) :
+  0 ≤ 2 * √(3 - x) * √(x + 1) := by
+  refine' mul_nonneg ?_ ?_
+  . refine mul_nonneg (by linarith) ?_
+    exact sqrt_nonneg (3 - x)
+  . exact sqrt_nonneg (x + 1)
+
+
+
+theorem imo_1962_p2_5
+  (x : ℝ)
+  (h₀ : 0 ≤ 3 - x)
+  (h₁ : 0 ≤ x + 1) :
+  -- (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) :
+  4 - 2 * √(3 - x) * √(x + 1) = (√(3 - x) - √(x + 1)) ^ 2 := by
+  rw [sub_sq]
+  rw [sq_sqrt h₀, sq_sqrt h₁]
+  ring_nf
+
+
+theorem imo_1962_p2_6
+  (x : ℝ)
+  -- (h₀ : 0 ≤ 3 - x)
+  -- (h₁ : 0 ≤ x + 1)
+  (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1))
+  (h₃: 4 - 2 * √(3 - x) * √(x + 1) = (√(3 - x) - √(x + 1)) ^ 2) :
+  1 / 4 < 4 - 2 * √(3 - x) * √(x + 1) := by
+  rw [h₃]
+  have g₁:  (1:ℝ) / 4 = (1/2) ^ 2 := by norm_num
+  rw [g₁]
+  exact pow_lt_pow_left₀ h₂ (by norm_num) (by norm_num)
+
+
+theorem imo_1962_p2_7
+  (x : ℝ)
+  (h₀ : 0 ≤ 3 - x)
+  (h₁ : 0 ≤ x + 1)
+  -- (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1))
+  (h₃: (2 *sqrt (3 - x) * sqrt (x + 1)) ^ 2 < (4 - 1 / 4) ^ 2) :
+  4 * (x + 1) * (3 - x) < 225 / 16 := by
+  norm_num at h₃
+  suffices g₀: 4 * (x + 1) * (3 - x) = (2 * sqrt (3 - x) * sqrt (x + 1)) ^ 2
+  . exact Eq.trans_lt g₀ h₃
+  . rw [mul_pow, mul_pow, sq_sqrt h₀, sq_sqrt h₁]
+    norm_num
+    exact mul_right_comm 4 (x + 1) (3 - x)
+
+
+theorem imo_1962_p2_8
+  (x : ℝ)
+  (h₀ : 0 ≤ 3 - x)
+  (h₁ : 0 ≤ x + 1)
+  (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) :
+  x < 1 := by
+  suffices g₀: x + 1 < 3 - x
+  . linarith
+  . rw [← sq_sqrt h₀, ← sq_sqrt h₁]
+    refine' pow_lt_pow_left₀ ?_ ?_ (by norm_num)
+    . linarith
+    . exact sqrt_nonneg (x + 1)
+
+
+theorem imo_1962_p2_9
+  (x : ℝ)
+  -- (h₀ : 0 ≤ 3 - x)
+  -- (h₁ : 0 ≤ x + 1)
+  -- (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1))
+  (h₄: 4 * (x + 1) * (3 - x) < 225 / 16) :
+  x < 1 - sqrt 31 / 8 ∨ 1 + sqrt 31 / 8 < x := by
+  ring_nf at h₄
+  have g₀: 0 < x * x + -2 * x + 33 / 64 := by linarith
+  let a:ℝ := sqrt 31 / 8
+  have g₁: x * x + -2 * x + 33 / 64 = (x - (1 + a)) * (x - (1 - a)) := by
+    simp
+    ring_nf
+    norm_num
+    linarith
+  rw [g₁] at g₀
+  by_cases g₂: (x - (1 - a)) < 0
+  . left
+    exact sub_neg.mp g₂
+  . push_neg at g₂
+    right
+    have g₃: 0 < (x - (1 + a)) := by exact pos_of_mul_pos_left g₀ g₂
+    exact sub_pos.mp g₃
+
+
+theorem imo_1962_p2_10
+  (x : ℝ)
+  -- (h₀ : 0 ≤ 3 - x)
+  -- (h₁ : 0 ≤ x + 1)
+  -- (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1))
+  (h₄: x < 1)
+  (h₅: x < 1 - sqrt 31 / 8 ∨ 1 + sqrt 31 / 8 < x) :
+  x < 1 - Real.sqrt 31 / 8 := by
+  cases h₅ with
+  | inl h₅ => exact h₅
+  | inr h₅ => linarith
+
+
+theorem imo_1962_p2_11
+  (x a : ℝ)
+  (ha: a = √31 / 8)
+  -- (h₀ : 0 ≤ 3 - x)
+  -- (h₁ : 0 ≤ x + 1)
+  -- (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1))
+  (h₃: 0 < (x - (1 + a)) * (x - (1 - a))) :
+  x < 1 - √31 / 8 ∨ 1 + √31 / 8 < x := by
+  by_cases g₂: (x - (1 - a)) < 0
+  . left
+    rw [ha] at g₂
+    exact sub_neg.mp g₂
+  . push_neg at g₂
+    right
+    have g₃: 0 < (x - (1 + a)) := by exact pos_of_mul_pos_left h₃ g₂
+    rw [ha] at g₃
+    exact sub_pos.mp g₃
+
+
+theorem imo_1962_p2_12
+  (x a : ℝ)
+  (ha: a = 0.5)
+  -- (h₀ : 0 ≤ 3 - x)
+  -- (h₁ : 0 ≤ x + 1)
+  -- (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1))
+  (h₃: 0 < (x - (1 + a)) * (x - (1 - a))) :
+  x < 1 - 0.5 ∨ 1 + 0.5 < x := by
+  by_cases g₂: (x - (1 - a)) < 0
+  . left
+    rw [ha] at g₂
+    exact sub_neg.mp g₂
+  . push_neg at g₂
+    right
+    have g₃: 0 < (x - (1 + a)) := by exact pos_of_mul_pos_left h₃ g₂
+    rw [ha] at g₃
+    exact sub_pos.mp g₃
+
+
+theorem imo_1962_p2_13
+  (x a : ℝ)
+  (ha: a = √31 / 8) :
+  -- h₀ : 0 ≤ 3 - x
+  -- h₁ : 0 ≤ x + 1
+  -- h₄ : 12 + (x * 8 - x ^ 2 * 4) < 225 / 16
+  -- g₀ : 0 < x * x + -2 * x + 33 / 64
+  x ^ 2 - 2 * x + 33 / 64 = (x - (1 + a)) * (x - (1 - a)) := by
+  rw [ha]
+  ring_nf
+  norm_num
+  linarith
+
+theorem imo_1962_p2_14
+  (x : ℝ)
+  -- (h₀ : 0 ≤ 3 - x)
+  -- (h₁ : 0 ≤ x + 1)
+  (h₄ : 12 + (x * 8 - x ^ 2 * 4) < 225 / 16) :
+  0 < x * x + -2 * x + 33 / 64 := by
+  ring_nf at h₄
+  linarith

Lemmas/imo_1964_p2_lemmas.lean

@@ -0,0 +1,183 @@
+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

Lemmas/imo_1983_p6_lemmas.lean

@@ -0,0 +1,1180 @@
+import Mathlib
+
+set_option linter.unusedVariables.analyzeTactics true
+
+
+lemma imo_1983_p6_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 imo_1983_p6_1_1
+  (a b c x y z : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₂ : c ≤ b ∧ b ≤ a)
+  (h₃ : z ≤ y ∧ y ≤ x) :
+  c * (y - z) + b * (x - y) ≤ a * (x - z) := by
+  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 imo_1983_p6_1_2
+  (a b c x y z : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₂ : c ≤ b ∧ b ≤ a)
+  (h₃ : z ≤ y ∧ y ≤ x) :
+  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
+
+
+lemma imo_1983_p6_1_3
+  (a b c x y z : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₂ : c ≤ b ∧ b ≤ a)
+  (h₃ : z ≤ y ∧ y ≤ x)
+  (h₅ : c * (y - z) + b * (x - y) ≤ b * (y - z) + b * (x - y)) :
+  c * (y - z) + b * (x - y) ≤ a * (x - z) := by
+  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 imo_1983_p6_1_4
+  (a b c x y z : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₂ : c ≤ b ∧ b ≤ a)
+  (h₃ : z ≤ y ∧ y ≤ x) :
+  -- (h₅ : c * (y - z) + b * (x - y) ≤ b * (y - z) + b * (x - y)) :
+  b * (y - z) + b * (x - y) ≤ a * (x - z) := by
+  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 imo_1983_p6_1_5
+  (a b c x y z : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₂ : c ≤ b ∧ b ≤ a)
+  (h₃ : z ≤ y ∧ y ≤ x) :
+  -- (h₅ : c * (y - z) + b * (x - y) ≤ b * (y - z) + b * (x - y)) :
+  b * (x - z) ≤ a * (x - z) := by
+  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 imo_1983_p6_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
+
+
+lemma imo_1983_p6_2_1
+  (a b c x y z : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₂ : c ≤ b ∧ b ≤ a)
+  (h₃ : z ≤ y ∧ y ≤ x) :
+  c * (x - z) + b * (z - y) ≤ a * (x - y) := by
+  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
+
+
+lemma imo_1983_p6_2_2
+  (a b c x y z : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₂ : c ≤ b ∧ b ≤ a)
+  (h₃ : z ≤ y ∧ y ≤ x) :
+  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
+
+
+lemma imo_1983_p6_2_3
+  (a b c x y z : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₂ : c ≤ b ∧ b ≤ a)
+  (h₃ : z ≤ y ∧ y ≤ x) :
+  c * (x - z) ≤ b * (x - z) := by
+  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
+
+
+lemma imo_1983_p6_2_4
+  -- (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) :
+  0 ≤ x - z := by
+  refine sub_nonneg_of_le ?_
+  exact le_trans h₃.1 h₃.2
+
+
+lemma imo_1983_p6_2_5
+  (a b c x y z : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₂ : c ≤ b ∧ b ≤ a)
+  (h₃ : z ≤ y ∧ y ≤ x) :
+  -- (h₅ : c * (x - z) + b * (z - y) ≤ b * (x - z) + b * (z - y)) :
+  b * (x - z) + b * (z - y) ≤ a * (x - y) := by
+  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
+
+
+lemma imo_1983_p6_2_6
+  (a b c x y z : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₂ : c ≤ b ∧ b ≤ a)
+  (h₃ : z ≤ y ∧ y ≤ x) :
+  -- (h₅ : c * (x - z) + b * (z - y) ≤ b * (x - z) + b * (z - y)) :
+  b * (x - y) ≤ a * (x - y) := by
+  refine mul_le_mul h₂.2 ?_ ?_ ?_
+  . exact le_rfl
+  . exact sub_nonneg_of_le h₃.2
+  . exact le_of_lt h₀.1
+
+
+
+lemma imo_1983_p6_3
+  (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 imo_1983_p6_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
+
+
+lemma imo_1983_p6_3_1
+  (a b c : ℝ)
+  -- (hap : 0 < a)
+  -- (hbp : 0 < b)
+  -- (hcp : 0 < c)
+  (h₁ : c < a + b)
+  -- (h₃ : a < b + c)
+  (hba : b ≤ a) :
+  -- (hcb : c ≤ b)
+  -- (g₀ : b * c ≤ a * c)
+  -- (g₁ : a * c ≤ a * b) :
+  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
+
+
+lemma imo_1983_p6_3_2
+  (a b c : ℝ)
+  -- (hap : 0 < a)
+  -- (hbp : 0 < b)
+  -- (hcp : 0 < c)
+  -- (h₁ : c < a + b)
+  (h₃ : a < b + c)
+  -- (hba : b ≤ a)
+  (hcb : c ≤ b) :
+  -- (g₀ : b * c ≤ a * c)
+  -- (g₁ : a * c ≤ a * b)
+  -- (g₂ : a * (b + c - a) ≤ b * (a + c - b)) :
+  b * (a + c - b) ≤ c * (a + b - c) := by
+  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
+  linarith
+
+
+lemma imo_1983_p6_3_3
+  (a b c : ℝ)
+  (hap : 0 < a)
+  (hbp : 0 < b)
+  (hcp : 0 < c)
+  -- (h₁ : c < a + b)
+  -- (h₃ : a < b + c)
+  -- (hba : b ≤ a)
+  -- (hcb : c ≤ b)
+  (g₀ : b * c ≤ a * c)
+  (g₁ : a * c ≤ a * b)
+  (g₂ : a * (b + c - a) ≤ b * (a + c - b))
+  (g₃ : b * (a + c - b) ≤ c * (a + b - c)) :
+  0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by
+  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 imo_1983_p6_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
+
+
+lemma imo_1983_p6_3_4
+  (a b c : ℝ)
+  (hap : 0 < a)
+  (hbp : 0 < b)
+  (hcp : 0 < c)
+  -- (h₁ : c < a + b)
+  -- (h₃ : a < b + c)
+  -- (hba : b ≤ a)
+  -- (hcb : c ≤ b)
+  (g₀ : b * c ≤ a * c)
+  (g₁ : a * c ≤ a * b)
+  (g₂ : a * (b + c - a) ≤ b * (a + c - b))
+  (g₃ : b * (a + c - b) ≤ c * (a + b - c)) :
+  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 imo_1983_p6_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 }
+
+
+
+lemma imo_1983_p6_4
+  (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 imo_1983_p6_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
+
+
+lemma imo_1983_p6_4_1
+  (a b c : ℝ)
+  (hap : 0 < a)
+  (hbp : 0 < b)
+  (hcp : 0 < c)
+  (h₁ : c < a + b)
+  (h₃ : a < b + c)
+  (hba : b ≤ a)
+  (hcb : c ≤ b)
+  (g₀ : b * c ≤ a * c)
+  (g₁ : a * c ≤ a * b) :
+  0 ≤ a ^ 2 * c * (a - c) + c ^ 2 * b * (c - b) + b ^ 2 * a * (b - a) := by
+  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 imo_1983_p6_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
+
+
+lemma imo_1983_p6_4_2
+  (a b c : ℝ)
+  (hap : 0 < a)
+  (hbp : 0 < b)
+  (hcp : 0 < c)
+  -- (h₁ : c < a + b)
+  (h₃ : a < b + c)
+  -- (hba : b ≤ a)
+  (hcb : c ≤ b)
+  (g₀ : b * c ≤ a * c)
+  (g₁ : a * c ≤ a * b)
+  (g₂ : a * (b + c - a) ≤ b * (a + c - b)) :
+  0 ≤ a ^ 2 * c * (a - c) + c ^ 2 * b * (c - b) + b ^ 2 * a * (b - a) := by
+  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 imo_1983_p6_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
+
+
+lemma imo_1983_p6_4_3
+  (a b c : ℝ)
+  (hap : 0 < a)
+  (hbp : 0 < b)
+  (hcp : 0 < c)
+  -- (h₁ : c < a + b)
+  -- (h₃ : a < b + c)
+  -- (hba : b ≤ a)
+  -- (hcb : c ≤ b)
+  (g₀ : b * c ≤ a * c)
+  (g₁ : a * c ≤ a * b)
+  (g₂ : a * (b + c - a) ≤ b * (a + c - b))
+  (g₃ : b * (a + c - b) ≤ c * (a + b - c)) :
+  0 ≤ a ^ 2 * c * (a - c) + c ^ 2 * b * (c - b) + b ^ 2 * a * (b - a) := by
+  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 imo_1983_p6_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
+
+
+lemma imo_1983_p6_4_4
+  (a b c : ℝ)
+  -- (hap : 0 < a)
+  -- (hbp : 0 < b)
+  -- (hcp : 0 < c)
+  (h₁ : c < a + b)
+  -- (h₃ : a < b + c)
+  (hba : b ≤ a) :
+  -- (hcb : c ≤ b)
+  -- (g₀ : b * c ≤ a * c)
+  -- (g₁ : a * c ≤ a * b) :
+  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
+
+
+lemma imo_1983_p6_4_5
+  (a b c : ℝ)
+  -- (hap : 0 < a)
+  -- (hbp : 0 < b)
+  -- (hcp : 0 < c)
+  (h₁ : c < a + b)
+  -- (h₃ : a < b + c)
+  (hba : b ≤ a) :
+  -- (hcb : c ≤ b)
+  -- (g₀ : b * c ≤ a * c)
+  -- (g₁ : a * c ≤ a * b) :
+  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
+
+
+lemma imo_1983_p6_4_6
+  (a b c : ℝ)
+  -- (hap : 0 < a)
+  -- (hbp : 0 < b)
+  -- (hcp : 0 < c)
+  -- (h₁ : c < a + b)
+  (h₃ : a < b + c)
+  -- (hba : b ≤ a)
+  (hcb : c ≤ b) :
+  -- (g₀ : b * c ≤ a * c)
+  -- (g₁ : a * c ≤ a * b)
+  -- (g₂ : a * (b + c - a) ≤ b * (a + c - b)) :
+  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
+
+
+lemma imo_1983_p6_4_7
+  (a b c : ℝ)
+  -- (hap : 0 < a)
+  -- (hbp : 0 < b)
+  -- (hcp : 0 < c)
+  -- (h₁ : c < a + b)
+  (h₃ : a < b + c)
+  -- (hba : b ≤ a)
+  (hcb : c ≤ b) :
+  -- (g₀ : b * c ≤ a * c)
+  -- (g₁ : a * c ≤ a * b)
+  -- (g₂ : a * (b + c - a) ≤ b * (a + c - b)) :
+  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
+
+
+lemma imo_1983_p6_4_8
+  (a b c : ℝ)
+  (hap : 0 < a)
+  (hbp : 0 < b)
+  (hcp : 0 < c)
+  -- (h₁ : c < a + b)
+  -- (h₃ : a < b + c)
+  -- (hba : b ≤ a)
+  -- (hcb : c ≤ b)
+  (g₀ : b * c ≤ a * c)
+  (g₁ : a * c ≤ a * b)
+  (g₂ : a * (b + c - a) ≤ b * (a + c - b))
+  (g₃ : b * (a + c - b) ≤ c * (a + b - c)) :
+  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 imo_1983_p6_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 }
+
+
+lemma imo_1983_p6_5_1
+  (a b c : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₁ : c < a + b)
+  (h₂ : b < a + c)
+  (h₃ : a < b + c)
+  (ho₀ : a < b)
+  (ho₁ : b < c) :
+  0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by
+  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 imo_1983_p6_4 c b a h₀.2.2 h₀.2.1 h₀.1 h₃ h₁ (le_of_lt ho₁) (le_of_lt ho₀)
+  linarith
+
+
+lemma imo_1983_p6_5_2
+  (a b c : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₁ : c < a + b)
+  (h₂ : b < a + c)
+  (h₃ : a < b + c)
+  (ho₀ : a < b)
+  (ho₁ : c ≤ b)
+  (ho₂ : a < c) :
+  0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by
+  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 imo_1983_p6_3 b c a h₀.2.1 h₀.2.2 h₀.1 h₃ h₂ ho₁ (le_of_lt ho₂)
+  linarith
+
+
+lemma imo_1983_p6_5_3
+  (a b c : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₁ : c < a + b)
+  (h₂ : b < a + c)
+  (h₃ : a < b + c)
+  (ho₀ : a < b)
+  (ho₁ : c ≤ b)
+  (ho₂ : c ≤ a) :
+  0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by
+  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 imo_1983_p6_4 b a c h₀.2.1 h₀.1 h₀.2.2 h₁ h₂ (le_of_lt ho₀) ho₂
+  linarith
+
+
+lemma imo_1983_p6_5_4
+  (a b c : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₁ : c < a + b)
+  (h₂ : b < a + c)
+  (h₃ : a < b + c)
+  (ho₀ : b ≤ a)
+  (ho₁ : b < c)
+  (ho₂ : a < c) :
+  0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by
+  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 imo_1983_p6_3 c a b h₀.2.2 h₀.1 h₀.2.1 h₂ h₁ (le_of_lt ho₂) ho₀
+  linarith
+
+
+lemma imo_1983_p6_5_5
+  (a b c : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  -- (h₁ : c < a + b)
+  (h₂ : b < a + c)
+  (h₃ : a < b + c)
+  (ho₀ : b ≤ a)
+  (ho₁ : b < c)
+  (ho₂ : c ≤ a) :
+  0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by
+  rw [add_comm] at h₃
+  exact imo_1983_p6_4 a c b h₀.1 h₀.2.2 h₀.2.1 h₂ h₃ ho₂ (le_of_lt ho₁)
+
+
+lemma imo_1983_p6_5_6
+  (a b c : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₁ : c < a + b)
+  (h₂ : b < a + c)
+  (h₃ : a < b + c)
+  (ho₀ : a < b) :
+  0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by
+  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 imo_1983_p6_4 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 imo_1983_p6_3 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 imo_1983_p6_4 b a c h₀.2.1 h₀.1 h₀.2.2 h₁ h₂ (le_of_lt ho₀) ho₂
+      linarith
+
+
+lemma imo_1983_p6_5_7
+  (a b c : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₁ : c < a + b)
+  (h₂ : b < a + c)
+  (h₃ : a < b + c)
+  (ho₀ : a < b)
+  (ho₁ : c ≤ b) :
+  0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by
+  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 imo_1983_p6_3 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 imo_1983_p6_4 b a c h₀.2.1 h₀.1 h₀.2.2 h₁ h₂ (le_of_lt ho₀) ho₂
+    linarith
+
+
+lemma imo_1983_p6_5_8
+  (a b c : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₁ : c < a + b)
+  (h₂ : b < a + c)
+  (h₃ : a < b + c)
+  (ho₀ : b ≤ a) :
+  0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by
+  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 imo_1983_p6_3 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 imo_1983_p6_4 a c b h₀.1 h₀.2.2 h₀.2.1 h₂ h₃ ho₂ (le_of_lt ho₁)
+  . exact imo_1983_p6_3 a b c h₀.1 h₀.2.1 h₀.2.2 h₁ h₃ ho₀ ho₁
+
+
+
+lemma imo_1983_p6_5_9
+  (a b c : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₁ : c < a + b)
+  (h₂ : b < a + c)
+  (h₃ : a < b + c)
+  (ho₀ : b ≤ a)
+  (ho₁ : b < c) :
+  0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by
+  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 imo_1983_p6_3 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 imo_1983_p6_4 a c b h₀.1 h₀.2.2 h₀.2.1 h₂ h₃ ho₂ (le_of_lt ho₁)
+
+
+lemma imo_1983_p6_6
+  (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) :
+  a^2 * c * (a - c) + c^2 * b * (c - b) + b^2 * a * (b - a) ≤
+      a^2 * b * (a - b) + b^2 * c * (b - c) + c^2 * a * (c - a) := by
+  have h₄ : 0 ≤ (a + b + c) * (a - b) * (a - c) * (b - c) := by
+    refine mul_nonneg ?_ (by linarith)
+    refine mul_nonneg ?_ (by linarith)
+    refine mul_nonneg ?_ (by linarith)
+    linarith
+  linarith
+
+
+-- give the tight as a hypothesis, use it to prove each of the 6 cases
+lemma imo_1983_p6_7_1
+  (a b c : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₁ : c < a + b)
+  -- (h₂ : b < a + c)
+  (h₃ : a < b + c)
+  (ho₀ : a < b)
+  (ho₁ : b < c)
+  (ht : ∀ a b c :ℝ, (0 < a ∧ 0 < b ∧ 0 < c) → (c < a + b ∧ a < b + c) → (c ≤ b ∧ b ≤ a)
+      → 0 ≤ a^2 * c * (a - c) + c^2 * b * (c - b) + b^2 * a * (b - a)) :
+  0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by
+  have h₄: 0 ≤ c ^ 2 * a * (c - a) + a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) := by
+    refine ht c b a ?_ ?_ ?_
+    . simp_all only [and_self]
+    . constructor
+      . rw [add_comm]
+        exact h₃
+      . rw [add_comm]
+        exact h₁
+    . constructor
+      . exact le_of_lt ho₀
+      . exact le_of_lt ho₁
+  linarith
+
+
+lemma imo_1983_p6_7_2
+  (a b c : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  -- (h₁ : c < a + b)
+  (h₂ : b < a + c)
+  (h₃ : a < b + c)
+  -- (ho₀ : a < b)
+  (ho₁ : c ≤ b)
+  (ho₂ : a < c)
+  (ht : ∀ a b c :ℝ, (0 < a ∧ 0 < b ∧ 0 < c) → (c < a + b ∧ a < b + c) → (c ≤ b ∧ b ≤ a)
+      → 0 ≤ a^2 * c * (a - c) + c^2 * b * (c - b) + b^2 * a * (b - a)) :
+  0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by
+  have h₄: 0 ≤ b ^ 2 * a * (b - a) + a ^ 2 * c * (a - c) + c ^ 2 * b * (c - b) := by
+    refine ht b c a ?_ ?_ ?_
+    . simp_all only [and_self]
+    . constructor
+      . exact h₃
+      . rw [add_comm]
+        exact h₂
+    . constructor
+      . exact le_of_lt ho₂
+      . exact ho₁
+  refine le_trans h₄ ?_
+  have h₅: b ^ 2 * a * (b - a) + a ^ 2 * c * (a - c) + c ^ 2 * b * (c - b) ≤
+    b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) + a ^ 2 * b * (a - b) := by
+    rw [add_comm] at h₂
+    refine imo_1983_p6_6 b c a h₀.1 ho₁ ?_
+    exact le_of_lt ho₂
+  linarith
+
+
+lemma imo_1983_p6_7_3
+  (a b c : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₁ : c < a + b)
+  (h₂ : b < a + c)
+  -- (h₃ : a < b + c)
+  (ho₀ : a < b)
+  -- (ho₁ : c ≤ b)
+  (ho₂ : c ≤ a)
+  (ht : ∀ a b c :ℝ, (0 < a ∧ 0 < b ∧ 0 < c) → (c < a + b ∧ a < b + c) → (c ≤ b ∧ b ≤ a)
+      → 0 ≤ a^2 * c * (a - c) + c^2 * b * (c - b) + b^2 * a * (b - a)) :
+  0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by
+  have h₄: 0 ≤ b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) + a ^ 2 * b * (a - b) := by
+    refine ht b a c ?_ ?_ ?_
+    . simp_all only [and_self]
+    . constructor
+      . rw [add_comm]
+        exact h₁
+      . exact h₂
+    . constructor
+      . exact ho₂
+      . exact le_of_lt ho₀
+  linarith
+
+
+lemma imo_1983_p6_7_4
+  (a b c : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₁ : c < a + b)
+  (h₂ : b < a + c)
+  -- (h₃ : a < b + c)
+  (ho₀ : b ≤ a)
+  -- (ho₁ : b < c)
+  (ho₂ : a < c)
+  (ht : ∀ a b c :ℝ, (0 < a ∧ 0 < b ∧ 0 < c) → (c < a + b ∧ a < b + c) → (c ≤ b ∧ b ≤ a)
+      → 0 ≤ a^2 * c * (a - c) + c^2 * b * (c - b) + b^2 * a * (b - a)) :
+  0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by
+  have h₄: 0 ≤ c ^ 2 * b * (c - b) + b ^ 2 * a * (b - a) + a ^ 2 * c * (a - c) := by
+    refine ht c a b ?_ ?_ ?_
+    . simp_all only [and_self]
+    . constructor
+      . rw [add_comm]
+        exact h₂
+      . exact h₁
+    . constructor
+      . exact ho₀
+      . exact le_of_lt ho₂
+  refine le_trans h₄ ?_
+  have h₅: c ^ 2 * b * (c - b) + b ^ 2 * a * (b - a) + a ^ 2 * c * (a - c) ≤
+    c ^ 2 * a * (c - a) + a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) := by
+    rw [add_comm] at h₂
+    refine imo_1983_p6_6 c a b h₀.2.1 ?_ ho₀
+    exact le_of_lt ho₂
+  linarith
+
+
+lemma imo_1983_p6_7_5
+  (a b c : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  -- (h₁ : c < a + b)
+  (h₂ : b < a + c)
+  (h₃ : a < b + c)
+  -- (ho₀ : b ≤ a)
+  (ho₁ : b < c)
+  (ho₂ : c ≤ a)
+  (ht : ∀ a b c :ℝ, (0 < a ∧ 0 < b ∧ 0 < c) → (c < a + b ∧ a < b + c) → (c ≤ b ∧ b ≤ a)
+      → 0 ≤ a^2 * c * (a - c) + c^2 * b * (c - b) + b^2 * a * (b - a)) :
+  0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by
+  refine ht a c b ?_ ?_ ?_
+  . simp_all only [and_self]
+  . simp_all only [true_and]
+    rw [add_comm]
+    exact h₃
+  . constructor
+    . exact le_of_lt ho₁
+    . exact ho₂
+
+
+lemma imo_1983_p6_7_6
+  (a b c : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₁ : c < a + b)
+  -- (h₂ : b < a + c)
+  (h₃ : a < b + c)
+  (ho₀ : b ≤ a)
+  (ho₁ : c ≤ b)
+  (ht : ∀ a b c :ℝ, (0 < a ∧ 0 < b ∧ 0 < c) → (c < a + b ∧ a < b + c) → (c ≤ b ∧ b ≤ a)
+      → 0 ≤ a^2 * c * (a - c) + c^2 * b * (c - b) + b^2 * a * (b - a)) :
+  0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by
+  have h₄: 0 ≤ a^2 * c * (a - c) + c^2 * b * (c - b) + b^2 * a * (b - a) := by
+    refine ht a b c h₀ ?_ ?_
+    . simp_all only [true_and]
+    . constructor
+      . exact ho₁
+      . exact ho₀
+  refine le_trans h₄ ?_
+  refine imo_1983_p6_6 a b c h₀.2.2 ho₀ ho₁
+
+
+lemma imo_1983_p6_8_1
+  (a b c : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  -- (h₁ : c < a + b)
+  (h₂ : b < a + c)
+  (h₃ : a < b + c)
+  -- (ho₀ : a < b)
+  (ho₁ : c ≤ b)
+  (ho₂ : a < c)
+  (ht : ∀ a b c :ℝ, (0 < a ∧ 0 < b ∧ 0 < c) → (c < a + b ∧ a < b + c) → (c ≤ b ∧ b ≤ a)
+      → 0 ≤ a^2 * b * (a - b) + b^2 * c * (b - c) + c^2 * a * (c - a)) :
+  0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by
+  have h₄: 0 ≤ b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) + a ^ 2 * b * (a - b) := by
+    refine ht b c a ?_ ?_ ?_
+    . exact and_rotate.mp h₀
+    . simp_all only [true_and]
+      linarith
+    . constructor
+      . exact le_of_lt ho₂
+      . exact ho₁
+  linarith
+
+
+lemma imo_1983_p6_8_2
+  (a b c : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₁ : c < a + b)
+  (h₂ : b < a + c)
+  -- (h₃ : a < b + c)
+  (ho₀ : b ≤ a)
+  -- (ho₁ : b < c)
+  (ho₂ : a < c)
+  (ht : ∀ a b c :ℝ, (0 < a ∧ 0 < b ∧ 0 < c) → (c < a + b ∧ a < b + c) → (c ≤ b ∧ b ≤ a)
+      → 0 ≤ a^2 * b * (a - b) + b^2 * c * (b - c) + c^2 * a * (c - a)) :
+  0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by
+  have h₄: 0 ≤ c ^ 2 * a * (c - a) + a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) := by
+    refine ht c a b ?_ ?_ ?_
+    . exact and_rotate.mp (and_rotate.mp h₀)
+    . constructor
+      . rw [add_comm]
+        exact h₂
+      . exact h₁
+    . constructor
+      . exact ho₀
+      . exact le_of_lt ho₂
+  linarith
+
+
+lemma imo_1983_p6_8_3
+  (a b c : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₁ : c < a + b)
+  -- (h₂ : b < a + c)
+  (h₃ : a < b + c)
+  (ho₀ : b ≤ a)
+  (ho₁ : c ≤ b)
+  (ht : ∀ a b c :ℝ, (0 < a ∧ 0 < b ∧ 0 < c) → (c < a + b ∧ a < b + c) → (c ≤ b ∧ b ≤ a)
+      → 0 ≤ a^2 * b * (a - b) + b^2 * c * (b - c) + c^2 * a * (c - a)) :
+  0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by
+  refine ht a b c h₀ ?_ ?_
+  . simp_all only [true_and]
+  . constructor
+    . exact ho₁
+    . exact ho₀
+
+
+lemma mylemma_1x
+  (a b c : ℝ)
+  (x y z : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  -- (h₁ : 0 < x ∧ 0 < y ∧ 0 < z)
+  (h₂: c ≤ b ∧ b ≤ a)
+  (h₃: x ≤ y ∧ y ≤ z) :
+  x / c + y / a + z / b ≤ x / a + y / b + z / c := by
+  have g3: (z - x) / b ≤ (z - x) / c := by
+    have g31: 0 ≤ (z-x) := by
+      refine sub_nonneg_of_le ?_
+      exact le_trans h₃.1 h₃.2
+    exact div_le_div_of_nonneg_left g31 (by linarith) h₂.1
+  have g4: (y-x)/a + (z-y)/b ≤ (z-x)/b := by
+    have g41: (y-x)/a + (z-y)/b ≤ (y-x)/b + (z-y)/b := by
+      rw [add_le_add_iff_right ((z-y)/b)]
+      have g411: 0 ≤ (y-x) := by linarith
+      exact div_le_div_of_nonneg_left g411 (by linarith) h₂.2
+    have g42: (y-x)/b + (z-y)/b = (z-x)/b := by ring
+    linarith
+  have g5: (y-x)/a + (z-y)/b ≤ (z-x)/c := by
+    exact le_trans g4 g3
+  ring_nf at g5
+  ring_nf
+  linarith
+
+
+lemma my_lemma_2x
+  (a b c : ℝ)
+  (x y z : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  -- (h₁ : 0 < x ∧ 0 < y ∧ 0 < z)
+  (h₂: c ≤ b ∧ b ≤ a)
+  (h₃: x ≤ y ∧ y ≤ z) :
+  x/c + y/a + z/b ≤ x/a + y/b + z/c := by
+  have g3: (z-x)/b ≤ (z-x)/c := by
+    have g31: 0 ≤ (z-x) := by linarith
+    exact div_le_div_of_nonneg_left g31 (by linarith) h₂.1
+  have g4: (y-x)/a + (z-y)/b ≤ (z-x)/b := by
+      have g41: (y-x)/a + (z-y)/b ≤ (y-x)/b + (z-y)/b := by
+          rw [add_le_add_iff_right ((z-y)/b)]
+          have g411: 0 ≤ (y-x) := by linarith
+          exact div_le_div_of_nonneg_left g411 (by linarith) h₂.2
+      have g42: (y-x)/b + (z-y)/b = (z-x)/b := by ring_nf
+      linarith
+  have g5: (y-x)/a + (z-y)/b ≤ (z-x)/c := by exact le_trans g4 g3
+  ring_nf at g5
+  ring_nf
+  linarith
+
+
+lemma my_lemma_3x
+  (a b c : ℝ)
+  (x y z : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  -- (h₁ : 0 < x ∧ 0 < y ∧ 0 < z)
+  (h₂: c ≤ b ∧ b ≤ a)
+  (h₃: x ≤ y ∧ y ≤ z) :
+  x/b + y/c + z/a ≤ x/a + y/b + z/c := by
+  have g3: (z-y)/b ≤ (z-y)/c := by
+    have g31: 0 ≤ (z-y) := by linarith
+    exact div_le_div_of_nonneg_left g31 (by linarith) h₂.1
+  have g4: (x-y)/b + (z-x)/a ≤ (z-y)/b := by
+      have g41: (x-y)/b + (z-x)/a ≤ (x-y)/b + (z-x)/b := by
+          rw [add_le_add_iff_left ((x-y)/b)]
+          have g411: 0 ≤ (z-x) := by linarith
+          exact div_le_div_of_nonneg_left g411 (by linarith) h₂.2
+      have g42: (x-y)/b + (z-x)/b = (z-y)/b := by ring_nf
+      linarith
+  have g5: (x-y)/b + (z-x)/a ≤ (z-y)/c := by
+    exact le_trans g4 g3
+  ring_nf at g5
+  ring_nf
+  linarith
+
+
+lemma my_lemma_4x
+  (a b c : ℝ)
+  (x y z : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  -- (h₁ : 0 < x ∧ 0 < y ∧ 0 < z)
+  (h₂: c ≤ b ∧ b ≤ a)
+  (h₃: x ≤ y ∧ y ≤ z) :
+  x/b + y/a + z/c ≤ x/a + y/b + z/c := by
+  rw [add_le_add_iff_right (z/c)]
+  have g2: (y-x)/a ≤ (y-x)/b := by
+    exact div_le_div_of_nonneg_left (by linarith) h₀.2.1 h₂.2
+  rw [sub_div] at g2
+  rw [sub_div] at g2
+  linarith
+
+
+lemma my_lemma_5x
+  (a b c : ℝ)
+  (x y z : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  -- (h₁ : 0 < x ∧ 0 < y ∧ 0 < z)
+  (h₂: c ≤ b ∧ b ≤ a)
+  (h₃: x ≤ y ∧ y ≤ z) :
+  x/a + y/c + z/b ≤ x/a + y/b + z/c := by
+  rw [add_assoc (x/a)]
+  rw [add_assoc (x/a)]
+  rw [add_le_add_iff_left (x/a)]
+  have g1: (z-y)/b ≤ (z-y)/c := by
+    exact div_le_div_of_nonneg_left (by linarith) h₀.2.2 h₂.1
+  rw [sub_div] at g1
+  rw [sub_div] at g1
+  linarith
+
+
+lemma my_lemma_6x
+  (a b c : ℝ)
+  (x y z : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  -- (h₁ : 0 < x ∧ 0 < y ∧ 0 < z)
+  (h₂: c ≤ b ∧ b ≤ a)
+  (h₃: x ≤ y ∧ y ≤ z) :
+  x/c + y/b + z/a ≤ x/a + y/b + z/c := by
+  have g1: (z-x)/a ≤ (z-x)/c := by
+    exact div_le_div_of_nonneg_left (by linarith) h₀.2.2 (by linarith)
+  have g2: x/c + z/a ≤ x/a + z/c := by
+    rw [sub_div] at g1
+    rw [sub_div] at g1
+    linarith
+  linarith
+
+
+lemma mylemma_7x
+  (a b c : ℝ)
+  (x y z : ℝ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₂: c ≤ b ∧ b ≤ a)
+  (h₃: x ≤ y ∧ y ≤ z) :
+  x / c + y / a + z / b ≤ x / a + y / b + z / c := by
+  have g3: (z - x) / b ≤ (z - x) / c := by
+    have g31: 0 ≤ (z-x) := by
+      refine sub_nonneg_of_le ?_
+      exact le_trans h₃.1 h₃.2
+    exact div_le_div_of_nonneg_left g31 (by linarith) h₂.1
+  have g4: (y-x)/a + (z-y)/b ≤ (z-x)/b := by
+    have g41: (y-x)/a + (z-y)/b ≤ (y-x)/b + (z-y)/b := by
+      rw [add_le_add_iff_right ((z-y)/b)]
+      have g411: 0 ≤ (y-x) := by linarith
+      exact div_le_div_of_nonneg_left g411 (by linarith) h₂.2
+    have g42: (y-x)/b + (z-y)/b = (z-x)/b := by ring
+    linarith
+  have g5: (y-x)/a + (z-y)/b ≤ (z-x)/c := by
+    exact le_trans g4 g3
+  ring_nf at g5
+  ring_nf
+  linarith

Lemmas/imo_1984_p6_lemmas.lean

@@ -0,0 +1,1601 @@
+import Mathlib
+set_option linter.unusedVariables.analyzeTactics true
+
+open Nat
+
+lemma imo_1984_p6_1
+  (a b : ℕ)
+  -- (hap: 0 < a)
+  -- (hbp: 0 < b)
+  (h₀: b < a) :
+  ((a - b) ^ 2 = a ^ 2 + b ^ 2 - 2 * a * b) := by
+  have h₁: b^2 ≤ a * b := by
+    rw [pow_two]
+    refine Nat.mul_le_mul_right ?_ ?_
+    exact Nat.le_of_lt h₀
+  have h₂: a * b ≤ a ^ 2 := by
+    rw [pow_two]
+    refine Nat.mul_le_mul_left ?_ ?_
+    exact Nat.le_of_lt h₀
+  repeat rw [pow_two]
+  repeat rw [Nat.mul_sub_left_distrib]
+  repeat rw [Nat.mul_sub_right_distrib a b a]
+  rw [Nat.sub_right_comm]
+  repeat rw [Nat.mul_sub_right_distrib a b b]
+  ring_nf
+  have h₃: a ^ 2 - (a * b - b ^ 2) = a ^ 2 - a * b + b ^ 2 := by
+    refine tsub_tsub_assoc ?h₁ h₁
+    exact h₂
+  rw [h₃]
+  rw [← Nat.sub_add_comm h₂]
+  . rw [← Nat.sub_add_eq, ← mul_two]
+
+
+lemma imo_1984_p6_2
+  (a b c d k m : ℕ)
+  (h₂ : a < b ∧ b < c ∧ c < d)
+  (h₃ : a * d = b * c)
+  (h₄ : a + d = 2 ^ k)
+  (h₅ : b + c = 2 ^ m) :
+  (m < k) := by
+  have h₆: (c - b) ^ 2 < (d - a) ^ 2 := by
+    refine Nat.pow_lt_pow_left ?_ (by norm_num)
+    have h₈₀: c - a < d - a := by
+      have g₀: c - a + a < d - a + a := by
+        rw [Nat.sub_add_cancel ?_]
+        rw [Nat.sub_add_cancel ?_]
+        . exact h₂.2.2
+        . linarith
+        . linarith
+      exact Nat.lt_of_add_lt_add_right g₀
+    refine lt_trans ?_ h₈₀
+    refine Nat.sub_lt_sub_left ?_ h₂.1
+    exact lt_trans h₂.1 h₂.2.1
+  have h₇: (b + c) ^ 2 < (a + d) ^ 2 := by
+    rw [add_sq b c, add_sq a d]
+    have hda: a < d := by
+      refine lt_trans h₂.1 ?_
+      exact lt_trans h₂.2.1 h₂.2.2
+    rw [imo_1984_p6_1 d a hda] at h₆
+    rw [imo_1984_p6_1 c b h₂.2.1] at h₆
+    rw [mul_assoc 2 b c, ← h₃, ← mul_assoc]
+    rw [mul_assoc 2 c b, mul_comm c b, ← h₃, ← mul_assoc] at h₆
+    rw [add_assoc, add_comm _ (c ^ 2), ← add_assoc]
+    rw [add_assoc (a ^ 2), add_comm _ (d ^ 2), ← add_assoc]
+    rw [mul_assoc 2 d a, mul_comm d a, ← mul_assoc] at h₆
+    rw [add_comm (d ^ 2) (a ^ 2)] at h₆
+    rw [add_comm (c ^ 2) (b ^ 2)] at h₆
+    have g₀: 2 * a * d ≤ 4 * a * d := by
+      ring_nf
+      exact Nat.mul_le_mul_left (a * d) (by norm_num)
+    have g₁: 2 * a * d = 4 * a * d - 2 * a * d := by
+      ring_nf
+      rw [← Nat.mul_sub_left_distrib]
+      norm_num
+    have g₂: 2 * a * d ≤ b ^ 2 + c ^ 2 := by
+      rw [mul_assoc, h₃, ← mul_assoc]
+      exact two_mul_le_add_sq b c
+    have g₃: 2 * a * d ≤ a ^ 2 + d ^ 2 := by
+      exact two_mul_le_add_sq a d
+    rw [g₁, ← Nat.add_sub_assoc (g₀) (b ^ 2 + c ^ 2)]
+    rw [← Nat.add_sub_assoc (g₀) (a ^ 2 + d ^ 2)]
+    rw [Nat.sub_add_comm g₂, Nat.sub_add_comm g₃]
+    exact (Nat.add_lt_add_iff_right).mpr h₆
+  have h2 : 1 < 2 := by norm_num
+  refine (Nat.pow_lt_pow_iff_right h2).mp ?_
+  rw [← h₄, ← h₅]
+  exact (Nat.pow_lt_pow_iff_left (by norm_num) ).mp h₇
+
+
+lemma imo_1984_p6_3
+  (a b c d : ℕ)
+  (h₀ : a < b ∧ b < c ∧ c < d) :
+  (c - b) ^ 2 < (d - a) ^ 2 := by
+  refine Nat.pow_lt_pow_left ?_ (by norm_num)
+  have h₁: c - a < d - a := by
+    have g₀: c - a + a < d - a + a := by
+      rw [Nat.sub_add_cancel ?_]
+      rw [Nat.sub_add_cancel ?_]
+      . exact h₀.2.2
+      . linarith
+      . linarith
+    exact Nat.lt_of_add_lt_add_right g₀
+  refine lt_trans ?_ h₁
+  refine Nat.sub_lt_sub_left ?_ h₀.1
+  exact lt_trans h₀.1 h₀.2.1
+
+
+lemma imo_1984_p6_4
+  (a b c d : ℕ)
+  (h₀ : a < b ∧ b < c ∧ c < d)
+  (h₁ : a * d = b * c)
+  (h₂ : (c - b) ^ 2 < (d - a) ^ 2) :
+  (b + c) ^ 2 < (a + d) ^ 2 := by
+  rw [add_sq b c, add_sq a d]
+  have hda: a < d := by
+    refine lt_trans h₀.1 ?_
+    exact lt_trans h₀.2.1 h₀.2.2
+  rw [imo_1984_p6_1 d a hda] at h₂
+  rw [imo_1984_p6_1 c b h₀.2.1] at h₂
+  rw [mul_assoc 2 b c, ← h₁, ← mul_assoc]
+  rw [mul_assoc 2 c b, mul_comm c b, ← h₁, ← mul_assoc] at h₂
+  rw [add_assoc, add_comm _ (c ^ 2), ← add_assoc]
+  rw [add_assoc (a ^ 2), add_comm _ (d ^ 2), ← add_assoc]
+  rw [mul_assoc 2 d a, mul_comm d a, ← mul_assoc] at h₂
+  rw [add_comm (d ^ 2) (a ^ 2)] at h₂
+  rw [add_comm (c ^ 2) (b ^ 2)] at h₂
+  have g₀: 2 * a * d ≤ 4 * a * d := by
+    ring_nf
+    exact Nat.mul_le_mul_left (a * d) (by norm_num)
+  have g₁: 2 * a * d = 4 * a * d - 2 * a * d := by
+    ring_nf
+    rw [← Nat.mul_sub_left_distrib]
+    norm_num
+  have g₂: 2 * a * d ≤ b ^ 2 + c ^ 2 := by
+    rw [mul_assoc, h₁, ← mul_assoc]
+    exact two_mul_le_add_sq b c
+  have g₃: 2 * a * d ≤ a ^ 2 + d ^ 2 := by
+    exact two_mul_le_add_sq a d
+  rw [g₁, ← Nat.add_sub_assoc (g₀) (b ^ 2 + c ^ 2)]
+  rw [← Nat.add_sub_assoc (g₀) (a ^ 2 + d ^ 2)]
+  rw [Nat.sub_add_comm g₂, Nat.sub_add_comm g₃]
+  exact (Nat.add_lt_add_iff_right).mpr h₂
+
+
+lemma imo_1984_p6_5
+  (a b c d : ℕ)
+  -- (h₀ : a < b ∧ b < c ∧ c < d)
+  (h₁ : a * d = b * c)
+  (h₂ : b ^ 2 + c ^ 2 - 2 * a * d < a ^ 2 + d ^ 2 - 2 * a * d) :
+  b ^ 2 + c ^ 2 + 2 * a * d < a ^ 2 + d ^ 2 + 2 * a * d := by
+  have g₀: 2 * a * d ≤ 4 * a * d := by
+    ring_nf
+    exact Nat.mul_le_mul_left (a * d) (by norm_num)
+  have g₁: 2 * a * d = 4 * a * d - 2 * a * d := by
+    ring_nf
+    rw [← Nat.mul_sub_left_distrib]
+    norm_num
+  have g₂: 2 * a * d ≤ b ^ 2 + c ^ 2 := by
+    rw [mul_assoc, h₁, ← mul_assoc]
+    exact two_mul_le_add_sq b c
+  have g₃: 2 * a * d ≤ a ^ 2 + d ^ 2 := by
+    exact two_mul_le_add_sq a d
+  rw [g₁, ← Nat.add_sub_assoc (g₀) (b ^ 2 + c ^ 2)]
+  rw [← Nat.add_sub_assoc (g₀) (a ^ 2 + d ^ 2)]
+  rw [Nat.sub_add_comm g₂, Nat.sub_add_comm g₃]
+  exact (Nat.add_lt_add_iff_right).mpr h₂
+
+
+lemma imo_1984_p6_6
+  (a b c d : ℕ)
+  (h₁ : a * d = b * c) :
+  -- (h₂ : b ^ 2 + c ^ 2 - 2 * a * d < a ^ 2 + d ^ 2 - 2 * a * d)
+  -- (g₀ : 2 * a * d ≤ 4 * a * d)
+  -- (g₁ : 2 * a * d = 4 * a * d - 2 * a * d) :
+  (2 * a * d ≤ b ^ 2 + c ^ 2) := by
+  rw [mul_assoc, h₁, ← mul_assoc]
+  exact two_mul_le_add_sq b c
+
+
+lemma imo_1984_p6_7
+  (a b c d k m : ℕ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  (h₁ : m < k)
+  (h₂ : a < b ∧ b < c ∧ c < d)
+  (h₃ : a * d = b * c)
+  (h₄ : a + d = 2 ^ k)
+  (h₅ : b + c = 2 ^ m)
+  (hkm : k ≤ m) :
+  a = 99 := by
+  linarith [h₁, hkm]
+
+
+
+lemma imo_1984_p6_8
+  (a b c d k m : ℕ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  -- (h₁ : Odd a ∧ Odd b ∧ Odd c ∧ Odd d)
+  (h₂ : a < b ∧ b < c ∧ c < d)
+  (h₃ : a * d = b * c)
+  (h₄ : a + d = 2 ^ k)
+  (h₅ : b + c = 2 ^ m) :
+  -- (hkm : m < k) :
+  b * 2 ^ m - a * 2 ^ k = (b - a) * (b + a) := by
+  have h₆₀: c = 2 ^ m - b := by exact (tsub_eq_of_eq_add_rev (id h₅.symm)).symm
+  have h₆₁: d = 2 ^ k - a := by exact (tsub_eq_of_eq_add_rev (id h₄.symm)).symm
+  rw [h₆₀, h₆₁] at h₃
+  repeat rw [Nat.mul_sub_left_distrib, ← pow_two] at h₃
+  have h₆₂: b * 2 ^ m - a * 2 ^ k =  b ^ 2 - a ^ 2 := by
+    symm at h₃
+    refine Nat.sub_eq_of_eq_add ?_
+    rw [add_comm, ← Nat.add_sub_assoc]
+    . rw [Nat.sub_add_comm]
+      . refine Nat.eq_add_of_sub_eq ?_ h₃
+        rw [pow_two]
+        refine le_of_lt ?_
+        refine mul_lt_mul' (by linarith) ?_ (le_of_lt h₀.2.1) h₀.2.1
+        linarith
+      . rw [pow_two]
+        refine le_of_lt ?_
+        refine mul_lt_mul' (by linarith) ?_ (le_of_lt h₀.1) h₀.1
+        linarith
+    . refine le_of_lt ?_
+      rw [pow_two, pow_two]
+      exact mul_lt_mul h₂.1 (le_of_lt h₂.1) h₀.1 (le_of_lt h₀.2.1)
+  rw [Nat.sq_sub_sq b a] at h₆₂
+  rw [mul_comm (b - a) _]
+  exact h₆₂
+
+
+lemma imo_1984_p6_8_1
+(a b c d k m : ℕ)
+-- (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+-- (h₂ : a < b ∧ b < c ∧ c < d)
+(h₃ : a * d = b * c)
+(h₄ : a + d = 2 ^ k)
+(h₅ : b + c = 2 ^ m) :
+-- (h₆₀ : c = 2 ^ m - b)
+-- (h₆₁ : d = 2 ^ k - a) :
+a * (2 ^ k - a) = b * (2 ^ m - b) := by
+have h₆₀: c = 2 ^ m - b := by exact (tsub_eq_of_eq_add_rev (id h₅.symm)).symm
+have h₆₁: d = 2 ^ k - a := by exact (tsub_eq_of_eq_add_rev (id h₄.symm)).symm
+rw [h₆₀, h₆₁] at h₃
+exact h₃
+
+
+
+lemma imo_1984_p6_8_2
+  (a b c d k m : ℕ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  (h₂ : a < b)
+  (h₃ : a * 2 ^ k - a ^ 2 = b * 2 ^ m - b ^ 2)
+  (h₄ : a + d = 2 ^ k)
+  (h₅ : b + c = 2 ^ m) :
+  -- (h₆₀ : c = 2 ^ m - b)
+  -- (h₆₁ : d = 2 ^ k - a) :
+  b * 2 ^ m - a * 2 ^ k = b ^ 2 - a ^ 2 := by
+  symm at h₃
+  refine Nat.sub_eq_of_eq_add ?_
+  rw [add_comm, ← Nat.add_sub_assoc]
+  . rw [Nat.sub_add_comm]
+    . refine Nat.eq_add_of_sub_eq ?_ h₃
+      rw [pow_two]
+      refine le_of_lt ?_
+      refine mul_lt_mul' (by linarith) ?_ (le_of_lt h₀.2.1) h₀.2.1
+      linarith
+    . rw [pow_two]
+      refine le_of_lt ?_
+      refine mul_lt_mul' (by linarith) ?_ (le_of_lt h₀.1) h₀.1
+      linarith
+  . refine le_of_lt ?_
+    rw [pow_two, pow_two]
+    exact mul_lt_mul h₂ (le_of_lt h₂) h₀.1 (le_of_lt h₀.2.1)
+
+
+lemma imo_1984_p6_8_3
+  (a b c d k m : ℕ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  -- (h₂ : a < b)
+  (h₃ : b * 2 ^ m - b ^ 2 = a * 2 ^ k - a ^ 2)
+  -- (h₄ : a + d = 2 ^ k)
+  (h₅ : b + c = 2 ^ m) :
+  -- (h₆₀ : c = 2 ^ m - b)
+  -- (h₆₁ : d = 2 ^ k - a) :
+  b * 2 ^ m = a * 2 ^ k - a ^ 2 + b ^ 2 := by
+  refine Nat.eq_add_of_sub_eq ?_ h₃
+  rw [pow_two]
+  refine le_of_lt ?_
+  refine mul_lt_mul' (by linarith) ?_ (le_of_lt h₀.2.1) h₀.2.1
+  linarith
+
+
+lemma imo_1984_p6_8_4
+  (a b c d k : ℕ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  -- (h₂ : a < b)
+  -- (h₃ : b * 2 ^ m - b ^ 2 = a * 2 ^ k - a ^ 2)
+  (h₄ : a + d = 2 ^ k) :
+  -- (h₅ : b + c = 2 ^ m) :
+  a ^ 2 ≤ a * 2 ^ k := by
+  rw [pow_two]
+  refine le_of_lt ?_
+  refine mul_lt_mul' (by linarith) ?_ (le_of_lt h₀.1) h₀.1
+  linarith
+
+
+lemma imo_1984_p6_8_5
+  (a b : ℕ)
+  -- (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  (h₂ : a < b) :
+  -- h₃ : b * 2 ^ m - b ^ 2 = a * 2 ^ k - a ^ 2
+  -- h₄ : a + d = 2 ^ k
+  -- h₅ : b + c = 2 ^ m
+  -- h₆₀ : c = 2 ^ m - b
+  -- h₆₁ : d = 2 ^ k - a
+  a ^ 2 < b ^ 2 := by
+  exact Nat.pow_lt_pow_left h₂ (by norm_num)
+
+
+lemma imo_1984_p6_8_6
+  (a b k m : ℕ)
+  -- (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  -- (h₂ : a < b ∧ b < c ∧ c < d)
+  -- (h₃ : a * 2 ^ k - a ^ 2 = b * 2 ^ m - b ^ 2)
+  -- (h₄ : a + d = 2 ^ k)
+  -- (h₅ : b + c = 2 ^ m)
+  -- (h₆₀ : c = 2 ^ m - b)
+  -- (h₆₁ : d = 2 ^ k - a)
+  (h₆₂ : b * 2 ^ m - a * 2 ^ k = b ^ 2 - a ^ 2) :
+  b * 2 ^ m - a * 2 ^ k = (b - a) * (b + a) := by
+  rw [Nat.sq_sub_sq b a] at h₆₂
+  rw [mul_comm (b - a) _]
+  exact h₆₂
+
+
+
+
+lemma imo_1984_p6_9
+  (a b k m : ℕ)
+  (hkm : m < k)
+  (h₆ : b * 2 ^ m - a * 2 ^ k = (b - a) * (b + a)) :
+  2 ^ m ∣ (b - a) * (b + a) := by
+  have h₇₀: k = (k - m) + m := by exact (Nat.sub_add_cancel (le_of_lt hkm)).symm
+  rw [h₇₀, pow_add] at h₆
+  have h₇₁: (b - a * 2 ^ (k - m)) * (2 ^ m) = (b - a) * (b + a) := by
+    rw [Nat.mul_sub_right_distrib]
+    rw [mul_assoc a _ _]
+    exact h₆
+  exact Dvd.intro_left (b - a * 2 ^ (k - m)) h₇₁
+
+
+
+
+lemma imo_1984_p6_10
+  (a b c d k m : ℕ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  (h₁ : Odd a ∧ Odd b ∧ Odd c ∧ Odd d)
+  (h₂ : a < b ∧ b < c ∧ c < d)
+  -- (h₃ : a * d = b * c)
+  -- (h₄ : a + d = 2 ^ k)
+  (h₅ : b + c = 2 ^ m)
+  (hkm : m < k)
+  (h₆ : b * 2 ^ m - a * 2 ^ k = (b - a) * (b + a))
+  (h₇ : 2 ^ m ∣ (b - a) * (b + a)) :
+  b + a = 2 ^ (m - 1) := by
+  have h₇₁: ∃ y z, y ∣ b - a ∧ z ∣ b + a ∧ y * z = 2 ^ m := by
+    exact Nat.dvd_mul.mp h₇
+  let ⟨p, q, hpd⟩ := h₇₁
+  cases' hpd with hpd hqd
+  cases' hqd with hqd hpq
+  have hm1: 1 ≤ m := by
+    by_contra! hc
+    interval_cases m
+    linarith
+  have h₈₀: b - a < 2 ^ (m - 1) := by
+    have g₀: b < (b + c) / 2 := by
+      refine (Nat.lt_div_iff_mul_lt' ?_ b).mpr ?_
+      . refine even_iff_two_dvd.mp ?_
+        exact Odd.add_odd h₁.2.1 h₁.2.2.1
+      . linarith
+    have g₁: (b + c) / 2 = 2 ^ (m-1) := by
+      rw [h₅]
+      rw [← Nat.pow_sub_mul_pow 2 hm1]
+      simp
+    rw [← g₁]
+    refine lt_trans ?_ g₀
+    exact Nat.sub_lt h₀.2.1 h₀.1
+  have hp: p = 2 := by
+    have hp₀: 2 * b < 2 ^ m := by
+      rw [← h₅, two_mul]
+      exact Nat.add_lt_add_left h₂.2.1 b
+    have hp₁: b + a < 2 ^ (m) := by
+      have g₀: b + a < b + b := by
+        exact Nat.add_lt_add_left h₂.1 b
+      refine Nat.lt_trans g₀ ?_
+      rw [← two_mul]
+      exact hp₀
+    have hp₂: q < 2 ^ m := by
+      refine Nat.lt_of_le_of_lt (Nat.le_of_dvd ?_ hqd) hp₁
+      exact Nat.add_pos_right b h₀.1
+    have hp₃: 1 < p := by
+      rw [← hpq] at hp₂
+      exact one_lt_of_lt_mul_left hp₂
+    have h2prime: Nat.Prime 2 := by exact prime_two
+    have hp₅: ∀ i j:ℕ , 2 ^ i ∣ (b - a) ∧ 2 ^ j ∣ (b + a) → (i < 2 ∨ j < 2) := by
+      by_contra! hc
+      let ⟨i, j, hi⟩ := hc
+      have hti: 2 ^ 2 ∣ 2 ^ i := by exact Nat.pow_dvd_pow 2 hi.2.1
+      have htj: 2 ^ 2 ∣ 2 ^ j := by exact Nat.pow_dvd_pow 2 hi.2.2
+      norm_num at hti htj
+      have hi₄: 4 ∣ b - a := by exact Nat.dvd_trans hti hi.1.1
+      have hi₅: 4 ∣ b + a := by exact Nat.dvd_trans htj hi.1.2
+      have hi₆: 4 ∣ (b - a) + (b + a) := by exact Nat.dvd_add hi₄ hi₅
+      have hi₇: 2 ∣ b := by
+        have g₀: 0 < 2 := by norm_num
+        refine Nat.dvd_of_mul_dvd_mul_left g₀ ?_
+        rw [← Nat.add_sub_cancel (2 * b) a, Nat.two_mul b]
+        rw [add_assoc, Nat.sub_add_comm (le_of_lt h₂.1)]
+        exact hi₆
+      have hi₈: Even b := by
+        exact even_iff_two_dvd.mpr hi₇
+      apply Nat.not_odd_iff_even.mpr hi₈
+      exact h₁.2.1
+    have hp₆: ∀ i j:ℕ , i + j = m ∧ 2 ^ i ∣ (b - a) ∧ 2 ^ j ∣ (b + a) → (¬ j < 2) := by
+      by_contra! hc
+      let ⟨i, j, hi⟩ := hc
+      have hi₀: m - 1 ≤ i := by
+        rw [← hi.1.1]
+        simp
+        exact Nat.le_pred_of_lt hi.2
+      have hi₁: 2 ^ (m - 1) ≤ 2 ^ i := by exact Nat.pow_le_pow_right (by norm_num) hi₀
+      have hi₂: 2 ^ i < 2 ^ (m - 1) := by
+        refine lt_of_le_of_lt ?_ h₈₀
+        refine Nat.le_of_dvd ?_ hi.1.2.1
+        exact Nat.sub_pos_of_lt h₂.1
+      -- j must be ≤ 1 which gives i ≥ m - 1
+      -- however from h₈₀ we have i < m - 1 leading to a contradiction
+      linarith [hi₁, hi₂]
+    have hi₀: ∃ i ≤ m, p = 2 ^ i := by
+      have g₀: p ∣ 2 ^ m := by
+        rw [← hpq]
+        exact Nat.dvd_mul_right p q
+      exact (Nat.dvd_prime_pow h2prime).mp g₀
+    let ⟨i, hp⟩ := hi₀
+    cases' hp with him hp
+    let j:ℕ := m - i
+    have hj₀: j = m - i := by linarith
+    have hj₁: i + j = m := by
+      rw [add_comm, ← Nat.sub_add_cancel him]
+    have hq: q = 2 ^ j := by
+      rw [hp] at hpq
+      rw [hj₀, ← Nat.pow_div him (by norm_num)]
+      refine Nat.eq_div_of_mul_eq_right ?_ hpq
+      refine Nat.ne_of_gt ?_
+      rw [← hp]
+      linarith [hp₃]
+    rw [hp] at hpd
+    rw [hq] at hqd
+    have hj₃: ¬ j < 2 := by
+      exact hp₆ i j {left:= hj₁ , right:= { left := hpd , right:= hqd} }
+    have hi₂: i < 2 := by
+      have g₀: i < 2 ∨ j < 2 := by
+        exact hp₅ i j { left := hpd , right:= hqd }
+      omega
+    have hi₃: 0 < i := by
+      rw [hp] at hp₃
+      refine Nat.zero_lt_of_ne_zero ?_
+      exact (Nat.one_lt_two_pow_iff).mp hp₃
+    have hi₄: i = 1 := by
+      interval_cases i
+      rfl
+    rw [hi₄] at hp
+    exact hp
+  have hq: q = 2 ^ (m - 1) := by
+    rw [hp, ← Nat.pow_sub_mul_pow 2 hm1, pow_one, mul_comm] at hpq
+    exact Nat.mul_right_cancel (by norm_num) hpq
+  rw [hq] at hqd
+  have h₈₂: ∃ c, (b + a) = c * 2 ^ (m - 1) := by
+    exact exists_eq_mul_left_of_dvd hqd
+  let ⟨f, hf⟩ := h₈₂
+  have hfeq1: f = 1 := by
+    have hf₀: f * 2 ^ (m - 1) < 2 * 2 ^ (m - 1) := by
+      rw [← hf, ← Nat.pow_succ', ← Nat.succ_sub hm1]
+      rw [Nat.succ_sub_one, ← h₅]
+      refine Nat.add_lt_add_left ?_ b
+      exact lt_trans h₂.1 h₂.2.1
+    have hf₁: f < 2 := by
+      exact Nat.lt_of_mul_lt_mul_right hf₀
+    interval_cases f
+    . simp at hf
+      exfalso
+      linarith [hf]
+    . linarith
+  rw [hfeq1, one_mul] at hf
+  exact hf
+
+
+lemma imo_1984_p6_10_1
+  (a b c d m : ℕ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  (h₁ : Odd a ∧ Odd b ∧ Odd c ∧ Odd d)
+  (h₂ : a < b ∧ b < c ∧ c < d)
+  (h₅ : b + c = 2 ^ m)
+  -- hkm : m < k
+  -- h₆ : b * 2 ^ m - a * 2 ^ k = (b - a) * (b + a)
+  -- h₇ : 2 ^ m ∣ (b - a) * (b + a)
+  -- h₇₁ : ∃ y z, y ∣ b - a ∧ z ∣ b + a ∧ y * z = 2 ^ m
+  -- p q : ℕ
+  -- hpd : p ∣ b - a
+  -- hqd : q ∣ b + a
+  -- hpq : p * q = 2 ^ m
+  (hm1 : 1 ≤ m) :
+  b - a < 2 ^ (m - 1) := by
+  have g₀: b < (b + c) / 2 := by
+    refine (Nat.lt_div_iff_mul_lt' ?_ b).mpr ?_
+    . refine even_iff_two_dvd.mp ?_
+      exact Odd.add_odd h₁.2.1 h₁.2.2.1
+    . linarith
+  have g₁: (b + c) / 2 = 2 ^ (m-1) := by
+    rw [h₅]
+    rw [← Nat.pow_sub_mul_pow 2 hm1]
+    simp
+  rw [← g₁]
+  refine lt_trans ?_ g₀
+  exact Nat.sub_lt h₀.2.1 h₀.1
+
+
+lemma imo_1984_p6_10_2
+  (b c: ℕ)
+  (h₁ : Odd b ∧ Odd c)
+  (h₂ : b < c) :
+  b < (b + c) / 2 := by
+  refine (Nat.lt_div_iff_mul_lt' ?_ b).mpr ?_
+  . refine even_iff_two_dvd.mp ?_
+    exact Odd.add_odd h₁.1 h₁.2
+  . linarith
+
+lemma imo_1984_p6_10_3
+  (b c m : ℕ)
+  (h₅ : b + c = 2 ^ m)
+  (hm1 : 1 ≤ m) :
+  (b + c) / 2 = 2 ^ (m - 1) := by
+  rw [h₅]
+  rw [← Nat.pow_sub_mul_pow 2 hm1]
+  simp
+
+
+lemma imo_1984_p6_10_4
+  (a b c m : ℕ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (g₀ : b < (b + c) / 2)
+  (g₁ : (b + c) / 2 = 2 ^ (m - 1)) :
+  b - a < 2 ^ (m - 1) := by
+  rw [← g₁]
+  refine lt_trans ?_ g₀
+  exact Nat.sub_lt h₀.2.1 h₀.1
+
+lemma imo_1984_p6_10_5
+  (a b c m : ℕ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₅ : b + c = 2 ^ m) :
+  1 ≤ m := by
+  by_contra! hc
+  interval_cases m
+  linarith
+
+
+lemma imo_1984_p6_10_6
+  (a b c m : ℕ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
+  (h₁ : Odd a ∧ Odd b ∧ Odd c)
+  (h₂ : a < b ∧ b < c)
+  (h₅ : b + c = 2 ^ m)
+  (p q : ℕ)
+  (hpd : p ∣ b - a)
+  (hqd : q ∣ b + a)
+  (hpq : p * q = 2 ^ m)
+  (h₈₀ : b - a < 2 ^ (m - 1)) :
+  p = 2 := by
+  have hp₀: 2 * b < 2 ^ m := by
+    rw [← h₅, two_mul]
+    exact Nat.add_lt_add_left h₂.2 b
+  have hp₁: b + a < 2 ^ (m) := by
+    have g₀: b + a < b + b := by
+      exact Nat.add_lt_add_left h₂.1 b
+    refine Nat.lt_trans g₀ ?_
+    rw [← two_mul]
+    exact hp₀
+  have hp₂: q < 2 ^ m := by
+    refine Nat.lt_of_le_of_lt (Nat.le_of_dvd ?_ hqd) hp₁
+    exact Nat.add_pos_right b h₀.1
+  have hp₃: 1 < p := by
+    rw [← hpq] at hp₂
+    exact one_lt_of_lt_mul_left hp₂
+  have h2prime: Nat.Prime 2 := by exact prime_two
+  have hp₅: ∀ i j:ℕ , 2 ^ i ∣ (b - a) ∧ 2 ^ j ∣ (b + a) → (i < 2 ∨ j < 2) := by
+    by_contra! hc
+    let ⟨i, j, hi⟩ := hc
+    have hti: 2 ^ 2 ∣ 2 ^ i := by exact Nat.pow_dvd_pow 2 hi.2.1
+    have htj: 2 ^ 2 ∣ 2 ^ j := by exact Nat.pow_dvd_pow 2 hi.2.2
+    norm_num at hti htj
+    have hi₄: 4 ∣ b - a := by exact Nat.dvd_trans hti hi.1.1
+    have hi₅: 4 ∣ b + a := by exact Nat.dvd_trans htj hi.1.2
+    have hi₆: 4 ∣ (b - a) + (b + a) := by exact Nat.dvd_add hi₄ hi₅
+    have hi₇: 2 ∣ b := by
+      have g₀: 0 < 2 := by norm_num
+      refine Nat.dvd_of_mul_dvd_mul_left g₀ ?_
+      rw [← Nat.add_sub_cancel (2 * b) a, Nat.two_mul b]
+      rw [add_assoc, Nat.sub_add_comm (le_of_lt h₂.1)]
+      exact hi₆
+    have hi₈: Even b := by
+      exact even_iff_two_dvd.mpr hi₇
+    apply Nat.not_odd_iff_even.mpr hi₈
+    exact h₁.2.1
+  have hp₆: ∀ i j:ℕ , i + j = m ∧ 2 ^ i ∣ (b - a) ∧ 2 ^ j ∣ (b + a) → (¬ j < 2) := by
+    by_contra! hc
+    let ⟨i, j, hi⟩ := hc
+    have hi₀: m - 1 ≤ i := by
+      rw [← hi.1.1]
+      simp
+      exact Nat.le_pred_of_lt hi.2
+    have hi₁: 2 ^ (m - 1) ≤ 2 ^ i := by exact Nat.pow_le_pow_right (by norm_num) hi₀
+    have hi₂: 2 ^ i < 2 ^ (m - 1) := by
+      refine lt_of_le_of_lt ?_ h₈₀
+      refine Nat.le_of_dvd ?_ hi.1.2.1
+      exact Nat.sub_pos_of_lt h₂.1
+    -- j must be ≤ 1 which gives i ≥ m - 1
+    -- however from h₈₀ we have i < m - 1 leading to a contradiction
+    linarith [hi₁, hi₂]
+  have hi₀: ∃ i ≤ m, p = 2 ^ i := by
+    have g₀: p ∣ 2 ^ m := by
+      rw [← hpq]
+      exact Nat.dvd_mul_right p q
+    exact (Nat.dvd_prime_pow h2prime).mp g₀
+  let ⟨i, hp⟩ := hi₀
+  cases' hp with him hp
+  let j:ℕ := m - i
+  have hj₀: j = m - i := by linarith
+  have hj₁: i + j = m := by
+    rw [add_comm, ← Nat.sub_add_cancel him]
+  have hq: q = 2 ^ j := by
+    rw [hp] at hpq
+    rw [hj₀, ← Nat.pow_div him (by norm_num)]
+    refine Nat.eq_div_of_mul_eq_right ?_ hpq
+    refine Nat.ne_of_gt ?_
+    rw [← hp]
+    linarith [hp₃]
+  rw [hp] at hpd
+  rw [hq] at hqd
+  have hj₃: ¬ j < 2 := by
+    exact hp₆ i j {left:= hj₁ , right:= { left := hpd , right:= hqd} }
+  have hi₂: i < 2 := by
+    have g₀: i < 2 ∨ j < 2 := by
+      exact hp₅ i j { left := hpd , right:= hqd }
+    omega
+  have hi₃: 0 < i := by
+    rw [hp] at hp₃
+    refine Nat.zero_lt_of_ne_zero ?_
+    exact (Nat.one_lt_two_pow_iff).mp hp₃
+  have hi₄: i = 1 := by
+    interval_cases i
+    rfl
+  rw [hi₄] at hp
+  exact hp
+
+lemma imo_1984_p6_10_6_1
+  (a b c m : ℕ)
+  (h₂ : a < b ∧ b < c)
+  (h₅ : b + c = 2 ^ m) :
+  2 * b < 2 ^ m := by
+  rw [← h₅, two_mul]
+  exact Nat.add_lt_add_left h₂.2 b
+
+
+lemma imo_1984_p6_10_6_2
+  (a b c m : ℕ)
+  -- h₀ : 0 < a ∧ 0 < b ∧ 0 < c
+  -- h₁ : Odd a ∧ Odd b ∧ Odd c
+  (h₂ : a < b ∧ b < c)
+  -- h₅ : b + c = 2 ^ m
+  -- p q : ℕ
+  -- hpd : p ∣ b - a
+  -- hqd : q ∣ b + a
+  -- hpq : p * q = 2 ^ m
+  -- h₈₀ : b - a < 2 ^ (m - 1)
+  (hp₀ : 2 * b < 2 ^ m) :
+  b + a < 2 ^ m := by
+  have g₀: b + a < b + b := by
+    exact Nat.add_lt_add_left h₂.1 b
+  refine Nat.lt_trans g₀ ?_
+  rw [← two_mul]
+  exact hp₀
+
+lemma imo_1984_p6_10_6_3
+  -- (a b c m : ℕ)
+  -- h₀ : 0 < a ∧ 0 < b ∧ 0 < c
+  -- h₁ : Odd a ∧ Odd b ∧ Odd c
+  -- h₂ : a < b ∧ b < c
+  -- h₅ : b + c = 2 ^ m
+  (m p q : ℕ)
+  -- hpd : p ∣ b - a
+  -- hqd : q ∣ b + a
+  (hpq : p * q = 2 ^ m)
+  -- h₈₀ : b - a < 2 ^ (m - 1)
+  -- hp₀ : 2 * b < 2 ^ m
+  -- hp₁ : b + a < 2 ^ m
+  (hp₂ : q < 2 ^ m) :
+  1 < p := by
+  rw [← hpq] at hp₂
+  exact one_lt_of_lt_mul_left hp₂
+
+
+lemma imo_1984_p6_10_6_4
+  (a b: ℕ)
+  -- h₀ : 0 < a ∧ 0 < b ∧ 0 < c
+  (h₁ : Odd a ∧ Odd b)
+  (h₂ : a < b) :
+  -- h₅ : b + c = 2 ^ m
+  -- p q : ℕ
+  -- hpd : p ∣ b - a
+  -- hqd : q ∣ b + a
+  -- hpq : p * q = 2 ^ m
+  -- h₈₀ : b - a < 2 ^ (m - 1)
+  -- hp₀ : 2 * b < 2 ^ m
+  -- hp₁ : b + a < 2 ^ m
+  -- hp₂ : q < 2 ^ m
+  -- hp₃ : 1 < p
+  -- h2prime : Nat.Prime 2
+  ∀ (i j : ℕ), 2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a → i < 2 ∨ j < 2 := by
+  by_contra! hc
+  let ⟨i, j, hi⟩ := hc
+  have hti: 2 ^ 2 ∣ 2 ^ i := by exact Nat.pow_dvd_pow 2 hi.2.1
+  have htj: 2 ^ 2 ∣ 2 ^ j := by exact Nat.pow_dvd_pow 2 hi.2.2
+  norm_num at hti htj
+  have hi₄: 4 ∣ b - a := by exact Nat.dvd_trans hti hi.1.1
+  have hi₅: 4 ∣ b + a := by exact Nat.dvd_trans htj hi.1.2
+  have hi₆: 4 ∣ (b - a) + (b + a) := by exact Nat.dvd_add hi₄ hi₅
+  have hi₇: 2 ∣ b := by
+    have g₀: 0 < 2 := by norm_num
+    refine Nat.dvd_of_mul_dvd_mul_left g₀ ?_
+    rw [← Nat.add_sub_cancel (2 * b) a, Nat.two_mul b]
+    rw [add_assoc, Nat.sub_add_comm (le_of_lt h₂)]
+    exact hi₆
+  have hi₈: Even b := by
+    exact even_iff_two_dvd.mpr hi₇
+  apply Nat.not_odd_iff_even.mpr hi₈
+  exact h₁.2
+
+lemma imo_1984_p6_10_6_5
+  (a b c : ℕ)
+  (h₁ : Odd a ∧ Odd b ∧ Odd c)
+  (h₂ : a < b ∧ b < c)
+  -- (hc : ∃ i j, (2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a) ∧ 2 ≤ i ∧ 2 ≤ j)
+  (i j : ℕ)
+  (hi : (2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a) ∧ 2 ≤ i ∧ 2 ≤ j)
+  (hti : 4 ∣ 2 ^ i)
+  (htj : 4 ∣ 2 ^ j) :
+  False := by
+  have hi₄: 4 ∣ b - a := by exact Nat.dvd_trans hti hi.1.1
+  have hi₅: 4 ∣ b + a := by exact Nat.dvd_trans htj hi.1.2
+  have hi₆: 4 ∣ (b - a) + (b + a) := by exact Nat.dvd_add hi₄ hi₅
+  have hi₇: 2 ∣ b := by
+    have g₀: 0 < 2 := by norm_num
+    refine Nat.dvd_of_mul_dvd_mul_left g₀ ?_
+    rw [← Nat.add_sub_cancel (2 * b) a, Nat.two_mul b]
+    rw [add_assoc, Nat.sub_add_comm (le_of_lt h₂.1)]
+    exact hi₆
+  have hi₈: Even b := by
+    exact even_iff_two_dvd.mpr hi₇
+  apply Nat.not_odd_iff_even.mpr hi₈
+  exact h₁.2.1
+
+lemma imo_1984_p6_10_6_6
+  (a b: ℕ)
+  -- (h₁ : Odd a ∧ Odd b ∧ Odd c)
+  -- (h₂ : a < b ∧ b < c)
+  -- hc : ∃ i j, (2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a) ∧ 2 ≤ i ∧ 2 ≤ j
+  (i j : ℕ)
+  (hi : (2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a) ∧ 2 ≤ i ∧ 2 ≤ j)
+  (hti : 4 ∣ 2 ^ i)
+  (htj : 4 ∣ 2 ^ j) :
+  4 ∣ b - a + (b + a) := by
+  have hi₄: 4 ∣ b - a := by exact Nat.dvd_trans hti hi.1.1
+  have hi₅: 4 ∣ b + a := by exact Nat.dvd_trans htj hi.1.2
+  exact Nat.dvd_add hi₄ hi₅
+
+
+lemma imo_1984_p6_10_6_7
+  (a b c : ℕ)
+  -- (h₁ : Odd a ∧ Odd b ∧ Odd c)
+  (h₂ : a < b ∧ b < c)
+  -- hc : ∃ i j, (2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a) ∧ 2 ≤ i ∧ 2 ≤ j
+  (i j : ℕ)
+  (hi : (2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a) ∧ 2 ≤ i ∧ 2 ≤ j)
+  (hti : 4 ∣ 2 ^ i)
+  (htj : 4 ∣ 2 ^ j) :
+  2 ∣ b := by
+  have hi₄: 4 ∣ b - a := by exact Nat.dvd_trans hti hi.1.1
+  have hi₅: 4 ∣ b + a := by exact Nat.dvd_trans htj hi.1.2
+  have hi₆: 4 ∣ (b - a) + (b + a) := by exact Nat.dvd_add hi₄ hi₅
+  have g₀: 0 < 2 := by norm_num
+  refine Nat.dvd_of_mul_dvd_mul_left g₀ ?_
+  rw [← Nat.add_sub_cancel (2 * b) a, Nat.two_mul b]
+  rw [add_assoc, Nat.sub_add_comm (le_of_lt h₂.1)]
+  exact hi₆
+
+
+lemma imo_1984_p6_10_6_8
+  (a b c : ℕ)
+  -- (h₁ : Odd a ∧ Odd b ∧ Odd c)
+  (h₂ : a < b ∧ b < c)
+  -- hc : ∃ i j, (2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a) ∧ 2 ≤ i ∧ 2 ≤ j
+  -- i j : ℕ
+  -- hi : (2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a) ∧ 2 ≤ i ∧ 2 ≤ j
+  -- hti : 4 ∣ 2 ^ i
+  -- htj : 4 ∣ 2 ^ j
+  -- (hi₄ : 4 ∣ b - a)
+  -- (hi₅ : 4 ∣ b + a)
+  (hi₆ : 4 ∣ b - a + (b + a)) :
+  2 ∣ b := by
+  have g₀: 0 < 2 := by norm_num
+  refine Nat.dvd_of_mul_dvd_mul_left g₀ ?_
+  rw [← Nat.add_sub_cancel (2 * b) a, Nat.two_mul b]
+  rw [add_assoc, Nat.sub_add_comm (le_of_lt h₂.1)]
+  exact hi₆
+
+lemma imo_1984_p6_10_6_9
+  (a b c : ℕ)
+  -- (h₁ : Odd a ∧ Odd b ∧ Odd c)
+  (h₂ : a < b ∧ b < c)
+  -- hc : ∃ i j, (2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a) ∧ 2 ≤ i ∧ 2 ≤ j
+  -- i j : ℕ
+  -- hi : (2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a) ∧ 2 ≤ i ∧ 2 ≤ j
+  -- hti : 4 ∣ 2 ^ i
+  -- htj : 4 ∣ 2 ^ j
+  -- (hi₄ : 4 ∣ b - a)
+  -- (hi₅ : 4 ∣ b + a)
+  (hi₆ : 4 ∣ b - a + (b + a)) :
+  Even b := by
+  refine even_iff_two_dvd.mpr ?_
+  have g₀: 0 < 2 := by norm_num
+  refine Nat.dvd_of_mul_dvd_mul_left g₀ ?_
+  rw [← Nat.add_sub_cancel (2 * b) a, Nat.two_mul b]
+  rw [add_assoc, Nat.sub_add_comm (le_of_lt h₂.1)]
+  exact hi₆
+
+lemma imo_1984_p6_10_6_10
+  (a b m : ℕ)
+  -- h₀ : 0 < a ∧ 0 < b ∧ 0 < c
+  -- (h₁ : Odd a ∧ Odd b)
+  (h₂ : a < b)
+  -- (a b c m : ℕ)
+  -- h₀ : 0 < a ∧ 0 < b ∧ 0 < c
+  -- h₁ : Odd a ∧ Odd b ∧ Odd c
+  -- h₂ : a < b ∧ b < c
+  -- h₅ : b + c = 2 ^ m
+  -- p q : ℕ
+  -- hpd : p ∣ b - a
+  -- hqd : q ∣ b + a
+  -- hpq : p * q = 2 ^ m
+  (h₈₀ : b - a < 2 ^ (m - 1)) :
+  -- hp₀ : 2 * b < 2 ^ m
+  -- hp₁ : b + a < 2 ^ m
+  -- hp₂ : q < 2 ^ m
+  -- hp₃ : 1 < p
+  -- h2prime : Nat.Prime 2
+  -- hp₅ : ∀ (i j : ℕ), 2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a → i < 2 ∨ j < 2
+  ∀ (i j : ℕ), i + j = m ∧ 2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a → ¬j < 2 := by
+  by_contra! hc
+  let ⟨i, j, hi⟩ := hc
+  have hi₀: m - 1 ≤ i := by
+    rw [← hi.1.1]
+    simp
+    exact Nat.le_pred_of_lt hi.2
+  have hi₁: 2 ^ (m - 1) ≤ 2 ^ i := by exact Nat.pow_le_pow_right (by norm_num) hi₀
+  have hi₂: 2 ^ i < 2 ^ (m - 1) := by
+    refine lt_of_le_of_lt ?_ h₈₀
+    refine Nat.le_of_dvd ?_ hi.1.2.1
+    exact Nat.sub_pos_of_lt h₂
+  linarith [hi₁, hi₂]
+
+
+lemma imo_1984_p6_10_6_11
+  (m a b : ℕ)
+  -- h₁ : Odd a ∧ Odd b
+  -- h₂ : a < b
+  -- h₈₀ : b - a < 2 ^ (m - 1)
+  -- hc : ∃ i j, (i + j = m ∧ 2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a) ∧ j < 2
+  (i j : ℕ)
+  (hi : (i + j = m ∧ 2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a) ∧ j < 2) :
+  2 ^ (m - 1) ≤ 2 ^ i := by
+  refine Nat.pow_le_pow_right (by norm_num) ?_
+  rw [← hi.1.1]
+  simp
+  exact Nat.le_pred_of_lt hi.2
+
+
+lemma imo_1984_p6_10_6_12
+  (m a b : ℕ)
+  -- h₁ : Odd a ∧ Odd b
+  (h₂ : a < b)
+  (h₈₀ : b - a < 2 ^ (m - 1))
+  -- hc : ∃ i j, (i + j = m ∧ 2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a) ∧ j < 2
+  (i j : ℕ)
+  (hi : (i + j = m ∧ 2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a) ∧ j < 2) :
+  -- hi₀ : m - 1 ≤ i
+  -- (hi₁ : 2 ^ (m - 1) ≤ 2 ^ i) :
+  2 ^ i < 2 ^ (m - 1) := by
+  refine lt_of_le_of_lt ?_ h₈₀
+  refine Nat.le_of_dvd ?_ hi.1.2.1
+  exact Nat.sub_pos_of_lt h₂
+
+
+
+lemma imo_1984_p6_10_6_13
+  -- (a b c : ℕ)
+  (m p q : ℕ)
+  (hpq : p * q = 2 ^ m)
+  -- h₈₀ : b - a < 2 ^ (m - 1)
+  -- hp₀ : 2 * b < 2 ^ m
+  -- hp₁ : b + a < 2 ^ m
+  -- hp₂ : q < 2 ^ m
+  -- hp₃ : 1 < p
+  (h2prime : Nat.Prime 2) :
+  -- hp₅ : ∀ (i j : ℕ), 2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a → i < 2 ∨ j < 2
+  -- hp₆ : ∀ (i j : ℕ), i + j = m ∧ 2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a → ¬j < 2
+  ∃ i ≤ m, p = 2 ^ i := by
+  have g₀: p ∣ 2 ^ m := by
+    rw [← hpq]
+    exact Nat.dvd_mul_right p q
+  exact (Nat.dvd_prime_pow h2prime).mp g₀
+
+
+lemma imo_1984_p6_10_6_14
+  (i m : ℕ)
+  (him : i ≤ m)
+  (j : ℕ := m - i)
+  (hj₀ : j = m - i) :
+  i + j = m := by
+  rw [add_comm, hj₀]
+  exact Nat.sub_add_cancel him
+
+
+lemma imo_1984_p6_10_6_15
+  (p q m j : ℕ)
+  (hpq : p * q = 2 ^ m)
+  (i : ℕ)
+  (him : i ≤ m)
+  (hp : p = 2 ^ i)
+  (hj₀ : j = m - i) :
+  q = 2 ^ j := by
+  rw [hp] at hpq
+  rw [hj₀, ← Nat.pow_div him (by norm_num)]
+  refine Nat.eq_div_of_mul_eq_right ?_ hpq
+  refine Nat.ne_of_gt ?_
+  exact Nat.two_pow_pos i
+
+
+lemma imo_1984_p6_10_6_16
+  (a b p q m : ℕ)
+  (hp₃ : 1 < p)
+  (hp₅ : ∀ (i j : ℕ), 2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a → i < 2 ∨ j < 2)
+  (hp₆ : ∀ (i j : ℕ), i + j = m ∧ 2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a → ¬j < 2)
+  (i j : ℕ)
+  (hp : p = 2 ^ i)
+  (hq : q = 2 ^ j)
+  (hpd : 2 ^ i ∣ b - a)
+  (hqd : 2 ^ j ∣ b + a)
+  (hij : i + j = m) :
+  p = 2 := by
+  have hj₃: ¬ j < 2 := by
+    exact hp₆ i j {left:= hij , right:= { left := hpd , right:= hqd} }
+  have hi₂: i < 2 := by
+    have g₀: i < 2 ∨ j < 2 := by
+      exact hp₅ i j { left := hpd , right:= hqd }
+    omega
+  have hi₃: 0 < i := by
+    rw [hp] at hp₃
+    refine Nat.zero_lt_of_ne_zero ?_
+    exact (Nat.one_lt_two_pow_iff).mp hp₃
+  have hi₄: i = 1 := by
+    exact Nat.eq_of_le_of_lt_succ hi₃ hi₂
+  rw [hi₄] at hp
+  exact hp
+
+
+
+lemma imo_1984_p6_10_6_17
+  (a b m : ℕ)
+  (hp₆ : ∀ (i j : ℕ), i + j = m ∧ 2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a → ¬j < 2)
+  (i j : ℕ)
+  (hpd : 2 ^ i ∣ b - a)
+  (hqd : 2 ^ j ∣ b + a)
+  (hij : i + j = m) :
+  ¬j < 2 := by
+  exact hp₆ i j {left:= hij , right:= { left := hpd , right:= hqd} }
+
+
+
+lemma imo_1984_p6_10_6_18
+  (a b : ℕ)
+  (hp₅ : ∀ (i j : ℕ), 2 ^ i ∣ b - a ∧ 2 ^ j ∣ b + a → i < 2 ∨ j < 2)
+  (i j : ℕ)
+  (hpd : 2 ^ i ∣ b - a)
+  (hqd : 2 ^ j ∣ b + a)
+  (hj : ¬j < 2) :
+  i < 2 := by
+  have g₀: i < 2 ∨ j < 2 := by
+    exact hp₅ i j { left := hpd , right:= hqd }
+  omega
+
+lemma imo_1984_p6_10_6_19
+  (p i : ℕ)
+  (hp₃ : 1 < p)
+  (hp : p = 2 ^ i) :
+  0 < i := by
+  rw [hp] at hp₃
+  refine Nat.zero_lt_of_ne_zero ?_
+  exact (Nat.one_lt_two_pow_iff).mp hp₃
+
+
+lemma imo_1984_p6_10_6_20
+  (p q i j m a b : ℕ)
+  (hp : p = 2 ^ i)
+  (hq : q = 2 ^ j)
+  (hpd : 2 ^ i ∣ b - a)
+  (hqd : 2 ^ j ∣ b + a)
+  (hij : i + j = m)
+  (hj₃ : ¬j < 2)
+  (hi₂ : i < 2)
+  (hi₃ : 0 < i) :
+  p = 2 := by
+  suffices hi: i = 1
+  . rw [hi] at hp
+    exact hp
+  . exact Nat.eq_of_le_of_lt_succ hi₃ hi₂
+
+
+lemma imo_1984_p6_10_7
+  (m p q : ℕ)
+  (hpq : p * q = 2 ^ m)
+  (hm1 : 1 ≤ m)
+  (hp : p = 2) :
+  q = 2 ^ (m - 1) := by
+  rw [hp, ← Nat.pow_sub_mul_pow 2 hm1, pow_one, mul_comm] at hpq
+  exact Nat.mul_right_cancel (by norm_num) hpq
+
+
+lemma imo_1984_p6_10_8
+  (a b c m : ℕ)
+  -- h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d
+  -- h₁ : Odd a ∧ Odd b ∧ Odd c ∧ Odd d
+  (h₂ : a < b ∧ b < c)
+  (h₅ : b + c = 2 ^ m)
+  -- hkm : m < k
+  -- h₆ : b * 2 ^ m - a * 2 ^ k = (b - a) * (b + a)
+  -- h₇ : 2 ^ m ∣ (b - a) * (b + a)
+  -- h₇₁ : ∃ y z, y ∣ b - a ∧ z ∣ b + a ∧ y * z = 2 ^ m
+  (q : ℕ)
+  -- (hpd : p ∣ b - a)
+  (hqd : q ∣ b + a)
+  -- (hpq : p * q = 2 ^ m)
+  (hm1 : 1 ≤ m)
+  (h₈₀ : b - a < 2 ^ (m - 1))
+  -- (hp : p = 2)
+  (hq : q = 2 ^ (m - 1)) :
+  b + a = 2 ^ (m - 1) := by
+  rw [hq] at hqd
+  have h₈₂: ∃ c, (b + a) = c * 2 ^ (m - 1) := by
+    exact exists_eq_mul_left_of_dvd hqd
+  obtain ⟨f, hf⟩ := h₈₂
+  have hfeq1: f = 1 := by
+    have hf₀: f * 2 ^ (m - 1) < 2 * 2 ^ (m - 1) := by
+      rw [← hf, ← Nat.pow_succ', ← Nat.succ_sub hm1]
+      rw [Nat.succ_sub_one, ← h₅]
+      refine Nat.add_lt_add_left ?_ b
+      exact lt_trans h₂.1 h₂.2
+    have hf₁: f < 2 := by
+      exact Nat.lt_of_mul_lt_mul_right hf₀
+    interval_cases f
+    . simp at hf
+      exfalso
+      linarith [hf]
+    . linarith
+  rw [hfeq1, one_mul] at hf
+  exact hf
+
+
+lemma imo_1984_p6_10_8_1
+  (a b m q: ℕ)
+  (hqd : q ∣ b + a)
+  (hq : q = 2 ^ (m - 1)) :
+  ∃ c, b + a = c * 2 ^ (m - 1) := by
+  refine exists_eq_mul_left_of_dvd ?_
+  rw [hq] at hqd
+  exact hqd
+
+
+lemma imo_1984_p6_10_8_2
+  (a b c m : ℕ)
+  (h₂ : a < b ∧ b < c)
+  (h₅ : b + c = 2 ^ m)
+  (hm1 : 1 ≤ m)
+  (f : ℕ)
+  (hf : b + a = f * 2 ^ (m - 1)) :
+  f = 1 := by
+  have hf₀: f * 2 ^ (m - 1) < 2 * 2 ^ (m - 1) := by
+    rw [← hf, ← Nat.pow_succ', ← Nat.succ_sub hm1]
+    rw [Nat.succ_sub_one, ← h₅]
+    refine Nat.add_lt_add_left ?_ b
+    exact lt_trans h₂.1 h₂.2
+  have hf₁: f < 2 := by
+    exact Nat.lt_of_mul_lt_mul_right hf₀
+  interval_cases f
+  . simp at hf
+    exfalso
+    linarith [hf]
+  . linarith
+
+
+lemma imo_1984_p6_10_8_3
+  (a b c m : ℕ)
+  (h₂ : a < b ∧ b < c)
+  (h₅ : b + c = 2 ^ m)
+  (hm1 : 1 ≤ m)
+  (f : ℕ)
+  (hf : b + a = f * 2 ^ (m - 1)) :
+  f < 2 := by
+  have hf₀: f * 2 ^ (m - 1) < 2 * 2 ^ (m - 1) := by
+    rw [← hf, ← Nat.pow_succ', ← Nat.succ_sub hm1]
+    rw [Nat.succ_sub_one, ← h₅]
+    refine Nat.add_lt_add_left ?_ b
+    exact lt_trans h₂.1 h₂.2
+  exact Nat.lt_of_mul_lt_mul_right hf₀
+
+
+lemma imo_1984_p6_10_8_4
+  (a b c m : ℕ)
+  -- (h₀ : 0 < a ∧ 0 < b)
+  (h₂ : a < b ∧ b < c)
+  (f : ℕ)
+  (hf : b + a = f * 2 ^ (m - 1))
+  -- (hf₀ : f * 2 ^ (m - 1) < 2 * 2 ^ (m - 1))
+  (hf₁ : f < 2) :
+  f = 1 := by
+  interval_cases f
+  . simp at hf
+    exfalso
+    linarith [hf]
+  . linarith
+
+
+lemma imo_1984_p6_11
+  (a b c d k m : ℕ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  (h₁ : Odd a ∧ Odd b ∧ Odd c ∧ Odd d)
+  (h₂ : a < b ∧ b < c ∧ c < d)
+  (h₃ : a * d = b * c)
+  -- (h₄ : a + d = 2 ^ k)
+  (h₅ : b + c = 2 ^ m)
+  -- (hkm : m < k)
+  (h₆ : b * 2 ^ m - a * 2 ^ k = (b - a) * (b + a))
+  (h₇ : 2 ^ m ∣ (b - a) * (b + a))
+  (h₈ : b + a = 2 ^ (m - 1)) :
+  a = 2 ^ (2 * m - 2) / 2 ^ k := by
+  have ga: 1 ≤ a := by exact Nat.succ_le_of_lt h₀.1
+  have gb: 3 ≤ b := by
+    by_contra! hc
+    interval_cases b
+    . linarith
+    . linarith [ga, h₂.1]
+    . have hc₁: Odd 2 := by exact h₁.2.1
+      have hc₂: Even 2 := by exact even_iff.mpr rfl
+      have hc₃: ¬ Even 2 := by exact not_even_iff_odd.mpr hc₁
+      exact hc₃ hc₂
+  have gm: 3 ≤ m := by
+    have gm₀: 2 ^ 2 ≤ 2 ^ (m - 1) := by
+      norm_num
+      rw [← h₈]
+      linarith
+    have gm₁: 2 ≤ m - 1 := by
+      exact (Nat.pow_le_pow_iff_right (by norm_num)).mp gm₀
+    omega
+  have g₀: a < 2 ^ (m - 2) := by
+    have g₀₀: a + a < b + a := by simp [h₂.1]
+    rw [h₈, ← mul_two a] at g₀₀
+    have g₀₁: m - 1 = Nat.succ (m - 2) := by
+      rw [← Nat.succ_sub ?_]
+      . rw [succ_eq_add_one]
+        omega
+      . linarith
+    rw [g₀₁, Nat.pow_succ 2 _] at g₀₀
+    exact Nat.lt_of_mul_lt_mul_right g₀₀
+  have h₉₀: b = 2 ^ (m - 1) - a := by
+    symm
+    exact Nat.sub_eq_of_eq_add h₈.symm
+  rw [h₈, h₉₀] at h₆
+  repeat rw [Nat.mul_sub_right_distrib] at h₆
+  repeat rw [← Nat.pow_add] at h₆
+  have hm1: 1 ≤ m := by
+    linarith
+  repeat rw [← Nat.sub_add_comm hm1] at h₆
+  repeat rw [← Nat.add_sub_assoc hm1] at h₆
+  ring_nf at h₆
+  rw [← Nat.sub_add_eq _ 1 1] at h₆
+  norm_num at h₆
+  rw [← Nat.sub_add_eq _ (a * 2 ^ (m - 1)) (a * 2 ^ (m - 1))] at h₆
+  rw [← two_mul (a * 2 ^ (m - 1))] at h₆
+  rw [mul_comm 2 _] at h₆
+  rw [mul_assoc a (2 ^ (m - 1)) 2] at h₆
+  rw [← Nat.pow_succ, succ_eq_add_one] at h₆
+  rw [Nat.sub_add_cancel hm1] at h₆
+  rw [← Nat.sub_add_eq ] at h₆
+  have h₉₁: 2 ^ (m * 2 - 1) = 2 ^ (m * 2 - 2) - a * 2 ^ m + (a * 2 ^ m + a * 2 ^ k) := by
+    refine Nat.eq_add_of_sub_eq ?_ h₆
+    by_contra! hc
+    have g₁: 2 ^ (m * 2 - 1) - (a * 2 ^ m + a * 2 ^ k) = 0 := by
+      exact Nat.sub_eq_zero_of_le (le_of_lt hc)
+    rw [g₁] at h₆
+    have g₂: 2 ^ (m * 2 - 2) ≤ a * 2 ^ m := by exact Nat.le_of_sub_eq_zero h₆.symm
+    have g₃: 2 ^ (m - 2) ≤ a := by
+      rw [mul_two, Nat.add_sub_assoc (by linarith) m] at g₂
+      rw [Nat.pow_add, mul_comm] at g₂
+      refine Nat.le_of_mul_le_mul_right g₂ ?_
+      exact Nat.two_pow_pos m
+    linarith [g₀, g₃]
+  rw [← Nat.add_assoc] at h₉₁
+  have h₉₂: a * 2 ^ k = 2 * 2 ^ (2 * m - 2) - 2 ^ (2 * m - 2) := by
+    rw [Nat.sub_add_cancel ?_] at h₉₁
+    . rw [add_comm] at h₉₁
+      symm
+      rw [← Nat.pow_succ', succ_eq_add_one]
+      rw [← Nat.sub_add_comm ?_]
+      . simp
+        rw [mul_comm 2 m]
+        exact Nat.sub_eq_of_eq_add h₉₁
+      . linarith [hm1]
+    . refine le_of_lt ?_
+      rw [mul_two, Nat.add_sub_assoc, Nat.pow_add, mul_comm (2 ^ m) _]
+      refine (Nat.mul_lt_mul_right (by linarith)).mpr g₀
+      linarith
+  nth_rewrite 2 [← Nat.one_mul (2 ^ (2 * m - 2))] at h₉₂
+  rw [← Nat.mul_sub_right_distrib 2 1 (2 ^ (2 * m - 2))] at h₉₂
+  norm_num at h₉₂
+  refine Nat.eq_div_of_mul_eq_left ?_ h₉₂
+  exact Ne.symm (NeZero.ne' (2 ^ k))
+
+
+lemma imo_1984_p6_11_1
+  (a b c d: ℕ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  (h₁ : Odd a ∧ Odd b ∧ Odd c ∧ Odd d)
+  (h₂ : a < b ∧ b < c ∧ c < d) :
+  -- (h₃ : a * d = b * c)
+  -- (h₅ : b + c = 2 ^ m)
+  -- (h₆ : b * 2 ^ m - a * 2 ^ k = (b - a) * (b + a))
+  -- (h₇ : 2 ^ m ∣ (b - a) * (b + a))
+  -- (h₈ : b + a = 2 ^ (m - 1))
+  3 ≤ b := by
+  by_contra! hc
+  interval_cases b
+  . linarith
+  . linarith [h₀.1, h₂.1]
+  . have hc₀: Odd 2 := by exact h₁.2.1
+    have hc₁: ¬ Odd 2 := by decide
+    exact hc₁ hc₀
+
+
+lemma imo_1984_p6_11_2
+  (a b m : ℕ)
+  -- (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  -- h₁ : Odd a ∧ Odd b ∧ Odd c ∧ Odd d
+  -- h₂ : a < b ∧ b < c ∧ c < d
+  -- h₃ : a * d = b * c
+  -- h₅ : b + c = 2 ^ m
+  -- h₆ : b * 2 ^ m - a * 2 ^ k = (b - a) * (b + a)
+  -- h₇ : 2 ^ m ∣ (b - a) * (b + a)
+  (h₈ : b + a = 2 ^ (m - 1))
+  (ga : 1 ≤ a)
+  (gb : 3 ≤ b) :
+  3 ≤ m := by
+  have gm₀: 2 ^ 2 ≤ 2 ^ (m - 1) := by
+    norm_num
+    rw [← h₈]
+    linarith
+  have gm₁: 2 ≤ m - 1 := by
+    exact (Nat.pow_le_pow_iff_right (by norm_num)).mp gm₀
+  omega
+
+
+lemma imo_1984_p6_11_3
+  (a b m : ℕ)
+  (h₂ : a < b)
+  (h₈ : b + a = 2 ^ (m - 1))
+  (gm : 3 ≤ m) :
+  a < 2 ^ (m - 2) := by
+  have g₀₀: a + a < b + a := by simp [h₂]
+  rw [h₈, ← mul_two a] at g₀₀
+  have g₀₁: m - 1 = Nat.succ (m - 2) := by
+    rw [← Nat.succ_sub ?_]
+    . rw [succ_eq_add_one]
+      omega
+    . linarith
+  rw [g₀₁, Nat.pow_succ 2 _] at g₀₀
+  exact Nat.lt_of_mul_lt_mul_right g₀₀
+
+
+lemma imo_1984_p6_11_4
+  (a b k m : ℕ)
+  (h₆ : b * 2 ^ m - a * 2 ^ k = (b - a) * (b + a))
+  (h₈ : b + a = 2 ^ (m - 1))
+  (h₉ : b = 2 ^ (m - 1) - a)
+  (hm1 : 1 ≤ m) :
+  2 ^ (m * 2 - 1) - (a * 2 ^ m + a * 2 ^ k) = 2 ^ (m * 2 - 2) - a * 2 ^ m := by
+  rw [h₈, h₉] at h₆
+  repeat rw [Nat.mul_sub_right_distrib] at h₆
+  repeat rw [← Nat.pow_add] at h₆
+  repeat rw [← Nat.sub_add_comm hm1] at h₆
+  repeat rw [← Nat.add_sub_assoc hm1] at h₆
+  ring_nf at h₆
+  rw [← Nat.sub_add_eq _ 1 1] at h₆
+  norm_num at h₆
+  rw [← Nat.sub_add_eq _ (a * 2 ^ (m - 1)) (a * 2 ^ (m - 1))] at h₆
+  rw [← two_mul (a * 2 ^ (m - 1))] at h₆
+  rw [mul_comm 2 _] at h₆
+  rw [mul_assoc a (2 ^ (m - 1)) 2] at h₆
+  rw [← Nat.pow_succ, succ_eq_add_one] at h₆
+  rw [Nat.sub_add_cancel hm1] at h₆
+  rw [← Nat.sub_add_eq ] at h₆
+  exact h₆
+
+
+lemma imo_1984_p6_11_5
+  (a k m : ℕ)
+  -- (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  -- (h₁ : Odd a ∧ Odd b ∧ Odd c ∧ Odd d)
+  -- (h₂ : a < b ∧ b < c ∧ c < d)
+  -- (h₃ : a * d = b * c)
+  -- (h₅ : b + c = 2 ^ m)
+  -- (h₇ : 2 ^ m ∣ (b - a) * (b + a))
+  -- (h₈ : b + a = 2 ^ (m - 1))
+  -- (ga : 1 ≤ a)
+  -- (gb : 3 ≤ b)
+  (gm : 3 ≤ m)
+  (g₀ : a < 2 ^ (m - 2))
+  -- (h₉ : b = 2 ^ (m - 1) - a)
+  -- (hm1 : 1 ≤ m)
+  (h₆ : 2 ^ (m * 2 - 1) - (a * 2 ^ m + a * 2 ^ k) = 2 ^ (m * 2 - 2) - a * 2 ^ m) :
+  2 ^ (m * 2 - 1) = 2 ^ (m * 2 - 2) - a * 2 ^ m + (a * 2 ^ m + a * 2 ^ k) := by
+  refine Nat.eq_add_of_sub_eq ?_ h₆
+  by_contra! hc
+  have g₁: 2 ^ (m * 2 - 1) - (a * 2 ^ m + a * 2 ^ k) = 0 := by
+    exact Nat.sub_eq_zero_of_le (le_of_lt hc)
+  rw [g₁] at h₆
+  have g₂: 2 ^ (m * 2 - 2) ≤ a * 2 ^ m := by exact Nat.le_of_sub_eq_zero h₆.symm
+  have g₃: 2 ^ (m - 2) ≤ a := by
+    rw [mul_two, Nat.add_sub_assoc (by linarith) m] at g₂
+    rw [Nat.pow_add, mul_comm] at g₂
+    refine Nat.le_of_mul_le_mul_right g₂ ?_
+    exact Nat.two_pow_pos m
+  linarith [g₀, g₃]
+
+
+lemma imo_1984_p6_11_6
+  (a b c d k m : ℕ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  -- h₁ : Odd a ∧ Odd b ∧ Odd c ∧ Odd d
+  -- h₂ : a < b ∧ b < c ∧ c < d
+  -- h₃ : a * d = b * c
+  (h₅ : b + c = 2 ^ m)
+  -- (h₇ : 2 ^ m ∣ (b - a) * (b + a))
+  -- h₈ : b + a = 2 ^ (m - 1)
+  -- ga : 1 ≤ a
+  -- gb : 3 ≤ b
+  (gm : 3 ≤ m)
+  (g₀ : a < 2 ^ (m - 2))
+  -- h₉₀ : b = 2 ^ (m - 1) - a
+  (hm1 : 1 ≤ m)
+  -- h₆ : 2 ^ (m * 2 - 1) - (a * 2 ^ m + a * 2 ^ k) = 2 ^ (m * 2 - 2) - a * 2 ^ m
+  (h₉₁ : 2 ^ (m * 2 - 1) = 2 ^ (m * 2 - 2) - a * 2 ^ m + a * 2 ^ m + a * 2 ^ k) :
+  a * 2 ^ k = 2 * 2 ^ (2 * m - 2) - 2 ^ (2 * m - 2) := by
+  rw [Nat.sub_add_cancel ?_] at h₉₁
+  . rw [add_comm] at h₉₁
+    symm
+    rw [← Nat.pow_succ', succ_eq_add_one]
+    rw [← Nat.sub_add_comm ?_]
+    . simp
+      rw [mul_comm 2 m]
+      exact Nat.sub_eq_of_eq_add h₉₁
+    . linarith [hm1]
+  . refine le_of_lt ?_
+    rw [mul_two, Nat.add_sub_assoc, Nat.pow_add, mul_comm (2 ^ m) _]
+    . refine (Nat.mul_lt_mul_right ?_).mpr g₀
+      linarith
+    . linarith
+
+lemma imo_1984_p6_11_7
+  (a k m : ℕ)
+  (h₉₂ : a * 2 ^ k = 2 * 2 ^ (2 * m - 2) - 2 ^ (2 * m - 2)) :
+  a = 2 ^ (2 * m - 2) / 2 ^ k := by
+  nth_rewrite 2 [← Nat.one_mul (2 ^ (2 * m - 2))] at h₉₂
+  rw [← Nat.mul_sub_right_distrib 2 1 (2 ^ (2 * m - 2))] at h₉₂
+  norm_num at h₉₂
+  refine Nat.eq_div_of_mul_eq_left ?_ h₉₂
+  exact Ne.symm (NeZero.ne' (2 ^ k))
+
+
+
+lemma imo_1984_p6_12
+  (a b c d k m : ℕ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  (h₁ : Odd a ∧ Odd b ∧ Odd c ∧ Odd d)
+  (h₂ : a < b ∧ b < c ∧ c < d)
+  (h₃ : a * d = b * c)
+  (h₄ : a + d = 2 ^ k)
+  -- (h₅ : b + c = 2 ^ m)
+  -- (hkm : m < k)
+  (h₆ : b * 2 ^ m - a * 2 ^ k = (b - a) * (b + a))
+  (h₇ : 2 ^ m ∣ (b - a) * (b + a))
+  (h₈ : b + a = 2 ^ (m - 1))
+  (h₉ : a = 2 ^ (2 * m - 2) / 2 ^ k) :
+  a = 1 := by
+  by_cases h₁₀: k ≤ 2 * m - 2
+  . rw [Nat.pow_div h₁₀ (by norm_num)] at h₉
+    rw [Nat.sub_right_comm (2*m) 2 k] at h₉
+    by_contra! hc
+    cases' (lt_or_gt_of_ne hc) with hc₀ hc₁
+    . interval_cases a
+      linarith
+    . have hc₂: ¬ Odd a := by
+        refine (not_odd_iff_even).mpr ?_
+        have hc₃: 1 ≤ 2 * m - k - 2 := by
+          by_contra! hc₄
+          interval_cases (2 * m - k - 2)
+          simp at h₉
+          rw [h₉] at hc₁
+          contradiction
+        have hc₄: 2 * m - k - 2 = succ (2 * m - k - 3) := by
+          rw [succ_eq_add_one]
+          exact Nat.eq_add_of_sub_eq hc₃ rfl
+        rw [h₉, hc₄, Nat.pow_succ']
+        exact even_two_mul (2 ^ (2 * m - k - 3))
+      exact hc₂ h₁.1
+  . push_neg at h₁₀
+    exfalso
+    have ha: a = 0 := by
+      rw [h₉]
+      refine (Nat.div_eq_zero_iff).mpr ?_
+      right
+      refine Nat.pow_lt_pow_right ?_ h₁₀
+      exact Nat.one_lt_two
+    linarith [ha, h₀.1]
+
+
+
+lemma imo_1984_p6_13
+  (a b c d k m : ℕ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  (h₁ : Odd a ∧ Odd b ∧ Odd c ∧ Odd d)
+  (h₂ : a < b ∧ b < c ∧ c < d)
+  (h₃ : a * d = b * c)
+  (h₄ : a + d = 2 ^ k)
+  -- (h₅ : b + c = 2 ^ m)
+  -- (hkm : m < k)
+  (h₆ : b * 2 ^ m - a * 2 ^ k = (b - a) * (b + a))
+  (h₇ : 2 ^ m ∣ (b - a) * (b + a))
+  (h₈ : b + a = 2 ^ (m - 1))
+  (h₉ : a = 2 ^ (2 * m - 2) / 2 ^ k)
+  (h₁₀: k ≤ 2 * m - 2) :
+  a = 1 := by
+  rw [Nat.pow_div h₁₀ (by norm_num)] at h₉
+  rw [Nat.sub_right_comm (2*m) 2 k] at h₉
+  by_contra! hc
+  cases' (lt_or_gt_of_ne hc) with hc₀ hc₁
+  . interval_cases a
+    linarith
+  . have hc₂: ¬ Odd a := by
+      refine (not_odd_iff_even).mpr ?_
+      have hc₃: 1 ≤ 2 * m - k - 2 := by
+        by_contra! hc₄
+        interval_cases (2 * m - k - 2)
+        simp at h₉
+        rw [h₉] at hc₁
+        contradiction
+      have hc₄: 2 * m - k - 2 = succ (2 * m - k - 3) := by
+        rw [succ_eq_add_one]
+        exact Nat.eq_add_of_sub_eq hc₃ rfl
+      rw [h₉, hc₄, Nat.pow_succ']
+      exact even_two_mul (2 ^ (2 * m - k - 3))
+    exact hc₂ h₁.1
+
+
+lemma imo_1984_p6_13_1
+  (a b c d k m : ℕ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  (h₁ : Odd a ∧ Odd b ∧ Odd c ∧ Odd d)
+  (h₂ : a < b ∧ b < c ∧ c < d)
+  (h₃ : a * d = b * c)
+  (h₄ : a + d = 2 ^ k)
+  (h₆ : b * 2 ^ m - a * 2 ^ k = (b - a) * (b + a))
+  (h₇ : 2 ^ m ∣ (b - a) * (b + a))
+  (h₈ : b + a = 2 ^ (m - 1))
+  (h₉ : a = 2 ^ (2 * m - k - 2))
+  -- (h₁₀ : k ≤ 2 * m - 2)
+  (hc : a < 1) :
+  False := by
+  interval_cases a
+  linarith
+
+
+lemma imo_1984_p6_13_2
+  (a b c d k m : ℕ)
+  -- (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  (h₁ : Odd a ∧ Odd b ∧ Odd c ∧ Odd d)
+  -- (h₂ : a < b ∧ b < c ∧ c < d)
+  -- (h₃ : a * d = b * c)
+  -- (h₄ : a + d = 2 ^ k)
+  -- (h₆ : b * 2 ^ m - a * 2 ^ k = (b - a) * (b + a))
+  -- (h₇ : 2 ^ m ∣ (b - a) * (b + a))
+  -- (h₈ : b + a = 2 ^ (m - 1))
+  (h₉ : a = 2 ^ (2 * m - k - 2))
+  -- (h₁₀ : k ≤ 2 * m - 2)
+  (hc : 1 < a) :
+  False := by
+  have hc₂: ¬ Odd a := by
+    refine (not_odd_iff_even).mpr ?_
+    have hc₃: 1 ≤ 2 * m - k - 2 := by
+      by_contra! hc₄
+      interval_cases (2 * m - k - 2)
+      simp at h₉
+      rw [h₉] at hc
+      contradiction
+    have hc₄: 2 * m - k - 2 = succ (2 * m - k - 3) := by
+      rw [succ_eq_add_one]
+      exact Nat.eq_add_of_sub_eq hc₃ rfl
+    rw [h₉, hc₄, Nat.pow_succ']
+    exact even_two_mul (2 ^ (2 * m - k - 3))
+  exact hc₂ h₁.1
+
+
+lemma imo_1984_p6_13_3
+  (a k m : ℕ)
+  (h₉ : a = 2 ^ (2 * m - k - 2))
+  (hc : 1 < a) :
+  1 ≤ 2 * m - k - 2 := by
+  by_contra! hc₄
+  interval_cases (2 * m - k - 2)
+  simp at h₉
+  rw [h₉] at hc
+  contradiction
+
+
+lemma imo_1984_p6_13_4
+  (a k m : ℕ)
+  (h₉ : a = 2 ^ (2 * m - k - 2))
+  (hc : 1 < a) :
+  Even a := by
+  have hc₃: 1 ≤ 2 * m - k - 2 := by
+    by_contra! hc₄
+    interval_cases (2 * m - k - 2)
+    simp at h₉
+    rw [h₉] at hc
+    contradiction
+  have hc₄: 2 * m - k - 2 = succ (2 * m - k - 3) := by
+    rw [succ_eq_add_one]
+    exact Nat.eq_add_of_sub_eq hc₃ rfl
+  rw [h₉, hc₄, Nat.pow_succ']
+  exact even_two_mul (2 ^ (2 * m - k - 3))
+
+
+lemma imo_1984_p6_14
+  (a k m : ℕ)
+  (h₀ : 0 < a)
+  -- (h₁ : Odd a ∧ Odd b ∧ Odd c ∧ Odd d)
+  -- (h₂ : a < b ∧ b < c ∧ c < d)
+  -- (h₃ : a * d = b * c)
+  -- (h₄ : a + d = 2 ^ k)
+  -- (h₅ : b + c = 2 ^ m)
+  -- (hkm : m < k)
+  -- (h₆ : b * 2 ^ m - a * 2 ^ k = (b - a) * (b + a))
+  -- (h₇ : 2 ^ m ∣ (b - a) * (b + a))
+  -- (h₈ : b + a = 2 ^ (m - 1))
+  (h₉ : a = 2 ^ (2 * m - 2) / 2 ^ k)
+  (hk2m : 2 * m - 2 < k) :
+  False := by
+  have ha: a = 0 := by
+    rw [h₉]
+    refine (Nat.div_eq_zero_iff).mpr ?_
+    right
+    exact Nat.pow_lt_pow_right (by norm_num) hk2m
+  linarith [ha, h₀]
+
+
+lemma imo_1984_p6_15
+  (a k m : ℕ)
+  (h₉ : a = 2 ^ (2 * m - 2) / 2 ^ k)
+  (hk2m : 2 * m - 2 < k) :
+  a = 0 := by
+  rw [h₉]
+  refine (Nat.div_eq_zero_iff).mpr ?_
+  right
+  exact Nat.pow_lt_pow_right (by norm_num) hk2m

Lemmas/imo_1992_p1_lemmas.lean

@@ -0,0 +1,2081 @@
+import Mathlib
+set_option linter.unusedVariables.analyzeTactics true
+
+open Int Rat
+
+
+lemma imo_1992_p1_1
+  (p q r: ℤ)
+  (hpl: 4 ≤ p)
+  (hql: 5 ≤ q)
+  (hrl: 6 ≤ r) :
+  (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ ↑2 := by
+  have h₁: (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ)
+    = (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)) := by
+    norm_cast
+    simp
+  have hp: (↑p/↑(p-1):ℚ) ≤ ((4/3):ℚ) := by
+    have g₁: 0 < (↑(p - 1):ℚ) := by
+      norm_cast
+      linarith [hpl]
+    have g₂: ↑p * ↑(3:ℚ) ≤ ↑(4:ℚ) * (↑(p - 1):ℚ) := by
+      norm_cast
+      linarith
+    refine (div_le_iff₀ g₁).mpr ?_
+    rw [div_mul_eq_mul_div]
+    refine (le_div_iff₀ ?_).mpr g₂
+    norm_num
+  have hq: (↑q/↑(q-1)) ≤ ((5/4):ℚ) := by
+    have g₁: 0 < (↑(q - 1):ℚ) := by
+      norm_cast
+      linarith[hql]
+    have g₂: ↑q * ↑(4:ℚ) ≤ ↑(5:ℚ) * (↑(q - 1):ℚ) := by
+      norm_cast
+      linarith
+    refine (div_le_iff₀ g₁).mpr ?_
+    rw [div_mul_eq_mul_div]
+    refine (le_div_iff₀ ?_).mpr g₂
+    norm_num
+  have hr: (↑r/↑(r-1)) ≤ ((6/5):ℚ) := by
+    have g₁: 0 < (↑(r - 1):ℚ) := by
+      norm_cast
+      linarith[hql]
+    have g₂: ↑r * ↑(5:ℚ) ≤ ↑(6:ℚ) * (↑(r - 1):ℚ) := by
+      norm_cast
+      linarith
+    refine (div_le_iff₀ g₁).mpr ?_
+    rw [div_mul_eq_mul_div]
+    refine (le_div_iff₀ ?_).mpr g₂
+    norm_num
+  have hub: (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)) ≤ (4/3:ℚ) * ((5/4):ℚ) * ((6/5):ℚ) := by
+    have hq_nonneg: 0 ≤ (↑q:ℚ) := by
+      norm_cast
+      linarith
+    have hq_1_nonneg: 0 ≤ (↑(q - 1):ℚ) := by
+      norm_cast
+      linarith
+    have h₂: 0 ≤ (((q:ℚ) / ↑(q - 1)):ℚ) := by
+      exact div_nonneg hq_nonneg hq_1_nonneg
+    have hub1: (↑p/↑(p-1)) * (↑q/↑(q-1)) ≤ ((4/3):ℚ) * ((5/4):ℚ) := by
+      exact mul_le_mul hp hq h₂ (by norm_num)
+    have hr_nonneg: 0 ≤ (↑r:ℚ) := by
+      norm_cast
+      linarith
+    have hr_1_nonneg: 0 ≤ (↑(r - 1):ℚ) := by
+      norm_cast
+      linarith
+    have h₃: 0 ≤ (((r:ℚ) / ↑(r - 1)):ℚ) := by
+      exact div_nonneg hr_nonneg hr_1_nonneg
+    exact mul_le_mul hub1 hr h₃ (by norm_num)
+  norm_num at hub
+  rw [h₁]
+  norm_num
+  exact hub
+
+
+lemma imo_1992_p1_1_1
+  (p : ℤ)
+  (hpl : 4 ≤ p) :
+  ↑p / ↑(p - 1) ≤ ((4/3):ℚ) := by
+  have g₁: 0 < (↑(p - 1):ℚ) := by
+    norm_cast
+    linarith [hpl]
+  have g₂: ↑p * ↑(3:ℚ) ≤ ↑(4:ℚ) * (↑(p - 1):ℚ) := by
+    norm_cast
+    linarith
+  refine (div_le_iff₀ g₁).mpr ?_
+  rw [div_mul_eq_mul_div]
+  refine (le_div_iff₀ ?_).mpr g₂
+  norm_num
+
+
+lemma imo_1992_p1_1_2
+  (p : ℤ)
+  -- (q r : ℤ)
+  -- (hpl : 4 ≤ p)
+  -- (hql : 5 ≤ q)
+  -- (hrl : 6 ≤ r)
+  -- (h₁ : ↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)) = ↑p / ↑(p - 1) * (↑q / ↑(q - 1)) * (↑r / ↑(r - 1)))
+  (g₁ : 0 < (↑(p - 1):ℚ))
+  (g₂ : ↑p * ↑(3:ℚ) ≤ ↑(4:ℚ) * (↑(p - 1):ℚ)) :
+  ↑p / ↑(p - 1) ≤ ((4/3):ℚ) := by
+  refine (div_le_iff₀ g₁).mpr ?_
+  rw [div_mul_eq_mul_div]
+  refine (le_div_iff₀ ?_).mpr g₂
+  norm_num
+
+
+lemma imo_1992_p1_1_3
+  -- (p r : ℤ)
+  (q: ℤ)
+  -- (hpl : 4 ≤ p)
+  (hql : 5 ≤ q) :
+  -- (hrl : 6 ≤ r)
+  -- (h₁ : ↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)) = ↑p / ↑(p - 1) * (↑q / ↑(q - 1)) * (↑r / ↑(r - 1)))
+  -- (hp : ↑p / ↑(p - 1) ≤ 4 / 3) :
+  ↑q / ↑(q - 1) ≤ ((5 / 4):ℚ) := by
+  have g₁: 0 < (↑(q - 1):ℚ) := by
+    norm_cast
+    linarith[hql]
+  have g₂: ↑q * ↑(4:ℚ) ≤ ↑(5:ℚ) * (↑(q - 1):ℚ) := by
+    norm_cast
+    linarith
+  refine (div_le_iff₀ g₁).mpr ?_
+  rw [div_mul_eq_mul_div]
+  refine (le_div_iff₀ ?_).mpr g₂
+  norm_num
+
+
+lemma imo_1992_p1_1_4
+  -- (p r : ℤ)
+  (q: ℤ)
+  -- (hpl : 4 ≤ p)
+  -- (hql : 5 ≤ q)
+  -- (hrl : 6 ≤ r)
+  -- (h₁ : ↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)) = ↑p / ↑(p - 1) * (↑q / ↑(q - 1)) * (↑r / ↑(r - 1)))
+  -- (hp : ↑p / ↑(p - 1) ≤ 4 / 3)
+  (g₁ : 0 < (↑(q - 1):ℚ))
+  (g₂ : ↑q * ↑(4:ℚ) ≤ ↑(5:ℚ) * (↑(q - 1):ℚ)) :
+  ↑q / ↑(q - 1) ≤ ((5 / 4):ℚ) := by
+  refine (div_le_iff₀ g₁).mpr ?_
+  rw [div_mul_eq_mul_div]
+  refine (le_div_iff₀ ?_).mpr g₂
+  norm_num
+
+
+lemma imo_1992_p1_1_5
+  (p q r : ℤ)
+  -- (hpl : 4 ≤ p)
+  (hql : 5 ≤ q)
+  (hrl : 6 ≤ r)
+  (h₁ : (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ)
+          = (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)))
+  (hp : ↑p / ↑(p - 1) ≤ ((4 / 3):ℚ))
+  (hq : ↑q / ↑(q - 1) ≤ ((5 / 4):ℚ)) :
+  (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ ↑2 := by
+  have hr: (↑r/↑(r-1)) ≤ ((6/5):ℚ) := by
+    have g₁: 0 < (↑(r - 1):ℚ) := by
+      norm_cast
+      linarith[hql]
+    have g₂: ↑r * ↑(5:ℚ) ≤ ↑(6:ℚ) * (↑(r - 1):ℚ) := by
+      norm_cast
+      linarith
+    refine (div_le_iff₀ g₁).mpr ?_
+    rw [div_mul_eq_mul_div]
+    refine (le_div_iff₀ ?_).mpr g₂
+    norm_num
+  have hub: (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)) ≤ (4/3:ℚ) * ((5/4):ℚ) * ((6/5):ℚ) := by
+    have hq_nonneg: 0 ≤ (↑q:ℚ) := by
+      norm_cast
+      linarith
+    have hq_1_nonneg: 0 ≤ (↑(q - 1):ℚ) := by
+      norm_cast
+      linarith
+    have h₂: 0 ≤ (((q:ℚ) / ↑(q - 1)):ℚ) := by
+      exact div_nonneg hq_nonneg hq_1_nonneg
+    have hub1: (↑p/↑(p-1)) * (↑q/↑(q-1)) ≤ ((4/3):ℚ) * ((5/4):ℚ) := by
+      exact mul_le_mul hp hq h₂ (by norm_num)
+    have hr_nonneg: 0 ≤ (↑r:ℚ) := by
+      norm_cast
+      linarith
+    have hr_1_nonneg: 0 ≤ (↑(r - 1):ℚ) := by
+      norm_cast
+      linarith
+    have h₃: 0 ≤ (((r:ℚ) / ↑(r - 1)):ℚ) := by
+      exact div_nonneg hr_nonneg hr_1_nonneg
+    exact mul_le_mul hub1 hr h₃ (by norm_num)
+  norm_num at hub
+  rw [h₁]
+  norm_num
+  exact hub
+
+
+lemma imo_1992_p1_1_6
+  -- (p : ℤ)
+  (q r : ℤ)
+  -- (hpl : 4 ≤ p)
+  (hql : 5 ≤ q)
+  (hrl : 6 ≤ r) :
+  -- (h₁ : ↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)) = ↑p / ↑(p - 1) * (↑q / ↑(q - 1)) * (↑r / ↑(r - 1)))
+  -- (hp : ↑p / ↑(p - 1) ≤ 4 / 3)
+  -- (hq : ↑q / ↑(q - 1) ≤ 5 / 4) :
+  ↑r / ↑(r - 1) ≤ ((6/5):ℚ) := by
+  have g₁: 0 < (↑(r - 1):ℚ) := by
+    norm_cast
+    linarith[hql]
+  have g₂: ↑r * ↑(5:ℚ) ≤ ↑(6:ℚ) * (↑(r - 1):ℚ) := by
+    norm_cast
+    linarith
+  refine (div_le_iff₀ g₁).mpr ?_
+  rw [div_mul_eq_mul_div]
+  refine (le_div_iff₀ ?_).mpr g₂
+  norm_num
+
+
+lemma imo_1992_p1_1_7
+  -- (p q : ℤ)
+  (r : ℤ)
+  -- (hpl : 4 ≤ p)
+  -- (hql : 5 ≤ q)
+  -- (hrl : 6 ≤ r)
+  -- (h₁ : ↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)) = ↑p / ↑(p - 1) * (↑q / ↑(q - 1)) * (↑r / ↑(r - 1)))
+  -- (hp : ↑p / ↑(p - 1) ≤ 4 / 3)
+  -- (hq : ↑q / ↑(q - 1) ≤ 5 / 4)
+  (g₁ : 0 < (↑(r - 1):ℚ))
+  (g₂ : ↑r * ↑(5:ℚ) ≤ ↑(6:ℚ) * (↑(r - 1):ℚ)) :
+  ↑r / ↑(r - 1) ≤ ((6/5):ℚ) := by
+  refine (div_le_iff₀ g₁).mpr ?_
+  rw [div_mul_eq_mul_div]
+  refine (le_div_iff₀ ?_).mpr g₂
+  norm_num
+
+
+lemma imo_1992_p1_1_8
+  (p q r : ℤ)
+  -- (hpl : 4 ≤ p)
+  (hql : 5 ≤ q)
+  (hrl : 6 ≤ r)
+  (h₁ : (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ)
+          = (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)))
+  (hp : ↑p / ↑(p - 1) ≤ ((4/3):ℚ))
+  (hq : ↑q / ↑(q - 1) ≤ ((5/4):ℚ))
+  (hr : ↑r / ↑(r - 1) ≤ ((6/5):ℚ)) :
+  (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ ↑2 := by
+  have hub: (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)) ≤ (4/3:ℚ) * ((5/4):ℚ) * ((6/5):ℚ) := by
+    have hq_nonneg: 0 ≤ (↑q:ℚ) := by
+      norm_cast
+      linarith
+    have hq_1_nonneg: 0 ≤ (↑(q - 1):ℚ) := by
+      norm_cast
+      linarith
+    have h₂: 0 ≤ (((q:ℚ) / ↑(q - 1)):ℚ) := by
+      exact div_nonneg hq_nonneg hq_1_nonneg
+    have hub1: (↑p/↑(p-1)) * (↑q/↑(q-1)) ≤ ((4/3):ℚ) * ((5/4):ℚ) := by
+      exact mul_le_mul hp hq h₂ (by norm_num)
+    have hr_nonneg: 0 ≤ (↑r:ℚ) := by
+      norm_cast
+      linarith
+    have hr_1_nonneg: 0 ≤ (↑(r - 1):ℚ) := by
+      norm_cast
+      linarith
+    have h₃: 0 ≤ (((r:ℚ) / ↑(r - 1)):ℚ) := by
+      exact div_nonneg hr_nonneg hr_1_nonneg
+    exact mul_le_mul hub1 hr h₃ (by norm_num)
+  norm_num at hub
+  rw [h₁]
+  norm_num
+  exact hub
+
+
+lemma imo_1992_p1_1_9
+  (p q r : ℤ)
+  -- (hpl : 4 ≤ p)
+  (hql : 5 ≤ q)
+  (hrl : 6 ≤ r)
+  -- (h₁ : ↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)) = ↑p / ↑(p - 1) * (↑q / ↑(q - 1)) * (↑r / ↑(r - 1)))
+  (hp : ↑p / ↑(p - 1) ≤ ((4 / 3):ℚ))
+  (hq : ↑q / ↑(q - 1) ≤ ((5 / 4):ℚ))
+  (hr : ↑r / ↑(r - 1) ≤ ((6 / 5):ℚ)) :
+  (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)) ≤ (4/3:ℚ) * ((5/4):ℚ) * ((6/5):ℚ) := by
+  have hq_nonneg: 0 ≤ (↑q:ℚ) := by
+    norm_cast
+    linarith
+  have hq_1_nonneg: 0 ≤ (↑(q - 1):ℚ) := by
+    norm_cast
+    linarith
+  have h₂: 0 ≤ (((q:ℚ) / ↑(q - 1)):ℚ) := by
+    exact div_nonneg hq_nonneg hq_1_nonneg
+  have hub1: (↑p/↑(p-1)) * (↑q/↑(q-1)) ≤ ((4/3):ℚ) * ((5/4):ℚ) := by
+    exact mul_le_mul hp hq h₂ (by norm_num)
+  have hr_nonneg: 0 ≤ (↑r:ℚ) := by
+    norm_cast
+    linarith
+  have hr_1_nonneg: 0 ≤ (↑(r - 1):ℚ) := by
+    norm_cast
+    linarith
+  have h₃: 0 ≤ (((r:ℚ) / ↑(r - 1)):ℚ) := by
+    exact div_nonneg hr_nonneg hr_1_nonneg
+  exact mul_le_mul hub1 hr h₃ (by norm_num)
+
+
+lemma imo_1992_p1_1_10
+  -- (p r : ℤ)
+  (q : ℤ)
+  -- (hpl : 4 ≤ p)
+  (hql : 5 ≤ q) :
+  -- (hrl : 6 ≤ r)
+  -- (h₁ : ↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)) = ↑p / ↑(p - 1) * (↑q / ↑(q - 1)) * (↑r / ↑(r - 1)))
+  -- (hp : ↑p / ↑(p - 1) ≤ 4 / 3)
+  -- (hq : ↑q / ↑(q - 1) ≤ 5 / 4)
+  -- (hr : ↑r / ↑(r - 1) ≤ 6 / 5) :
+  -- hq_nonneg : 0 ≤ ↑q
+  -- hq_1_nonneg : 0 ≤ ↑(q - 1)
+  0 ≤ (((q:ℚ) / ↑(q - 1)):ℚ) := by
+  have hq_nonneg: 0 ≤ (↑q:ℚ) := by
+    norm_cast
+    linarith
+  have hq_1_nonneg: 0 ≤ (↑(q - 1):ℚ) := by
+    norm_cast
+    linarith
+  exact div_nonneg hq_nonneg hq_1_nonneg
+
+
+lemma imo_1992_p1_1_11
+  (p q r : ℤ)
+  -- (hpl : 4 ≤ p)
+  -- (hql : 5 ≤ q)
+  -- (hrl : 6 ≤ r)
+  (h₁ : (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ)
+          = (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)))
+  -- (hp : ↑p / ↑(p - 1) ≤ 4 / 3)
+  -- (hq : ↑q / ↑(q - 1) ≤ 5 / 4)
+  -- (hr : ↑r / ↑(r - 1) ≤ 6 / 5)
+  (hub : (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)) ≤ (4/3:ℚ) * ((5/4):ℚ) * ((6/5):ℚ)) :
+  (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ ↑2 := by
+  rw [h₁]
+  norm_num
+  norm_num at hub
+  exact hub
+
+
+lemma imo_1992_p1_1_12
+  (p q r : ℤ)
+  -- (hpl : 4 ≤ p)
+  -- (hql : 5 ≤ q)
+  -- (hrl : 6 ≤ r)
+  -- -- (h₁ : ↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)) = ↑p / ↑(p - 1) * (↑q / ↑(q - 1)) * (↑r / ↑(r - 1)))
+  -- (hp : ↑p / ↑(p - 1) ≤ 4 / 3)
+  -- (hq : ↑q / ↑(q - 1) ≤ 5 / 4)
+  -- (hr : ↑r / ↑(r - 1) ≤ 6 / 5)
+  (hub : (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)) ≤ (4/3:ℚ) * ((5/4):ℚ) * ((6/5):ℚ)) :
+  (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ ↑2 := by
+  have h₁: (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ)
+    = (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)) := by
+    norm_cast
+    simp
+  rw [h₁]
+  norm_num
+  norm_num at hub
+  exact hub
+
+
+lemma imo_1992_p1_2
+  (p q r k: ℤ)
+  (hk: p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  (hpl: 4 ≤ p)
+  (hql: 5 ≤ q)
+  (hrl: 6 ≤ r)
+  (hden: 0 < (p - 1) * (q - 1) * (r - 1) ) :
+  (k < 2) := by
+  have h₁: (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ ↑2 := by
+    exact imo_1992_p1_1 p q r hpl hql hrl
+  have h₂: ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+    have g₁: ↑(p * q * r - 1) = ↑k * (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+      norm_cast
+      linarith
+    symm
+    have g₂: (↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≠ 0 := by
+      norm_cast
+      linarith[hden]
+    exact (div_eq_iff g₂).mpr g₁
+  have h₃: ↑k < (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+    rw [h₂]
+    have g₁: (↑(p * q * r - 1):ℚ) < (↑(p * q * r):ℚ) := by
+      norm_cast
+      exact sub_one_lt (p * q * r)
+    have g₂: 0 < (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+      norm_cast
+    exact div_lt_div_of_pos_right g₁ g₂
+  have h₄: (↑k:ℚ) < ↑2 := by
+    exact lt_of_lt_of_le h₃ h₁
+  norm_cast at h₄
+
+
+lemma imo_1992_p1_2_1
+  (p q r k : ℤ)
+  (hk : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  -- (hpl : 4 ≤ p)
+  -- (hql : 5 ≤ q)
+  -- (hrl : 6 ≤ r)
+  (hden : 0 < (p - 1) * (q - 1) * (r - 1))
+  (h₁ : (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ 2) :
+  k < 2 := by
+  have h₂: ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+    have g₁: ↑(p * q * r - 1) = ↑k * (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+      norm_cast
+      linarith
+    symm
+    have g₂: (↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≠ 0 := by
+      norm_cast
+      linarith[hden]
+    exact (div_eq_iff g₂).mpr g₁
+  have h₃: ↑k < (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+    rw [h₂]
+    have g₁: (↑(p * q * r - 1):ℚ) < (↑(p * q * r):ℚ) := by
+      norm_cast
+      exact sub_one_lt (p * q * r)
+    have g₂: 0 < (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+      norm_cast
+    exact div_lt_div_of_pos_right g₁ g₂
+  have h₄: (↑k:ℚ) < ↑2 := by
+    exact lt_of_lt_of_le h₃ h₁
+  norm_cast at h₄
+
+
+lemma imo_1992_p1_2_2
+  (p q r k : ℤ)
+  (hk : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  -- (hpl : 4 ≤ p)
+  -- (hql : 5 ≤ q)
+  -- (hrl : 6 ≤ r)
+  (hden : 0 < (p - 1) * (q - 1) * (r - 1)) :
+  -- (h₁ : ↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)) ≤ 2) :
+  ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+  have g₁: ↑(p * q * r - 1) = ↑k * (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+    norm_cast
+    linarith
+  symm
+  have g₂: (↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≠ 0 := by
+    norm_cast
+    linarith[hden]
+  exact (div_eq_iff g₂).mpr g₁
+
+
+lemma imo_1992_p1_2_3
+  (p q r k : ℤ)
+  -- (hk : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  -- (hpl : 4 ≤ p)
+  -- (hql : 5 ≤ q)
+  -- (hrl : 6 ≤ r)
+  (hden : 0 < (p - 1) * (q - 1) * (r - 1))
+  (h₁ : (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ ↑2)
+  (h₂ : ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ)) :
+  k < 2 := by
+  have h₃: ↑k < (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+    rw [h₂]
+    have g₁: (↑(p * q * r - 1):ℚ) < (↑(p * q * r):ℚ) := by
+      norm_cast
+      exact sub_one_lt (p * q * r)
+    have g₂: 0 < (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+      norm_cast
+    exact div_lt_div_of_pos_right g₁ g₂
+  have h₄: (↑k:ℚ) < ↑2 := by
+    exact lt_of_lt_of_le h₃ h₁
+  norm_cast at h₄
+
+
+lemma imo_1992_p1_2_4
+  (p q r k : ℤ)
+  -- (hk : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  -- (hpl : 4 ≤ p)
+  -- (hql : 5 ≤ q)
+  -- (hrl : 6 ≤ r)
+  (hden : 0 < (p - 1) * (q - 1) * (r - 1))
+  -- (h₁ : ↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)) ≤ 2)
+  (h₂ : ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ)) :
+  ↑k < (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+  rw [h₂]
+  have g₁: (↑(p * q * r - 1):ℚ) < (↑(p * q * r):ℚ) := by
+    norm_cast
+    exact sub_one_lt (p * q * r)
+  have g₂: 0 < (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+    norm_cast
+  exact div_lt_div_of_pos_right g₁ g₂
+
+
+lemma imo_1992_p1_2_5
+  (p q r k : ℤ)
+  -- (hk : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  -- (hpl : 4 ≤ p)
+  -- (hql : 5 ≤ q)
+  -- (hrl : 6 ≤ r)
+  -- (hden : 0 < (p - 1) * (q - 1) * (r - 1))
+  (h₁ : (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ ↑2)
+  -- (h₂ : ↑k = ↑(p * q * r - 1) / ↑((p - 1) * (q - 1) * (r - 1)))
+  (h₃ : ↑k < (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ)) :
+  k < 2 := by
+  have h₄: (↑k:ℚ) < ↑2 := by
+    exact lt_of_lt_of_le h₃ h₁
+  norm_cast at h₄
+
+
+lemma imo_1992_p1_3
+  (p q r: ℤ)
+  (hpl: 2 ≤ p)
+  (hql: 3 ≤ q)
+  (hrl: 4 ≤ r) :
+  (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ ↑4 := by
+  have h₁: (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ)
+      = (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)) := by
+    norm_cast
+    simp
+  have hp: (↑p/↑(p-1):ℚ) ≤ ↑(2:ℚ) := by
+    have g₁: 0 < (↑(p - 1):ℚ) := by
+      norm_cast
+      linarith[hpl]
+    have g₂: ↑p ≤ ↑(2:ℚ) * (↑(p - 1):ℚ) := by
+      norm_cast
+      linarith
+    exact (div_le_iff₀ g₁).mpr g₂
+  have hq: (↑q/↑(q-1)) ≤ ((3/2):ℚ) := by
+    have g₁: 0 < (↑(q - 1):ℚ) := by
+      norm_cast
+      linarith[hql]
+    have g₂: ↑q * ↑(2:ℚ) ≤ ↑(3:ℚ) * (↑(q - 1):ℚ) := by
+      norm_cast
+      linarith
+    refine (div_le_iff₀ g₁).mpr ?_
+    rw [div_mul_eq_mul_div]
+    refine (le_div_iff₀ ?_).mpr g₂
+    norm_num
+  have hr: (↑r/↑(r-1)) ≤ ((4/3):ℚ) := by
+    have g₁: 0 < (↑(r - 1):ℚ) := by
+      norm_cast
+      linarith[hql]
+    have g₂: ↑r * ↑(3:ℚ) ≤ ↑(4:ℚ) * (↑(r - 1):ℚ) := by
+      norm_cast
+      linarith
+    refine (div_le_iff₀ g₁).mpr ?_
+    rw [div_mul_eq_mul_div]
+    refine (le_div_iff₀ ?_).mpr g₂
+    norm_num
+  have hub: (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)) ≤ (2:ℚ) * ((3/2):ℚ) * ((4/3):ℚ) := by
+    have hq_nonneg: 0 ≤ (↑q:ℚ) := by
+      norm_cast
+      linarith
+    have hq_1_nonneg: 0 ≤ (↑(q - 1):ℚ) := by
+      norm_cast
+      linarith
+    have h₂: 0 ≤ (((q:ℚ) / ↑(q - 1)):ℚ) := by
+      exact div_nonneg hq_nonneg hq_1_nonneg
+    have hub1: (↑p/↑(p-1)) * (↑q/↑(q-1)) ≤ (2:ℚ) * ((3/2):ℚ) := by
+      exact mul_le_mul hp hq h₂ (by norm_num)
+    have hr_nonneg: 0 ≤ (↑r:ℚ) := by
+      norm_cast
+      linarith
+    have hr_1_nonneg: 0 ≤ (↑(r - 1):ℚ) := by
+      norm_cast
+      linarith
+    have h₃: 0 ≤ (((r:ℚ) / ↑(r - 1)):ℚ) := by
+      exact div_nonneg hr_nonneg hr_1_nonneg
+    exact mul_le_mul hub1 hr h₃ (by norm_num)
+  norm_num at hub
+  rw [h₁]
+  norm_num
+  exact hub
+
+
+lemma imo_1992_p1_3_1
+  (p : ℤ)
+  -- (q r : ℤ)
+  (hpl : 2 ≤ p) :
+  -- (hql : 3 ≤ q)
+  -- (hrl : 4 ≤ r)
+  -- (h₁ : (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ)
+  --     = (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1))) :
+  (↑p/↑(p-1):ℚ) ≤ ↑(2:ℚ) := by
+  have g₁: 0 < (↑(p - 1):ℚ) := by
+    norm_cast
+    linarith[hpl]
+  have g₂: ↑p ≤ ↑(2:ℚ) * (↑(p - 1):ℚ) := by
+    norm_cast
+    linarith
+  exact (div_le_iff₀ g₁).mpr g₂
+
+
+lemma imo_1992_p1_3_2
+  (p : ℤ)
+  -- (q r : ℤ)
+  (hpl : 2 ≤ p)
+  -- (hql : 3 ≤ q)
+  -- (hrl : 4 ≤ r)
+  -- (h₁ : ↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)) = ↑p / ↑(p - 1) * (↑q / ↑(q - 1)) * (↑r / ↑(r - 1)))
+  (g₁ : 0 < (↑(p - 1):ℚ)) :
+  (↑p/↑(p-1):ℚ) ≤ ↑(2:ℚ) := by
+  have g₂: ↑p ≤ ↑(2:ℚ) * (↑(p - 1):ℚ) := by
+    norm_cast
+    linarith
+  exact (div_le_iff₀ g₁).mpr g₂
+
+
+lemma imo_1992_p1_3_3
+  -- (p r : ℤ)
+  (q : ℤ)
+  -- (hpl : 2 ≤ p)
+  (hql : 3 ≤ q) :
+  -- (hrl : 4 ≤ r)
+  -- (h₁ : ↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)) = ↑p / ↑(p - 1) * (↑q / ↑(q - 1)) * (↑r / ↑(r - 1)))
+  -- (hp : ↑p / ↑(p - 1) ≤ 2) :
+  (↑q/↑(q-1)) ≤ ((3/2):ℚ) := by
+  have g₁: 0 < (↑(q - 1):ℚ) := by
+    norm_cast
+    linarith[hql]
+  have g₂: ↑q * ↑(2:ℚ) ≤ ↑(3:ℚ) * (↑(q - 1):ℚ) := by
+    norm_cast
+    linarith
+  refine (div_le_iff₀ g₁).mpr ?_
+  rw [div_mul_eq_mul_div]
+  refine (le_div_iff₀ ?_).mpr g₂
+  norm_num
+
+
+lemma imo_1992_p1_3_4
+  -- (p r : ℤ)
+  (q : ℤ)
+  -- (hpl : 2 ≤ p)
+  -- (hql : 3 ≤ q)
+  -- (hrl : 4 ≤ r)
+  -- (h₁ : ↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)) = ↑p / ↑(p - 1) * (↑q / ↑(q - 1)) * (↑r / ↑(r - 1)))
+  -- (hp : ↑p / ↑(p - 1) ≤ 2)
+  (g₁ : 0 < (↑(q - 1):ℚ))
+  (g₂ : ↑q * ↑(2:ℚ) ≤ ↑(3:ℚ) * (↑(q - 1):ℚ)) :
+  (↑q/↑(q-1)) ≤ ((3/2):ℚ) := by
+  refine (div_le_iff₀ g₁).mpr ?_
+  rw [div_mul_eq_mul_div]
+  refine (le_div_iff₀ ?_).mpr g₂
+  norm_num
+
+
+lemma imo_1992_p1_3_5
+  -- (p q : ℤ)
+  (r : ℤ)
+  -- (hpl : 2 ≤ p)
+  -- (hql : 3 ≤ q)
+  (hrl : 4 ≤ r) :
+  -- (h₁ : ↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)) = ↑p / ↑(p - 1) * (↑q / ↑(q - 1)) * (↑r / ↑(r - 1)))
+  -- (hp : ↑p / ↑(p - 1) ≤ 2)
+  -- (hq : ↑q / ↑(q - 1) ≤ 3 / 2) :
+  ↑r / ↑(r - 1) ≤ ((4 / 3):ℚ) := by
+  have g₁: 0 < (↑(r - 1):ℚ) := by
+    norm_cast
+    linarith
+  have g₂: ↑r * ↑(3:ℚ) ≤ ↑(4:ℚ) * (↑(r - 1):ℚ) := by
+    norm_cast
+    linarith
+  refine (div_le_iff₀ g₁).mpr ?_
+  rw [div_mul_eq_mul_div]
+  refine (le_div_iff₀ ?_).mpr g₂
+  norm_num
+
+
+lemma imo_1992_p1_3_6
+  -- (p q : ℤ)
+  (r : ℤ)
+  -- (hpl : 2 ≤ p)
+  -- (hql : 3 ≤ q)
+  -- (hrl : 4 ≤ r)
+  -- (h₁ : ↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)) = ↑p / ↑(p - 1) * (↑q / ↑(q - 1)) * (↑r / ↑(r - 1)))
+  -- (hp : ↑p / ↑(p - 1) ≤ 2)
+  -- (hq : ↑q / ↑(q - 1) ≤ 3 / 2)
+  (g₁ : 0 < (↑(r - 1):ℚ))
+  (g₂ : ↑r * ↑(3:ℚ) ≤ ↑(4:ℚ) * (↑(r - 1):ℚ)) :
+  ↑r / ↑(r - 1) ≤ ((4 / 3):ℚ) := by
+  refine (div_le_iff₀ g₁).mpr ?_
+  rw [div_mul_eq_mul_div]
+  refine (le_div_iff₀ ?_).mpr g₂
+  norm_num
+
+
+lemma imo_1992_p1_3_7
+  (p q r : ℤ)
+  -- (hpl : 2 ≤ p)
+  (hql : 3 ≤ q)
+  (hrl : 4 ≤ r)
+  -- (h₁ : ↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)) = ↑p / ↑(p - 1) * (↑q / ↑(q - 1)) * (↑r / ↑(r - 1)))
+  (hp : (↑p/↑(p-1):ℚ) ≤ ↑(2:ℚ))
+  (hq : ↑q / ↑(q - 1) ≤ ((3 / 2):ℚ))
+  (hr : ↑r / ↑(r - 1) ≤ ((4 / 3):ℚ)) :
+  (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)) ≤ (2:ℚ) * ((3/2):ℚ) * ((4/3):ℚ) := by
+  have hq_nonneg: 0 ≤ (↑q:ℚ) := by
+    norm_cast
+    linarith
+  have hq_1_nonneg: 0 ≤ (↑(q - 1):ℚ) := by
+    norm_cast
+    linarith
+  have h₂: 0 ≤ (((q:ℚ) / ↑(q - 1)):ℚ) := by
+    exact div_nonneg hq_nonneg hq_1_nonneg
+  have hub1: (↑p/↑(p-1)) * (↑q/↑(q-1)) ≤ (2:ℚ) * ((3/2):ℚ) := by
+    exact mul_le_mul hp hq h₂ (by norm_num)
+  have hr_nonneg: 0 ≤ (↑r:ℚ) := by
+    norm_cast
+    linarith
+  have hr_1_nonneg: 0 ≤ (↑(r - 1):ℚ) := by
+    norm_cast
+    linarith
+  have h₃: 0 ≤ (((r:ℚ) / ↑(r - 1)):ℚ) := by
+    exact div_nonneg hr_nonneg hr_1_nonneg
+  exact mul_le_mul hub1 hr h₃ (by norm_num)
+
+
+lemma imo_1992_p1_3_8
+  (p q r : ℤ)
+  -- (hpl : 2 ≤ p)
+  -- (hql : 3 ≤ q)
+  -- (hrl : 4 ≤ r)
+  (h₁ : ↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)) = ↑p / ↑(p - 1) * (↑q / ↑(q - 1)) * (↑r / ↑(r - 1)))
+  -- (hp : ↑p / ↑(p - 1) ≤ 2)
+  -- (hq : ↑q / ↑(q - 1) ≤ 3 / 2)
+  -- (hr : ↑r / ↑(r - 1) ≤ 4 / 3)
+  (hub : ↑p / (↑p - 1) * (↑q / (↑q - 1)) * (↑r / (↑r - 1)) ≤ 4) :
+  ↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)) ≤ 4 := by
+  rw [h₁]
+  exact hub
+
+
+lemma imo_1992_p1_4
+  (p q r k: ℤ)
+  (hk: p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  (hpl: 2 ≤ p)
+  (hql: 3 ≤ q)
+  (hrl: 4 ≤ r)
+  (hden: 0 < (p - 1) * (q - 1) * (r - 1) ) :
+  (k < 4) := by
+  have h₁: (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ ↑4 := by
+    exact imo_1992_p1_3 p q r hpl hql hrl
+  have h₂: ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+    have g₁: ↑(p * q * r - 1) = ↑k * (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+      norm_cast
+      linarith
+    symm
+    have g₂: (↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≠ 0 := by
+      norm_cast
+      linarith [hden]
+    exact (div_eq_iff g₂).mpr g₁
+  have h₃: ↑k < (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+    rw [h₂]
+    have g₁: (↑(p * q * r - 1):ℚ) < (↑(p * q * r):ℚ) := by
+      norm_cast
+      exact sub_one_lt (p * q * r)
+    have g₂: 0 < (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+      norm_cast
+    exact div_lt_div_of_pos_right g₁ g₂
+  have h₄: (↑k:ℚ) < ↑4 := by
+    exact lt_of_lt_of_le h₃ h₁
+  norm_cast at h₄
+
+
+lemma imo_1992_p1_4_1
+  (p q r k : ℤ)
+  -- (hk : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  -- (hpl : 2 ≤ p)
+  -- (hql : 3 ≤ q)
+  -- (hrl : 4 ≤ r)
+  -- (hden : 0 < (p - 1) * (q - 1) * (r - 1))
+  (h₁ : (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ ↑4)
+  -- (h₂ : ↑k = ↑(p * q * r - 1) / ↑((p - 1) * (q - 1) * (r - 1)))
+  (h₃ : ↑k < (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ)) :
+  k < 4 := by
+  have h₄: (↑k:ℚ) < ↑4 := by
+    exact lt_of_lt_of_le h₃ h₁
+  norm_cast at h₄
+
+
+lemma imo_1992_p1_5
+  (p q r k: ℤ)
+  (hk: p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  (h₁: ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ))
+  (hpl: 2 ≤ p)
+  (hql: 3 ≤ q)
+  (hrl: 4 ≤ r)
+  (hden: 0 < (p - 1) * (q - 1) * (r - 1)) :
+  (1 < k) := by
+  have hk0: 0 < (↑k:ℚ) := by
+    have g₁: 2 * 3 * 4 ≤ p * q * r := by
+      have g₂: 2 * 3 ≤ p * q := by
+        exact mul_le_mul hpl hql (by norm_num) (by linarith[hpl])
+      exact mul_le_mul g₂ hrl (by norm_num) (by linarith[g₂])
+    have g₂: 0 < (↑(p * q * r - 1):ℚ) := by
+      norm_cast
+      linarith[g₁]
+    have g₃: 0 < (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+      norm_cast
+    rw [h₁]
+    exact div_pos g₂ g₃
+  norm_cast at hk0
+  by_contra! hc
+  interval_cases k
+  simp at hk
+  have g₁: p*q + q*r + r*p = p+q+r := by linarith
+  have g₂: p < p*q := by exact lt_mul_right (by linarith) (by linarith)
+  have g₃: q < q*r := by exact lt_mul_right (by linarith) (by linarith)
+  have g₄: r < r*p := by exact lt_mul_right (by linarith) (by linarith)
+  have g₅: p+q+r < p*q + q*r + r*p := by linarith[g₂,g₃,g₄]
+  linarith [g₁,g₅]
+
+
+lemma imo_1992_p1_5_1
+  (p q r k : ℤ)
+  -- (hk : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  (h₁ : ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ))
+  (hpl : 2 ≤ p)
+  (hql : 3 ≤ q)
+  (hrl : 4 ≤ r)
+  (hden: 0 < (p - 1) * (q - 1) * (r - 1)) :
+  0 < (↑k:ℚ) := by
+  have g₁: 2 * 3 * 4 ≤ p * q * r := by
+    have g₂: 2 * 3 ≤ p * q := by
+      exact mul_le_mul hpl hql (by norm_num) (by linarith[hpl])
+    exact mul_le_mul g₂ hrl (by norm_num) (by linarith[g₂])
+  have g₂: 0 < (↑(p * q * r - 1):ℚ) := by
+    norm_cast
+    linarith[g₁]
+  have g₃: 0 < (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+    norm_cast
+  rw [h₁]
+  exact div_pos g₂ g₃
+
+
+lemma imo_1992_p1_5_2
+  (p q r : ℤ)
+  -- (k : ℤ)
+  -- (hk : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  -- (h₁ : ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ))
+  (hpl : 0 < (p - 1))
+  (hql : 0 < (q - 1))
+  (hrl : 0 < (r - 1)) :
+  -- (hden: 0 < (p - 1) * (q - 1) * (r - 1)) :
+  -- (g₁ : 2 * 3 * 4 ≤ p * q * r)
+  -- (g₂ : 0 < (↑(p * q * r - 1):ℚ)) :
+  0 < (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+  norm_cast
+  refine mul_pos ?_ hrl
+  exact mul_pos hpl hql
+
+
+lemma imo_1992_p1_5_3
+  (p q r : ℤ)
+  -- (k : ℤ)
+  -- (hk : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  -- (h₁ : ↑k = ↑(p * q * r - 1) / ↑((p - 1) * (q - 1) * (r - 1)))
+  (hpl : 2 ≤ p)
+  (hql : 3 ≤ q)
+  (hrl : 4 ≤ r) :
+  0 < ↑(p * q * r - 1) := by
+  have g₁: 2 * 3 * 4 ≤ p * q * r := by
+    have g₂: 2 * 3 ≤ p * q := by
+      exact mul_le_mul hpl hql (by norm_num) (by linarith[hpl])
+    exact mul_le_mul g₂ hrl (by norm_num) (by linarith[g₂])
+  norm_cast
+  linarith[g₁]
+
+
+lemma imo_1992_p1_5_4
+  (p q r k : ℤ)
+  (hk : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  (h₁ : ↑k = ↑(p * q * r - 1) / ↑((p - 1) * (q - 1) * (r - 1)))
+  (hpl : 2 ≤ p)
+  (hql : 3 ≤ q)
+  (hrl : 4 ≤ r)
+  -- (hden : 0 < (p - 1) * (q - 1) * (r - 1))
+  (hk0 : 0 < k) :
+  1 < k := by
+  by_contra! hc
+  interval_cases k
+  simp at hk
+  have g₁: p*q + q*r + r*p = p+q+r := by linarith
+  have g₂: p < p*q := by exact lt_mul_right (by linarith) (by linarith)
+  have g₃: q < q*r := by exact lt_mul_right (by linarith) (by linarith)
+  have g₄: r < r*p := by exact lt_mul_right (by linarith) (by linarith)
+  have g₅: p+q+r < p*q + q*r + r*p := by linarith[g₂,g₃,g₄]
+  linarith [g₁,g₅]
+
+
+lemma imo_1992_p1_5_5
+  (p q r : ℤ)
+  -- (k : ℤ)
+  (hpl : 2 ≤ p)
+  (hql : 3 ≤ q)
+  (hrl : 4 ≤ r)
+  -- (hden : 0 < (p - 1) * (q - 1) * (r - 1))
+  (hk : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * 1) :
+  -- (h₁ : ↑1 = ↑(p * q * r - 1) / ↑((p - 1) * (q - 1) * (r - 1)))
+  -- (hk0 : 0 < 1)
+  -- (hc : 1 ≤ 1) :
+  False := by
+  simp at hk
+  have g₁: p * q + q * r + r * p = p + q + r := by linarith
+  have g₂: p < p * q := by exact lt_mul_right (by linarith) (by linarith)
+  have g₃: q < q * r := by exact lt_mul_right (by linarith) (by linarith)
+  have g₄: r < r * p := by exact lt_mul_right (by linarith) (by linarith)
+  have g₅: p + q + r < p * q + q * r + r * p := by linarith[g₂,g₃,g₄]
+  linarith [g₁,g₅]
+
+
+lemma imo_1992_p1_5_6
+  (p q r : ℤ)
+  -- (k : ℤ)
+  (hpl : 2 ≤ p)
+  (hql : 3 ≤ q)
+  (hrl : 4 ≤ r)
+  -- (hden : 0 < (p - 1) * (q - 1) * (r - 1))
+  -- (h₁ : ↑1 = ↑(p * q * r - 1) / ↑((p - 1) * (q - 1) * (r - 1)))
+  -- (hk0 : 0 < 1)
+  -- (hc : 1 ≤ 1)
+  -- (hk : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1))
+  (g₁ : p * q + q * r + r * p = p + q + r) :
+  False := by
+  have g₂: p < p * q := by exact lt_mul_right (by linarith) (by linarith)
+  have g₃: q < q * r := by exact lt_mul_right (by linarith) (by linarith)
+  have g₄: r < r * p := by exact lt_mul_right (by linarith) (by linarith)
+  have g₅: p + q + r < p * q + q * r + r * p := by linarith[g₂,g₃,g₄]
+  linarith [g₁,g₅]
+
+
+lemma imo_1992_p1_5_7
+  (p q r : ℤ)
+  -- (k : ℤ)
+  (hpl : 2 ≤ p)
+  -- (hql : 3 ≤ q)
+  (hrl : 4 ≤ r)
+  -- (hden : 0 < (p - 1) * (q - 1) * (r - 1))
+  -- (h₁ : ↑1 = ↑(p * q * r - 1) / ↑((p - 1) * (q - 1) * (r - 1)))
+  -- (hk0 : 0 < 1)
+  -- (hc : 1 ≤ 1)
+  -- (hk : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1))
+  (g₁ : p * q + q * r + r * p = p + q + r)
+  (g₂: p < p * q)
+  (g₃: q < q * r) :
+  False := by
+  have g₄: r < r * p := by exact lt_mul_right (by linarith) (by linarith)
+  have g₅: p + q + r < p * q + q * r + r * p := by linarith[g₂,g₃,g₄]
+  linarith [g₁,g₅]
+
+
+lemma imo_1992_p1_6
+  (p q r k: ℤ)
+  (h₀ : 1 < p ∧ p < q ∧ q < r)
+  (hk: p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  (h₁: ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ))
+  (hpl: 2 ≤ p)
+  (hql: 3 ≤ q)
+  (hrl: 4 ≤ r)
+  (hden: 0 < (p - 1) * (q - 1) * (r - 1) ) :
+  (p < 4) := by
+  by_contra! hcp
+  have hcq: 5 ≤ q := by linarith
+  have hcr: 6 ≤ r := by linarith
+  have h₃: k < 2 := by exact imo_1992_p1_2 p q r k hk hcp hcq hcr hden
+  have h₄: 1 < k := by exact imo_1992_p1_5 p q r k hk h₁ hpl hql hrl hden
+  linarith
+
+
+lemma imo_1992_p1_6_1
+  (p q r k : ℤ)
+  (h₀ : 1 < p ∧ p < q ∧ q < r)
+  (hk : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  (h₁ : ↑k = ↑(p * q * r - 1) / ↑((p - 1) * (q - 1) * (r - 1)))
+  (hpl : 2 ≤ p)
+  (hql : 3 ≤ q)
+  (hrl : 4 ≤ r)
+  (hden : 0 < (p - 1) * (q - 1) * (r - 1))
+  (hcp : 4 ≤ p)
+  (hcq : 5 ≤ q)
+  (hcr : 6 ≤ r)
+  (h₃ : k < 2)
+  (h₄ : 1 < k) :
+  p < 4 := by
+  linarith
+
+
+lemma imo_1992_p1_7
+  (q r : ℤ)
+  (p: ℕ)
+  (h₀ : q * r = ↑p)
+  (h₁: Nat.Prime p) :
+  q = -1 ∨ q = 1 ∨ q = -p ∨ q = p := by
+  have hq : q ≠ 0 := by
+    intro h
+    rw [h] at h₀
+    simp at h₀
+    symm at h₀
+    norm_cast at h₀
+    rw [h₀] at h₁
+    exact Nat.not_prime_zero h₁
+  have hr : r ≠ 0 := by
+    intro h
+    rw [h] at h₀
+    simp at h₀
+    norm_cast at h₀
+    rw [← h₀] at h₁
+    exact Nat.not_prime_zero h₁
+  have hqr : abs q * abs r = p := by
+    have h₃: abs q = q.natAbs := by exact abs_eq_natAbs q
+    have h₄: abs r = r.natAbs := by exact abs_eq_natAbs r
+    rw [h₃,h₄]
+    norm_cast
+    exact Int.natAbs_mul_natAbs_eq h₀
+  have h_abs: abs (↑(q.natAbs):ℤ) = 1 ∨ abs q = p := by
+    cases' Int.natAbs_eq q with h_1 h_2
+    . rw [h_1] at hqr
+      have h₂: abs (↑(q.natAbs):ℤ) ∣ p := by exact Dvd.intro (abs r) hqr
+      have h₃: (↑(q.natAbs):ℕ) ∣ p := by
+        norm_cast at *
+      have h₄: (↑(q.natAbs):ℕ) = 1 ∨ (↑(q.natAbs):ℕ) = p := by
+        exact Nat.Prime.eq_one_or_self_of_dvd h₁ (↑(q.natAbs):ℕ) h₃
+      cases' h₄ with h₄₀ h₄₁
+      . left
+        norm_cast at *
+      . have h₅: abs q = q.natAbs := by exact abs_eq_natAbs q
+        right
+        rw [h₅]
+        norm_cast at *
+    . rw [h_2] at hqr
+      rw [abs_neg _] at hqr
+      have h₂: abs (↑(q.natAbs):ℤ) ∣ p := by exact Dvd.intro (abs r) hqr
+      have h₃: (↑(q.natAbs):ℕ) ∣ p := by
+        norm_cast at *
+      have h₄: (↑(q.natAbs):ℕ) = 1 ∨ (↑(q.natAbs):ℕ) = p := by
+        exact Nat.Prime.eq_one_or_self_of_dvd h₁ (↑(q.natAbs):ℕ) h₃
+      cases' h₄ with h₄₀ h₄₁
+      . left
+        norm_cast at *
+      . have h₅: abs q = q.natAbs := by exact abs_eq_natAbs q
+        right
+        rw [h₅]
+        norm_cast
+  cases' h_abs with hq_abs hq_abs
+  . norm_cast at *
+    have h₄: q = ↑(q.natAbs) ∨ q = -↑(q.natAbs) := by
+      exact Int.natAbs_eq q
+    rw [hq_abs] at h₄
+    norm_cast at h₄
+    cases' h₄ with h₄₀ h₄₁
+    . right
+      left
+      exact h₄₀
+    . left
+      exact h₄₁
+  . right
+    right
+    have h₂: abs q = q.natAbs := by exact abs_eq_natAbs q
+    rw [h₂] at hq_abs
+    norm_cast at hq_abs
+    refine or_comm.mp ?_
+    refine (Int.natAbs_eq_natAbs_iff).mp ?_
+    norm_cast
+
+
+lemma imo_1992_p1_7_1
+  (q r : ℤ)
+  (p : ℕ)
+  (h₀ : q * r = ↑p)
+  (h₁ : Nat.Prime p) :
+  q ≠ 0 := by
+  intro h
+  rw [h] at h₀
+  simp at h₀
+  symm at h₀
+  norm_cast at h₀
+  rw [h₀] at h₁
+  exact Nat.not_prime_zero h₁
+
+
+lemma imo_1992_p1_7_2
+  (q r : ℤ)
+  (p : ℕ)
+  (h₀ : q * r = ↑p)
+  (h₁ : Nat.Prime p)
+  (hq : q ≠ 0) :
+  r ≠ 0 := by
+  intro h
+  rw [h] at h₀
+  simp at h₀
+  norm_cast at h₀
+  rw [← h₀] at h₁
+  exact Nat.not_prime_zero h₁
+
+
+lemma imo_1992_p1_7_3
+  (q r : ℤ)
+  (p : ���)
+  (h₀ : q * r = ↑p) :
+  -- (h₁ : Nat.Prime p)
+  -- (hq : q ≠ 0)
+  -- (hr : r ≠ 0) :
+  |q| * |r| = ↑p := by
+  have h₃: abs q = q.natAbs := by exact abs_eq_natAbs q
+  have h₄: abs r = r.natAbs := by exact abs_eq_natAbs r
+  rw [h₃,h₄]
+  norm_cast
+  exact Int.natAbs_mul_natAbs_eq h₀
+
+
+lemma imo_1992_p1_7_4
+  (q r : ℤ)
+  (p : ℕ)
+  (h₀ : q * r = ↑p)
+  -- (h₁ : Nat.Prime p)
+  -- (hq : q ≠ 0)
+  -- (hr : r ≠ 0)
+  (h₃ : |q| = ↑(natAbs q))
+  (h₄ : |r| = ↑(natAbs r)) :
+  |q| * |r| = ↑p := by
+  rw [h₃,h₄]
+  norm_cast
+  exact Int.natAbs_mul_natAbs_eq h₀
+
+
+lemma imo_1992_p1_7_5
+  (q r : ℤ)
+  (p : ℕ)
+  -- (h₀ : q * r = ↑p)
+  (h₁ : Nat.Prime p)
+  (hq : q ≠ 0)
+  (hr : r ≠ 0)
+  (hqr : |q| * |r| = ↑p) :
+  |(↑(natAbs q):ℤ)| = 1 ∨ |q| = ↑p := by
+  cases' Int.natAbs_eq q with h_1 h_2
+  . rw [h_1] at hqr
+    have h₂: abs (↑(q.natAbs):ℤ) ∣ p := by exact Dvd.intro (abs r) hqr
+    have h₃: (↑(q.natAbs):ℕ) ∣ p := by
+      norm_cast at *
+    have h₄: (↑(q.natAbs):ℕ) = 1 ∨ (↑(q.natAbs):ℕ) = p := by
+      exact Nat.Prime.eq_one_or_self_of_dvd h₁ (↑(q.natAbs):ℕ) h₃
+    cases' h₄ with h₄₀ h₄₁
+    . left
+      norm_cast at *
+    . have h₅: abs q = q.natAbs := by exact abs_eq_natAbs q
+      right
+      rw [h₅]
+      norm_cast at *
+  . rw [h_2] at hqr
+    rw [abs_neg _] at hqr
+    have h₂: abs (↑(q.natAbs):ℤ) ∣ p := by exact Dvd.intro (abs r) hqr
+    have h₃: (↑(q.natAbs):ℕ) ∣ p := by
+      norm_cast at *
+    have h₄: (↑(q.natAbs):ℕ) = 1 ∨ (↑(q.natAbs):ℕ) = p := by
+      exact Nat.Prime.eq_one_or_self_of_dvd h₁ (↑(q.natAbs):ℕ) h₃
+    cases' h₄ with h₄₀ h₄₁
+    . left
+      norm_cast at *
+    . have h₅: abs q = q.natAbs := by exact abs_eq_natAbs q
+      right
+      rw [h₅]
+      norm_cast
+
+
+lemma imo_1992_p1_7_6
+  (q r : ℤ)
+  (p : ℕ)
+  -- (h₀ : q * r = ↑p)
+  (h₁ : Nat.Prime p)
+  (hq : q ≠ 0)
+  (hr : r ≠ 0)
+  (hqr : |q| * |r| = ↑p)
+  (h_1 : q = ↑(natAbs q)) :
+  |(↑(natAbs q):ℤ)| = 1 ∨ |q| = ↑p := by
+  rw [h_1] at hqr
+  have h₂: abs (↑(q.natAbs):ℤ) ∣ p := by exact Dvd.intro (abs r) hqr
+  have h₃: (↑(q.natAbs):ℕ) ∣ p := by
+    norm_cast at *
+  have h₄: (↑(q.natAbs):ℕ) = 1 ∨ (↑(q.natAbs):ℕ) = p := by
+    exact Nat.Prime.eq_one_or_self_of_dvd h₁ (↑(q.natAbs):ℕ) h₃
+  cases' h₄ with h₄₀ h₄₁
+  . left
+    norm_cast at *
+  . have h₅: abs q = q.natAbs := by exact abs_eq_natAbs q
+    right
+    rw [h₅]
+    norm_cast at *
+
+
+lemma imo_1992_p1_7_7
+  (q r : ℤ)
+  (p : ℕ)
+  -- (h₀ : q * r = ↑p)
+  -- (h₁ : Nat.Prime p)
+  (hq : q ≠ 0)
+  (hr : r ≠ 0)
+  (hqr : |↑(natAbs q)| * |r| = ↑p)
+  (h_1 : q = ↑(natAbs q))
+  (h₂ : |(↑(natAbs q):ℤ)| ∣ ↑p)
+  -- (h₃ : natAbs q ∣ p)
+  (h₄ : natAbs q = 1 ∨ natAbs q = p) :
+  |(↑(natAbs q):ℤ)| = 1 ∨ |q| = ↑p := by
+  cases' h₄ with h₄₀ h₄₁
+  . left
+    norm_cast at *
+  . have h₅: abs q = q.natAbs := by exact abs_eq_natAbs q
+    right
+    rw [h₅]
+    norm_cast at *
+
+
+lemma imo_1992_p1_7_8
+  (q r : ℤ)
+  (p : ℕ)
+  -- (h₀ : q * r = ↑p)
+  -- (h₁ : Nat.Prime p)
+  (hq : q ≠ 0)
+  (hr : r ≠ 0)
+  (hqr : |↑(natAbs q)| * |r| = ↑p)
+  (h_1 : q = ↑(natAbs q))
+  (h₂ : |(↑(natAbs q):ℤ)| ∣ ↑p)
+  -- (h₃ : natAbs q ∣ p)
+  (h₄₁ : natAbs q = p) :
+  |(↑(natAbs q):ℤ)| = 1 ∨ |q| = ↑p := by
+  have h₅: abs q = q.natAbs := by exact abs_eq_natAbs q
+  right
+  rw [h₅]
+  norm_cast at *
+
+
+lemma imo_1992_p1_7_9
+  (q r : ℤ)
+  (p : ℕ)
+  -- (h₀ : q * r = ↑p)
+  (h₁ : Nat.Prime p)
+  (hq : q ≠ 0)
+  (hr : r ≠ 0)
+  (hqr : |q| * |r| = ↑p)
+  (h_2 : q = -↑(natAbs q)) :
+  |(↑(natAbs q):ℤ)| = 1 ∨ |q| = ↑p := by
+  rw [h_2] at hqr
+  rw [abs_neg _] at hqr
+  have h₂: abs (↑(q.natAbs):ℤ) ∣ p := by exact Dvd.intro (abs r) hqr
+  have h₃: (↑(q.natAbs):ℕ) ∣ p := by
+    norm_cast at *
+  have h₄: (↑(q.natAbs):ℕ) = 1 ∨ (↑(q.natAbs):ℕ) = p := by
+    exact Nat.Prime.eq_one_or_self_of_dvd h₁ (↑(q.natAbs):ℕ) h₃
+  cases' h₄ with h₄₀ h₄₁
+  . left
+    norm_cast at *
+  . have h₅: abs q = q.natAbs := by exact abs_eq_natAbs q
+    right
+    rw [h₅]
+    norm_cast
+
+
+lemma imo_1992_p1_7_10
+  (q r : ℤ)
+  (p : ℕ)
+  -- (h₀ : q * r = ↑p)
+  -- (h₁ : Nat.Prime p)
+  -- (hq : q ≠ 0)
+  -- (hr : r ≠ 0)
+  (hqr : |(↑(natAbs q):ℤ)| * |r| = ↑p)
+  (h_2 : q = (-↑(q.natAbs):ℤ)) :
+  |(↑(natAbs q):ℤ)| ∣ ↑p := by
+  refine Dvd.intro (abs r) ?_
+  simp at *
+  exact hqr
+
+
+lemma imo_1992_p1_7_11
+  (q : ℤ)
+  -- (r : ℤ)
+  (p : ℕ)
+  -- (h₀ : q * r = ↑p)
+  (h₁ : Nat.Prime p)
+  -- (hq : q ≠ 0)
+  -- (hr : r ≠ 0)
+  -- (hqr : |↑(natAbs q)| * |r| = ↑p)
+  -- (h_2 : q = -↑(natAbs q))
+  (h₂ : |(↑(natAbs q):ℤ)| ∣ ↑p) :
+  natAbs q = 1 ∨ natAbs q = p := by
+  have h₃: (↑(q.natAbs):ℕ) ∣ p := by
+    norm_cast at *
+  exact Nat.Prime.eq_one_or_self_of_dvd h₁ (↑(q.natAbs):ℕ) h₃
+
+
+lemma imo_1992_p1_7_12
+  (q : ℤ)
+  -- (r : ℤ)
+  (p : ℕ)
+  -- (h₀ : q * r = ↑p)
+  -- (h₁ : Nat.Prime p)
+  -- (hq : q ≠ 0)
+  -- (hr : r ≠ 0)
+  -- (hqr : |↑(natAbs q)| * |r| = ↑p)
+  -- (h_2 : q = -↑(natAbs q))
+  -- (h₂ : |(↑(natAbs q):ℤ)| ∣ ↑p)
+  -- (h₃ : natAbs q ∣ p)
+  (h₄₁ : natAbs q = p) :
+  |q| = ↑p := by
+  have h₅: abs q = q.natAbs := by exact abs_eq_natAbs q
+  rw [h₅]
+  norm_cast
+
+
+lemma imo_1992_p1_7_13
+  (q r : ℤ)
+  (p : ℕ)
+  -- (h₀ : q * r = ↑p)
+  -- (h₁ : Nat.Prime p)
+  (hq : q ≠ 0)
+  (hr : r ≠ 0)
+  -- (hqr : |q| * |r| = ↑p)
+  (h_abs : |(↑(natAbs q):ℤ)| = 1 ∨ |q| = ↑p) :
+  q = -1 ∨ q = 1 ∨ q = -↑p ∨ q = ↑p := by
+  cases' h_abs with hq_abs hq_abs
+  . norm_cast at *
+    have h₄: q = ↑(q.natAbs) ∨ q = -↑(q.natAbs) := by
+      exact Int.natAbs_eq q
+    rw [hq_abs] at h₄
+    norm_cast at h₄
+    cases' h₄ with h₄₀ h₄₁
+    . right
+      left
+      exact h₄₀
+    . left
+      exact h₄₁
+  . right
+    right
+    have h₂: abs q = q.natAbs := by exact abs_eq_natAbs q
+    rw [h₂] at hq_abs
+    norm_cast at hq_abs
+    refine or_comm.mp ?_
+    refine (Int.natAbs_eq_natAbs_iff).mp ?_
+    norm_cast
+
+
+lemma imo_1992_p1_7_14
+  (q r : ℤ)
+  (p : ℕ)
+  -- (h₀ : q * r = ↑p)
+  -- (h₁ : Nat.Prime p)
+  (hq : q ≠ 0)
+  (hr : r ≠ 0)
+  -- (hqr : |q| * |r| = ↑p)
+  (hq_abs : |(↑(natAbs q):ℤ)| = 1) :
+  q = -1 ∨ q = 1 ∨ q = -↑p ∨ q = ↑p := by
+  norm_cast at *
+  have h₄: q = ↑(q.natAbs) ∨ q = -↑(q.natAbs) := by
+    exact Int.natAbs_eq q
+  rw [hq_abs] at h₄
+  norm_cast at h₄
+  cases' h₄ with h₄₀ h₄₁
+  . right
+    left
+    exact h₄₀
+  . left
+    exact h₄₁
+
+
+lemma imo_1992_p1_7_15
+  (q r : ℤ)
+  -- (p : ℕ)
+  (hrq: r = q) :
+  -- (h₀ : q * r = ↑p)
+  -- (h₁ : Nat.Prime p)
+  -- (hqr : |q| * |r| = ↑p)
+  -- (hq : ¬q = 0)
+  -- (hr : ¬r = 0)
+  -- (hq_abs : natAbs q = 1) :
+  r = ↑(natAbs q) ∨ r = -↑(natAbs q) := by
+  rw [← hrq]
+  exact Int.natAbs_eq r
+
+
+lemma imo_1992_p1_7_16
+  (q : ℤ)
+  -- (r : ℤ)
+  (p : ℕ)
+  -- (h₀ : q * r = ↑p)
+  -- (h₁ : Nat.Prime p)
+  -- (hq : q ≠ 0)
+  -- (hr : r ≠ 0)
+  -- (hqr : |q| * |r| = ↑p)
+  (hq_abs : |q| = ↑p) :
+  q = -1 ∨ q = 1 ∨ q = -↑p ∨ q = ↑p := by
+  right
+  right
+  have h₂: abs q = q.natAbs := by exact abs_eq_natAbs q
+  rw [h₂] at hq_abs
+  norm_cast at hq_abs
+  refine or_comm.mp ?_
+  refine (Int.natAbs_eq_natAbs_iff).mp ?_
+  norm_cast
+
+
+lemma imo_1992_p1_7_17
+  (q : ℤ)
+  -- (r : ℤ)
+  (p : ℕ)
+  -- (h₀ : q * r = ↑p)
+  -- (h₁ : Nat.Prime p)
+  -- (hq : q ≠ 0)
+  -- (hr : r ≠ 0)
+  -- (hqr : |q| * |r| = ↑p)
+  (hq_abs : |q| = ↑p) :
+  q = -↑p ∨ q = ↑p := by
+  have h₂: abs q = q.natAbs := by exact abs_eq_natAbs q
+  rw [h₂] at hq_abs
+  norm_cast at hq_abs
+  refine or_comm.mp ?_
+  refine (Int.natAbs_eq_natAbs_iff).mp ?_
+  norm_cast
+
+
+lemma imo_1992_p1_7_18
+  (q : ℤ)
+  -- (r : ℤ)
+  (p : ℕ)
+  -- (h₀ : q * r = ↑p)
+  -- (h₁ : Nat.Prime p)
+  -- (hq : q ≠ 0)
+  -- (hr : r ≠ 0)
+  -- (hqr : |q| * |r| = ↑p)
+  -- (h₂ : |q| = ↑(natAbs q))
+  (hq_abs : natAbs q = p) :
+  q = -↑p ∨ q = ↑p := by
+  refine or_comm.mp ?_
+  refine (Int.natAbs_eq_natAbs_iff).mp ?_
+  norm_cast
+
+
+
+-- my_case_k_2
+lemma imo_1992_p1_8
+  (p q r: ℤ)
+  (h₀: 1 < p ∧ p < q ∧ q < r)
+  (hpl: 2 ≤ p)
+  (hql: 3 ≤ q)
+  (hrl: 4 ≤ r)
+  (hpu: p < 4)
+  (hk: p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * 2) :
+  (p, q, r) = (2, 4, 8) ∨ (p, q, r) = (3, 5, 15) := by
+  interval_cases p
+  . exfalso
+    norm_num at *
+    have g₁: 2*q + 2*r = 3 := by linarith
+    linarith [g₁,hql,hrl]
+  . right
+    norm_num at *
+    have g₂: (4-q)*(4-r) = 11 := by linarith
+    have g₃: (4-q) = -1 ∨ (4-q) = 1 ∨ (4-q) = -11 ∨ (4-q) = 11 := by
+      refine imo_1992_p1_7 (4-q) (4-r) 11 g₂ ?_
+      decide
+    cases' g₃ with g₃₁ g₃₂
+    . have hq: q = 5 := by linarith
+      constructor
+      . exact hq
+      . rw [hq] at g₂
+        linarith[g₂]
+    . exfalso
+      cases' g₃₂ with g₃₂ g₃₃
+      . have hq: q = 3 := by linarith[g₃₂]
+        rw [hq] at g₂
+        have hr: r = -7 := by linarith[g₂]
+        linarith[hrl,hr]
+      . cases' g₃₃ with g₃₃ g₃₄
+        . have hq: q = 15 := by linarith[g₃₃]
+          rw [hq] at g₂
+          have hr: r = 5 := by linarith[g₂]
+          linarith[hq,hr,h₀.2]
+        . have hq: q = -7 := by linarith[g₃₄]
+          linarith[hq,hql]
+
+
+lemma imo_1992_p1_8_1
+  (p q r : ℤ)
+  (h₀ : 1 < p ∧ p < q ∧ q < r)
+  (hpl : p = 2)
+  (hql : 3 ≤ q)
+  (hrl : 4 ≤ r)
+  (hk : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * 2) :
+  False := by
+  rw [hpl] at *
+  norm_num at *
+  have g₁: 2 * q + 2 * r = 3 := by
+    linarith
+  linarith [g₁,hql,hrl]
+
+
+lemma imo_1992_p1_8_2
+  -- (p : ℤ)
+  (q r : ℤ)
+  -- (hql : 3 s≤ q)
+  (hrl : 4 ≤ r)
+  (h₀ : 1 < 3 ∧ 3 < q ∧ q < r)
+  -- (hpl : 2 ≤ 3)
+  -- (hpu : 3 < 4)
+  (hk : 3 * q * r - 1 = (3 - 1) * (q - 1) * (r - 1) * 2) :
+  (3, q, r) = (3, 5, 15) := by
+  norm_num at *
+  have g₂: (4-q)*(4-r) = 11 := by linarith
+  have g₃: (4-q) = -1 ∨ (4-q) = 1 ∨ (4-q) = -11 ∨ (4-q) = 11 := by
+    refine imo_1992_p1_7 (4-q) (4-r) 11 g₂ ?_
+    decide
+  cases' g₃ with g₃₁ g₃₂
+  . have hq: q = 5 := by linarith
+    constructor
+    . exact hq
+    . rw [hq] at g₂
+      linarith[g₂]
+  . exfalso
+    cases' g₃₂ with g₃₂ g₃₃
+    . have hq: q = 3 := by linarith[g₃₂]
+      rw [hq] at g₂
+      have hr: r = -7 := by linarith[g₂]
+      linarith[hrl,hr]
+    . cases' g₃₃ with g₃₃ g₃₄
+      . have hq: q = 15 := by linarith[g₃₃]
+        rw [hq] at g₂
+        have hr: r = 5 := by linarith[g₂]
+        linarith[hq,hr,h₀.2]
+      . linarith
+
+
+lemma imo_1992_p1_8_3
+  -- (p : ℤ)
+  (q r : ℤ)
+  -- (hql : 3 ≤ q)
+  -- (hrl : 4 ≤ r)
+  -- (h₀ : 3 < q ∧ q < r)
+  -- (hk : 3 * q * r - 1 = 2 * (q - 1) * (r - 1) * 2)
+  -- g₁ : q * r - 4 * q - 4 * r + 5 = 0
+  (g₂ : (4 - q) * (4 - r) = 11) :
+  4 - q = -1 ∨ 4 - q = 1 ∨ 4 - q = -11 ∨ 4 - q = 11 := by
+  refine imo_1992_p1_7 (4-q) (4-r) 11 g₂ ?_
+  decide
+
+
+lemma imo_1992_p1_8_4
+  -- (p : ℤ)
+  (q r : ℤ)
+  -- (hql : 3 ≤ q)
+  -- (hrl : 4 ≤ r)
+  -- (h₀ : 3 < q ∧ q < r)
+  -- (hk : 3 * q * r - 1 = 2 * (q - 1) * (r - 1) * 2)
+  -- (g₁ : q * r - 4 * q - 4 * r + 5 = 0)
+  (g₂ : (4 - q) * (4 - r) = 11)
+  (g₃₁ : 4 - q = -1) :
+  q = 5 ∧ r = 15 := by
+  have hq: q = 5 := by linarith
+  constructor
+  . exact hq
+  . rw [hq] at g₂
+    linarith[g₂]
+
+
+lemma imo_1992_p1_8_5
+  -- (p : ℤ)
+  (q r : ℤ)
+  -- (hql : 3 ≤ q)
+  (hrl : 4 ≤ r)
+  (h₀ : 3 < q ∧ q < r)
+  -- (hk : 3 * q * r - 1 = 2 * (q - 1) * (r - 1) * 2)
+  -- (g₁ : q * r - 4 * q - 4 * r + 5 = 0)
+  (g₂ : (4 - q) * (4 - r) = 11)
+  (g₃₂ : 4 - q = 1 ∨ 4 - q = -11 ∨ 4 - q = 11) :
+  False := by
+  cases' g₃₂ with g₃₂ g₃₃
+  . have hq: q = 3 := by linarith[g₃₂]
+    rw [hq] at g₂
+    have hr: r = -7 := by linarith[g₂]
+    linarith[hrl,hr]
+  . cases' g₃₃ with g₃₃ g₃₄
+    . have hq: q = 15 := by linarith[g₃₃]
+      rw [hq] at g₂
+      have hr: r = 5 := by linarith[g₂]
+      linarith[hq,hr,h₀.2]
+    . linarith
+
+
+lemma imo_1992_p1_8_6
+  -- (p : ℤ)
+  (q r : ℤ)
+  -- (hql : 3 ≤ q)
+  (hrl : 4 ≤ r)
+  (h₀ : 3 < q ∧ q < r)
+  -- (hk : 3 * q * r - 1 = 2 * (q - 1) * (r - 1) * 2)
+  -- (g₁ : q * r - 4 * q - 4 * r + 5 = 0)
+  (g₂ : (4 - q) * (4 - r) = 11)
+  (g₃₂ : 4 - q = 1) :
+  False := by
+  have hq: q = 3 := by linarith[g₃₂]
+  rw [hq] at g₂
+  have hr: r = -7 := by linarith[g₂]
+  linarith[hrl,hr]
+
+
+lemma imo_1992_p1_8_7
+  -- (p : ℤ)
+  (q r : ℤ)
+  -- (hql : 3 ≤ q)
+  -- (hrl : 4 ≤ r)
+  (h₀ : 3 < q ∧ q < r)
+  -- (hk : 3 * q * r - 1 = 2 * (q - 1) * (r - 1) * 2)
+  -- (g₁ : q * r - 4 * q - 4 * r + 5 = 0)
+  (g₂ : (4 - q) * (4 - r) = 11)
+  (g₃₃ : 4 - q = -11) :
+  False := by
+  have hq: q = 15 := by linarith[g₃₃]
+  rw [hq] at g₂
+  have hr: r = 5 := by linarith[g₂]
+  linarith[hq,hr,h₀.2]
+
+
+lemma imo_1992_p1_8_8
+  -- (p : ℤ)
+  (q r : ℤ)
+  -- (hql : 3 ≤ q)
+  -- (hrl : 4 ≤ r)
+  (h₀ : q < r)
+  (h₁ : 6 < -r)
+  -- (hk : 3 * q * r - 1 = 2 * (q - 1) * (r - 1) * 2)
+  -- (g₁ : q * r - 4 * q - 4 * r + 5 = 0)
+  -- (g₂ : (4 - q) * (4 - r) = 11)
+  (g₃₄ : 4 - q = 11) :
+  False := by
+  have h₂: q = -7 := by
+    exact (Int.sub_right_inj 4).mp g₃₄
+  have h₃: -6 ≤ r := by
+    rw [h₂] at h₀
+    exact h₀
+  apply neg_le_neg at h₃
+  exact Lean.Omega.Int.le_lt_asymm h₃ h₁
+
+
+lemma imo_1992_p1_9
+  (p q r: ℤ)
+  (h₀: 1 < p ∧ p < q ∧ q < r)
+  (hpl: 2 ≤ p)
+  (hql: 3 ≤ q)
+  (hrl: 4 ≤ r)
+  (hpu: p < 4)
+  (hk: p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * 3) :
+  (p, q, r) = (2, 4, 8) ∨ (p, q, r) = (3, 5, 15) := by
+  interval_cases p
+  -- p = 2
+  . norm_num at *
+    have g₂: (q - 3) * (r - 3) = 5 := by linarith
+    have g₃: (q - 3) = -1 ∨ (q - 3) = 1 ∨ (q - 3) = -5 ∨ (q - 3) = 5 := by
+      refine imo_1992_p1_7 (q - 3) (r - 3) 5 g₂ ?_
+      decide
+    cases' g₃ with g₃₁ g₃₂
+    . exfalso
+      linarith [hql,g₃₁]
+    . cases' g₃₂ with g₃₂ g₃₃
+      . have hq: q = 4 := by linarith
+        rw [hq] at g₂
+        have hr: r = 8 := by linarith[g₂]
+        exact { left := hq, right := hr }
+      . exfalso
+        cases' g₃₃ with g₃₃ g₃₄
+        . linarith[hql,g₃₃]
+        . have hq: q = 8 := by linarith
+          rw [hq] at g₂
+          norm_num at g₂
+          have hr: r = 4 := by linarith
+          linarith[hrl,hr]
+  . right
+    norm_num at *
+    have g₂: (6 - 3*q) * (2 - r) = 5 := by linarith
+    have g₃: (6 - 3*q) = -1 ∨ (6 - 3*q) = 1 ∨ (6 - 3*q) = -5 ∨ (6 - 3*q) = 5 := by
+      refine imo_1992_p1_7 (6 - 3*q) (2 - r) 5 g₂ ?_
+      decide
+    exfalso
+    cases' g₃ with g₃₁ g₃₂
+    . linarith[g₃₁,q]
+    . cases' g₃₂ with g₃₂ g₃₃
+      . linarith[g₃₂,q]
+      . cases' g₃₃ with g₃₃ g₃₄
+        . linarith[g₃₃,q]
+        . linarith[g₃₄,q]
+
+
+
+lemma imo_1992_p1_9_1
+  (q r : ℤ)
+  (hql : 3 ≤ q)
+  (hrl : 4 ≤ r)
+  (h₀ : 2 < q ∧ q < r)
+  (hk : 2 * q * r - 1 = (q - 1) * (r - 1) * 3) :
+  q = 4 ∧ r = 8 := by
+  have g₂: (q - 3) * (r - 3) = 5 := by linarith
+  have g₃: (q - 3) = -1 ∨ (q - 3) = 1 ∨ (q - 3) = -5 ∨ (q - 3) = 5 := by
+    refine imo_1992_p1_7 (q - 3) (r - 3) 5 g₂ ?_
+    decide
+  cases' g₃ with g₃₁ g₃₂
+  . exfalso
+    linarith [hql,g₃₁]
+  . cases' g₃₂ with g₃₂ g₃₃
+    . have hq: q = 4 := by linarith
+      rw [hq] at g₂
+      have hr: r = 8 := by linarith[g₂]
+      exact { left := hq, right := hr }
+    . exfalso
+      cases' g₃₃ with g₃₃ g₃₄
+      . linarith[hql,g₃₃]
+      . have hq: q = 8 := by linarith
+        rw [hq] at g₂
+        norm_num at g₂
+        have hr: r = 4 := by linarith
+        linarith[hrl,hr]
+
+
+
+lemma imo_1992_p1_9_2
+  (q r : ℤ)
+  (hql : 3 ≤ q)
+  (hrl : 4 ≤ r)
+  (h₀ : 2 < q ∧ q < r)
+  (g₂ : (q - 3) * (r - 3) = 5) :
+  q = 4 ∧ r = 8 := by
+  have g₃: (q - 3) = -1 ∨ (q - 3) = 1 ∨ (q - 3) = -5 ∨ (q - 3) = 5 := by
+    refine imo_1992_p1_7 (q - 3) (r - 3) 5 g₂ ?_
+    decide
+  cases' g₃ with g₃₁ g₃₂
+  . exfalso
+    linarith [hql,g₃₁]
+  . cases' g₃₂ with g₃₂ g₃₃
+    . have hq: q = 4 := by linarith
+      rw [hq] at g₂
+      have hr: r = 8 := by linarith[g₂]
+      exact { left := hq, right := hr }
+    . exfalso
+      cases' g₃₃ with g₃₃ g₃₄
+      . linarith[hql,g₃₃]
+      . have hq: q = 8 := by linarith
+        rw [hq] at g₂
+        norm_num at g₂
+        have hr: r = 4 := by linarith
+        linarith[hrl,hr]
+
+
+lemma imo_1992_p1_9_3
+  (q r : ℤ)
+  (g₂ : (q - 3) * (r - 3) = 5) :
+  q - 3 = -1 ∨ q - 3 = 1 ∨ q - 3 = -5 ∨ q - 3 = 5 := by
+  refine imo_1992_p1_7 (q - 3) (r - 3) 5 g₂ ?_
+  decide
+
+
+lemma imo_1992_p1_9_4
+  -- (p : ℤ)
+  (q r : ℤ)
+  (hql : 3 ≤ q)
+  (hrl : 4 ≤ r)
+  (h₀ : 2 < q ∧ q < r)
+  -- (hk : 2 * q * r - 1 = (q - 1) * (r - 1) * 3)
+  -- (g₁ : q * r - 3 * q - 3 * r + 4 = 0)
+  (g₂ : (q - 3) * (r - 3) = 5)
+  (g₃ : q - 3 = -1 ∨ q - 3 = 1 ∨ q - 3 = -5 ∨ q - 3 = 5) :
+  q = 4 ∧ r = 8 := by
+  cases' g₃ with g₃₁ g₃₂
+  . exfalso
+    linarith [hql,g₃₁]
+  . cases' g₃₂ with g₃₂ g₃₃
+    . have hq: q = 4 := by linarith
+      rw [hq] at g₂
+      have hr: r = 8 := by linarith[g₂]
+      exact { left := hq, right := hr }
+    . exfalso
+      cases' g₃₃ with g₃₃ g₃₄
+      . linarith[hql,g₃₃]
+      . have hq: q = 8 := by linarith
+        rw [hq] at g₂
+        norm_num at g₂
+        have hr: r = 4 := by linarith
+        linarith[hrl,hr]
+
+
+lemma imo_1992_p1_9_5
+  -- (p : ℤ)
+  (q r : ℤ)
+  (hql : 3 ≤ q)
+  -- (hrl : 4 ≤ r)
+  -- (h₀ : 2 < q ∧ q < r)
+  -- (hk : 2 * q * r - 1 = (q - 1) * (r - 1) * 3)
+  -- (g₁ : q * r - 3 * q - 3 * r + 4 = 0)
+  -- (g₂ : (q - 3) * (r - 3) = 5)
+  (g₃₁ : q - 3 = -1) :
+  q = 4 ∧ r = 8 := by
+  exfalso
+  linarith [hql,g₃₁]
+
+
+lemma imo_1992_p1_9_6
+  -- (p r : ℤ)
+  (q r : ℤ)
+  (hql : 3 ≤ q)
+  (hrl : 4 ≤ r)
+  -- (h₀ : 2 < q ∧ q < r)
+  -- (hk : 2 * q * r - 1 = (q - 1) * (r - 1) * 3)
+  -- (g₁ : q * r - 3 * q - 3 * r + 4 = 0)
+  -- (g₂ : (q - 3) * (r - 3) = 5)
+  (g₃₁ : r * (q - 4) < r * (3 - r)) :
+  False := by
+  have h₀: 3 - r ≤ q - 4 := by
+    exact sub_le_sub hql hrl
+  have h₀: r * (3 - r) ≤ r * (q - 4) := by
+    refine (mul_le_mul_left ?_).mpr h₀
+    linarith
+  linarith
+
+
+lemma imo_1992_p1_9_7
+  -- (p : ℤ)
+  (q r : ℤ)
+  (hql : 3 ≤ q)
+  (hrl : 4 ≤ r)
+  (h₀ : 2 < q ∧ q < r)
+  -- (hk : 2 * q * r - 1 = (q - 1) * (r - 1) * 3)
+  -- (g₁ : q * r - 3 * q - 3 * r + 4 = 0)
+  (g₂ : (q - 3) * (r - 3) = 5)
+  (g₃₂ : q - 3 = 1 ∨ q - 3 = -5 ∨ q - 3 = 5) :
+  q = 4 ∧ r = 8 := by
+  cases' g₃₂ with g₃₂ g₃₃
+  . have hq: q = 4 := by linarith
+    rw [hq] at g₂
+    have hr: r = 8 := by linarith[g₂]
+    exact { left := hq, right := hr }
+  . exfalso
+    cases' g₃₃ with g₃₃ g₃₄
+    . linarith[hql,g₃₃]
+    . have hq: q = 8 := by linarith
+      rw [hq] at g₂
+      norm_num at g₂
+      have hr: r = 4 := by linarith
+      linarith[hrl,hr]
+
+
+lemma imo_1992_p1_9_8
+  -- (p : ℤ)
+  (q r : ℤ)
+  -- (hql : 3 ≤ q)
+  -- (hrl : 4 ≤ r)
+  -- (h₀ : 2 < q ∧ q < r)
+  -- (hk : 2 * q * r - 1 = (q - 1) * (r - 1) * 3)
+  -- (g₁ : q * r - 3 * q - 3 * r + 4 = 0)
+  (g₂ : (q - 3) * (r - 3) = 5)
+  (g₃₂ : q - 3 = 1) :
+  q = 4 ∧ r = 8 := by
+  have hq: q = 4 := by linarith
+  rw [hq] at g₂
+  have hr: r = 8 := by linarith[g₂]
+  exact { left := hq, right := hr }
+
+
+lemma imo_1992_p1_9_9
+  -- (p : ℤ)
+  (q r : ℤ)
+  -- (hql : 3 ≤ q)
+  -- (hrl : 4 ≤ r)
+  -- (h₀ : 2 < q ∧ q < r)
+  -- (hk : 2 * q * r - 1 = (q - 1) * (r - 1) * 3)
+  -- (g₁ : q * r - 3 * q - 3 * r + 4 = 0)
+  (g₂ : (q - 3) * (r - 3) = 5)
+  (g₃₂ : q - 3 = 1)
+  (hq : q = 4) :
+  q = 4 ∧ r = 8 := by
+  rw [hq] at g₂
+  have hr: r = 8 := by linarith[g₂]
+  exact { left := hq, right := hr }
+
+
+lemma imo_1992_p1_9_10
+  -- (p : ℤ)
+  (q r : ℤ)
+  (hql : 3 ≤ q)
+  (hrl : 4 ≤ r)
+  (h₀ : 2 < q ∧ q < r)
+  -- (hk : 2 * q * r - 1 = (q - 1) * (r - 1) * 3)
+  -- (g₁ : q * r - 3 * q - 3 * r + 4 = 0)
+  (g₂ : (q - 3) * (r - 3) = 5)
+  (g₃₃ : q - 3 = -5 ∨ q - 3 = 5) :
+  False := by
+  cases' g₃₃ with g₃₃ g₃₄
+  . linarith[hql,g₃₃]
+  . have hq: q = 8 := by linarith
+    rw [hq] at g₂
+    norm_num at g₂
+    have hr: r = 4 := by linarith
+    linarith[hrl,hr]
+
+
+lemma imo_1992_p1_9_11
+  -- (p : ℤ)
+  (q r : ℤ)
+  -- (hql : 3 ≤ q)
+  (hrl : 4 ≤ r)
+  (h₀ : 2 < q ∧ q < r)
+  -- (hk : 2 * q * r - 1 = (q - 1) * (r - 1) * 3)
+  -- (g₁ : q * r - 3 * q - 3 * r + 4 = 0)
+  (g₂ : (q - 3) * (r - 3) = 5)
+  (g₃₄ : q - 3 = 5) :
+  False := by
+  have hq: q = 8 := by linarith
+  rw [hq] at g₂
+  norm_num at g₂
+  have hr: r = 4 := by linarith
+  linarith[hrl,hr]
+
+
+lemma imo_1992_p1_9_12
+  -- (p : ℤ)
+  (q r : ℤ)
+  -- (hql : 3 ≤ q)
+  -- (hrl : 4 ≤ r)
+  (h₀ : 3 < q ∧ q < r)
+  (hk : 3 * q * r - 1 = 2 * (q - 1) * (r - 1) * 3) :
+  q = 5 ∧ r = 15 := by
+  have g₂: (6 - 3*q) * (2 - r) = 5 := by linarith
+  have g₃: (6 - 3*q) = -1 ∨ (6 - 3*q) = 1 ∨ (6 - 3*q) = -5 ∨ (6 - 3*q) = 5 := by
+    refine imo_1992_p1_7 (6 - 3*q) (2 - r) 5 g₂ ?_
+    decide
+  exfalso
+  cases' g₃ with g₃₁ g₃₂
+  . linarith[g₃₁,q]
+  . cases' g₃₂ with g₃₂ g₃₃
+    . linarith[g₃₂,q]
+    . cases' g₃₃ with g₃₃ g₃₄
+      . linarith[g₃₃,q]
+      . linarith[g₃₄,q]
+
+
+lemma imo_1992_p1_9_13
+  -- (p : ℤ)
+  (q r : ℤ)
+  -- (hql : 3 ≤ q)
+  -- (hrl : 4 ≤ r)
+  (h₀ : 3 < q ∧ q < r)
+  -- (hk : 3 * q * r - 1 = 2 * (q - 1) * (r - 1) * 3)
+  -- (g₁ : 3 * q * r - 6 * q - 6 * r + 7 = 0)
+  (g₂ : (6 - 3 * q) * (2 - r) = 5) :
+  False := by
+  have g₃: (6 - 3*q) = -1 ∨ (6 - 3*q) = 1 ∨ (6 - 3*q) = -5 ∨ (6 - 3*q) = 5 := by
+    refine imo_1992_p1_7 (6 - 3*q) (2 - r) 5 g₂ ?_
+    decide
+  exfalso
+  cases' g₃ with g₃₁ g₃₂
+  . linarith[g₃₁,q]
+  . cases' g₃₂ with g₃₂ g₃₃
+    . linarith[g₃₂,q]
+    . cases' g₃₃ with g₃₃ g₃₄
+      . linarith[g₃₃,q]
+      . linarith[g₃₄,q]
+
+
+lemma imo_1992_p1_9_14
+  -- (p : ℤ)
+  (q r : ℤ)
+  -- (hql : 3 ≤ q)
+  -- (hrl : 4 ≤ r)
+  -- (h₀ : 3 < q ∧ q < r)
+  -- (hk : 3 * q * r - 1 = 2 * (q - 1) * (r - 1) * 3)
+  -- (g₁ : 3 * q * r - 6 * q - 6 * r + 7 = 0)
+  (g₂ : (6 - 3 * q) * (2 - r) = 5) :
+  6 - 3 * q = -1 ∨ 6 - 3 * q = 1 ∨ 6 - 3 * q = -5 ∨ 6 - 3 * q = 5 := by
+  refine imo_1992_p1_7 (6 - 3*q) (2 - r) 5 g₂ ?_
+  decide
+
+lemma imo_1992_p1_9_15
+  -- (p : ℤ)
+  (q r : ℤ)
+  -- (hql : 3 ≤ q)
+  -- (hrl : 4 ≤ r)
+  (h₀ : 3 < q ∧ q < r)
+  -- (hk : 3 * q * r - 1 = 2 * (q - 1) * (r - 1) * 3)
+  -- (g₁ : 3 * q * r - 6 * q - 6 * r + 7 = 0)
+  -- (g₂ : (6 - 3 * q) * (2 - r) = 5)
+  (g₃ : 6 - 3 * q = -1 ∨ 6 - 3 * q = 1 ∨ 6 - 3 * q = -5 ∨ 6 - 3 * q = 5) :
+  False := by
+  exfalso
+  cases' g₃ with g₃₁ g₃₂
+  . linarith[g₃₁,q]
+  . cases' g₃₂ with g₃₂ g₃₃
+    . linarith[g₃₂,q]
+    . cases' g₃₃ with g₃₃ g₃₄
+      . linarith[g₃₃,q]
+      . linarith[g₃₄,q]
+
+
+lemma q_of_qr_eq_11_nat
+  (q r : ℕ)
+  (h₀ : q * r = 11) :
+  q = 1 ∨ q = 11 := by
+  have h₁: Nat.Prime (11:ℕ) := by decide
+  have h₂: ↑q ∣ 11 := by
+    exact Dvd.intro r h₀
+  exact Nat.Prime.eq_one_or_self_of_dvd h₁ q h₂
+
+
+lemma abs_q_r_product
+  (q r : ℤ)
+  (h₀ : q * r = 11) :
+  q.natAbs * r.natAbs = (11:ℕ)  := by
+  exact Int.natAbs_mul_natAbs_eq h₀
+  -- Since q * r = 11, taking the absolute value of both sides gives |q * r| = 11.
+  -- By properties of absolute values, |q * r| = |q| * |r|.
+
+
+lemma myprime5 : Nat.Prime 5 := by
+  rw [Nat.prime_def_lt']
+  constructor
+  . norm_num
+  . intros m hm mu
+    interval_cases m
+    all_goals {try norm_num }
+
+
+
+lemma abs_q_r_product_2
+  (q r : ℤ)
+  (h₀ : q * r = (11:ℕ)) :
+  abs q * abs r = 11 := by
+  have h₁: q.natAbs * r.natAbs = (11:ℕ) := by
+    exact Int.natAbs_mul_natAbs_eq h₀
+  have h₃: abs q = q.natAbs := by exact abs_eq_natAbs q
+  have h₄: abs r = r.natAbs := by exact abs_eq_natAbs r
+  rw [h₃,h₄]
+  norm_cast
+
+
+lemma imo_1992_p1_19_1
+  (p q r : ℤ)
+  -- (h₀ : 1 < p ∧ p < q ∧ q < r)
+  (k : ℤ)
+  (hk : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  -- (hpl : 2 ≤ p)
+  -- (hql : 3 ≤ q)
+  -- (hrl : 4 ≤ r)
+  (hden : 0 < (p - 1) * (q - 1) * (r - 1)) :
+  ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+  have g₁: ↑(p * q * r - 1) = ↑k * (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+    norm_cast
+    linarith
+  symm
+  have g₂: (↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≠ 0 := by
+    norm_cast
+    linarith[hden]
+  exact (div_eq_iff g₂).mpr g₁
+
+
+lemma imo_1992_p1_19_2
+  (p q r : ℤ)
+  -- (h₀ : 1 < p ∧ p < q ∧ q < r)
+  (k : ℤ)
+  -- (hk : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  -- (hpl : 2 ≤ p)
+  -- (hql : 3 ≤ q)
+  -- (hrl : 4 ≤ r)
+  (hden : 0 < (p - 1) * (q - 1) * (r - 1))
+  (g₁ : ↑(p * q * r - 1) = ↑k * (↑((p - 1) * (q - 1) * (r - 1)):ℚ)) :
+  ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+  symm
+  have g₂: (↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≠ 0 := by
+    norm_cast
+    linarith[hden]
+  exact (div_eq_iff g₂).mpr g₁
+
+
+lemma imo_1992_p1_19_3
+  (p q r : ℤ)
+  (h₀ : 1 < p ∧ p < q ∧ q < r)
+  (k : ℤ)
+  (hk : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  (hpl : 2 ≤ p)
+  (hql : 3 ≤ q)
+  (hrl : 4 ≤ r)
+  -- (hden : 0 < (p - 1) * (q - 1) * (r - 1))
+  (h₁ : ↑k = ↑(p * q * r - 1) / ↑((p - 1) * (q - 1) * (r - 1)))
+  (hk4 : k < 4)
+  (hk1 : 1 < k)
+  (hpu : p < 4) :
+  (p, q, r) = (2, 4, 8) ∨ (p, q, r) = (3, 5, 15) := by
+  interval_cases k
+  . exact imo_1992_p1_8 p q r h₀ hpl hql hrl hpu hk
+  . exact imo_1992_p1_9 p q r h₀ hpl hql hrl hpu hk

Lemmas/imo_1997_p5_lemmas.lean

@@ -0,0 +1,2926 @@
+import Mathlib
+set_option linter.unusedVariables.analyzeTactics true
+
+open Nat Real
+
+
+lemma imo_1997_p5_1
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (g : x ^ y ^ 2 = (x ^ y) ^ y)
+  (hxy : x ≤ y)
+  (h₁ : (x ^ y) ^ y = y ^ x) :
+  x ^ y ≤ y := by
+  by_contra! hc
+  have h₂: y^x ≤ y^y := by
+    { exact Nat.pow_le_pow_of_le_right h₀.2 hxy }
+  have h₃: y^y < (x^y)^y := by
+    refine Nat.pow_lt_pow_left hc ?_
+    refine Nat.pos_iff_ne_zero.mp h₀.2
+  rw [h₁] at h₃
+  linarith [h₂, h₃]
+
+
+lemma imo_1997_p5_1_1
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (hxy : x ≤ y)
+  (h₁ : (x ^ y) ^ y = y ^ x)
+  (hc : y < x ^ y) :
+  False := by
+  have h₂: y^x ≤ y^y := by
+    { exact Nat.pow_le_pow_of_le_right h₀.2 hxy }
+  have h₃: y^y < (x^y)^y := by
+    refine Nat.pow_lt_pow_left hc ?_
+    refine Nat.pos_iff_ne_zero.mp h₀.2
+  rw [h₁] at h₃
+  linarith [h₂, h₃]
+
+
+lemma imo_1997_p5_1_2
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (hxy : x ≤ y)
+  (h₁ : (x ^ y) ^ y = y ^ x)
+  (hc : y < x ^ y)
+  (h₂ : y ^ x ≤ y ^ y) :
+  False := by
+  have h₃: y^y < (x^y)^y := by
+    refine Nat.pow_lt_pow_left hc ?_
+    refine Nat.pos_iff_ne_zero.mp h₀.2
+  rw [h₁] at h₃
+  linarith [h₂, h₃]
+
+
+lemma imo_1997_p5_1_3
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (hxy : x ≤ y)
+  -- (h₁ : (x ^ y) ^ y = y ^ x)
+  (hc : y < x ^ y) :
+  -- (h₂ : y ^ x ≤ y ^ y) :
+  y ^ y < (x ^ y) ^ y := by
+  refine Nat.pow_lt_pow_left hc ?_
+  exact Nat.pos_iff_ne_zero.mp h₀.2
+
+
+lemma imo_1997_p5_2
+  (k : ℕ)
+  (hk : 5 ≤ k) :
+  4 * k < 2 ^ k := by
+  -- Proceed by induction on k
+  induction' k using Nat.case_strong_induction_on with n ih
+  -- Base case: k = 0 is not possible since 5 ≤ k
+  -- so we start directly with k = 5 as the base case
+  . norm_num
+  by_cases h₀ : n < 5
+  . have hn: n = 4 := by linarith
+    rw [hn]
+    norm_num
+  . push_neg at h₀
+    have ih₁ : 4 * n < 2 ^ n := ih n (le_refl n) h₀
+    rw [mul_add, pow_add, mul_one, pow_one, mul_two]
+    refine Nat.add_lt_add ih₁ ?_
+    refine lt_trans ?_ ih₁
+    refine (Nat.lt_mul_iff_one_lt_right (by norm_num)).mpr ?_
+    refine Nat.lt_of_lt_of_le ?_ h₀
+    norm_num
+
+
+lemma imo_1997_p5_2_1
+  (n : ℕ)
+  (ih : ∀ m ≤ n, 5 ≤ m → 4 * m < 2 ^ m)
+  (hk : 5 ≤ succ n) :
+  4 * succ n < 2 ^ succ n := by
+  by_cases h₀ : n < 5
+  . rw [succ_eq_add_one] at hk
+    have hn: n = 4 := by linarith
+    rw [hn]
+    norm_num
+  . push_neg at h₀
+    have ih₁ : 4 * n < 2 ^ n := ih n (le_refl n) h₀
+    rw [succ_eq_add_one, mul_add, pow_add, mul_one, pow_one, mul_two]
+    refine Nat.add_lt_add ih₁ ?_
+    refine lt_trans ?_ ih₁
+    refine (Nat.lt_mul_iff_one_lt_right (by norm_num)).mpr ?_
+    refine Nat.lt_of_lt_of_le ?_ h₀
+    norm_num
+
+
+lemma imo_1997_p5_2_2
+  (n : ℕ)
+  -- (ih : ∀ m ≤ n, 5 ≤ m → 4 * m < 2 ^ m)
+  (hk : 5 ≤ succ n)
+  (h₀ : n < 5) :
+  4 * succ n < 2 ^ succ n := by
+  rw [succ_eq_add_one] at hk
+  have hn: n = 4 := by linarith
+  rw [hn]
+  norm_num
+
+
+lemma imo_1997_p5_2_3
+  (n : ℕ)
+  (ih : ∀ m ≤ n, 5 ≤ m → 4 * m < 2 ^ m)
+  -- (hk : 5 ≤ succ n)
+  (h₀ : 5 ≤ n) :
+  4 * succ n < 2 ^ succ n := by
+  have ih₁ : 4 * n < 2 ^ n := ih n (le_refl n) h₀
+  rw [succ_eq_add_one, mul_add, pow_add, mul_one, pow_one, mul_two]
+  refine Nat.add_lt_add ih₁ ?_
+  refine lt_trans ?_ ih₁
+  refine (Nat.lt_mul_iff_one_lt_right (by norm_num)).mpr ?_
+  refine Nat.lt_of_lt_of_le ?_ h₀
+  norm_num
+
+
+lemma imo_1997_p5_2_4
+  (n : ℕ)
+  -- (ih : ∀ m ≤ n, 5 ≤ m → 4 * m < 2 ^ m)
+  -- (hk : 5 ≤ succ n)
+  (h₀ : 5 ≤ n)
+  (ih₁ : 4 * n < 2 ^ n) :
+  4 * succ n < 2 ^ succ n := by
+  rw [succ_eq_add_one, mul_add, pow_add, mul_one, pow_one, mul_two]
+  refine Nat.add_lt_add ih₁ ?_
+  refine lt_trans ?_ ih₁
+  refine (Nat.lt_mul_iff_one_lt_right (by norm_num)).mpr ?_
+  refine Nat.lt_of_lt_of_le ?_ h₀
+  norm_num
+
+
+lemma imo_1997_p5_2_5
+  (n : ℕ)
+  -- (ih : ∀ m ≤ n, 5 ≤ m → 4 * m < 2 ^ m)
+  -- (hk : 5 ≤ succ n)
+  (h₀ : 5 ≤ n)
+  (ih₁ : 4 * n < 2 ^ n) :
+  4 * n + 4 < 2 ^ n + 2 ^ n := by
+  refine Nat.add_lt_add ih₁ ?_
+  refine lt_trans ?_ ih₁
+  refine (Nat.lt_mul_iff_one_lt_right (by norm_num)).mpr ?_
+  refine Nat.lt_of_lt_of_le ?_ h₀
+  norm_num
+
+
+lemma imo_1997_p5_2_6
+  (n : ℕ)
+  -- (ih : ∀ m ≤ n, 5 ≤ m → 4 * m < 2 ^ m)
+  -- (hk : 5 ≤ succ n)
+  (h₀ : 5 ≤ n)
+  (ih₁ : 4 * n < 2 ^ n) :
+  4 < 2 ^ n := by
+  refine lt_trans ?_ ih₁
+  refine (Nat.lt_mul_iff_one_lt_right (by norm_num)).mpr ?_
+  refine Nat.lt_of_lt_of_le ?_ h₀
+  norm_num
+
+
+lemma imo_1997_p5_2_7
+  (n : ℕ)
+  -- (ih : ∀ m ≤ n, 5 ≤ m → 4 * m < 2 ^ m)
+  -- (hk : 5 ≤ succ n)
+  (h₀ : 5 ≤ n) :
+  -- (ih₁ : 4 * n < 2 ^ n) :
+  4 < 4 * n := by
+  refine (Nat.lt_mul_iff_one_lt_right (by norm_num)).mpr ?_
+  refine Nat.lt_of_lt_of_le ?_ h₀
+  norm_num
+
+
+lemma imo_1997_p5_3
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x^(y^2) = y^x)
+  (g₁ : x^(y^2) = (x^y)^y)
+  (hxy : x ≤ y) :
+  (x, y) = (1, 1) ∨ (x, y) = (16, 2) ∨ (x, y) = (27, 3) := by
+  rw [g₁] at h₁
+  have g2: x^y ≤ y := by
+    exact imo_1997_p5_1 x y h₀ hxy h₁
+  have g3: x = 1 := by
+    by_contra! hc
+    have g3: 2 ≤ x := by
+      by_contra! gc
+      interval_cases x
+      . linarith
+      . omega
+    have g4: 2 ^ y ≤ x ^ y := by { exact Nat.pow_le_pow_of_le_left g3 y }
+    have g5: y < 2 ^ y := by exact Nat.lt_two_pow_self
+    linarith
+  rw [g3] at h₁
+  simp at h₁
+  left
+  norm_num
+  exact { left := g3, right := id h₁.symm }
+
+
+lemma imo_1997_p5_3_1
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : (x ^ y) ^ y = y ^ x)
+  (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  (hxy : x ≤ y)
+  (g₂ : x ^ y ≤ y) :
+  (x, y) = (1, 1) := by
+  have g₃: x = 1 := by
+    by_contra! hc
+    have g3: 2 ≤ x := by
+      by_contra! gc
+      interval_cases x
+      . linarith
+      . omega
+    have g4: 2^y ≤ x^y := by { exact Nat.pow_le_pow_of_le_left g3 y }
+    have g5: y < 2^y := by exact Nat.lt_two_pow_self
+    linarith
+  rw [g₃] at h₁
+  simp at h₁
+  norm_num
+  exact { left := g₃, right := id h₁.symm }
+
+
+lemma imo_1997_p5_3_2
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : (x ^ y) ^ y = y ^ x)
+  (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  (hxy : x ≤ y)
+  (g2 : x ^ y ≤ y) :
+  x = 1 := by
+  by_contra! hc
+  have g₃: 2 ≤ x := by
+    by_contra! gc
+    interval_cases x
+    . linarith
+    . omega
+  have g₄: 2^y ≤ x ^ y := by { exact Nat.pow_le_pow_of_le_left g₃ y }
+  have g₅: y < 2 ^ y := by exact Nat.lt_two_pow_self
+  linarith
+
+
+lemma imo_1997_p5_3_3
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : (x ^ y) ^ y = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : x ≤ y)
+  (g₂ : x ^ y ≤ y)
+  -- (hc : ¬x = 1)
+  (g₃ : 2 ≤ x) :
+  False := by
+  have g₄: 2^y ≤ x ^ y := by { exact Nat.pow_le_pow_of_le_left g₃ y }
+  have g₅: y < 2 ^ y := by exact Nat.lt_two_pow_self
+  linarith
+
+
+lemma imo_1997_p5_3_4
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : (x ^ y) ^ y = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : x ≤ y)
+  (g2 : x ^ y ≤ y)
+  -- (hc : ¬x = 1)
+  -- (g₃ : 2 ≤ x)
+  (g₄ : 2 ^ y ≤ x ^ y) :
+  False := by
+  have g₅: y < 2 ^ y := by exact Nat.lt_two_pow_self
+  linarith
+
+
+lemma imo_1997_p5_3_5
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : (x ^ y) ^ y = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : x ≤ y)
+  -- (g2 : x ^ y ≤ y)
+  -- (hc : ¬x = 1)
+  (g₃ : 2 ≤ x) :
+  -- (g4 : 2 ^ y ≤ x ^ y) :
+  y + 2 < 2 ^ y + x := by
+  refine lt_add_of_lt_add_left ?_ g₃
+  refine add_lt_add_right ?_ 2
+  exact Nat.lt_two_pow_self
+
+
+lemma imo_1997_p5_3_6
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : (x ^ y) ^ y = y ^ x)
+  (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  (hxy : x ≤ y)
+  (g₂ : x ^ y ≤ y)
+  (g₃ : x = 1) :
+  y = 1 := by
+  rw [g₃] at h₁
+  simp at h₁
+  exact id h₁.symm
+
+
+lemma imo_1997_p5_4
+  (x: ℕ)
+  (h₀: 0 < x):
+  (↑x = Real.exp (Real.log ↑x)):= by
+  have hx_pos : 0 < (↑x : ℝ) := by exact Nat.cast_pos.mpr h₀
+  symm
+  exact Real.exp_log hx_pos
+
+
+lemma imo_1997_p5_5
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (hxy : y < x) :
+  y ^ 2 < x := by
+  by_cases hy: 1 < y
+  . have hx: 2 ≤ x := by linarith
+    have h₂: y ^ x < x ^ x := by
+      refine Nat.pow_lt_pow_left hxy ?_
+      exact Nat.ne_of_lt' h₀.1
+    rw [← h₁] at h₂
+    exact (Nat.pow_lt_pow_iff_right hx).mp h₂
+  . push_neg at hy
+    interval_cases y
+    . simp
+      exact h₀.1
+    . simp at *
+      assumption
+
+
+lemma imo_1997_p5_5_1
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (hxy : y < x)
+  (hy : 1 < y) :
+  y ^ 2 < x := by
+  have hx: 2 ≤ x := by linarith
+  have h₂: y ^ x < x ^ x := by
+    refine Nat.pow_lt_pow_left hxy ?_
+    exact Nat.ne_of_lt' h₀.1
+  rw [← h₁] at h₂
+  exact (Nat.pow_lt_pow_iff_right hx).mp h₂
+
+
+lemma imo_1997_p5_5_2
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  (hxy : y < x) :
+  -- (hy : 1 < y)
+  -- (hx : 2 ≤ x) :
+  y ^ x < x ^ x := by
+  refine Nat.pow_lt_pow_left hxy ?_
+  exact Nat.ne_of_lt' h₀.1
+
+
+lemma imo_1997_p5_5_3
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  -- (hy : 1 < y)
+  (hx : 2 ≤ x)
+  (h₂ : y ^ x < x ^ x) :
+  y ^ 2 < x := by
+  rw [← h₁] at h₂
+  exact (Nat.pow_lt_pow_iff_right hx).mp h₂
+
+
+lemma imo_1997_p5_5_4
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (hxy : y < x)
+  (hy : ¬1 < y) :
+  y ^ 2 < x := by
+  push_neg at hy
+  interval_cases y
+  . simp
+    exact h₀.1
+  . simp at *
+    assumption
+
+
+lemma imo_1997_p5_6
+  (x y: ℕ)
+  (h₀: 0 < x ∧ 0 < y)
+  (h₁: x ^ y ^ 2 = y ^ x) :
+  (↑x / ↑y^2) ^ y ^ 2 = (↑y:ℝ)^ ((↑x:ℝ) - 2 * ↑y ^ 2) := by
+  have g₁: (↑x:ℝ) ^ (↑y:ℝ) ^ 2 = (↑y:ℝ) ^ (↑x:ℝ) := by
+    norm_cast
+  have g₂: 0 < ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2)) := by
+    norm_cast
+    exact pow_pos h₀.2 _
+  have g₃: ((↑x:ℝ) ^ (↑y:ℝ) ^ 2) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2))
+        = ((↑y:ℝ) ^ (↑x:ℝ)) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2)) := by
+    refine (div_left_inj' ?_).mpr g₁
+    norm_cast
+    refine pow_ne_zero _ ?_
+    linarith [h₀.2]
+  have gy: 0 < (↑y:ℝ) := by
+    norm_cast
+    exact h₀.2
+  rw [← Real.rpow_sub gy (↑x) (2 * ↑y ^ 2)] at g₃
+  have g₄: ((↑x:ℝ) ^ (↑y:ℝ) ^ 2) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2))
+        = (↑x / ↑y^2) ^ y ^ 2 := by
+    have g₅: (↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2) = ((↑y:ℝ) ^ 2) ^ ((↑y:ℝ) ^ 2) := by
+      norm_cast
+      refine pow_mul y 2 (y^2)
+    rw [g₅]
+    symm
+    norm_cast
+    have g₆: ((↑x:ℝ) / ↑y ^ 2) ^ y ^ 2 = ↑x ^ y ^ 2 / (↑y ^ 2) ^ y ^ 2 := by
+      refine div_pow (↑x:ℝ) ((↑y:ℝ) ^ 2) (y^2)
+    norm_cast at *
+  rw [g₄] at g₃
+  norm_cast at *
+
+
+lemma imo_1997_p5_6_1
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y) :
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : ↑x ^ ↑y ^ 2 = ↑y ^ ↑x) :
+  0 < ↑y ^ (2 * ↑y ^ 2) := by
+  exact pow_pos h₀.2 _
+
+
+lemma imo_1997_p5_6_2
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  (g₁ : (↑x:ℝ) ^ (↑y:ℝ) ^ 2 = (↑y:ℝ) ^ (↑x:ℝ)) :
+  -- (g₂ : 0 < ↑y ^ (2 * ↑y ^ 2)) :
+  (↑x / ↑y ^ 2) ^ y ^ 2 = (↑y:ℝ) ^ ((↑x:ℝ) - 2 * ↑y ^ 2) := by
+  have g₃: ((↑x:ℝ) ^ (↑y:ℝ) ^ 2) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2))
+        = ((↑y:ℝ) ^ (↑x:ℝ)) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2)) := by
+    refine (div_left_inj' ?_).mpr g₁
+    norm_cast
+    refine pow_ne_zero _ ?_
+    linarith [h₀.2]
+  have gy: 0 < (↑y:ℝ) := by
+    norm_cast
+    exact h₀.2
+  rw [← Real.rpow_sub gy (↑x) (2 * ↑y ^ 2)] at g₃
+  have g₄: ((↑x:ℝ) ^ (↑y:ℝ) ^ 2) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2))
+        = (↑x / ↑y^2) ^ y ^ 2 := by
+    have g₅: (↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2) = ((↑y:ℝ) ^ 2) ^ ((↑y:ℝ) ^ 2) := by
+      norm_cast
+      refine pow_mul y 2 (y^2)
+    rw [g₅]
+    symm
+    norm_cast
+    have g₆: ((↑x:ℝ) / ↑y ^ 2) ^ y ^ 2 = ↑x ^ y ^ 2 / (↑y ^ 2) ^ y ^ 2 := by
+      refine div_pow (↑x:ℝ) ((↑y:ℝ) ^ 2) (y^2)
+    norm_cast at *
+  rw [g₄] at g₃
+  norm_cast at *
+
+
+lemma imo_1997_p5_6_3
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y) :
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : ↑x ^ ↑y ^ 2 = ↑y ^ ↑x)
+  -- (g₂ : 0 < ↑y ^ (2 * ↑y ^ 2)) :
+  ↑y ^ (2 * ↑y ^ 2) ≠ 0 := by
+  norm_cast
+  refine pow_ne_zero _ ?_
+  linarith [h₀.2]
+
+
+lemma imo_1997_p5_6_4
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  (g₁ : (↑x:ℝ) ^ (↑y:ℝ) ^ 2 = (↑y:ℝ) ^ (↑x:ℝ)) :
+  -- (g₂ : 0 < ↑y ^ (2 * ↑y ^ 2))
+  -- (g₃ : ↑x ^ ↑y ^ 2 / ↑y ^ (2 * ↑y ^ 2) = ↑y ^ ↑x / ↑y ^ (2 * ↑y ^ 2))
+  -- (gy : 0 < ↑y) :
+  ((↑x:ℝ) ^ (↑y:ℝ) ^ 2) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2))
+        = ((↑y:ℝ) ^ (↑x:ℝ)) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2)) := by
+  refine (div_left_inj' ?_).mpr g₁
+  norm_cast
+  refine pow_ne_zero _ ?_
+  linarith [h₀.2]
+
+
+lemma imo_1997_p5_6_5
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : ↑x ^ ↑y ^ 2 = ↑y ^ ↑x)
+  -- (g₂ : 0 < ↑y ^ (2 * ↑y ^ 2))
+  (g₃ : ((↑x:ℝ) ^ (↑y:ℝ) ^ 2) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2))
+        = ((↑y:ℝ) ^ (↑x:ℝ)) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2)))
+  (gy : 0 < (↑y:ℝ)) :
+  (↑x / ↑y ^ 2) ^ y ^ 2 = (↑y:ℝ) ^ ((↑x:ℝ) - 2 * ↑y ^ 2) := by
+  rw [← Real.rpow_sub gy (↑x) (2 * ↑y ^ 2)] at g₃
+  have g₄: ((↑x:ℝ) ^ (↑y:ℝ) ^ 2) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2))
+        = (↑x / ↑y^2) ^ y ^ 2 := by
+    have g₅: (↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2) = ((↑y:ℝ) ^ 2) ^ ((↑y:ℝ) ^ 2) := by
+      norm_cast
+      refine pow_mul y 2 (y^2)
+    rw [g₅]
+    symm
+    norm_cast
+    have g₆: ((↑x:ℝ) / ↑y ^ 2) ^ y ^ 2 = ↑x ^ y ^ 2 / (↑y ^ 2) ^ y ^ 2 := by
+      refine div_pow (↑x:ℝ) ((↑y:ℝ) ^ 2) (y^2)
+    norm_cast at *
+  rw [g₄] at g₃
+  norm_cast at *
+
+
+lemma imo_1997_p5_6_6
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : ↑x ^ ↑y ^ 2 = ↑y ^ ↑x)
+  -- (g₂ : 0 < ↑y ^ (2 * ↑y ^ 2))
+  (g₃ : ((↑x:ℝ) ^ (↑y:ℝ) ^ 2) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2))
+        = ((↑y:ℝ) ^ (↑x:ℝ)) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2)))
+  (gy : 0 < (↑y:ℝ)) :
+  ((↑x:ℝ) ^ (↑y:ℝ) ^ 2) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2)) = (↑x / ↑y^2) ^ y ^ 2 := by
+  have g₅: (↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2) = ((↑y:ℝ) ^ 2) ^ ((↑y:ℝ) ^ 2) := by
+    norm_cast
+    refine pow_mul y 2 (y^2)
+  rw [g₅]
+  symm
+  norm_cast
+  have g₆: ((↑x:ℝ) / ↑y ^ 2) ^ y ^ 2 = ↑x ^ y ^ 2 / (↑y ^ 2) ^ y ^ 2 := by
+    refine div_pow (↑x:ℝ) ((↑y:ℝ) ^ 2) (y^2)
+  norm_cast at *
+
+
+lemma imo_1997_p5_6_7
+  -- (x : ℕ)
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : ↑x ^ ↑y ^ 2 = ↑y ^ ↑x)
+  -- (g₂ : 0 < ↑y ^ (2 * ↑y ^ 2))
+  -- (g₃ : ((↑x:ℝ) ^ (↑y:ℝ) ^ 2) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2))
+  --       = ((↑y:ℝ) ^ (↑x:ℝ)) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2)))
+  (hy : 0 < y)
+  (hxy : y < x) :
+  (↑y:ℝ) ^ (2 * (y ^ 2)) < ((↑x:ℝ) ^ 2) ^ (y ^ 2) := by
+  rw [pow_mul (↑y:ℝ) 2 (y ^ 2)]
+  refine pow_lt_pow_left₀ ?_ ?_ ?_
+  . norm_cast
+    exact Nat.pow_lt_pow_left hxy (by decide)
+  . exact sq_nonneg (↑y:ℝ)
+  . symm
+    refine Nat.ne_of_lt ?_
+    exact pos_pow_of_pos 2 hy
+
+
+lemma imo_1997_p5_6_8
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : ↑x ^ ↑y ^ 2 = ↑y ^ ↑x)
+  -- (g₂ : 0 < ↑y ^ (2 * ↑y ^ 2))
+  (g₃ : ((↑x:ℝ) ^ (↑y:ℝ) ^ 2) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2))
+        = ((↑y:ℝ) ^ (↑x:ℝ)) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2)))
+  (gy : 0 < (↑y:ℝ))
+  (g₅ : (↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2) = ((↑y:ℝ) ^ 2) ^ ((↑y:ℝ) ^ 2)) :
+  ((↑x:ℝ) ^ (↑y:ℝ) ^ 2) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2)) = (↑x / ↑y^2) ^ y ^ 2 := by
+  rw [g₅]
+  symm
+  norm_cast
+  have g₆: ((↑x:ℝ) / ↑y ^ 2) ^ y ^ 2 = ↑x ^ y ^ 2 / (↑y ^ 2) ^ y ^ 2 := by
+    refine div_pow (↑x:ℝ) ((↑y:ℝ) ^ 2) (y^2)
+  norm_cast at *
+
+
+lemma imo_1997_p5_6_9
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : ↑x ^ ↑y ^ 2 = ↑y ^ ↑x)
+  -- (g₂ : 0 < ↑y ^ (2 * ↑y ^ 2))
+  (g₃ : ((↑x:ℝ) ^ (↑y:ℝ) ^ 2) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2))
+        = ((↑y:ℝ) ^ (↑x:ℝ)) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2)))
+  (gy : 0 < (↑y:ℝ))
+  (g₅ : (↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2) = ((↑y:ℝ) ^ 2) ^ ((↑y:ℝ) ^ 2)) :
+  ((↑x:ℝ) ^ (↑y:ℝ) ^ 2) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2)) = (↑x / ↑y^2) ^ y ^ 2 := by
+  rw [g₅]
+  symm
+  norm_cast
+  have g₆: ((↑x:ℝ) / ↑y ^ 2) ^ y ^ 2 = ↑x ^ y ^ 2 / (↑y ^ 2) ^ y ^ 2 := by
+    refine div_pow (↑x:ℝ) ((↑y:ℝ) ^ 2) (y^2)
+  norm_cast at *
+
+
+lemma imo_1997_p5_6_10
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : ↑x ^ ↑y ^ 2 = ↑y ^ ↑x)
+  -- (g₂ : 0 < ↑y ^ (2 * ↑y ^ 2))
+  (g₃ : ↑x ^ ↑y ^ 2 / ↑y ^ (2 * ↑y ^ 2) = ↑y ^ (↑x - 2 * ↑y ^ 2))
+  (gy : 0 < ↑y)
+  (g₄ : ↑x ^ ↑y ^ 2 / ↑y ^ (2 * ↑y ^ 2) = (↑x / ↑y ^ 2) ^ y ^ 2) :
+  (↑x / ↑y ^ 2) ^ y ^ 2 = ↑y ^ (↑x - 2 * ↑y ^ 2) := by
+  rw [g₄] at g₃
+  norm_cast at *
+
+
+lemma imo_1997_p5_7
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (hxy : y < x) :
+  2 * y ^ 2 < x := by
+  by_cases hy1: y = 1
+  . rw [hy1]
+    norm_num
+    by_contra! hc
+    interval_cases x
+    . linarith
+    . linarith
+    . rw [hy1] at h₁
+      simp at h₁
+  . have hy: 1 < y := by
+      contrapose! hy1
+      linarith
+    clear hy1
+    have h₂: (↑y:ℝ) ^ 2 < ↑x := by
+      norm_cast
+      exact imo_1997_p5_5 x y h₀ h₁ hxy
+    have h₃: 1 < ↑x / (↑y:ℝ) ^ 2 := by
+      refine (one_lt_div ?_).mpr h₂
+      norm_cast
+      exact pow_pos h₀.2 2
+    have h₄: 1 < (↑x / (↑y:ℝ)^2)^(y^2) := by
+      refine one_lt_pow₀ h₃ ?_
+      refine Nat.ne_of_gt ?_
+      refine sq_pos_of_pos ?_
+      exact lt_of_succ_lt hy
+    have h₅: (↑x/ (↑y:ℝ)^2)^(y^2) = (↑y:ℝ)^((↑x:ℝ) - 2*(↑y:ℝ)^2) := by
+      exact imo_1997_p5_6 x y h₀ h₁
+    rw [h₅] at h₄
+    have h₆: 0 < (↑x:ℝ) - 2 * (↑y:ℝ) ^ 2 := by
+      by_contra! hc
+      cases' lt_or_eq_of_le hc with hlt heq
+      . have gy: 1 < (↑y:ℝ) := by
+          norm_cast
+        have glt: (↑x:ℝ) - 2*(↑y:ℝ)^2 < (↑0:ℝ) := by
+          norm_cast at *
+        have g₁: (↑y:ℝ) ^ ((↑x:ℝ) - 2*(↑y:ℝ)^2) < (↑y:ℝ) ^ (↑0:ℝ) := by
+          exact Real.rpow_lt_rpow_of_exponent_lt gy glt
+        simp at g₁
+        linarith[ h₄,g₁]
+      . rw [heq] at h₄
+        simp at h₄
+    simp at h₆
+    norm_cast at h₆
+
+
+lemma imo_1997_p5_7_1
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (hxy : y < x)
+  (hy1 : y = 1) :
+  2 * y ^ 2 < x := by
+  rw [hy1]
+  norm_num
+  by_contra! hc
+  interval_cases x
+  . linarith
+  . linarith
+  . rw [hy1] at h₁
+    simp at h₁
+
+
+lemma imo_1997_p5_7_2
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (hxy : y < x)
+  (hy1 : y = 1) :
+  2 < x := by
+  by_contra! hc
+  interval_cases x
+  . linarith
+  . linarith
+  . rw [hy1] at h₁
+    simp at h₁
+
+
+lemma imo_1997_p5_7_3
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (hxy : y < x)
+  (hy1 : y = 1)
+  (hc : x ≤ 2) :
+  False := by
+  interval_cases x
+  . linarith
+  . linarith
+  . rw [hy1] at h₁
+    simp at h₁
+
+
+lemma imo_1997_p5_7_4
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (hxy : y < x)
+  (hy : 1 < y) :
+  2 * y ^ 2 < x := by
+  have h₂: (↑y:ℝ) ^ 2 < ↑x := by
+    norm_cast
+    exact imo_1997_p5_5 x y h₀ h₁ hxy
+  have h₃: 1 < ↑x / (↑y:ℝ) ^ 2 := by
+    refine (one_lt_div ?_).mpr h₂
+    norm_cast
+    exact pow_pos h₀.2 2
+  have h₄: 1 < (↑x / (↑y:ℝ)^2)^(y^2) := by
+    refine one_lt_pow₀ h₃ ?_
+    refine Nat.ne_of_gt ?_
+    refine sq_pos_of_pos ?_
+    exact lt_of_succ_lt hy
+  have h₅: (↑x/ (↑y:ℝ)^2)^(y^2) = (↑y:ℝ)^((↑x:ℝ) - 2*(↑y:ℝ)^2) := by
+    exact imo_1997_p5_6 x y h₀ h₁
+  rw [h₅] at h₄
+  have h₆: 0 < (↑x:ℝ) - 2 * (↑y:ℝ) ^ 2 := by
+    by_contra! hc
+    cases' lt_or_eq_of_le hc with hlt heq
+    . have gy: 1 < (↑y:ℝ) := by
+        norm_cast
+      have glt: (↑x:ℝ) - 2*(↑y:ℝ)^2 < (↑0:ℝ) := by
+        norm_cast at *
+      have g₁: (↑y:ℝ) ^ ((↑x:ℝ) - 2*(↑y:ℝ)^2) < (↑y:ℝ) ^ (↑0:ℝ) := by
+        exact Real.rpow_lt_rpow_of_exponent_lt gy glt
+      simp at g₁
+      linarith[ h₄,g₁]
+    . rw [heq] at h₄
+      simp at h₄
+  simp at h₆
+  norm_cast at h₆
+
+
+lemma imo_1997_p5_7_5
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  (hy : 1 < y)
+  (h₂ : (↑y:ℝ) ^ 2 < ↑x) :
+  2 * y ^ 2 < x := by
+  have h₃: 1 < ↑x / (↑y:ℝ) ^ 2 := by
+    refine (one_lt_div ?_).mpr h₂
+    norm_cast
+    exact pow_pos h₀.2 2
+  have h₄: 1 < (↑x / (↑y:ℝ)^2)^(y^2) := by
+    refine one_lt_pow₀ h₃ ?_
+    refine Nat.ne_of_gt ?_
+    refine sq_pos_of_pos ?_
+    exact lt_of_succ_lt hy
+  have h₅: (↑x/ (↑y:ℝ)^2)^(y^2) = (↑y:ℝ)^((↑x:ℝ) - 2*(↑y:ℝ)^2) := by
+    exact imo_1997_p5_6 x y h₀ h₁
+  rw [h₅] at h₄
+  have h₆: 0 < (↑x:ℝ) - 2 * (↑y:ℝ) ^ 2 := by
+    by_contra! hc
+    cases' lt_or_eq_of_le hc with hlt heq
+    . have gy: 1 < (↑y:ℝ) := by
+        norm_cast
+      have glt: (↑x:ℝ) - 2*(↑y:ℝ)^2 < (↑0:ℝ) := by
+        norm_cast at *
+      have g₁: (↑y:ℝ) ^ ((↑x:ℝ) - 2*(↑y:ℝ)^2) < (↑y:ℝ) ^ (↑0:ℝ) := by
+        exact Real.rpow_lt_rpow_of_exponent_lt gy glt
+      simp at g₁
+      linarith[ h₄,g₁]
+    . rw [heq] at h₄
+      simp at h₄
+  simp at h₆
+  norm_cast at h₆
+
+
+lemma imo_1997_p5_7_6
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  -- (hy : 1 < y)
+  (h₂ : (↑y:ℝ) ^ 2 < ↑x) :
+  1 < ↑x / (↑y:ℝ) ^ 2 := by
+  refine (one_lt_div ?_).mpr h₂
+  norm_cast
+  exact pow_pos h₀.2 2
+
+
+lemma imo_1997_p5_7_7
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  (hy : 1 < y)
+  -- (h₂ : ↑y ^ 2 < ↑x)
+  (h₃ : 1 < ↑x / (↑y:ℝ) ^ 2) :
+  2 * y ^ 2 < x := by
+  have h₄: 1 < (↑x / (↑y:ℝ)^2)^(y^2) := by
+    refine one_lt_pow₀ h₃ ?_
+    refine Nat.ne_of_gt ?_
+    refine sq_pos_of_pos ?_
+    exact lt_of_succ_lt hy
+  have h₅: (↑x/ (↑y:ℝ)^2)^(y^2) = (↑y:ℝ)^((↑x:ℝ) - 2*(↑y:ℝ)^2) := by
+    exact imo_1997_p5_6 x y h₀ h₁
+  rw [h₅] at h₄
+  have h₆: 0 < (↑x:ℝ) - 2 * (↑y:ℝ) ^ 2 := by
+    by_contra! hc
+    cases' lt_or_eq_of_le hc with hlt heq
+    . have gy: 1 < (↑y:ℝ) := by
+        norm_cast
+      have glt: (↑x:ℝ) - 2*(↑y:ℝ)^2 < (↑0:ℝ) := by
+        norm_cast at *
+      have g₁: (↑y:ℝ) ^ ((↑x:ℝ) - 2*(↑y:ℝ)^2) < (↑y:ℝ) ^ (↑0:ℝ) := by
+        exact Real.rpow_lt_rpow_of_exponent_lt gy glt
+      simp at g₁
+      linarith[ h₄,g₁]
+    . rw [heq] at h₄
+      simp at h₄
+  simp at h₆
+  norm_cast at h₆
+
+
+lemma imo_1997_p5_7_8
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  (hy : 1 < y)
+  -- (h₂ : ↑y ^ 2 < ↑x)
+  (h₃ : 1 < ↑x / ↑y ^ 2) :
+  1 < (↑x / ↑y ^ 2) ^ y ^ 2 := by
+  refine one_lt_pow₀ h₃ ?_
+  refine Nat.ne_of_gt ?_
+  refine sq_pos_of_pos ?_
+  exact lt_of_succ_lt hy
+
+
+lemma imo_1997_p5_7_9
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  (hy : 1 < y)
+  -- (h₂ : ↑y ^ 2 < ↑x)
+  -- (h₃ : 1 < ↑x / ↑y ^ 2)
+  (h₄ : 1 < (↑x / (↑y:ℝ)^2)^(y^2))
+  (h₅ : (↑x/ (↑y:ℝ)^2)^(y^2) = (↑y:ℝ)^((↑x:ℝ) - 2*(↑y:ℝ)^2)) :
+  2 * y ^ 2 < x := by
+  rw [h₅] at h₄
+  have h₆: 0 < (↑x:ℝ) - 2 * (↑y:ℝ) ^ 2 := by
+    by_contra! hc
+    cases' lt_or_eq_of_le hc with hlt heq
+    . have gy: 1 < (↑y:ℝ) := by
+        norm_cast
+      have glt: (↑x:ℝ) - 2*(↑y:ℝ)^2 < (↑0:ℝ) := by
+        norm_cast at *
+      have g₁: (↑y:ℝ) ^ ((↑x:ℝ) - 2*(↑y:ℝ)^2) < (↑y:ℝ) ^ (↑0:ℝ) := by
+        exact Real.rpow_lt_rpow_of_exponent_lt gy glt
+      simp at g₁
+      linarith[ h₄,g₁]
+    . rw [heq] at h₄
+      simp at h₄
+  simp at h₆
+  norm_cast at h₆
+
+
+lemma imo_1997_p5_7_10
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  -- (hy : 1 < y)
+  -- (h₂ : ↑y ^ 2 < ↑x)
+  -- (h₃ : 1 < ↑x / ↑y ^ 2)
+  (h₄ : 1 < ↑y ^ (↑x - 2 * ↑y ^ 2))
+  (h₅ : (↑x / ↑y ^ 2) ^ y ^ 2 = ↑y ^ (↑x - 2 * ↑y ^ 2)) :
+  0 < ↑x - 2 * ↑y ^ 2 := by
+  by_contra! hc
+  cases' lt_or_eq_of_le hc with hlt heq
+  . have gy: 1 < (↑y:ℝ) := by
+      norm_cast
+    have glt: (↑x:ℝ) - 2*(↑y:ℝ)^2 < (↑0:ℝ) := by
+      norm_cast at *
+    have g₁: (↑y:ℝ) ^ ((↑x:ℝ) - 2*(↑y:ℝ)^2) < (↑y:ℝ) ^ (↑0:ℝ) := by
+      exact Real.rpow_lt_rpow_of_exponent_lt gy glt
+    simp at g₁
+    linarith[ h₄,g₁]
+  . rw [heq] at h₄
+    simp at h₄
+
+
+lemma imo_1997_p5_7_11
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  -- (hy : 1 < y)
+  -- (h₂ : ↑y ^ 2 < ↑x)
+  -- (h₃ : 1 < ↑x / ↑y ^ 2)
+  (h₄ : 1 < ↑y ^ (↑x - 2 * ↑y ^ 2))
+  (h₅ : (↑x / ↑y ^ 2) ^ y ^ 2 = ↑y ^ (↑x - 2 * ↑y ^ 2))
+  (hc : ↑x - 2 * ↑y ^ 2 ≤ 0) :
+  False := by
+  cases' lt_or_eq_of_le hc with hlt heq
+  . have gy: 1 < (↑y:ℝ) := by
+      norm_cast
+    have glt: (↑x:ℝ) - 2*(↑y:ℝ)^2 < (↑0:ℝ) := by
+      norm_cast at *
+    have g₁: (↑y:ℝ) ^ ((↑x:ℝ) - 2*(↑y:ℝ)^2) < (↑y:ℝ) ^ (↑0:ℝ) := by
+      exact Real.rpow_lt_rpow_of_exponent_lt gy glt
+    simp at g₁
+    linarith[ h₄,g₁]
+  . rw [heq] at h₄
+    simp at h₄
+
+
+lemma imo_1997_p5_7_12
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  -- (hy : 1 < y)
+  -- (h₂ : ↑y ^ 2 < ↑x)
+  -- (h₃ : 1 < ↑x / ↑y ^ 2)
+  (h₄ : 1 < ↑y ^ (↑x - 2 * ↑y ^ 2))
+  -- (h₅ : (↑x / ↑y ^ 2) ^ y ^ 2 = ↑y ^ (↑x - 2 * ↑y ^ 2))
+  -- (hc : ↑x - 2 * ↑y ^ 2 ≤ 0)
+  (hlt : ↑x - 2 * ↑y ^ 2 < 0) :
+  False := by
+  have gy: 1 < (↑y:ℝ) := by
+      norm_cast
+  have glt: (↑x:ℝ) - 2*(↑y:ℝ)^2 < (↑0:ℝ) := by
+    norm_cast at *
+  have g₁: (↑y:ℝ) ^ ((↑x:ℝ) - 2*(↑y:ℝ)^2) < (↑y:ℝ) ^ (↑0:ℝ) := by
+    exact Real.rpow_lt_rpow_of_exponent_lt gy glt
+  simp at g₁
+  linarith[ h₄,g₁]
+
+
+lemma imo_1997_p5_7_13
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  -- (hy : 1 < y)
+  -- (h₂ : ↑y ^ 2 < ↑x)
+  -- (h₃ : 1 < ↑x / ↑y ^ 2)
+  (h₄ : 1 < ↑y ^ (↑x - 2 * ↑y ^ 2))
+  -- (h₅ : (↑x / ↑y ^ 2) ^ y ^ 2 = ↑y ^ (↑x - 2 * ↑y ^ 2))
+  -- (hc : ↑x - 2 * ↑y ^ 2 ≤ 0)
+  -- (hlt : ↑x - 2 * ↑y ^ 2 < 0)
+  (gy : 1 < ↑y)
+  -- (glt : ↑x - 2 * ↑y ^ 2 < 0)
+  (g₁ : ↑y ^ (↑x - 2 * ↑y ^ 2) < ↑y ^ 0) :
+  False := by
+  simp at g₁
+  linarith[ h₄,g₁]
+
+
+lemma imo_1997_p5_7_14
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  -- (hy : 1 < y)
+  -- (h₂ : ↑y ^ 2 < ↑x)
+  -- (h₃ : 1 < ↑x / ↑y ^ 2)
+  (h₄ : 1 < ↑y ^ (↑x - 2 * ↑y ^ 2))
+  (h₅ : (↑x / ↑y ^ 2) ^ y ^ 2 = ↑y ^ (↑x - 2 * ↑y ^ 2))
+  (hc : ↑x - 2 * ↑y ^ 2 ≤ 0)
+  (heq : ↑x - 2 * ↑y ^ 2 = 0) :
+  False := by
+  rw [heq] at h₄
+  simp at h₄
+
+
+lemma imo_1997_p5_7_15
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  -- (hy : 1 < y)
+  -- (h₂ : ↑y ^ 2 < ↑x)
+  -- (h₃ : 1 < ↑x / ↑y ^ 2)
+  -- (h₄ : 1 < ↑y ^ (↑x - 2 * ↑y ^ 2))
+  -- (h₅ : (↑x / ↑y ^ 2) ^ y ^ 2 = ↑y ^ (↑x - 2 * ↑y ^ 2))
+  (h₆ : 0 < ↑x - 2 * ↑y ^ 2) :
+  2 * y ^ 2 < x := by
+  simp at h₆
+  norm_cast at h₆
+
+
+lemma imo_1997_p5_8
+  (x y: ℕ)
+  (h₀: 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (hyx: y < x) :
+  (y^2 ∣ x) := by
+  have h₂: (x ^ y ^ 2).factorization = (y^x).factorization := by
+    exact congr_arg Nat.factorization h₁
+  simp at h₂
+  symm at h₂
+  have hxy1: 2 * y^2 ≤ x := by exact le_of_lt (imo_1997_p5_7 x y h₀ h₁ hyx)
+  have hxy: 2 • y^2 ≤ x := by exact hxy1
+  have h₃: 2 • y^2 • x.factorization ≤ x • x.factorization := by
+    rw [← smul_assoc]
+    refine nsmul_le_nsmul_left ?_ hxy
+    norm_num
+  rw [← h₂] at h₃
+  have h₄: 2 • x • y.factorization = x • (2 • y.factorization) := by
+    rw [← smul_assoc, ← smul_assoc]
+    have g₄: 2 • x = x • 2 := by
+      simp
+      exact mul_comm 2 x
+    rw [g₄]
+  rw [h₄] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  have h₅: (y ^ 2) ^ x ∣ x^x := by
+    have g₁: (y ^ 2) ^ x ≠ 0 := by
+      refine pow_ne_zero x ?_
+      refine pow_ne_zero 2 ?_
+      linarith
+    have g₂: x ^ x ≠ 0 := by
+      refine pow_ne_zero x ?_
+      linarith
+    exact (Nat.factorization_le_iff_dvd g₁ g₂).mp h₃
+  refine (Nat.pow_dvd_pow_iff ?_).mp h₅
+  exact Nat.ne_of_gt h₀.1
+
+
+lemma imo_1997_p5_8_1
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (hyx : y < x)
+  (h₂ : Nat.factorization (x ^ y ^ 2) = Nat.factorization (y ^ x)) :
+  y ^ 2 ∣ x := by
+  simp at h₂
+  symm at h₂
+  have hxy1: 2 * y^2 ≤ x := by exact le_of_lt (imo_1997_p5_7 x y h₀ h₁ hyx)
+  have hxy: 2 • y^2 ≤ x := by exact hxy1
+  have h₃: 2 • y^2 • x.factorization ≤ x • x.factorization := by
+    rw [← smul_assoc]
+    refine nsmul_le_nsmul_left ?_ hxy
+    norm_num
+  rw [← h₂] at h₃
+  have h₄: 2 • x • y.factorization = x • (2 • y.factorization) := by
+    rw [← smul_assoc, ← smul_assoc]
+    have g₄: 2 • x = x • 2 := by
+      simp
+      exact mul_comm 2 x
+    rw [g₄]
+  rw [h₄] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  have h₅: (y ^ 2) ^ x ∣ x^x := by
+    have g₁: (y ^ 2) ^ x ≠ 0 := by
+      refine pow_ne_zero x ?_
+      refine pow_ne_zero 2 ?_
+      linarith
+    have g₂: x ^ x ≠ 0 := by
+      refine pow_ne_zero x ?_
+      linarith
+    exact (Nat.factorization_le_iff_dvd g₁ g₂).mp h₃
+  refine (Nat.pow_dvd_pow_iff ?_).mp h₅
+  exact Nat.ne_of_gt h₀.1
+
+
+lemma imo_1997_p5_8_2
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (hyx : y < x)
+  (h₂ : x • Nat.factorization y = y ^ 2 • Nat.factorization x) :
+  y ^ 2 ∣ x := by
+  have hxy1: 2 * y^2 ≤ x := by exact le_of_lt (imo_1997_p5_7 x y h₀ h₁ hyx)
+  have hxy: 2 • y^2 ≤ x := by exact hxy1
+  have h₃: 2 • y^2 • x.factorization ≤ x • x.factorization := by
+    rw [← smul_assoc]
+    refine nsmul_le_nsmul_left ?_ hxy
+    norm_num
+  rw [← h₂] at h₃
+  have h₄: 2 • x • y.factorization = x • (2 • y.factorization) := by
+    rw [← smul_assoc, ← smul_assoc]
+    have g₄: 2 • x = x • 2 := by
+      simp
+      exact mul_comm 2 x
+    rw [g₄]
+  rw [h₄] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  have h₅: (y ^ 2) ^ x ∣ x^x := by
+    have g₁: (y ^ 2) ^ x ≠ 0 := by
+      refine pow_ne_zero x ?_
+      refine pow_ne_zero 2 ?_
+      linarith
+    have g₂: x ^ x ≠ 0 := by
+      refine pow_ne_zero x ?_
+      linarith
+    exact (Nat.factorization_le_iff_dvd g₁ g₂).mp h₃
+  refine (Nat.pow_dvd_pow_iff ?_).mp h₅
+  exact Nat.ne_of_gt h₀.1
+
+
+lemma imo_1997_p5_8_3
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  (hyx : y < x)
+  (h₂ : x • Nat.factorization y = y ^ 2 • Nat.factorization x)
+  -- (hxy1 : 2 * y ^ 2 ≤ x)
+  (hxy : 2 • y ^ 2 ≤ x) :
+  y ^ 2 ∣ x := by
+  have h₃: 2 • y^2 • x.factorization ≤ x • x.factorization := by
+    rw [← smul_assoc]
+    refine nsmul_le_nsmul_left ?_ hxy
+    norm_num
+  rw [← h₂] at h₃
+  have h₄: 2 • x • y.factorization = x • (2 • y.factorization) := by
+    rw [← smul_assoc, ← smul_assoc]
+    have g₄: 2 • x = x • 2 := by
+      simp
+      exact mul_comm 2 x
+    rw [g₄]
+  rw [h₄] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  have h₅: (y ^ 2) ^ x ∣ x^x := by
+    have g₁: (y ^ 2) ^ x ≠ 0 := by
+      refine pow_ne_zero x ?_
+      refine pow_ne_zero 2 ?_
+      linarith
+    have g₂: x ^ x ≠ 0 := by
+      refine pow_ne_zero x ?_
+      linarith
+    exact (Nat.factorization_le_iff_dvd g₁ g₂).mp h₃
+  refine (Nat.pow_dvd_pow_iff ?_).mp h₅
+  exact Nat.ne_of_gt h₀.1
+
+
+lemma imo_1997_p5_8_4
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hyx : y < x)
+  -- (h₂ : x • Nat.factorization y = y ^ 2 • Nat.factorization x)
+  -- (hxy1 : 2 * y ^ 2 ≤ x)
+  (hxy : 2 • y ^ 2 ≤ x) :
+  2 • y ^ 2 • Nat.factorization x ≤ x • Nat.factorization x := by
+  rw [← smul_assoc]
+  refine nsmul_le_nsmul_left ?_ hxy
+  norm_num
+
+
+lemma imo_1997_p5_8_5
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hyx : y < x)
+  -- (h₂ : x • Nat.factorization y = y ^ 2 • Nat.factorization x)
+  -- (hxy1 : 2 * y ^ 2 ≤ x)
+  (hxy : 2 • y ^ 2 ≤ x) :
+  (2 • y ^ 2) • Nat.factorization x ≤ x • Nat.factorization x := by
+  refine nsmul_le_nsmul_left ?_ hxy
+  norm_num
+
+
+lemma imo_1997_p5_8_6
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (hyx : y < x)
+  (h₂ : x • Nat.factorization y = y ^ 2 • Nat.factorization x)
+  (hxy1 : 2 * y ^ 2 ≤ x)
+  (hxy : 2 • y ^ 2 ≤ x) :
+  0 ≤ Nat.factorization x := by
+  exact _root_.zero_le x.factorization
+
+lemma imo_1997_p5_8_7
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  (hyx : y < x)
+  (h₂ : x • Nat.factorization y = y ^ 2 • Nat.factorization x)
+  -- (hxy1 : 2 * y ^ 2 ≤ x)
+  -- (hxy : 2 • y ^ 2 ≤ x)
+  (h₃ : 2 • y ^ 2 • Nat.factorization x ≤ x • Nat.factorization x) :
+  y ^ 2 ∣ x := by
+  rw [← h₂] at h₃
+  have h₄: 2 • x • y.factorization = x • (2 • y.factorization) := by
+    rw [← smul_assoc, ← smul_assoc]
+    have g₄: 2 • x = x • 2 := by
+      simp
+      exact mul_comm 2 x
+    rw [g₄]
+  rw [h₄] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  have h₅: (y ^ 2) ^ x ∣ x^x := by
+    have g₁: (y ^ 2) ^ x ≠ 0 := by
+      refine pow_ne_zero x ?_
+      refine pow_ne_zero 2 ?_
+      linarith
+    have g₂: x ^ x ≠ 0 := by
+      refine pow_ne_zero x ?_
+      linarith
+    exact (Nat.factorization_le_iff_dvd g₁ g₂).mp h₃
+  refine (Nat.pow_dvd_pow_iff ?_).mp h₅
+  exact Nat.ne_of_gt h₀.1
+
+
+lemma imo_1997_p5_8_8
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  (hyx : y < x)
+  -- (h₂ : x • Nat.factorization y = y ^ 2 • Nat.factorization x)
+  -- (hxy1 : 2 * y ^ 2 ≤ x)
+  -- (hxy : 2 • y ^ 2 ≤ x)
+  (h₃ : 2 • x • Nat.factorization y ≤ x • Nat.factorization x) :
+  y ^ 2 ∣ x := by
+  have h₄: 2 • x • y.factorization = x • (2 • y.factorization) := by
+    rw [← smul_assoc, ← smul_assoc]
+    have g₄: 2 • x = x • 2 := by
+      simp
+      exact mul_comm 2 x
+    rw [g₄]
+  rw [h₄] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  have h₅: (y ^ 2) ^ x ∣ x^x := by
+    have g₁: (y ^ 2) ^ x ≠ 0 := by
+      refine pow_ne_zero x ?_
+      refine pow_ne_zero 2 ?_
+      linarith
+    have g₂: x ^ x ≠ 0 := by
+      refine pow_ne_zero x ?_
+      linarith
+    exact (Nat.factorization_le_iff_dvd g₁ g₂).mp h₃
+  refine (Nat.pow_dvd_pow_iff ?_).mp h₅
+  exact Nat.ne_of_gt h₀.1
+
+
+
+lemma imo_1997_p5_8_9
+  (x y : ℕ) :
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hyx : y < x)
+  -- (h₂ : x • Nat.factorization y = y ^ 2 • Nat.factorization x)
+  -- (hxy1 : 2 * y ^ 2 ≤ x)
+  -- (hxy : 2 • y ^ 2 ≤ x)
+  -- (h₃ : 2 • x • Nat.factorization y ≤ x • Nat.factorization x) :
+  2 • x • Nat.factorization y = x • 2 • Nat.factorization y := by
+  rw [← smul_assoc, ← smul_assoc]
+  have g₄: 2 • x = x • 2 := by
+    simp
+    exact mul_comm 2 x
+  rw [g₄]
+
+
+lemma imo_1997_p5_8_10
+  (x y : ℕ) :
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hyx : y < x)
+  -- (h₂ : x • Nat.factorization y = y ^ 2 • Nat.factorization x)
+  -- (hxy1 : 2 * y ^ 2 ≤ x)
+  -- (hxy : 2 • y ^ 2 ≤ x)
+  -- (h₃ : 2 • x • Nat.factorization y ≤ x • Nat.factorization x) :
+  (2 • x) • Nat.factorization y = (x • 2) • Nat.factorization y := by
+  have g₄: 2 • x = x • 2 := by
+    simp
+    exact mul_comm 2 x
+  rw [g₄]
+
+
+lemma imo_1997_p5_8_11
+  (x : ℕ) :
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hyx : y < x)
+  -- (h₂ : x • Nat.factorization y = y ^ 2 • Nat.factorization x)
+  -- (hxy1 : 2 * y ^ 2 ≤ x)
+  -- (hxy : 2 • y ^ 2 ≤ x)
+  -- (h₃ : 2 • x • Nat.factorization y ≤ x • Nat.factorization x) :
+  2 • x = x • 2 := by
+  rw [smul_eq_mul, smul_eq_mul]
+  exact Nat.mul_comm 2 x
+
+
+lemma imo_1997_p5_8_12
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  (hyx : y < x)
+  -- (h₂ : x • Nat.factorization y = y ^ 2 • Nat.factorization x)
+  -- (hxy1 : 2 * y ^ 2 ≤ x)
+  -- (hxy : 2 • y ^ 2 ≤ x)
+  (h₃ : 2 • x • Nat.factorization y ≤ x • Nat.factorization x)
+  (h₄ : 2 • x • Nat.factorization y = x • 2 • Nat.factorization y) :
+  y ^ 2 ∣ x := by
+  rw [h₄] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  have h₅: (y ^ 2) ^ x ∣ x^x := by
+    have g₁: (y ^ 2) ^ x ≠ 0 := by
+      refine pow_ne_zero x ?_
+      refine pow_ne_zero 2 ?_
+      linarith
+    have g₂: x ^ x ≠ 0 := by
+      refine pow_ne_zero x ?_
+      linarith
+    exact (Nat.factorization_le_iff_dvd g₁ g₂).mp h₃
+  refine (Nat.pow_dvd_pow_iff ?_).mp h₅
+  exact Nat.ne_of_gt h₀.1
+
+
+lemma imo_1997_p5_8_13
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  (hyx : y < x)
+  -- (h₂ : x • Nat.factorization y = y ^ 2 • Nat.factorization x)
+  -- (hxy1 : 2 * y ^ 2 ≤ x)
+  -- (hxy : 2 • y ^ 2 ≤ x)
+  (h₃ : Nat.factorization ((y ^ 2) ^ x) ≤ Nat.factorization (x ^ x)) :
+  -- (h₄ : 2 • x • Nat.factorization y = x • 2 • Nat.factorization y) :
+  y ^ 2 ∣ x := by
+  have h₅: (y ^ 2) ^ x ∣ x^x := by
+    have g₁: (y ^ 2) ^ x ≠ 0 := by
+      refine pow_ne_zero x ?_
+      refine pow_ne_zero 2 ?_
+      linarith
+    have g₂: x ^ x ≠ 0 := by
+      refine pow_ne_zero x ?_
+      linarith
+    exact (Nat.factorization_le_iff_dvd g₁ g₂).mp h₃
+  refine (Nat.pow_dvd_pow_iff ?_).mp h₅
+  exact Nat.ne_of_gt h₀.1
+
+
+lemma imo_1997_p5_8_14
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  (hyx : y < x)
+  -- (h₂ : x • Nat.factorization y = y ^ 2 • Nat.factorization x)
+  -- (hxy1 : 2 * y ^ 2 ≤ x)
+  -- (hxy : 2 • y ^ 2 ≤ x)
+  (h₃ : Nat.factorization ((y ^ 2) ^ x) ≤ Nat.factorization (x ^ x)) :
+  -- (h₄ : 2 • x • Nat.factorization y = x • 2 • Nat.factorization y) :
+  (y ^ 2) ^ x ∣ x ^ x := by
+  have g₁: (y ^ 2) ^ x ≠ 0 := by
+    refine pow_ne_zero x ?_
+    refine pow_ne_zero 2 ?_
+    linarith
+  have g₂: x ^ x ≠ 0 := by
+    refine pow_ne_zero x ?_
+    linarith
+  exact (Nat.factorization_le_iff_dvd g₁ g₂).mp h₃
+
+
+lemma imo_1997_p5_8_15
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  (hyx : y < x)
+  -- (h₂ : x • Nat.factorization y = y ^ 2 • Nat.factorization x)
+  -- (hxy1 : 2 * y ^ 2 ≤ x)
+  -- (hxy : 2 • y ^ 2 ≤ x)
+  (h₃ : Nat.factorization ((y ^ 2) ^ x) ≤ Nat.factorization (x ^ x))
+  -- (h₄ : 2 • x • Nat.factorization y = x • 2 • Nat.factorization y)
+  (g₁ : (y ^ 2) ^ x ≠ 0) :
+  (y ^ 2) ^ x ∣ x ^ x := by
+  have g₂: x ^ x ≠ 0 := by
+    refine pow_ne_zero x ?_
+    linarith
+  exact (Nat.factorization_le_iff_dvd g₁ g₂).mp h₃
+
+
+lemma imo_1997_p5_8_16
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hyx : y < x)
+  -- (h₂ : x • Nat.factorization y = y ^ 2 • Nat.factorization x)
+  -- (hxy1 : 2 * y ^ 2 ≤ x)
+  -- (hxy : 2 • y ^ 2 ≤ x)
+  (h₃ : Nat.factorization ((y ^ 2) ^ x) ≤ Nat.factorization (x ^ x))
+  -- (h₄ : 2 • x • Nat.factorization y = x • 2 • Nat.factorization y)
+  (g₁ : y = 0 → x = 0) :
+  (y ^ 2) ^ x ∣ x ^ x := by
+  refine (Nat.factorization_le_iff_dvd ?_ ?_).mp h₃
+  . simp_all only [Nat.factorization_pow, ne_eq, pow_eq_zero_iff', OfNat.ofNat_ne_zero, not_false_eq_true,]
+    omega
+  . simp_all only [ne_eq, pow_eq_zero_iff', and_not_self, not_false_eq_true]
+
+
+
+lemma imo_1997_p5_8_17
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hyx : y < x)
+  -- (h₂ : x • Nat.factorization y = y ^ 2 • Nat.factorization x)
+  -- (hxy1 : 2 * y ^ 2 ≤ x)
+  -- (hxy : 2 • y ^ 2 ≤ x)
+  -- (h₃ : Nat.factorization ((y ^ 2) ^ x) ≤ Nat.factorization (x ^ x))
+  -- (h₄ : 2 • x • Nat.factorization y = x • 2 • Nat.factorization y)
+  (h₅ : (y ^ 2) ^ x ∣ x ^ x) :
+  y ^ 2 ∣ x := by
+  refine (Nat.pow_dvd_pow_iff ?_).mp h₅
+  exact Nat.ne_of_gt h₀.1
+
+
+
+lemma imo_1997_p5_9
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (h₂ : Real.log (↑x:ℝ)  = Real.log ↑y * ↑x / (↑(y ^ 2:ℕ ):ℝ) )
+  (hxy : y < x) :
+  x = y ^ (x / y ^ 2) := by
+  have h_exp : Real.exp (Real.log ↑x)
+            = Real.exp (Real.log ↑y * (↑x:ℝ)  / ((↑y:ℝ)) ^ 2) := by
+    rw [h₂]
+    norm_cast
+  rw [← imo_1997_p5_4 x h₀.1] at h_exp
+  rw [← mul_div] at h_exp
+  rw [Real.exp_mul] at h_exp
+  rw [← imo_1997_p5_4 y h₀.2] at h_exp
+  have h₃: (↑x:ℝ) / ((↑y:ℝ)^2) = (↑(x / y^2:ℕ):ℝ) := by
+    norm_cast
+    symm
+    have g₂: y^2 ∣ x := by
+      exact imo_1997_p5_8 x y h₀ h₁ hxy
+    have h₃: (↑(y^(2:ℕ)):ℝ) ≠ 0 := by
+      norm_cast
+      exact pow_ne_zero 2 ( by linarith)
+    exact Nat.cast_div g₂ h₃
+  have h₄ : (↑(y ^ (x / y ^ (2:ℕ))):ℝ) = (↑y:ℝ)^((↑x:ℝ) / ((↑y:ℝ)^2)) := by
+    rw [Nat.cast_pow, h₃]
+    norm_cast
+  rw [←h₄] at h_exp
+  exact Nat.cast_inj.mp h_exp
+
+
+lemma imo_1997_p5_9_1
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (h₂ : Real.log ↑x = Real.log ↑y * ↑x / ↑(y ^ 2:ℕ))
+  (hxy : y < x)
+  (h_exp : rexp (Real.log ↑x) = rexp (Real.log ↑y * ↑x / ↑y ^ 2)) :
+  x = y ^ (x / y ^ 2) := by
+  rw [← imo_1997_p5_4 x h₀.1] at h_exp
+  rw [← mul_div] at h_exp
+  rw [Real.exp_mul] at h_exp
+  rw [← imo_1997_p5_4 y h₀.2] at h_exp
+  have h₃: (↑x:ℝ) / ((↑y:ℝ)^2) = (↑(x / y^2:ℕ):ℝ) := by
+    norm_cast
+    symm
+    have g₂: y^2 ∣ x := by
+      exact imo_1997_p5_8 x y h₀ h₁ hxy
+    have h₃: (↑(y^(2:ℕ)):ℝ) ≠ 0 := by
+      norm_cast
+      exact pow_ne_zero 2 ( by linarith)
+    exact Nat.cast_div g₂ h₃
+  have h₄ : (↑(y ^ (x / y ^ (2:ℕ))):ℝ) = (↑y:ℝ)^((↑x:ℝ) / ((↑y:ℝ)^2)) := by
+    rw [Nat.cast_pow, h₃]
+    norm_cast
+  rw [←h₄] at h_exp
+  exact Nat.cast_inj.mp h_exp
+
+
+lemma imo_1997_p5_9_2
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (h₂ : Real.log ↑x = Real.log ↑y * ↑x / ↑(y ^ 2:ℕ))
+  (hxy : y < x)
+  (h_exp : ↑x = rexp (Real.log ↑y * (↑x / ↑y ^ 2))) :
+  x = y ^ (x / y ^ 2) := by
+  rw [Real.exp_mul] at h_exp
+  rw [← imo_1997_p5_4 y h₀.2] at h_exp
+  have h₃: (↑x:ℝ) / ((↑y:ℝ)^2) = (↑(x / y^2:ℕ):ℝ) := by
+    norm_cast
+    symm
+    have g₂: y^2 ∣ x := by
+      exact imo_1997_p5_8 x y h₀ h₁ hxy
+    have h₃: (↑(y^(2:ℕ)):ℝ) ≠ 0 := by
+      norm_cast
+      exact pow_ne_zero 2 ( by linarith)
+    exact Nat.cast_div g₂ h₃
+  have h₄ : (↑(y ^ (x / y ^ (2:ℕ))):ℝ) = (↑y:ℝ)^((↑x:ℝ) / ((↑y:ℝ)^2)) := by
+    rw [Nat.cast_pow, h₃]
+    norm_cast
+  rw [←h₄] at h_exp
+  exact Nat.cast_inj.mp h_exp
+
+
+lemma imo_1997_p5_9_3
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (h₂ : Real.log ↑x = Real.log ↑y * ↑x / ↑(y ^ 2:ℕ))
+  (hxy : y < x)
+  (h_exp : ↑x = rexp (Real.log ↑y) ^ (↑x  / (↑y:ℝ) ^ 2)) :
+  x = y ^ (x / y ^ 2) := by
+  rw [← imo_1997_p5_4 y h₀.2] at h_exp
+  have h₃: (↑x:ℝ) / ((↑y:ℝ)^2) = (↑(x / y^2:ℕ):ℝ) := by
+    norm_cast
+    symm
+    have g₂: y^2 ∣ x := by
+      exact imo_1997_p5_8 x y h₀ h₁ hxy
+    have h₃: (↑(y^(2:ℕ)):ℝ) ≠ 0 := by
+      norm_cast
+      exact pow_ne_zero 2 ( by linarith)
+    exact Nat.cast_div g₂ h₃
+  have h₄ : (↑(y ^ (x / y ^ (2:ℕ))):ℝ) = (↑y:ℝ)^((↑x:ℝ) / ((↑y:ℝ)^2)) := by
+    rw [Nat.cast_pow, h₃]
+    norm_cast
+  rw [←h₄] at h_exp
+  exact Nat.cast_inj.mp h_exp
+
+
+lemma imo_1997_p5_9_4
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (h₂ : Real.log ↑x = Real.log ↑y * ↑x / ↑(y ^ 2:ℕ))
+  (hxy : y < x)
+  (h_exp : (↑x:ℝ) = (↑y:ℝ) ^ ((↑x:ℝ) / (↑y:ℝ) ^ 2)) :
+  x = y ^ (x / y ^ 2) := by
+  have h₃: (↑x:ℝ) / ((↑y:ℝ)^2) = (↑(x / y^2:ℕ)) := by
+    norm_cast
+    symm
+    have g₂: y^2 ∣ x := by
+      exact imo_1997_p5_8 x y h₀ h₁ hxy
+    have h₃: (↑(y^(2:ℕ)):ℝ) ≠ 0 := by
+      norm_cast
+      exact pow_ne_zero 2 ( by linarith)
+    exact Nat.cast_div g₂ h₃
+  have h₄ : (↑(y ^ (x / y ^ (2:ℕ))):ℝ) = (↑y:ℝ)^((↑x:ℝ) / ((↑y:ℝ)^2)) := by
+    rw [Nat.cast_pow, h₃]
+    norm_cast
+  rw [←h₄] at h_exp
+  exact Nat.cast_inj.mp h_exp
+
+
+lemma imo_1997_p5_9_5
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (h₂ : Real.log ↑x = Real.log ↑y * ↑x / ↑(y ^ 2:ℕ))
+  (hxy : y < x) :
+  -- (h_exp : ↑x = ↑y ^ (↑x / ↑y ^ 2:ℕ)) :
+  (↑x:ℝ) / ((↑y:ℝ)^2) = (↑(x / y^2:ℕ):ℝ) := by
+  norm_cast
+  symm
+  have g₂: y^2 ∣ x := by
+    exact imo_1997_p5_8 x y h₀ h₁ hxy
+  have h₃: (↑(y^(2:ℕ)):ℝ) ≠ 0 := by
+    norm_cast
+    exact pow_ne_zero 2 ( by linarith)
+  exact Nat.cast_div g₂ h₃
+
+
+lemma imo_1997_p5_9_6
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (h₂ : Real.log ↑x = Real.log ↑y * ↑x / ↑(y ^ 2:ℕ))
+  -- (hxy : y < x)
+  -- (h_exp : ↑x = ↑y ^ (↑x / ↑y ^ 2))
+  (g₂ : y ^ 2 ∣ x) :
+  (↑(x / y^2:ℕ):ℝ) = (↑x:ℝ) / (↑(y^2:ℕ)) := by
+  have h₃: (↑(y^(2:ℕ)):ℝ) ≠ 0 := by
+    norm_cast
+    exact pow_ne_zero 2 ( by linarith)
+  exact Nat.cast_div g₂ h₃
+
+
+lemma imo_1997_p5_9_7
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  (h₂ : Real.log (↑x:ℝ)  = Real.log ↑y * ↑x / (↑(y ^ 2:ℕ ):ℝ) ) :
+  (↑x:ℝ) = (↑y:ℝ) ^ ((↑x:ℝ) / (↑y:ℝ) ^ 2) := by
+  have h_exp : Real.exp (Real.log ↑x)
+            = Real.exp (Real.log ↑y * (↑x:ℝ)  / ((↑y:ℝ)) ^ 2) := by
+    rw [h₂]
+    norm_cast
+  rw [← imo_1997_p5_4 x h₀.1] at h_exp
+  rw [← mul_div] at h_exp
+  rw [Real.exp_mul] at h_exp
+  rw [← imo_1997_p5_4 y h₀.2] at h_exp
+  exact h_exp
+
+
+lemma imo_1997_p5_9_8
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  (h₂ : Real.log (↑x:ℝ)  = Real.log ↑y * ↑x / (↑(y ^ 2:ℕ ):ℝ) ) :
+  ↑x = rexp (Real.log ↑y * (↑x / ↑y ^ 2)) := by
+  have h_exp : Real.exp (Real.log ↑x)
+            = Real.exp (Real.log ↑y * (↑x:ℝ)  / ((↑y:ℝ)) ^ 2) := by
+    rw [h₂]
+    norm_cast
+  rw [← imo_1997_p5_4 x h₀.1] at h_exp
+  rw [← mul_div] at h_exp
+  exact h_exp
+
+
+lemma imo_1997_p5_9_9
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (h₂ : Real.log ↑x = Real.log ↑y * ↑x / ↑(y ^ 2:ℕ))
+  -- (hxy : y < x)
+  (h_exp : (↑x:ℝ) = (↑y:ℝ) ^ ((↑x:ℝ) / (↑y:ℝ) ^ 2))
+  (h₃ : (↑x:ℝ) / ((↑y:ℝ)^2) = (↑(x / y^2:ℕ))) :
+  x = y ^ (x / y ^ 2) := by
+  have h₄ : (↑(y ^ (x / y ^ (2:ℕ))):ℝ) = (↑y:ℝ)^((↑x:ℝ) / ((↑y:ℝ)^2)) := by
+    rw [Nat.cast_pow, h₃]
+    norm_cast
+  rw [←h₄] at h_exp
+  exact Nat.cast_inj.mp h_exp
+
+
+lemma imo_1997_p5_9_10
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (h₂ : Real.log ↑x = Real.log ↑y * ↑x / ↑(y ^ 2:ℕ))
+  -- (hxy : y < x)
+  -- (h_exp : ↑x = ↑y ^ (↑x / ↑y ^ 2))
+  (h₃ : (↑x:ℝ) / ((↑y:ℝ)^2) = ↑(x / y^2:ℕ)) :
+  (↑(y ^ (x / y ^ (2:ℕ))):ℝ) = (↑y:ℝ) ^ ((↑x:ℝ) / ((↑y:ℝ)^2)) := by
+  rw [Nat.cast_pow, h₃]
+  norm_cast
+
+
+lemma imo_1997_p5_9_11
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (h₂ : Real.log ↑x = Real.log ↑y * ↑x / ↑(y ^ 2:ℕ))
+  -- (hxy : y < x)
+  (h_exp : ↑x = ↑(y ^ (x / y ^ 2)))
+  (h₃ : ↑x / ↑y ^ 2 = ↑(x / y ^ 2))
+  (h₄ : ↑(y ^ (x / y ^ 2)) = ↑y ^ (↑x / ↑y ^ 2)) :
+  x = y ^ (x / y ^ 2) := by
+  rw [←h₄] at h_exp
+  exact Nat.cast_inj.mp h_exp
+
+
+
+
+  lemma imo_1997_p5_10
+    (x y : ℕ)
+    (h₀ : 0 < x ∧ 0 < y)
+    (h₁ : x ^ y ^ 2 = y ^ x)
+    (hxy : y < x) :
+    x = y ^ (x / y ^ 2) := by
+    -- sketch: y^2 * log x = x * log y
+    have h₃: Real.log (x^(y^2)) = Real.log (y^x) := by
+      norm_cast
+      rw [h₁]
+    have h₄: (↑(y ^ (2:ℕ)):ℝ)  * Real.log x = ↑x * Real.log y := by
+      have h41: Real.log (y^x) = (↑x:ℝ) * Real.log (y) := by
+        exact Real.log_pow y x
+      have h42: Real.log (x^(y^2)) = (↑(y ^ (2:ℕ)):ℝ) * Real.log x := by
+        exact Real.log_pow x (y^2)
+      rw [h41,h42] at h₃
+      exact h₃
+    have h₅: Real.log ↑x = Real.log ↑y * ↑x / (↑(y ^ (2:ℕ)):ℝ) := by
+      by_contra! hc
+      rw [mul_comm (Real.log ↑y) (↑x)] at hc
+      rw [← h₄, mul_comm, ← mul_div] at hc
+      rw [div_self, mul_one] at hc
+      . apply hc
+        norm_cast
+      . norm_cast
+        push_neg
+        refine pow_ne_zero 2 ?_
+        exact Nat.ne_of_gt h₀.2
+    have h₆: x = y ^ (x / y ^ 2) := by
+      exact imo_1997_p5_9 x y h₀ h₁ h₅ hxy
+    exact h₆
+
+lemma imo_1997_p5_10_1
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (hxy : y < x)
+  (h₃ : Real.log (↑x ^ y ^ 2) = Real.log (↑y ^ x)) :
+  x = y ^ (x / y ^ 2) := by
+  have h₄: (↑(y ^ (2:ℕ)):ℝ)  * Real.log x = ↑x * Real.log y := by
+    have h41: Real.log (y^x) = (↑x:ℝ) * Real.log (y) := by
+      exact Real.log_pow y x
+    have h42: Real.log (x^(y^2)) = (↑(y ^ (2:ℕ)):ℝ) * Real.log x := by
+      exact Real.log_pow x (y^2)
+    rw [h41,h42] at h₃
+    exact h₃
+  have h₅: Real.log ↑x = Real.log ↑y * ↑x / (↑(y ^ (2:ℕ)):ℝ) := by
+    by_contra! hc
+    rw [mul_comm (Real.log ↑y) (↑x)] at hc
+    rw [← h₄, mul_comm, ← mul_div] at hc
+    rw [div_self, mul_one] at hc
+    . apply hc
+      norm_cast
+    . norm_cast
+      push_neg
+      refine pow_ne_zero 2 ?_
+      exact Nat.ne_of_gt h₀.2
+  have h₆: x = y ^ (x / y ^ 2) := by
+    exact imo_1997_p5_9 x y h₀ h₁ h₅ hxy
+  exact h₆
+
+
+lemma imo_1997_p5_10_2
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  (h₃ : Real.log (↑x ^ y ^ 2) = Real.log (↑y ^ x)) :
+  ↑(y ^ 2:ℕ) * Real.log ↑x = ↑x * Real.log ↑y := by
+  have h41: Real.log (y^x) = (↑x:ℝ) * Real.log (y) := by
+    exact Real.log_pow y x
+  have h42: Real.log (x^(y^2)) = (↑(y ^ (2:ℕ)):ℝ) * Real.log x := by
+    exact Real.log_pow x (y^2)
+  rw [h41,h42] at h₃
+  exact h₃
+
+
+lemma imo_1997_p5_10_3
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  (h₃ : Real.log (↑x ^ y ^ 2) = Real.log (↑y ^ x))
+  (h₄₁ : Real.log (↑y ^ x) = ↑x * Real.log ↑y) :
+  ↑(y ^ 2:ℕ) * Real.log ↑x = ↑x * Real.log ↑y := by
+  have h₄₂: Real.log (x^(y^2)) = (↑(y ^ (2:ℕ)):ℝ) * Real.log x := by
+    exact Real.log_pow x (y^2)
+  rw [h₄₁,h₄₂] at h₃
+  exact h₃
+
+
+lemma imo_1997_p5_10_4
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  (h₃ : Real.log (↑x ^ y ^ 2) = Real.log (↑y ^ x))
+  (h₄₁ : Real.log (↑y ^ x) = ↑x * Real.log ↑y)
+  (h₄₂ : Real.log (↑x ^ y ^ 2) = ↑(y ^ 2:ℕ) * Real.log ↑x) :
+  ↑(y ^ 2:ℕ) * Real.log ↑x = ↑x * Real.log ↑y := by
+  rw [h₄₁,h₄₂] at h₃
+  exact h₃
+
+lemma imo_1997_p5_10_5
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (hxy : y < x)
+  -- (h₃ : Real.log (↑x ^ y ^ 2) = Real.log (↑y ^ x))
+  (h₄ : ↑(y ^ 2:ℕ) * Real.log ↑x = ↑x * Real.log ↑y) :
+  x = y ^ (x / y ^ 2) := by
+  have h₅: Real.log ↑x = Real.log ↑y * ↑x / (↑(y ^ (2:ℕ)):ℝ) := by
+    by_contra! hc
+    rw [mul_comm (Real.log ↑y) (↑x)] at hc
+    rw [← h₄, mul_comm, ← mul_div] at hc
+    rw [div_self, mul_one] at hc
+    . apply hc
+      norm_cast
+    . norm_cast
+      push_neg
+      refine pow_ne_zero 2 ?_
+      exact Nat.ne_of_gt h₀.2
+  have h₆: x = y ^ (x / y ^ 2) := by
+    exact imo_1997_p5_9 x y h₀ h₁ h₅ hxy
+  exact h₆
+
+
+lemma imo_1997_p5_10_6
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  -- (h₃ : Real.log (↑x ^ y ^ 2) = Real.log (↑y ^ x))
+  (h₄ : ↑(y ^ 2:ℕ) * Real.log ↑x = ↑x * Real.log ↑y) :
+  Real.log ↑x = Real.log ↑y * ↑x / ↑(y ^ 2:ℕ) := by
+  by_contra! hc
+  rw [mul_comm (Real.log ↑y) (↑x)] at hc
+  rw [← h₄, mul_comm, ← mul_div] at hc
+  rw [div_self, mul_one] at hc
+  . apply hc
+    norm_cast
+  . norm_cast
+    push_neg
+    refine pow_ne_zero 2 ?_
+    exact Nat.ne_of_gt h₀.2
+
+
+lemma imo_1997_p5_10_7
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  -- (h₃ : Real.log (↑x ^ y ^ 2) = Real.log (↑y ^ x))
+  (h₄ : ↑(y ^ 2:ℕ) * Real.log ↑x = ↑x * Real.log ↑y)
+  (hc : ¬Real.log ↑x = Real.log ↑y * ↑x / ↑(y ^ 2:ℕ)) :
+  False := by
+  rw [mul_comm (Real.log ↑y) (↑x)] at hc
+  rw [← h₄, mul_comm, ← mul_div] at hc
+  rw [div_self, mul_one] at hc
+  . apply hc
+    norm_cast
+  . norm_cast
+    push_neg
+    refine pow_ne_zero 2 ?_
+    exact Nat.ne_of_gt h₀.2
+
+
+lemma imo_1997_p5_10_8
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  -- (h₃ : Real.log (↑x ^ y ^ 2) = Real.log (↑y ^ x))
+  (h₄ : ↑(y ^ 2:ℕ) * Real.log ↑x = ↑x * Real.log ↑y)
+  (hc : ¬Real.log ↑x = ↑x * Real.log ↑y / ↑(y ^ 2:ℕ)) :
+  False := by
+  rw [← h₄, mul_comm, ← mul_div] at hc
+  rw [div_self, mul_one] at hc
+  . apply hc
+    norm_cast
+  . norm_cast
+    push_neg
+    refine pow_ne_zero 2 ?_
+    exact Nat.ne_of_gt h₀.2
+
+
+lemma imo_1997_p5_10_9
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  -- (h₃ : Real.log (↑x ^ y ^ 2) = Real.log (↑y ^ x))
+  -- (h₄ : ↑(y ^ 2:ℕ) * Real.log ↑x = ↑x * Real.log ↑y)
+  (hc : ¬Real.log ↑x = Real.log ↑x * (↑(y ^ 2:ℕ) / ↑(y ^ 2:ℕ))) :
+  False := by
+  rw [div_self, mul_one] at hc
+  . apply hc
+    norm_cast
+  . norm_cast
+    push_neg
+    refine pow_ne_zero 2 ?_
+    exact Nat.ne_of_gt h₀.2
+
+
+lemma imo_1997_p5_10_10
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (hxy : y < x)
+  -- (h₃ : Real.log (↑x ^ y ^ 2) = Real.log (↑y ^ x))
+  -- (h₄ : ↑(y ^ 2:ℕ) * Real.log ↑x = ↑x * Real.log ↑y)
+  (hc : ¬Real.log ↑x = Real.log ↑x * (↑(y ^ 2:ℕ) / ↑(y ^ 2:ℕ))) :
+  ↑((y ^ 2):ℝ) ≠ 0 := by
+  norm_cast
+  push_neg
+  refine pow_ne_zero 2 ?_
+  exact Nat.ne_of_gt h₀.2
+
+
+lemma imo_1997_p5_10_11
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (hxy : y < x)
+  -- (h₃ : Real.log (↑x ^ y ^ 2) = Real.log (↑y ^ x))
+  -- (h₄ : ↑(y ^ 2:ℕ) * Real.log ↑x = ↑x * Real.log ↑y)
+  (h₅ : Real.log ↑x = Real.log ↑y * ↑x / ↑(y ^ 2:ℕ)) :
+  x = y ^ (x / y ^ 2) := by
+  exact imo_1997_p5_9 x y h₀ h₁ h₅ hxy
+
+
+lemma imo_1997_p5_11_1
+  (x y : ℕ) :
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x) :
+  x ^ y ^ 2 = (x ^ y) ^ y := by
+  rw [Nat.pow_two]
+  exact Nat.pow_mul x y y
+
+
+lemma imo_1997_p5_11_2
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  (hxy : y < x) :
+  (x, y) = (16, 2) ∨ (x, y) = (27, 3) := by
+  have h₃: x = y ^ (x / y ^ 2) := by
+    exact imo_1997_p5_10 x y h₀ h₁ hxy
+  let k:ℕ  := x / y^2
+  have hk_def: k = x / y^2 := by exact rfl
+  by_cases hk: k < 2
+  . rw [← hk_def] at h₃
+    interval_cases k
+    . exfalso
+      simp at h₃
+      linarith
+    . exfalso
+      simp at *
+      linarith [hxy,h₃]
+  . push_neg at hk
+    rw [← hk_def] at h₃
+    have h₅: k = y^(k-2) := by
+      rw [h₃] at hk_def
+      nth_rewrite 1 [hk_def]
+      exact Nat.pow_div hk h₀.2
+    by_cases hk5: k < 5
+    . interval_cases k
+      . exfalso
+        simp at h₅
+      . right
+        norm_num
+        simp at h₅
+        symm at h₅
+        rw [h₅] at h₃
+        norm_num at h₃
+        exact { left := h₃, right := h₅ }
+      . simp at h₅
+        symm at h₅
+        have g₂: y^4 = y^2 * y^2 := by ring_nf
+        rw [g₂, h₅] at h₃
+        norm_num at h₃
+        left
+        norm_num
+        constructor
+        . exact h₃
+        . have h₆ : y ^ 2 = 2 ^ 2 := by
+            norm_num
+            exact h₅
+          have h₇: 0 ≤ y := by
+            linarith
+          exact (sq_eq_sq₀ h₇ (by linarith)).mp (h₆)
+    push_neg at hk5
+    by_cases hy: 2 ≤ y
+    . have h₅₁: k < y^(k-2) := by
+        have h₆: 2^(k-2) ≤ y^(k-2) := by
+          have hk1: 3 ≤ k - 2 := by exact Nat.sub_le_sub_right hk5 2
+          exact (Nat.pow_le_pow_iff_left (by linarith)).mpr hy
+        have h₇: 4*k < 2^k := by
+          exact imo_1997_p5_2 k hk5
+        have h₇: k < 2^(k-2) := by
+          have h₈ : k < 2 ^ k / 4 := by
+            have h81: 4 ∣ 2^k := by
+              have h82: 2^k = 4*2^(k-2) := by
+                have h83: k = 2 + (k -2) := by
+                  ring_nf
+                  exact (add_sub_of_le hk).symm
+                nth_rewrite 1 [h83]
+                rw [pow_add]
+                norm_num
+              rw [h82]
+              exact Nat.dvd_mul_right 4 (2^(k-2))
+            exact (Nat.lt_div_iff_mul_lt' h81 k).mpr h₇
+          have h₉: 2 ^ k / 4 = 2 ^ (k-2) := by
+            have g2: k = k - 2 + 2 := by
+              exact (Nat.sub_eq_iff_eq_add hk).mp rfl
+            have h1: 2^k = 2^(k - 2 + 2) := by
+              exact congrArg (HPow.hPow 2) g2
+            have h2: 2 ^ (k - 2 + 2) = 2 ^ (k - 2) * 2 ^ 2 := by rw [pow_add]
+            rw [h1, h2]
+            ring_nf
+            simp
+          linarith
+        linarith
+      exfalso
+      linarith
+    . push_neg at hy
+      interval_cases y
+      . linarith
+      . simp at h₅
+        simp at h₃
+        linarith
+
+
+lemma imo_1997_p5_11_3
+  (x y k : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  (hxy : y < x)
+  (h₃ : x = y ^ (x / y ^ 2))
+  (hk_def : k = x / y ^ 2) :
+  (x, y) = (16, 2) ∨ (x, y) = (27, 3) := by
+  by_cases hk: k < 2
+  . rw [← hk_def] at h₃
+    interval_cases k
+    . exfalso
+      simp at h₃
+      linarith
+    . exfalso
+      simp at *
+      linarith [hxy,h₃]
+  . push_neg at hk
+    rw [← hk_def] at h₃
+    have h₅: k = y^(k-2) := by
+      rw [h₃] at hk_def
+      nth_rewrite 1 [hk_def]
+      exact Nat.pow_div hk h₀.2
+    by_cases hk5: k < 5
+    . interval_cases k
+      . exfalso
+        simp at h₅
+      . right
+        norm_num
+        simp at h₅
+        symm at h₅
+        rw [h₅] at h₃
+        norm_num at h₃
+        exact { left := h₃, right := h₅ }
+      . simp at h₅
+        symm at h₅
+        have g₂: y^4 = y^2 * y^2 := by ring_nf
+        rw [g₂, h₅] at h₃
+        norm_num at h₃
+        left
+        norm_num
+        constructor
+        . exact h₃
+        . have h₆ : y ^ 2 = 2 ^ 2 := by
+            norm_num
+            exact h₅
+          have h₇: 0 ≤ y := by
+            linarith
+          exact (sq_eq_sq₀ h₇ (by linarith)).mp (h₆)
+    push_neg at hk5
+    by_cases hy: 2 ≤ y
+    . have h₅₁: k < y^(k-2) := by
+        have h₆: 2^(k-2) ≤ y^(k-2) := by
+          have hk1: 3 ≤ k - 2 := by exact Nat.sub_le_sub_right hk5 2
+          refine (Nat.pow_le_pow_iff_left ?_).mpr hy
+          have h₆₀: 2 < k - 2 := by exact hk1
+          exact Nat.not_eq_zero_of_lt h₆₀
+        have h₇: 4*k < 2^k := by
+          exact imo_1997_p5_2 k hk5
+        have h₇: k < 2^(k-2) := by
+          have h₈ : k < 2 ^ k / 4 := by
+            have h81: 4 ∣ 2^k := by
+              have h82: 2^k = 4*2^(k-2) := by
+                have h83: k = 2 + (k -2) := by
+                  ring_nf
+                  exact (add_sub_of_le hk).symm
+                nth_rewrite 1 [h83]
+                rw [pow_add]
+                norm_num
+              rw [h82]
+              exact Nat.dvd_mul_right 4 (2^(k-2))
+            exact (Nat.lt_div_iff_mul_lt' h81 k).mpr h₇
+          have h₉: 2 ^ k / 4 = 2 ^ (k-2) := by
+            have g2: k = k - 2 + 2 := by
+              exact (Nat.sub_eq_iff_eq_add hk).mp rfl
+            have h1: 2^k = 2^(k - 2 + 2) := by
+              exact congrArg (HPow.hPow 2) g2
+            have h2: 2 ^ (k - 2 + 2) = 2 ^ (k - 2) * 2 ^ 2 := by rw [pow_add]
+            rw [h1, h2]
+            ring_nf
+            simp
+          linarith
+        linarith
+      exfalso
+      linarith
+    . push_neg at hy
+      interval_cases y
+      . linarith
+      . simp at h₅
+        simp at h₃
+        linarith
+
+
+lemma imo_1997_p5_11_4
+  (x y k : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  (hxy : y < x)
+  (h₃ : x = y ^ (x / y ^ 2))
+  (hk_def : k = x / y ^ 2)
+  (hk : k < 2) :
+  (x, y) = (16, 2) ∨ (x, y) = (27, 3) := by
+  rw [← hk_def] at h₃
+  interval_cases k
+  . exfalso
+    simp at h₃
+    linarith
+  . exfalso
+    simp at *
+    linarith [hxy,h₃]
+
+
+lemma imo_1997_p5_11_5
+  (x y k : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  (hxy : y < x)
+  (h₃ : x = y ^ (x / y ^ 2))
+  (hk_def : k = x / y ^ 2)
+  (hk : k < 2) :
+  False := by
+  rw [← hk_def] at h₃
+  interval_cases k
+  . simp at h₃
+    linarith
+  . simp at *
+    linarith [hxy,h₃]
+
+
+lemma imo_1997_p5_11_6
+  (x y k : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  (hxy : y < x)
+  (h₃ : x = y ^ (x / y ^ 2))
+  (hk_def : k = x / y ^ 2)
+  (hk : 2 ≤ k) :
+  (x, y) = (16, 2) ∨ (x, y) = (27, 3) := by
+  rw [← hk_def] at h₃
+  have h₅: k = y^(k-2) := by
+    rw [h₃] at hk_def
+    nth_rewrite 1 [hk_def]
+    exact Nat.pow_div hk h₀.2
+  by_cases hk5: k < 5
+  . interval_cases k
+    . exfalso
+      simp at h₅
+    . right
+      norm_num
+      simp at h₅
+      symm at h₅
+      rw [h₅] at h₃
+      norm_num at h₃
+      exact { left := h₃, right := h₅ }
+    . simp at h₅
+      symm at h₅
+      have g₂: y^4 = y^2 * y^2 := by ring_nf
+      rw [g₂, h₅] at h₃
+      norm_num at h₃
+      left
+      norm_num
+      constructor
+      . exact h₃
+      . have h₆ : y ^ 2 = 2 ^ 2 := by
+          norm_num
+          exact h₅
+        have h₇: 0 ≤ y := by
+          linarith
+        exact (sq_eq_sq₀ h₇ (by linarith)).mp (h₆)
+  push_neg at hk5
+  by_cases hy: 2 ≤ y
+  . have h₅₁: k < y^(k-2) := by
+      have h₆: 2^(k-2) ≤ y^(k-2) := by
+        have hk1: 3 ≤ k - 2 := by exact Nat.sub_le_sub_right hk5 2
+        exact (Nat.pow_le_pow_iff_left (by linarith)).mpr hy
+      have h₇: 4*k < 2^k := by
+        exact imo_1997_p5_2 k hk5
+      have h₇: k < 2^(k-2) := by
+        have h₈ : k < 2 ^ k / 4 := by
+          have h81: 4 ∣ 2^k := by
+            have h82: 2^k = 4*2^(k-2) := by
+              have h83: k = 2 + (k -2) := by
+                ring_nf
+                exact (add_sub_of_le hk).symm
+              nth_rewrite 1 [h83]
+              rw [pow_add]
+              norm_num
+            rw [h82]
+            exact Nat.dvd_mul_right 4 (2^(k-2))
+          exact (Nat.lt_div_iff_mul_lt' h81 k).mpr h₇
+        have h₉: 2 ^ k / 4 = 2 ^ (k-2) := by
+          have g2: k = k - 2 + 2 := by
+            exact (Nat.sub_eq_iff_eq_add hk).mp rfl
+          have h1: 2^k = 2^(k - 2 + 2) := by
+            exact congrArg (HPow.hPow 2) g2
+          have h2: 2 ^ (k - 2 + 2) = 2 ^ (k - 2) * 2 ^ 2 := by rw [pow_add]
+          rw [h1, h2]
+          ring_nf
+          simp
+        linarith
+      linarith
+    exfalso
+    linarith
+  . push_neg at hy
+    interval_cases y
+    . linarith
+    . simp at h₅
+      simp at h₃
+      linarith
+
+
+lemma imo_1997_p5_11_7
+  (x y k : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  (h₃ : x = y ^ k)
+  (hk_def : k = x / y ^ 2)
+  (hk : 2 ≤ k) :
+  k = y ^ (k - 2) := by
+  rw [h₃] at hk_def
+  nth_rewrite 1 [hk_def]
+  exact Nat.pow_div hk h₀.2
+
+
+lemma imo_1997_p5_11_8
+  (x y k : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  (h₃ : x = y ^ k)
+  (hk_def : k = x / y ^ 2)
+  (hk : 2 ≤ k)
+  (h₅ : k = y ^ (k - 2))
+  (hk5 : k < 5) :
+  (x, y) = (16, 2) ∨ (x, y) = (27, 3) := by
+  interval_cases k
+  . exfalso
+    simp at h₅
+  . right
+    norm_num
+    simp at h₅
+    symm at h₅
+    rw [h₅] at h₃
+    norm_num at h₃
+    exact { left := h₃, right := h₅ }
+  . simp at h₅
+    symm at h₅
+    have g₂: y^4 = y^2 * y^2 := by ring_nf
+    rw [g₂, h₅] at h₃
+    norm_num at h₃
+    left
+    norm_num
+    constructor
+    . exact h₃
+    . have h₆ : y ^ 2 = 2 ^ 2 := by
+        norm_num
+        exact h₅
+      have h₇: 0 ≤ y := by
+        linarith
+      exact (sq_eq_sq₀ h₇ (by linarith)).mp (h₆)
+
+
+lemma imo_1997_p5_11_9
+  (x y : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  (h₃ : x = y ^ 3)
+  (hk_def : 3 = x / y ^ 2)
+  (hk : 2 ≤ 3)
+  (h₅ : 3 = y ^ (3 - 2))
+  (hk5 : 3 < 5) :
+  (x, y) = (27, 3) := by
+  norm_num
+  simp at h₅
+  symm at h₅
+  rw [h₅] at h₃
+  norm_num at h₃
+  exact { left := h₃, right := h₅ }
+
+
+lemma imo_1997_p5_11_10
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  (h₃ : x = y ^ 4)
+  (hk_def : 4 = x / y ^ 2)
+  (hk : 2 ≤ 4)
+  (h₅ : 4 = y ^ (4 - 2))
+  (hk5 : 4 < 5) :
+  (x, y) = (16, 2) := by
+  simp at h₅
+  symm at h₅
+  have g₂: y^4 = y^2 * y^2 := by ring_nf
+  rw [g₂, h₅] at h₃
+  norm_num at h₃
+  norm_num
+  constructor
+  . exact h₃
+  . have h₆ : y ^ 2 = 2 ^ 2 := by
+      norm_num
+      exact h₅
+    have h₇: 0 ≤ y := by
+      linarith
+    exact (sq_eq_sq₀ h₇ (by linarith)).mp (h₆)
+
+
+lemma imo_1997_p5_11_11
+  (y: ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  -- (hk_def : 4 = x / y ^ 2)
+  -- (hk : 2 ≤ 4)
+  -- (hk5 : 4 < 5)
+  (h₅ : y ^ 2 = 4)
+  (g₂ : y ^ 4 = y ^ 2 * y ^ 2) :
+  -- (h₃ : x = 16) :
+  y = 2 := by
+  rw [pow_two] at h₅
+  refine ((fun {m n} => Nat.mul_self_inj.mp) (?_)).symm
+  exact h₅.symm
+
+
+lemma imo_1997_p5_11_12
+  (x y k : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  (hxy : y < x)
+  (h₃ : x = y ^ k)
+  (hk_def : k = x / y ^ 2)
+  (hk : 2 ≤ k)
+  (h₅ : k = y ^ (k - 2))
+  (hk5 : 5 ≤ k) :
+  (x, y) = (16, 2) ∨ (x, y) = (27, 3) := by
+  by_cases hy: 2 ≤ y
+  . have h₅₁: k < y^(k-2) := by
+      have h₆: 2^(k-2) ≤ y^(k-2) := by
+        have hk1: 3 ≤ k - 2 := by exact Nat.sub_le_sub_right hk5 2
+        exact (Nat.pow_le_pow_iff_left (by linarith)).mpr hy
+      have h₇: 4*k < 2^k := by
+        exact imo_1997_p5_2 k hk5
+      have h₇: k < 2^(k-2) := by
+        have h₈ : k < 2 ^ k / 4 := by
+          have h81: 4 ∣ 2^k := by
+            have h82: 2^k = 4*2^(k-2) := by
+              have h83: k = 2 + (k -2) := by
+                ring_nf
+                exact (add_sub_of_le hk).symm
+              nth_rewrite 1 [h83]
+              rw [pow_add]
+              norm_num
+            rw [h82]
+            exact Nat.dvd_mul_right 4 (2^(k-2))
+          exact (Nat.lt_div_iff_mul_lt' h81 k).mpr h₇
+        have h₉: 2 ^ k / 4 = 2 ^ (k-2) := by
+          have g2: k = k - 2 + 2 := by
+            exact (Nat.sub_eq_iff_eq_add hk).mp rfl
+          have h1: 2^k = 2^(k - 2 + 2) := by
+            exact congrArg (HPow.hPow 2) g2
+          have h2: 2 ^ (k - 2 + 2) = 2 ^ (k - 2) * 2 ^ 2 := by rw [pow_add]
+          rw [h1, h2]
+          ring_nf
+          simp
+        linarith
+      linarith
+    exfalso
+    linarith
+  . push_neg at hy
+    interval_cases y
+    . linarith
+    . simp at h₅
+      simp at h₃
+      linarith
+
+
+lemma imo_1997_p5_11_13
+  (x y k : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  -- (h₃ : x = y ^ k)
+  -- (hk_def : k = x / y ^ 2)
+  (hk : 2 ≤ k)
+  (h₅ : k = y ^ (k - 2))
+  (hk5 : 5 ≤ k)
+  (hy : 2 ≤ y) :
+  (x, y) = (16, 2) ∨ (x, y) = (27, 3) := by
+  have h₅₁: k < y^(k-2) := by
+    have h₆: 2^(k-2) ≤ y^(k-2) := by
+      have hk1: 3 ≤ k - 2 := by exact Nat.sub_le_sub_right hk5 2
+      exact (Nat.pow_le_pow_iff_left (by linarith)).mpr hy
+    have h₇: 4*k < 2^k := by
+      exact imo_1997_p5_2 k hk5
+    have h₇: k < 2^(k-2) := by
+      have h₈ : k < 2 ^ k / 4 := by
+        have h81: 4 ∣ 2^k := by
+          have h82: 2^k = 4*2^(k-2) := by
+            have h83: k = 2 + (k -2) := by
+              ring_nf
+              exact (add_sub_of_le hk).symm
+            nth_rewrite 1 [h83]
+            rw [pow_add]
+            norm_num
+          rw [h82]
+          exact Nat.dvd_mul_right 4 (2^(k-2))
+        exact (Nat.lt_div_iff_mul_lt' h81 k).mpr h₇
+      have h₉: 2 ^ k / 4 = 2 ^ (k-2) := by
+        have g2: k = k - 2 + 2 := by
+          exact (Nat.sub_eq_iff_eq_add hk).mp rfl
+        have h1: 2^k = 2^(k - 2 + 2) := by
+          exact congrArg (HPow.hPow 2) g2
+        have h2: 2 ^ (k - 2 + 2) = 2 ^ (k - 2) * 2 ^ 2 := by rw [pow_add]
+        rw [h1, h2]
+        ring_nf
+        simp
+      linarith
+    linarith
+  exfalso
+  linarith
+
+
+lemma imo_1997_p5_11_14
+  (x y k : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  (hxy : y < x)
+  (h₃ : x = y ^ k)
+  (hk_def : k = x / y ^ 2)
+  -- (hk : 2 ≤ k)
+  (h₅ : k = y ^ (k - 2))
+  (hk5 : 5 ≤ k)
+  (hy : y < 2) :
+  (x, y) = (16, 2) ∨ (x, y) = (27, 3) := by
+  interval_cases y
+  . linarith
+  . simp at h₅
+    simp at h₃
+    linarith
+
+
+lemma imo_1997_p5_11_15
+  (x y k : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  -- (h₃ : x = y ^ k)
+  -- (hk_def : k = x / y ^ 2)
+  (hk : 2 ≤ k)
+  (h₅ : k = y ^ (k - 2))
+  (hk5 : 5 ≤ k)
+  (hy : 2 ≤ y) :
+  (x, y) = (16, 2) ∨ (x, y) = (27, 3) := by
+  have h₅₁: k < y^(k-2) := by
+    have h₆: 2^(k-2) ≤ y^(k-2) := by
+      have hk1: 3 ≤ k - 2 := by exact Nat.sub_le_sub_right hk5 2
+      exact (Nat.pow_le_pow_iff_left (by linarith)).mpr hy
+    have h₇: 4*k < 2^k := by
+      exact imo_1997_p5_2 k hk5
+    have h₈: k < 2^(k-2) := by
+      have h₈ : k < 2 ^ k / 4 := by
+        have h81: 4 ∣ 2^k := by
+          have h82: 2^k = 4*2^(k-2) := by
+            have h83: k = 2 + (k -2) := by
+              ring_nf
+              exact (add_sub_of_le hk).symm
+            nth_rewrite 1 [h83]
+            rw [pow_add]
+            norm_num
+          rw [h82]
+          exact Nat.dvd_mul_right 4 (2^(k-2))
+        exact (Nat.lt_div_iff_mul_lt' h81 k).mpr h₇
+      have h₉: 2 ^ k / 4 = 2 ^ (k-2) := by
+        have g2: k = k - 2 + 2 := by
+          exact (Nat.sub_eq_iff_eq_add hk).mp rfl
+        have h1: 2^k = 2^(k - 2 + 2) := by
+          exact congrArg (HPow.hPow 2) g2
+        have h2: 2 ^ (k - 2 + 2) = 2 ^ (k - 2) * 2 ^ 2 := by rw [pow_add]
+        rw [h1, h2]
+        ring_nf
+        simp
+      linarith
+    linarith
+  exfalso
+  linarith
+
+
+lemma imo_1997_p5_11_16
+  (y k : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  -- (h₃ : x = y ^ k)
+  -- (hk_def : k = x / y ^ 2)
+  (hk : 2 ≤ k)
+  (h₅ : k = y ^ (k - 2))
+  (hk5 : 5 ≤ k)
+  (hy : 2 ≤ y) :
+  False := by
+  have h₅₁: k < y^(k-2) := by
+    have h₆: 2^(k-2) ≤ y^(k-2) := by
+      have hk1: 3 ≤ k - 2 := by exact Nat.sub_le_sub_right hk5 2
+      exact (Nat.pow_le_pow_iff_left (by linarith)).mpr hy
+    have h₇: 4*k < 2^k := by
+      exact imo_1997_p5_2 k hk5
+    have h₈: k < 2^(k-2) := by
+      have h₈ : k < 2 ^ k / 4 := by
+        have h81: 4 ∣ 2^k := by
+          have h82: 2^k = 4*2^(k-2) := by
+            have h83: k = 2 + (k -2) := by
+              ring_nf
+              exact (add_sub_of_le hk).symm
+            nth_rewrite 1 [h83]
+            rw [pow_add]
+            norm_num
+          rw [h82]
+          exact Nat.dvd_mul_right 4 (2^(k-2))
+        exact (Nat.lt_div_iff_mul_lt' h81 k).mpr h₇
+      have h₉: 2 ^ k / 4 = 2 ^ (k-2) := by
+        have g2: k = k - 2 + 2 := by
+          exact (Nat.sub_eq_iff_eq_add hk).mp rfl
+        have h1: 2^k = 2^(k - 2 + 2) := by
+          exact congrArg (HPow.hPow 2) g2
+        have h2: 2 ^ (k - 2 + 2) = 2 ^ (k - 2) * 2 ^ 2 := by rw [pow_add]
+        rw [h1, h2]
+        ring_nf
+        simp
+      linarith
+    linarith
+  nth_rw 1 [← h₅] at h₅₁
+  apply Nat.ne_of_lt at h₅₁
+  refine false_of_ne h₅₁
+
+
+lemma imo_1997_p5_11_17
+  (y k : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  -- (h₃ : x = y ^ k)
+  -- (hk_def : k = x / y ^ 2)
+  (hk : 2 ≤ k)
+  (h₅ : k = y ^ (k - 2))
+  (hk5 : 5 ≤ k)
+  (hy : 2 ≤ y) :
+  k < y ^ (k - 2) := by
+  have h₆: 2^(k-2) ≤ y^(k-2) := by
+    have hk1: 3 ≤ k - 2 := by exact Nat.sub_le_sub_right hk5 2
+    exact (Nat.pow_le_pow_iff_left (by linarith)).mpr hy
+  have h₇: 4*k < 2^k := by
+    exact imo_1997_p5_2 k hk5
+  have h₈: k < 2^(k-2) := by
+    have h₈ : k < 2 ^ k / 4 := by
+      have h81: 4 ∣ 2^k := by
+        have h82: 2^k = 4*2^(k-2) := by
+          have h83: k = 2 + (k -2) := by
+            ring_nf
+            exact (add_sub_of_le hk).symm
+          nth_rewrite 1 [h83]
+          rw [pow_add]
+          norm_num
+        rw [h82]
+        exact Nat.dvd_mul_right 4 (2^(k-2))
+      exact (Nat.lt_div_iff_mul_lt' h81 k).mpr h₇
+    have h₉: 2 ^ k / 4 = 2 ^ (k-2) := by
+      have g2: k = k - 2 + 2 := by
+        exact (Nat.sub_eq_iff_eq_add hk).mp rfl
+      have h1: 2^k = 2^(k - 2 + 2) := by
+        exact congrArg (HPow.hPow 2) g2
+      have h2: 2 ^ (k - 2 + 2) = 2 ^ (k - 2) * 2 ^ 2 := by rw [pow_add]
+      rw [h1, h2]
+      ring_nf
+      simp
+    linarith
+  linarith
+
+
+lemma imo_1997_p5_11_18
+  (y k : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  -- (h₃ : x = y ^ k)
+  -- (hk_def : k = x / y ^ 2)
+  -- (hk : 2 ≤ k)
+  -- (h₅ : k = y ^ (k - 2))
+  (hk5 : 5 ≤ k)
+  (hy : 2 ≤ y) :
+  2 ^ (k - 2) ≤ y ^ (k - 2) := by
+  have hk1: 3 ≤ k - 2 := by exact Nat.sub_le_sub_right hk5 2
+  exact (Nat.pow_le_pow_iff_left (by linarith)).mpr hy
+
+
+lemma imo_1997_p5_11_19
+  (y k : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  -- (h₃ : x = y ^ k)
+  -- (hk_def : k = x / y ^ 2)
+  (hk : 2 ≤ k)
+  (h₅ : k = y ^ (k - 2))
+  -- (hk5 : 5 ≤ k)
+  -- (hy : 2 ≤ y)
+  (h₆ : 2 ^ (k - 2) ≤ y ^ (k - 2))
+  (h₇ : 4 * k < 2 ^ k) :
+  k < y ^ (k - 2) := by
+  have h₈: k < 2^(k-2) := by
+    have h₈ : k < 2 ^ k / 4 := by
+      have h81: 4 ∣ 2^k := by
+        have h82: 2^k = 4*2^(k-2) := by
+          have h83: k = 2 + (k -2) := by
+            ring_nf
+            exact (add_sub_of_le hk).symm
+          nth_rewrite 1 [h83]
+          rw [pow_add]
+          norm_num
+        rw [h82]
+        exact Nat.dvd_mul_right 4 (2^(k-2))
+      exact (Nat.lt_div_iff_mul_lt' h81 k).mpr h₇
+    have h₉: 2 ^ k / 4 = 2 ^ (k-2) := by
+      have g2: k = k - 2 + 2 := by
+        exact (Nat.sub_eq_iff_eq_add hk).mp rfl
+      have h1: 2^k = 2^(k - 2 + 2) := by
+        exact congrArg (HPow.hPow 2) g2
+      have h2: 2 ^ (k - 2 + 2) = 2 ^ (k - 2) * 2 ^ 2 := by rw [pow_add]
+      rw [h1, h2]
+      ring_nf
+      simp
+    linarith
+  linarith
+
+
+lemma imo_1997_p5_11_20
+  (y k : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  -- (h₃ : x = y ^ k)
+  -- (hk_def : k = x / y ^ 2)
+  (hk : 2 ≤ k)
+  (h₅ : k = y ^ (k - 2))
+  -- (hk5 : 5 ≤ k)
+  -- (hy : 2 ≤ y)
+  (h₆ : 2 ^ (k - 2) ≤ y ^ (k - 2))
+  (h₇ : 4 * k < 2 ^ k) :
+  k < 2 ^ (k - 2) := by
+  have h₈ : k < 2 ^ k / 4 := by
+    have h81: 4 ∣ 2^k := by
+      have h82: 2^k = 4*2^(k-2) := by
+        have h83: k = 2 + (k -2) := by
+          ring_nf
+          exact (add_sub_of_le hk).symm
+        nth_rewrite 1 [h83]
+        rw [pow_add]
+        norm_num
+      rw [h82]
+      exact Nat.dvd_mul_right 4 (2^(k-2))
+    exact (Nat.lt_div_iff_mul_lt' h81 k).mpr h₇
+  have h₉: 2 ^ k / 4 = 2 ^ (k-2) := by
+    have g2: k = k - 2 + 2 := by
+      exact (Nat.sub_eq_iff_eq_add hk).mp rfl
+    have h1: 2^k = 2^(k - 2 + 2) := by
+      exact congrArg (HPow.hPow 2) g2
+    have h2: 2 ^ (k - 2 + 2) = 2 ^ (k - 2) * 2 ^ 2 := by rw [pow_add]
+    rw [h1, h2]
+    ring_nf
+    simp
+  linarith
+
+
+lemma imo_1997_p5_11_21
+  (k : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  -- (h₃ : x = y ^ k)
+  -- (hk_def : k = x / y ^ 2)
+  (hk : 2 ≤ k)
+  -- (h₅ : k = y ^ (k - 2))
+  -- (hk5 : 5 ≤ k)
+  -- (hy : 2 ≤ y)
+  -- (h₆ : 2 ^ (k - 2) ≤ y ^ (k - 2))
+  (h₇ : 4 * k < 2 ^ k) :
+  k < 2 ^ k / 4 := by
+  have h81: 4 ∣ 2^k := by
+    have h82: 2^k = 4*2^(k-2) := by
+      have h83: k = 2 + (k -2) := by
+        ring_nf
+        exact (add_sub_of_le hk).symm
+      nth_rewrite 1 [h83]
+      rw [pow_add]
+      norm_num
+    rw [h82]
+    exact Nat.dvd_mul_right 4 (2^(k-2))
+  exact (Nat.lt_div_iff_mul_lt' h81 k).mpr h₇
+
+
+lemma imo_1997_p5_11_22
+  -- (x y : ℕ)
+  (k : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  -- (h₃ : x = y ^ k)
+  -- (hk_def : k = x / y ^ 2)
+  (hk : 2 ≤ k) :
+  -- (h₅ : k = y ^ (k - 2))
+  -- (hk5 : 5 ≤ k)
+  -- (hy : 2 ≤ y)
+  -- (h₆ : 2 ^ (k - 2) ≤ y ^ (k - 2))
+  -- (h₇ : 4 * k < 2 ^ k) :
+  4 ∣ 2 ^ k := by
+  have h82: 2^k = 4*2^(k-2) := by
+    have h83: k = 2 + (k -2) := by
+      ring_nf
+      exact (add_sub_of_le hk).symm
+    nth_rewrite 1 [h83]
+    rw [pow_add]
+    norm_num
+  rw [h82]
+  exact Nat.dvd_mul_right 4 (2^(k-2))
+
+
+
+lemma imo_1997_p5_11_23
+  -- (x y : ℕ)
+  (k : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  -- (h₃ : x = y ^ k)
+  -- (hk_def : k = x / y ^ 2)
+  (hk : 2 ≤ k) :
+  -- (h₅ : k = y ^ (k - 2))
+  -- (hk5 : 5 ≤ k)
+  -- (hy : 2 ≤ y)
+  -- (h₆ : 2 ^ (k - 2) ≤ y ^ (k - 2))
+  -- (h₇ : 4 * k < 2 ^ k) :
+  2 ^ k = 4 * 2 ^ (k - 2) := by
+  have h83: k = 2 + (k -2) := by
+    ring_nf
+    exact (add_sub_of_le hk).symm
+  nth_rewrite 1 [h83]
+  rw [pow_add]
+  norm_num
+
+
+lemma imo_1997_p5_11_24
+  -- (x y : ℕ)
+  (k : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  -- (h₃ : x = y ^ k)
+  -- (hk_def : k = x / y ^ 2)
+  -- (hk : 2 ≤ k)
+  -- (h₅ : k = y ^ (k - 2))
+  -- (hk5 : 5 ≤ k)
+  -- (hy : 2 ≤ y)
+  -- (h₆ : 2 ^ (k - 2) ≤ y ^ (k - 2))
+  -- (h₇ : 4 * k < 2 ^ k)
+  (h₈₃ : k = 2 + (k - 2)) :
+  2 ^ k = 4 * 2 ^ (k - 2) := by
+  nth_rewrite 1 [h₈₃]
+  rw [pow_add]
+  norm_num
+
+
+lemma imo_1997_p5_11_25
+  -- (x y : ℕ)
+  (k : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  -- (h₃ : x = y ^ k)
+  -- (hk_def : k = x / y ^ 2)
+  -- (hk : 2 ≤ k)
+  -- (h₅ : k = y ^ (k - 2))
+  -- (hk5 : 5 ≤ k)
+  -- (hy : 2 ≤ y)
+  -- (h₆ : 2 ^ (k - 2) ≤ y ^ (k - 2))
+  -- (h₇ : 4 * k < 2 ^ k)
+  (h82 : 2 ^ k = 4 * 2 ^ (k - 2)) :
+  4 ∣ 2 ^ k := by
+  rw [h82]
+  exact Nat.dvd_mul_right 4 (2^(k-2))
+
+
+lemma imo_1997_p5_11_26
+  -- (x : ℕ)
+  (y k : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  -- (h₃ : x = y ^ k)
+  -- (hk_def : k = x / y ^ 2)
+  (hk : 2 ≤ k)
+  (h₅ : k = y ^ (k - 2))
+  -- (hk5 : 5 ≤ k)
+  -- (hy : 2 ≤ y)
+  (h₆ : 2 ^ (k - 2) ≤ y ^ (k - 2))
+  -- (h₇ : 4 * k < 2 ^ k)
+  (h₈ : k < 2 ^ k / 4) :
+  k < 2 ^ (k - 2) := by
+  have h₉: 2 ^ k / 4 = 2 ^ (k-2) := by
+    have g2: k = k - 2 + 2 := by
+      exact (Nat.sub_eq_iff_eq_add hk).mp rfl
+    have h1: 2^k = 2^(k - 2 + 2) := by
+      exact congrArg (HPow.hPow 2) g2
+    have h2: 2 ^ (k - 2 + 2) = 2 ^ (k - 2) * 2 ^ 2 := by rw [pow_add]
+    rw [h1, h2]
+    ring_nf
+    simp
+  linarith
+
+
+lemma imo_1997_p5_11_27
+  -- (x y : ℕ)
+  (k : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  -- (h₃ : x = y ^ k)
+  -- (hk_def : k = x / y ^ 2)
+  (hk : 2 ≤ k) :
+  -- (h₅ : k = y ^ (k - 2))
+  -- (hk5 : 5 ≤ k)
+  -- (hy : 2 ≤ y)
+  -- (h₆ : 2 ^ (k - 2) ≤ y ^ (k - 2))
+  -- (h₇ : 4 * k < 2 ^ k)
+  -- (h₈ : k < 2 ^ k / 4) :
+  2 ^ k / 4 = 2 ^ (k - 2) := by
+  have g2: k = k - 2 + 2 := by
+    exact (Nat.sub_eq_iff_eq_add hk).mp rfl
+  have h1: 2^k = 2^(k - 2 + 2) := by
+    exact congrArg (HPow.hPow 2) g2
+  have h2: 2 ^ (k - 2 + 2) = 2 ^ (k - 2) * 2 ^ 2 := by rw [pow_add]
+  rw [h1, h2]
+  ring_nf
+  simp
+
+lemma imo_1997_p5_11_28
+  -- (x y : ℕ)
+  (k : ℕ)
+  -- (h₀ : 0 < x ∧ 0 < y)
+  -- (h₁ : x ^ y ^ 2 = y ^ x)
+  -- (g₁ : x ^ y ^ 2 = (x ^ y) ^ y)
+  -- (hxy : y < x)
+  -- (h₃ : x = y ^ k)
+  -- (hk_def : k = x / y ^ 2)
+  -- (hk : 2 ≤ k)
+  -- (h₅ : k = y ^ (k - 2))
+  -- (hk5 : 5 ≤ k)
+  -- (hy : 2 ≤ y)
+  -- (h₆ : 2 ^ (k - 2) ≤ y ^ (k - 2))
+  -- (h₇ : 4 * k < 2 ^ k)
+  -- (h₈ : k < 2 ^ k / 4)
+  -- (g2 : k = k - 2 + 2)
+  (h1 : 2 ^ k = 2 ^ (k - 2 + 2))
+  (h2 : 2 ^ (k - 2 + 2) = 2 ^ (k - 2) * 2 ^ 2) :
+  2 ^ k / 4 = 2 ^ (k - 2) := by
+  rw [h1, h2]
+  ring_nf
+  simp

Lemmas/imo_2022_p2_lemmas.lean

@@ -0,0 +1,1606 @@
+import Mathlib
+
+set_option linter.unusedVariables.analyzeTactics true
+
+open Real
+
+lemma imo_2022_p2_simp_1
+  (g : ℝ → ℝ)
+  (h₀ : ∀ (x : ℝ), 0 < x → ∃ y,
+  0 < y ∧ g x + g y ≤ 2 * x * y ∧ ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z) :
+  ∀ (x y : ℝ), 0 < x ∧ 0 < y → g x + g y ≤ 2 * x * y → x = y := by
+  intros x y hp h₁
+  by_contra! hc
+  have g₁: 2 * x * x < g x + g x := by
+    let ⟨p,h₁₁⟩ := h₀ x hp.1
+    cases' h₁₁ with h₁₁ h₁₂
+    cases' h₁₂ with h₁₂ h₁₃
+    by_cases hxp: x ≠ p
+    . have h₁₄: ¬ g x + g x ≤ 2 * x * x := by
+        refine h₁₃ x ?_
+        constructor
+        . exact hp.1
+        . exact hxp
+      exact not_le.mp h₁₄
+    . push_neg at hxp
+      exfalso
+      have hpy: y ≠ p := by exact Ne.trans_eq (id (Ne.symm hc)) hxp
+      have hcy: ¬g x + g y ≤ 2 * x * y := by
+        refine h₁₃ y ?_
+        constructor
+        . exact hp.2
+        . exact hpy
+      linarith
+  have g₂: 2 * y * y < g y + g y := by
+    let ⟨p,h₁₁⟩ := h₀ y hp.2
+    cases' h₁₁ with h₁₁ h₁₂
+    cases' h₁₂ with h₁₂ h₁₃
+    by_cases hyp: y ≠ p
+    . have h₁₄: ¬ g y + g y ≤ 2 * y * y := by
+        refine h₁₃ y ?_
+        constructor
+        . exact hp.2
+        . exact hyp
+      exact not_le.mp h₁₄
+    . push_neg at hyp
+      exfalso
+      have hpx: x ≠ p := by exact Ne.trans_eq hc hyp
+      have hcy: ¬g x + g y ≤ 2 * x * y := by
+        rw [add_comm, mul_right_comm]
+        refine h₁₃ x ?_
+        constructor
+        . exact hp.1
+        . exact hpx
+      linarith
+  ring_nf at g₁ g₂
+  simp at g₁ g₂
+  have g₆: (x - y) ≠ 0 := by exact sub_ne_zero.mpr hc
+  have g₇: 0 < (x - y) ^ 2 := by exact (sq_pos_iff).mpr g₆
+  linarith
+
+
+lemma imo_2022_p2_simp_1_1
+  (g : ℝ → ℝ)
+  (h₀ : ∀ (x : ℝ), 0 < x → ∃ y, 0 < y ∧ g x + g y ≤ 2 * x * y
+      ∧ ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z)
+  (x y : ℝ)
+  (hp : 0 < x ∧ 0 < y)
+  (h₁ : g x + g y ≤ 2 * x * y)
+  (hc : x ≠ y) :
+  2 * x * x < g x + g x := by
+  let ⟨p,h₁₁⟩ := h₀ x hp.1
+  cases' h₁₁ with h₁₁ h₁₂
+  cases' h₁₂ with h₁₂ h₁₃
+  by_cases hxp: x ≠ p
+  . have h₁₄: ¬ g x + g x ≤ 2 * x * x := by
+      refine h₁₃ x ?_
+      constructor
+      . exact hp.1
+      . exact hxp
+    exact not_le.mp h₁₄
+  . push_neg at hxp
+    exfalso
+    have hpy: y ≠ p := by exact Ne.trans_eq (id (Ne.symm hc)) hxp
+    have hcy: ¬g x + g y ≤ 2 * x * y := by
+      refine h₁₃ y ?_
+      constructor
+      . exact hp.2
+      . exact hpy
+    linarith
+
+
+
+lemma imo_2022_p2_simp_1_2
+  (g : ℝ → ℝ)
+  -- h₀ : ∀ (x : ℝ), 0 < x → ∃ y, 0 < y ∧ g x + g y ≤ 2 * x * y ∧ ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z
+  (x y : ℝ)
+  -- (hp : 0 < x ∧ 0 < y)
+  (h₁ : g x + g y ≤ 2 * x * y)
+  (hc : x ≠ y)
+  (g₁ : 2 * x * x < g x + g x)
+  (g₂ : 2 * y * y < g y + g y) :
+  False := by
+  ring_nf at g₁ g₂
+  simp at g₁ g₂
+  have g₆: (x - y) ≠ 0 := by exact sub_ne_zero.mpr hc
+  have g₇: 0 < (x - y) ^ 2 := by exact (sq_pos_iff).mpr g₆
+  linarith
+
+
+lemma imo_2022_p2_simp_1_3
+  -- (g : ℝ → ℝ)
+  (x y : ℝ)
+  -- (h₁ : g x + g y ≤ 2 * x * y)
+  (hc : x ≠ y) :
+  -- (g₁ : x ^ 2 < g x)
+  -- (g₂ : y ^ 2 < g y) :
+  0 < (x - y) ^ 2 := by
+  refine (sq_pos_iff).mpr ?_
+  exact sub_ne_zero.mpr hc
+
+
+lemma imo_2022_p2_simp_1_4
+  (g : ℝ → ℝ)
+  -- h₀ : ∀ (x : ℝ), 0 < x → ∃ y, 0 < y ∧ g x + g y ≤ 2 * x * y ∧ ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z
+  (x y : ℝ)
+  -- (hp : 0 < x ∧ 0 < y)
+  (h₁ : g x + g y ≤ 2 * x * y)
+  -- (hc : x ≠ y)
+  (g₁ : 2 * x * x < g x + g x)
+  (g₂ : 2 * y * y < g y + g y) :
+  (x - y) ^ 2 < 0 := by
+  linarith
+
+
+lemma imo_2022_p2_simp_1_5
+  (g : ℝ → ℝ)
+  -- h₀ : ∀ (x : ℝ), 0 < x → ∃ y, 0 < y ∧ g x + g y ≤ 2 * x * y ∧ ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z
+  (x y : ℝ)
+  (hp : 0 < x ∧ 0 < y)
+  (h₁ : g x + g y ≤ 2 * x * y)
+  (hc : x ≠ y)
+  (p : ℝ)
+  -- (h₁₁ : 0 < p)
+  -- (h₁₂ : g x + g p ≤ 2 * x * p)
+  (h₁₃ : ∀ (z : ℝ), 0 < z ∧ z ≠ p → ¬g x + g z ≤ 2 * x * z) :
+  2 * x * x < g x + g x := by
+  by_cases hxp: x ≠ p
+  . have h₁₄: ¬ g x + g x ≤ 2 * x * x := by
+      refine h₁₃ x ?_
+      constructor
+      . exact hp.1
+      . exact hxp
+    exact not_le.mp h₁₄
+  . push_neg at hxp
+    exfalso
+    have hpy: y ≠ p := by exact Ne.trans_eq (id (Ne.symm hc)) hxp
+    have hcy: ¬g x + g y ≤ 2 * x * y := by
+      refine h₁₃ y ?_
+      constructor
+      . exact hp.2
+      . exact hpy
+    linarith
+
+
+lemma imo_2022_p2_simp_1_6
+  (g : ℝ → ℝ)
+  (x y : ℝ)
+  (hxyp : 0 < x ∧ 0 < y)
+  -- h₁ : g x + g y ≤ 2 * x * y
+  -- hc : x ≠ y
+  (p : ℝ)
+  (h₁₃ : ∀ (z : ℝ), 0 < z ∧ z ≠ p → ¬g x + g z ≤ 2 * x * z)
+  (hxp : x ≠ p) :
+  2 * x * x < g x + g x := by
+  have h₁₄: ¬ g x + g x ≤ 2 * x * x := by
+    refine h₁₃ x ?_
+    constructor
+    . exact hxyp.1
+    . exact hxp
+  exact not_le.mp h₁₄
+
+
+lemma imo_2022_p2_simp_1_7
+  (g : ℝ → ℝ)
+  (x y : ℝ)
+  (hxyp : 0 < x ∧ 0 < y)
+  (p : ℝ)
+  (h₁₃ : ∀ (z : ℝ), 0 < z ∧ z ≠ p → ¬g x + g z ≤ 2 * x * z)
+  (hxp : x ≠ p) :
+  ¬g x + g x ≤ 2 * x * x := by
+  refine h₁₃ x ?_
+  constructor
+  . exact hxyp.1
+  . exact hxp
+
+
+
+lemma imo_2022_p2_simp_1_8
+  (g : ℝ → ℝ)
+  (x y : ℝ)
+  (hp : 0 < x ∧ 0 < y)
+  (h₁ : g x + g y ≤ 2 * x * y)
+  (hc : x ≠ y)
+  (p : ℝ)
+  (h₁₃ : ∀ (z : ℝ), 0 < z ∧ z ≠ p → ¬g x + g z ≤ 2 * x * z)
+  (hxp : ¬x ≠ p) :
+  2 * x * x < g x + g x := by
+  push_neg at hxp
+  exfalso
+  have hpy: y ≠ p := by exact Ne.trans_eq (id (Ne.symm hc)) hxp
+  have hcy: ¬g x + g y ≤ 2 * x * y := by
+    refine h₁₃ y ?_
+    constructor
+    . exact hp.2
+    . exact hpy
+  linarith
+
+
+
+lemma imo_2022_p2_simp_1_9
+  (g : ℝ → ℝ)
+  (x y : ℝ)
+  (hp : 0 < x ∧ 0 < y)
+  (h₁ : g x + g y ≤ 2 * x * y)
+  (hc : x ≠ y)
+  (p : ℝ)
+  (h₁₃ : ∀ (z : ℝ), 0 < z ∧ z ≠ p → ¬g x + g z ≤ 2 * x * z)
+  (hxp : x = p) :
+  False := by
+  have hpy: y ≠ p := by exact Ne.trans_eq (id (Ne.symm hc)) hxp
+  have hcy: ¬g x + g y ≤ 2 * x * y := by
+    refine h₁₃ y ?_
+    constructor
+    . exact hp.2
+    . exact hpy
+  linarith
+
+
+
+
+lemma imo_2022_p2_simp_2
+  (g : ℝ → ℝ)
+  (h₀ : ∀ (x : ℝ), 0 < x → ∃ y, 0 < y ∧ g x + g y ≤ 2 * x * y ∧
+      ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z)
+  (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → g x + g y ≤ 2 * x * y → x = y) :
+  ∀ (x : ℝ), 0 < x → g x ≤ x ^ 2 := by
+  intros x hxp
+  let ⟨y,h₁₁⟩ := h₀ x hxp
+  cases' h₁₁ with h₁₁ h₁₂
+  cases' h₁₂ with h₁₂ h₁₃
+  have hxy: x = y := by
+    apply h₁ x y
+    . exact { left := hxp, right := h₁₁ }
+    . exact h₁₂
+  rw [← hxy] at h₁₂
+  linarith
+
+
+lemma imo_2022_p2_simp_2_1
+  (g : ℝ → ℝ)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃ y, 0 < y ∧ g x + g y ≤ 2 * x * y
+  --   ∧ ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z)
+  (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → g x + g y ≤ 2 * x * y → x = y)
+  (x y: ℝ)
+  (hxp : 0 < x)
+  (h₁₁ : 0 < y)
+  (h₁₂ : g x + g y ≤ 2 * x * y) :
+  -- (h₁₃ : ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z) :
+  x = y := by
+  apply h₁ x y
+  . exact { left := hxp, right := h₁₁ }
+  . exact h₁₂
+
+
+lemma imo_2022_p2_simp_2_2
+  (g : ℝ → ℝ)
+  (x y : ℝ)
+  (h₁₂ : g x + g y ≤ 2 * x * y)
+  (hxy : x = y) :
+  g x ≤ x ^ 2 := by
+  rw [← hxy] at h₁₂
+  linarith
+
+
+lemma imo_2022_p2_simp_3
+  (g : ℝ → ℝ)
+  (h₀ : ∀ (x : ℝ), 0 < x → ∃ y, 0 < y ∧ g x + g y ≤ 2 * x * y
+    ∧ ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z)
+  (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → g x + g y ≤ 2 * x * y → x = y)
+  (h₂ : ∀ (x : ℝ), 0 < x → g x ≤ x ^ 2) :
+  ∀ (x : ℝ), 0 < x → ¬g x < x ^ 2 := by
+  simp
+  by_contra! hc
+  let ⟨x,hxp⟩ := hc
+  cases' hxp with hxp h₃
+  let d₁:ℝ := x ^ 2 - g x
+  have hd₁ : g x = x ^ 2 - d₁ := by exact (sub_sub_self (x ^ 2) (g x)).symm
+  let z:ℝ := x + Real.sqrt d₁
+  have hz: z = x + Real.sqrt d₁ := by exact rfl
+  have hzp: 0 < z := by
+    refine add_pos hxp ?_
+    refine Real.sqrt_pos_of_pos ?_
+    exact sub_pos.mpr h₃
+  have hxz: z ≠ x := by
+    rw [hz]
+    simp
+    push_neg
+    refine Real.sqrt_ne_zero'.mpr ?_
+    exact sub_pos.mpr h₃
+  -- have h₄: g z ≤ z ^ 2 := by exact h₂ z hzp
+  have h₅: g x + g z ≤ 2 * x * z := by
+    rw [hd₁]
+    have h₅₁: - d₁ + Real.sqrt (x ^ 2 - (x ^ 2 - d₁)) ^ 2 ≤ 0 := by
+      simp
+      rw [Real.sq_sqrt]
+      exact sub_nonneg_of_le (h₂ x hxp)
+    have h₅₂: x ^ 2 - d₁ + z ^ 2 ≤ 2 * x * z := by
+      rw [hz, mul_add, add_sq]
+      ring_nf
+      repeat rw [add_assoc]
+      refine add_le_add_left ?_ (x * Real.sqrt (x ^ 2 - g x) * 2)
+      rw [hd₁]
+      linarith
+    exact add_le_of_add_le_left h₅₂ (h₂ z hzp)
+  let ⟨y,hyp⟩ := h₀ x hxp
+  cases' hyp with hyp h₆
+  cases' h₆ with h₆ h₇
+  have hxy: x = y := by
+    apply h₁
+    . exact { left := hxp, right := hyp }
+    . exact h₆
+  have h₈: ¬g x + g z ≤ 2 * x * z := by
+      refine h₇ z ?_
+      constructor
+      . exact hzp
+      . exact Ne.trans_eq hxz hxy
+  linarith[h₅,h₈]
+
+
+lemma imo_2022_p2_simp_3_1
+  (g : ℝ → ℝ)
+  (h₀ : ∀ (x : ℝ), 0 < x → ∃ y, 0 < y ∧ g x + g y ≤ 2 * x * y
+      ∧ ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z)
+  (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → g x + g y ≤ 2 * x * y → x = y)
+  (h₂ : ∀ (x : ℝ), 0 < x → g x ≤ x ^ 2)
+  (hc : ∃ x, 0 < x ∧ g x < x ^ 2) :
+  False := by
+  let ⟨x,hxp⟩ := hc
+  cases' hxp with hxp h₃
+  let d₁:ℝ := x ^ 2 - g x
+  have hd₁ : g x = x ^ 2 - d₁ := by exact (sub_sub_self (x ^ 2) (g x)).symm
+  let z:ℝ := x + Real.sqrt d₁
+  have hz: z = x + Real.sqrt d₁ := by exact rfl
+  have hzp: 0 < z := by
+    refine add_pos hxp ?_
+    refine Real.sqrt_pos_of_pos ?_
+    exact sub_pos.mpr h₃
+  have hxz: z ≠ x := by
+    rw [hz]
+    simp
+    push_neg
+    refine Real.sqrt_ne_zero'.mpr ?_
+    exact sub_pos.mpr h₃
+  have h₅: g x + g z ≤ 2 * x * z := by
+    rw [hd₁]
+    have h₅₁: - d₁ + Real.sqrt (x ^ 2 - (x ^ 2 - d₁)) ^ 2 ≤ 0 := by
+      simp
+      rw [Real.sq_sqrt]
+      exact sub_nonneg_of_le (h₂ x hxp)
+    have h₅₂: x ^ 2 - d₁ + z ^ 2 ≤ 2 * x * z := by
+      rw [hz, mul_add, add_sq]
+      ring_nf
+      repeat rw [add_assoc]
+      refine add_le_add_left ?_ (x * Real.sqrt (x ^ 2 - g x) * 2)
+      rw [hd₁]
+      linarith
+    exact add_le_of_add_le_left h₅₂ (h₂ z hzp)
+  let ⟨y,hyp⟩ := h₀ x hxp
+  cases' hyp with hyp h₆
+  cases' h₆ with h₆ h₇
+  have hxy: x = y := by
+    apply h₁
+    . exact { left := hxp, right := hyp }
+    . exact h₆
+  have h₈: ¬g x + g z ≤ 2 * x * z := by
+      refine h₇ z ?_
+      constructor
+      . exact hzp
+      . exact Ne.trans_eq hxz hxy
+  linarith[h₅,h₈]
+
+
+lemma imo_2022_p2_simp_3_2
+  (g : ℝ → ℝ)
+  (h₀ : ∀ (x : ℝ), 0 < x → ∃ y, 0 < y ∧ g x + g y ≤ 2 * x * y ∧ ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z)
+  (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → g x + g y ≤ 2 * x * y → x = y)
+  (h₂ : ∀ (x : ℝ), 0 < x → g x ≤ x ^ 2)
+  -- (hc : ∃ x, 0 < x ∧ g x < x ^ 2)
+  (x z d₁ : ℝ)
+  (hxp : 0 < x)
+  (h₃ : g x < x ^ 2)
+  (hd₀ : d₁ = x ^ 2 - g x)
+  (hd₁ : g x = x ^ 2 - d₁)
+  (hz : z = x + √d₁) :
+  False := by
+  have hzp: 0 < z := by
+    rw [hz]
+    refine add_pos hxp ?_
+    refine Real.sqrt_pos_of_pos ?_
+    rw [hd₀]
+    exact sub_pos.mpr h₃
+  have hxz: z ≠ x := by
+    rw [hz]
+    simp
+    push_neg
+    refine Real.sqrt_ne_zero'.mpr ?_
+    rw [hd₀]
+    exact sub_pos.mpr h₃
+  have h₅: g x + g z ≤ 2 * x * z := by
+    rw [hd₁]
+    have h₅₂: x ^ 2 - d₁ + z ^ 2 ≤ 2 * x * z := by
+      rw [hz, mul_add, add_sq]
+      ring_nf
+      repeat rw [add_assoc]
+      refine add_le_add_left ?_ (x * √d₁ * 2)
+      rw [sq_sqrt]
+      simp
+      linarith
+    exact add_le_of_add_le_left h₅₂ (h₂ z hzp)
+  let ⟨y,hyp⟩ := h₀ x hxp
+  cases' hyp with hyp h₆
+  cases' h₆ with h₆ h₇
+  have hxy: x = y := by
+    apply h₁
+    . exact { left := hxp, right := hyp }
+    . exact h₆
+  have h₈: ¬g x + g z ≤ 2 * x * z := by
+      refine h₇ z ?_
+      constructor
+      . exact hzp
+      . exact Ne.trans_eq hxz hxy
+  linarith[h₅,h₈]
+
+
+lemma imo_2022_p2_simp_3_3
+  (g : ℝ → ℝ)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃ y, 0 < y ∧ g x + g y ≤ 2 * x * y ∧ ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → g x + g y ≤ 2 * x * y → x = y)
+  -- (h₂ : ∀ (x : ℝ), 0 < x → g x ≤ x ^ 2)
+  -- (hc : ∃ x, 0 < x ∧ g x < x ^ 2)
+  (x z d₁ : ℝ)
+  (hxp : 0 < x)
+  (h₃ : g x < x ^ 2)
+  (hd₀ : d₁ = x ^ 2 - g x)
+  -- (hd₁ : g x = x ^ 2 - d₁)
+  (hz : z = x + √d₁) :
+  0 < z := by
+  rw [hz]
+  refine add_pos hxp ?_
+  refine Real.sqrt_pos_of_pos ?_
+  rw [hd₀]
+  exact sub_pos.mpr h₃
+
+
+lemma imo_2022_p2_simp_3_4
+  (g : ℝ → ℝ)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃ y, 0 < y ∧ g x + g y ≤ 2 * x * y ∧ ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → g x + g y ≤ 2 * x * y → x = y)
+  -- (h₂ : ∀ (x : ℝ), 0 < x → g x ≤ x ^ 2)
+  -- (hc : ∃ x, 0 < x ∧ g x < x ^ 2)
+  (x z d₁: ℝ)
+  -- (hxp : 0 < x)
+  (h₃ : g x < x ^ 2)
+  (hd₀ : d₁ = x ^ 2 - g x)
+  -- (hd₁ : g x = x ^ 2 - d₁)
+  (hz : z = x + √d₁) :
+  -- (hzp : 0 < z) :
+  z ≠ x := by
+  rw [hz]
+  simp
+  push_neg
+  refine Real.sqrt_ne_zero'.mpr ?_
+  rw [hd₀]
+  exact sub_pos.mpr h₃
+
+
+lemma imo_2022_p2_simp_3_5
+  (g : ℝ → ℝ)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃ y, 0 < y ∧ g x + g y ≤ 2 * x * y ∧ ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → g x + g y ≤ 2 * x * y → x = y)
+  (h₂ : ∀ (x : ℝ), 0 < x → g x ≤ x ^ 2)
+  -- (hc : ∃ x, 0 < x ∧ g x < x ^ 2)
+  (x z d₁: ℝ)
+  -- (hxp : 0 < x)
+  (h₃ : g x < x ^ 2)
+  -- (hd₀ : d₁ = x ^ 2 - g x)
+  (hd₁ : g x = x ^ 2 - d₁)
+  (hz : z = x + √d₁)
+  (hzp : 0 < z) :
+  -- (hxz : z ≠ x) :
+  g x + g z ≤ 2 * x * z := by
+  rw [hd₁]
+  have h₅₂: x ^ 2 - d₁ + z ^ 2 ≤ 2 * x * z := by
+    rw [hz, mul_add, add_sq]
+    ring_nf
+    repeat rw [add_assoc]
+    refine add_le_add_left ?_ (x * √d₁ * 2)
+    rw [sq_sqrt]
+    simp
+    linarith
+  exact add_le_of_add_le_left h₅₂ (h₂ z hzp)
+
+
+lemma imo_2022_p2_simp_3_6
+  (g : ℝ → ℝ)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃ y, 0 < y ∧ g x + g y ≤ 2 * x * y ∧ ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z)
+  (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → g x + g y ≤ 2 * x * y → x = y)
+  -- (h₂ : ∀ (x : ℝ), 0 < x → g x ≤ x ^ 2)
+  -- (hc : ∃ x, 0 < x ∧ g x < x ^ 2)
+  (x z : ℝ)
+  (hxp : 0 < x)
+  -- (h₃ : g x < x ^ 2)
+  -- (hd₀ : d₁ = x ^ 2 - g x)
+  -- (hd₁ : g x = x ^ 2 - d₁)
+  -- (hz : z = x + √d₁)
+  (hzp : 0 < z)
+  (hxz : z ≠ x)
+  (h₅ : g x + g z ≤ 2 * x * z)
+  (y : ℝ)
+  (hyp : 0 < y)
+  (h₆ : g x + g y ≤ 2 * x * y)
+  (h₇ : ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z) :
+  False := by
+  have hxy: x = y := by
+    apply h₁
+    . exact { left := hxp, right := hyp }
+    . exact h₆
+  have h₈: ¬g x + g z ≤ 2 * x * z := by
+      refine h₇ z ?_
+      constructor
+      . exact hzp
+      . exact Ne.trans_eq hxz hxy
+  linarith[h₅,h₈]
+
+
+lemma imo_2022_p2_simp_3_7
+  (g : ℝ → ℝ)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃ y, 0 < y ∧ g x + g y ≤ 2 * x * y ∧ ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z)
+  (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → g x + g y ≤ 2 * x * y → x = y)
+  -- (h₂ : ∀ (x : ℝ), 0 < x → g x ≤ x ^ 2)
+  -- (hc : ∃ x, 0 < x ∧ g x < x ^ 2)
+  (x : ℝ)
+  (hxp : 0 < x)
+  -- (h₃ : g x < x ^ 2)
+  -- (hd₀ : d₁ = x ^ 2 - g x)
+  -- (hd₁ : g x = x ^ 2 - d₁)
+  -- (hz : z = x + √d₁)
+  -- (hzp : 0 < z)
+  -- (hxz : z ≠ x)
+  -- (h₅ : g x + g z ≤ 2 * x * z)
+  (y : ℝ)
+  (hyp : 0 < y)
+  (h₆ : g x + g y ≤ 2 * x * y) :
+  -- (h₇ : ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z) :
+  x = y := by
+  apply h₁
+  . exact { left := hxp, right := hyp }
+  . exact h₆
+
+
+lemma imo_2022_p2_simp_3_8
+  (g : ℝ → ℝ)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃ y, 0 < y ∧ g x + g y ≤ 2 * x * y ∧ ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → g x + g y ≤ 2 * x * y → x = y)
+  -- (h₂ : ∀ (x : ℝ), 0 < x → g x ≤ x ^ 2)
+  -- (hc : ∃ x, 0 < x ∧ g x < x ^ 2)
+  (x z : ℝ)
+  -- (hxp : 0 < x)
+  -- (h₃ : g x < x ^ 2)
+  -- (hd₀ : d₁ = x ^ 2 - g x)
+  -- (hd₁ : g x = x ^ 2 - d₁)
+  -- (hz : z = x + √d₁)
+  (hzp : 0 < z)
+  (hxz : z ≠ x)
+  -- (h₅ : g x + g z ≤ 2 * x * z)
+  (y : ℝ)
+  -- (hyp : 0 < y)
+  -- (h₆ : g x + g y ≤ 2 * x * y)
+  (h₇ : ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z)
+  (hxy : x = y) :
+  ¬g x + g z ≤ 2 * x * z := by
+  refine h₇ z ?_
+  constructor
+  . exact hzp
+  . exact Ne.trans_eq hxz hxy
+
+
+lemma imo_2022_p2_simp_3_9
+  (g : ℝ → ℝ)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃ y, 0 < y ∧ g x + g y ≤ 2 * x * y ∧ ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → g x + g y ≤ 2 * x * y → x = y)
+  (h₂ : ∀ (x : ℝ), 0 < x → g x ≤ x ^ 2)
+  -- (hc : ∃ x, 0 < x ∧ g x < x ^ 2)
+  (x d₁ : ℝ)
+  (hxp : 0 < x)
+  -- (h₃ : g x < x ^ 2)
+  (hd₀ : d₁ = x ^ 2 - g x) :
+  -- (hd₁ : g x = x ^ 2 - d₁)
+  -- (hz : z = x + √d₁)
+  -- (hzp : 0 < z)
+  -- (hxz : z ≠ x) :
+  -d₁ + √(x ^ 2 - (x ^ 2 - d₁)) ^ 2 ≤ 0 := by
+  simp
+  rw [Real.sq_sqrt]
+  rw [hd₀]
+  exact sub_nonneg_of_le (h₂ x hxp)
+
+
+lemma imo_2022_p2_simp_3_10
+  (g : ℝ → ℝ)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃ y, 0 < y ∧ g x + g y ≤ 2 * x * y ∧ ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → g x + g y ≤ 2 * x * y → x = y)
+  -- (h₂ : ∀ (x : ℝ), 0 < x → g x ≤ x ^ 2)
+  -- (hc : ∃ x, 0 < x ∧ g x < x ^ 2)
+  (x z d₁ : ℝ)
+  -- (hxp : 0 < x)
+  (h₃ : g x < x ^ 2)
+  -- (hd₀ : d₁ = x ^ 2 - g x)
+  (hd₁ : g x = x ^ 2 - d₁)
+  (hz : z = x + √d₁) :
+  -- (hzp : 0 < z)
+  -- (hxz : z ≠ x)
+  -- (h₅₁ : -d₁ + √(x ^ 2 - (x ^ 2 - d₁)) ^ 2 ≤ 0) :
+  x ^ 2 - d₁ + z ^ 2 ≤ 2 * x * z := by
+  rw [hz, mul_add, add_sq]
+  ring_nf
+  repeat rw [add_assoc]
+  refine add_le_add_left ?_ (x * √d₁ * 2)
+  rw [sq_sqrt]
+  simp
+  linarith
+
+
+lemma imo_2022_p2_simp_3_11
+  (g : ℝ → ℝ)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃ y, 0 < y ∧ g x + g y ≤ 2 * x * y ∧ ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → g x + g y ≤ 2 * x * y → x = y)
+  (h₂ : ∀ (x : ℝ), 0 < x → g x ≤ x ^ 2)
+  -- (hc : ∃ x, 0 < x ∧ g x < x ^ 2)
+  (x z d₁ : ℝ)
+  -- (hxp : 0 < x)
+  -- (h₃ : g x < x ^ 2)
+  -- (hd₀ : d₁ = x ^ 2 - g x)
+  -- (hd₁ : g x = x ^ 2 - d₁)
+  -- (hz : z = x + √d₁)
+  (hzp : 0 < z)
+  -- (hxz : z ≠ x)
+  -- (h₅₁ : -d₁ + √(x ^ 2 - (x ^ 2 - d₁)) ^ 2 ≤ 0)
+  (h₅₂ : x ^ 2 - d₁ + z ^ 2 ≤ 2 * x * z) :
+  x ^ 2 - d₁ + g z ≤ 2 * x * z := by
+  refine add_le_of_add_le_left h₅₂ ?_
+  exact h₂ z hzp
+
+
+lemma imo_2022_p2_simp_4
+  (g : ℝ → ℝ)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃ y, 0 < y ∧ g x + g y ≤ 2 * x * y
+  --     ∧ ∀ (z : ℝ), 0 < z ∧ z ≠ y → ¬g x + g z ≤ 2 * x * z)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → g x + g y ≤ 2 * x * y → x = y)
+  (h₂ : ∀ (x : ℝ), 0 < x → g x ≤ x ^ 2)
+  (h₃ : ∀ (x : ℝ), 0 < x → ¬g x < x ^ 2) :
+  ∀ (x : ℝ), 0 < x → g x = x ^ 2 := by
+  intros x hxp
+  have g₂: g x ≤ x ^ 2 := by exact h₂ x hxp
+  have g₃: ¬ g x < x ^ 2 := by exact h₃ x hxp
+  linarith
+
+
+
+lemma imo_2022_p2_1
+  (f : ℝ → ℝ)
+  -- (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2) :
+  ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y := by
+  intros x y hp h₁
+  by_contra! hc
+  have h₁₀: x * f x + x * f x > 2 := by
+    let ⟨z,h₁₁⟩ := h₀ x hp.1
+    cases' h₁₁ with h₁₁ h₁₂
+    have h₁₄: y = z := by
+      apply h₁₂ y
+      constructor
+      . exact hp.2
+      . exact h₁
+    have hxz: ¬ x = z := by exact Ne.trans_eq hc h₁₄
+    have h₁₆: ¬ (fun y => 0 < y ∧ x * f y + y * f x ≤ 2) x := by
+      exact mt (h₁₂ x) hxz
+    have h₁₇: ¬ (0 < x ∧ x * f x + x * f x ≤ 2) := by exact h₁₆
+    push_neg at h₁₇
+    exact h₁₇ hp.1
+  have h₁₁: y * f y + y * f y > 2 := by
+    let ⟨z,h₁₁⟩ := h₀ y hp.2
+    cases' h₁₁ with h₁₁ h₁₂
+    have h₁₄: x = z := by
+      apply h₁₂ x
+      constructor
+      . exact hp.1
+      . rw [add_comm]
+        exact h₁
+    have hxz: ¬ y = z := by exact Ne.trans_eq (id (Ne.symm hc)) h₁₄
+    have h₁₆: ¬ (fun y_2 => 0 < y_2 ∧ y * f y_2 + y_2 * f y ≤ 2) y := by
+      exact mt (h₁₂ y) hxz
+    have h₁₇: ¬ (0 < y ∧ y * f y + y * f y ≤ 2) := by exact h₁₆
+    push_neg at h₁₇
+    exact h₁₇ hp.2
+  ring_nf at h₁₀ h₁₁
+  simp at h₁₀ h₁₁
+  have h₁₅: 1 / x < f x := by exact (div_lt_iff₀' hp.1).mpr (h₁₀)
+  have h₁₆: 1 / y < f y := by exact (div_lt_iff₀' hp.2).mpr (h₁₁)
+  have h₁₂: x / y + y / x < 2 := by
+    refine lt_of_le_of_lt' h₁ ?_
+    refine add_lt_add ?_ ?_
+    . rw [← mul_one_div]
+      exact (mul_lt_mul_left hp.1).mpr h₁₆
+    . rw [← mul_one_div]
+      exact (mul_lt_mul_left hp.2).mpr h₁₅
+  have h₁₃: 2 < x / y + y / x := by
+    refine lt_of_mul_lt_mul_right ?_ (le_of_lt hp.1)
+    refine lt_of_mul_lt_mul_right ?_ (le_of_lt hp.2)
+    repeat rw [add_mul, mul_assoc]
+    rw [mul_comm x y, ← mul_assoc (x/y)]
+    rw [div_mul_comm x y y, div_mul_comm y x x, div_self, div_self]
+    . ring_nf
+      refine lt_of_sub_pos ?_
+      rw [mul_comm _ 2, ← mul_assoc]
+      rw [← sub_sq']
+      refine sq_pos_of_ne_zero ?_
+      exact sub_ne_zero.mpr hc.symm
+    . exact ne_of_gt hp.1
+    . exact ne_of_gt hp.2
+  linarith
+
+
+lemma imo_2022_p2_1_1
+  (f : ℝ → ℝ)
+  -- (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  (x y : ℝ)
+  (hp : 0 < x ∧ 0 < y)
+  (h₁ : x * f y + y * f x ≤ 2)
+  (hc : x ≠ y) :
+  x * f x + x * f x > 2 := by
+  let ⟨z,h₁₁⟩ := h₀ x hp.1
+  cases' h₁₁ with h₁₁ h₁₂
+  have h₁₄: y = z := by
+    apply h₁₂ y
+    constructor
+    . exact hp.2
+    . exact h₁
+  have hxz: ¬ x = z := by exact Ne.trans_eq hc h₁₄
+  have h₁₆: ¬ (fun y => 0 < y ∧ x * f y + y * f x ≤ 2) x := by
+    exact mt (h₁₂ x) hxz
+  have h₁₇: ¬ (0 < x ∧ x * f x + x * f x ≤ 2) := by exact h₁₆
+  push_neg at h₁₇
+  exact h₁₇ hp.1
+
+
+lemma imo_2022_p2_1_2
+  (f : ℝ → ℝ)
+  -- (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  (x y : ℝ)
+  (hp : 0 < x ∧ 0 < y)
+  (h₁ : x * f y + y * f x ≤ 2)
+  (hc : x ≠ y)
+  (z : ℝ)
+  -- (h₁₁ : (fun y => 0 < y ∧ x * f y + y * f x ≤ 2) z)
+  (h₁₂ : ∀ (y : ℝ), (fun y => 0 < y ∧ x * f y + y * f x ≤ 2) y → y = z) :
+  x * f x + x * f x > 2 := by
+  have h₁₄: y = z := by
+    apply h₁₂ y
+    constructor
+    . exact hp.2
+    . exact h₁
+  have hxz: ¬ x = z := by exact Ne.trans_eq hc h₁₄
+  have h₁₆: ¬ (fun y => 0 < y ∧ x * f y + y * f x ≤ 2) x := by
+    exact mt (h₁₂ x) hxz
+  have h₁₇: ¬ (0 < x ∧ x * f x + x * f x ≤ 2) := by exact h₁₆
+  push_neg at h₁₇
+  exact h₁₇ hp.1
+
+
+lemma imo_2022_p2_1_3
+  (f : ℝ → ℝ)
+  -- (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  (x y : ℝ)
+  (hp : 0 < x ∧ 0 < y)
+  (h₁ : x * f y + y * f x ≤ 2)
+  -- (hc : x ≠ y)
+  (z : ℝ)
+  -- (h₁₁ : (fun y => 0 < y ∧ x * f y + y * f x ≤ 2) z)
+  (h₁₂ : ∀ (y : ℝ), (fun y => 0 < y ∧ x * f y + y * f x ≤ 2) y → y = z) :
+  y = z := by
+  apply h₁₂ y
+  constructor
+  . exact hp.2
+  . exact h₁
+
+
+lemma imo_2022_p2_1_4
+  (f : ℝ → ℝ)
+  -- (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  (x z : ℝ)
+  -- (y : ℝ)
+  -- (hp : 0 < x ∧ 0 < y)
+  -- (h₁ : x * f y + y * f x ≤ 2)
+  -- (hc : x ≠ y)
+  -- (h₁₁ : (fun y => 0 < y ∧ x * f y + y * f x ≤ 2) z)
+  (h₁₂ : ∀ (y : ℝ), (fun y => 0 < y ∧ x * f y + y * f x ≤ 2) y → y = z)
+  -- (h₁₄ : y = z)
+  (hxz : ¬x = z) :
+  ¬(fun y => 0 < y ∧ x * f y + y * f x ≤ 2) x := by
+  exact mt (h₁₂ x) hxz
+
+
+lemma imo_2022_p2_1_5
+  (f : ℝ → ℝ)
+  -- (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  (x y : ℝ)
+  (hp : 0 < x ∧ 0 < y)
+  -- (h₁ : x * f y + y * f x ≤ 2)
+  -- (hc : x ≠ y)
+  -- (z : ℝ)
+  -- (h₁₁ : (fun y => 0 < y ∧ x * f y + y * f x ≤ 2) z)
+  -- (h₁₂ : ∀ (y : ℝ), (fun y => 0 < y ∧ x * f y + y * f x ≤ 2) y → y = z)
+  -- (h₁₄ : y = z)
+  -- (hxz : ¬x = z)
+  -- (h₁₆ : ¬(fun y => 0 < y ∧ x * f y + y * f x ≤ 2) x)
+  (h₁₇ : ¬(0 < x ∧ x * f x + x * f x ≤ 2)) :
+  x * f x + x * f x > 2 := by
+  push_neg at h₁₇
+  refine h₁₇ ?_
+  exact hp.1
+
+
+lemma imo_2022_p2_1_6
+  (f : ℝ → ℝ)
+  -- (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  (x y : ℝ)
+  (hp : 0 < x ∧ 0 < y)
+  -- (h₁ : x * f y + y * f x ≤ 2)
+  -- (hc : x ≠ y)
+  -- (z : ℝ)
+  -- (h₁₁ : (fun y => 0 < y ∧ x * f y + y * f x ≤ 2) z)
+  -- (h₁₂ : ∀ (y : ℝ), (fun y => 0 < y ∧ x * f y + y * f x ≤ 2) y → y = z)
+  -- (h₁₄ : y = z)
+  -- (hxz : ¬x = z)
+  -- (h₁₆ : ¬(fun y => 0 < y ∧ x * f y + y * f x ≤ 2) x)
+  (h₁₇ : 0 < x → 2 < x * f x + x * f x) :
+  x * f x + x * f x > 2 := by
+  refine h₁₇ ?_
+  exact hp.1
+
+
+lemma imo_2022_p2_1_7
+  (f : ℝ → ℝ)
+  -- (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  (x y : ℝ)
+  (hp : 0 < x ∧ 0 < y)
+  (h₁ : x * f y + y * f x ≤ 2)
+  (hc : x ≠ y)
+  (h₁₀ : 1 < x * f x)
+  (h₁₁ : 1 < y * f y) :
+  False := by
+  have h₁₅: 1 / x < f x := by exact (div_lt_iff₀' hp.1).mpr (h₁₀)
+  have h₁₆: 1 / y < f y := by exact (div_lt_iff₀' hp.2).mpr (h₁₁)
+  have h₁₂: x / y + y / x < 2 := by
+    refine lt_of_le_of_lt' h₁ ?_
+    refine add_lt_add ?_ ?_
+    . rw [← mul_one_div]
+      exact (mul_lt_mul_left hp.1).mpr h₁₆
+    . rw [← mul_one_div]
+      exact (mul_lt_mul_left hp.2).mpr h₁₅
+  have h₁₃: 2 < x / y + y / x := by
+    refine lt_of_mul_lt_mul_right ?_ (le_of_lt hp.1)
+    refine lt_of_mul_lt_mul_right ?_ (le_of_lt hp.2)
+    repeat rw [add_mul, mul_assoc]
+    -- rw [div_mul_mul_cancel x x y]
+    rw [mul_comm x y, ← mul_assoc (x/y)]
+    -- rw [mul_comm (x / y * y) x]
+    rw [div_mul_comm x y y, div_mul_comm y x x, div_self, div_self]
+    . ring_nf
+      refine lt_of_sub_pos ?_
+      rw [mul_comm _ 2, ← mul_assoc]
+      rw [← sub_sq']
+      refine sq_pos_of_ne_zero ?_
+      exact sub_ne_zero.mpr hc.symm
+    . exact ne_of_gt hp.1
+    . exact ne_of_gt hp.2
+  linarith
+
+
+lemma imo_2022_p2_1_8
+  (f : ℝ → ℝ)
+  -- (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  (x y : ℝ)
+  (hp : 0 < x ∧ 0 < y)
+  (h₁ : x * f y + y * f x ≤ 2)
+  -- (hc : x ≠ y)
+  -- (h₁₀ : 1 < x * f x)
+  -- (h₁₁ : 1 < y * f y)
+  (h₁₅ : 1 / x < f x)
+  (h₁₆ : 1 / y < f y) :
+  x / y + y / x < 2 := by
+  refine lt_of_le_of_lt' h₁ ?_
+  refine add_lt_add ?_ ?_
+  . rw [← mul_one_div]
+    exact (mul_lt_mul_left hp.1).mpr h₁₆
+  . rw [← mul_one_div]
+    exact (mul_lt_mul_left hp.2).mpr h₁₅
+
+
+lemma imo_2022_p2_1_9
+  (f : ℝ → ℝ)
+  -- (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  (x y : ℝ)
+  (hp : 0 < x ∧ 0 < y)
+  -- (h₁ : x * f y + y * f x ≤ 2)
+  -- (hc : x ≠ y)
+  -- (h₁₀ : 1 < x * f x)
+  -- (h₁₁ : 1 < y * f y)
+  -- (h₁₅ : 1 / x < f x)
+  (h₁₆ : 1 / y < f y) :
+  x / y < x * f y := by
+  rw [← mul_one_div]
+  exact (mul_lt_mul_left hp.1).mpr h₁₆
+
+
+lemma imo_2022_p2_1_10
+  -- (f : ℝ → ℝ)
+  -- hfp : ∀ (x : ℝ), 0 < x → 0 < f x
+  -- h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2
+  (x y : ℝ)
+  (hp : 0 < x ∧ 0 < y)
+  -- h₁ : x * f y + y * f x ≤ 2
+  (hc : x ≠ y) :
+  -- h₁₀ : 1 < x * f x
+  -- h₁₁ : 1 < y * f y
+  -- h₁₅ : 1 / x < f x
+  -- h₁₆ : 1 / y < f y
+  -- (h₁₂ : x / y + y / x < 2) :
+  2 < x / y + y / x := by
+  refine lt_of_mul_lt_mul_right ?_ (le_of_lt hp.1)
+  refine lt_of_mul_lt_mul_right ?_ (le_of_lt hp.2)
+  repeat rw [add_mul, mul_assoc]
+  -- rw [div_mul_mul_cancel x x y]
+  rw [mul_comm x y, ← mul_assoc (x/y)]
+  -- rw [mul_comm (x / y * y) x]
+  rw [div_mul_comm x y y, div_mul_comm y x x, div_self, div_self]
+  . ring_nf
+    refine lt_of_sub_pos ?_
+    rw [mul_comm _ 2, ← mul_assoc]
+    rw [← sub_sq']
+    refine sq_pos_of_ne_zero ?_
+    exact sub_ne_zero.mpr hc.symm
+  . exact ne_of_gt hp.1
+  . exact ne_of_gt hp.2
+
+
+lemma imo_2022_p2_1_11
+  -- (f : ℝ → ℝ)
+  (x y : ℝ)
+  (hp : 0 < x ∧ 0 < y)
+  (hc : x ≠ y) :
+  2 * x * y < (x / y + y / x) * x * y := by
+  repeat rw [add_mul, mul_assoc]
+  -- rw [div_mul_mul_cancel x x y]
+  rw [mul_comm x y, ← mul_assoc (x/y)]
+  -- rw [mul_comm (x / y * y) x]
+  rw [div_mul_comm x y y, div_mul_comm y x x, div_self, div_self]
+  . ring_nf
+    refine lt_of_sub_pos ?_
+    rw [mul_comm _ 2, ← mul_assoc]
+    rw [← sub_sq']
+    refine sq_pos_of_ne_zero ?_
+    exact sub_ne_zero.mpr hc.symm
+  . exact ne_of_gt hp.1
+  . exact ne_of_gt hp.2
+
+
+lemma imo_2022_p2_1_12
+  -- (f : ℝ → ℝ)
+  (x y : ℝ)
+  -- (hp : 0 < x ∧ 0 < y)
+  (hc : x ≠ y) :
+  y * x * 2 < y ^ 2 + x ^ 2 := by
+  refine lt_of_sub_pos ?_
+  rw [mul_comm _ 2, ← mul_assoc]
+  rw [← sub_sq']
+  refine sq_pos_of_ne_zero ?_
+  exact sub_ne_zero.mpr hc.symm
+
+
+
+lemma imo_2022_p2_2
+  (f : ℝ → ℝ)
+  -- (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y) :
+  ∀ (x : ℝ), 0 < x → x * f x ≤ 1 := by
+  intros x hxp
+  obtain ⟨y,h₂₁⟩ := h₀ x hxp
+  cases' h₂₁ with h₂₁ h₂₂
+  have hxy: x = y := by
+    have h₂₃: 0 < y ∧ x * f y + y * f x ≤ 2 := by exact h₂₁
+    apply h₁ x y
+    . constructor
+      . exact hxp
+      . exact h₂₃.1
+    . exact h₂₃.2
+  rw [← hxy] at h₂₁
+  linarith
+
+
+lemma imo_2022_p2_2_1
+  (f : ℝ → ℝ)
+  (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y)
+  (x : ℝ)
+  (hxp : 0 < x)
+  (y : ℝ)
+  (h₂ : (fun y => 0 < y ∧ x * f y + y * f x ≤ 2) y) :
+  x * f x ≤ 1 := by
+  have hxy: x = y := by
+    apply h₁ x y
+    . constructor
+      . exact hxp
+      . exact h₂.1
+    . exact h₂.2
+  rw [← hxy] at h₂
+  linarith
+
+
+lemma imo_2022_p2_2_2
+  (f : ℝ → ℝ)
+  (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y)
+  (x : ℝ)
+  (hxp : 0 < x)
+  (y : ℝ)
+  (h₂ : (fun y => 0 < y ∧ x * f y + y * f x ≤ 2) y) :
+  x = y := by
+  apply h₁ x y
+  . constructor
+    . exact hxp
+    . exact h₂.1
+  . exact h₂.2
+
+
+lemma imo_2022_p2_2_3
+  (f : ℝ → ℝ)
+  -- h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y
+  (x y : ℝ)
+  -- (hxp : 0 < x)
+  (h₂ : 0 < y ∧ x * f y + y * f x ≤ 2)
+  (hxy : x = y) :
+  x * f x ≤ 1 := by
+  rw [← hxy] at h₂
+  linarith
+
+
+
+lemma imo_2022_p2_3
+  (f : ℝ → ℝ)
+  (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y)
+  (h₂ : ∀ (x : ℝ), 0 < x → x * f x ≤ 1) :
+  ∀ (x : ℝ), 0 < x → ¬x * f x < 1 := by
+  by_contra! hc
+  let ⟨x,hxp⟩ := hc
+  cases' hxp with hxp h₃
+  let d₁:ℝ := 1 - x * f x
+  have hd₁ : x * f x = 1 - d₁ := by exact (sub_sub_self 1 (x * f x)).symm
+  let z:ℝ := x + d₁ / f x
+  have hz: z = x + d₁ / f x := by exact rfl
+  have hzp: 0 < z := by
+    refine add_pos hxp ?_
+    refine div_pos ?_ ?_
+    . exact sub_pos.mpr h₃
+    . exact hfp x hxp
+  have hxz: ¬ x = z := by
+    by_contra! hcz₀
+    rw [← hcz₀] at hz
+    have hcz₁: 0 < d₁ / f x := by
+      refine div_pos ?_ (hfp x hxp)
+      exact sub_pos.mpr h₃
+    linarith
+  have h₄: ¬ (x * f z + z * f x ≤ 2) := by
+    have h₄₁: x * f z + z * f x ≤ 2 → x = z := by
+      exact h₁ x z { left := hxp, right := hzp }
+    exact mt h₄₁ hxz
+  have h₅: x * f z < 1 := by
+    suffices h₅₁: z * f z ≤ 1 by
+      refine lt_of_lt_of_le ?_ h₅₁
+      refine (mul_lt_mul_right (hfp z hzp)).mpr ?_
+      rw [hz]
+      refine lt_add_of_pos_right x ?_
+      refine div_pos ?_ (hfp x hxp)
+      exact sub_pos.mpr h₃
+    exact h₂ z hzp
+  have h₆: x * f z + z * f x < 2 := by
+    suffices h₇: z * f x ≤ 1 by
+      linarith
+    rw [hz, add_mul, hd₁]
+    rw [div_mul_comm d₁ (f x) (f x)]
+    rw [div_self]
+    . rw [one_mul, sub_add_cancel]
+    . exact Ne.symm (ne_of_lt (hfp x hxp))
+  linarith
+
+
+lemma imo_2022_p2_3_1
+  (f : ℝ → ℝ)
+  (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y)
+  (h₂ : ∀ (x : ℝ), 0 < x → x * f x ≤ 1)
+  (hc : ∃ x, 0 < x ∧ x * f x < 1) :
+  -- (x : ℝ)
+  -- (hxp : 0 < x)
+  -- (h₃ : x * f x < 1) :
+  False := by
+  let ⟨x,hxp⟩ := hc
+  cases' hxp with hxp h₃
+  let d₁:ℝ := 1 - x * f x
+  have hd₁ : x * f x = 1 - d₁ := by exact (sub_sub_self 1 (x * f x)).symm
+  let z:ℝ := x + d₁ / f x
+  have hz: z = x + d₁ / f x := by exact rfl
+  have hzp: 0 < z := by
+    refine add_pos hxp ?_
+    refine div_pos ?_ ?_
+    . exact sub_pos.mpr h₃
+    . exact hfp x hxp
+  have hxz: ¬ x = z := by
+    by_contra! hcz₀
+    rw [← hcz₀] at hz
+    have hcz₁: 0 < d₁ / f x := by
+      refine div_pos ?_ (hfp x hxp)
+      exact sub_pos.mpr h₃
+    linarith
+  have h₄: ¬ (x * f z + z * f x ≤ 2) := by
+    have h₄₁: x * f z + z * f x ≤ 2 → x = z := by
+      exact h₁ x z { left := hxp, right := hzp }
+    exact mt h₄₁ hxz
+  have h₅: x * f z < 1 := by
+    suffices h₅₁: z * f z ≤ 1 by
+      refine lt_of_lt_of_le ?_ h₅₁
+      refine (mul_lt_mul_right (hfp z hzp)).mpr ?_
+      rw [hz]
+      refine lt_add_of_pos_right x ?_
+      refine div_pos ?_ (hfp x hxp)
+      exact sub_pos.mpr h₃
+    exact h₂ z hzp
+  have h₆: x * f z + z * f x < 2 := by
+    suffices h₇: z * f x ≤ 1 by
+      linarith
+    rw [hz, add_mul, hd₁]
+    rw [div_mul_comm d₁ (f x) (f x)]
+    rw [div_self]
+    . rw [one_mul, sub_add_cancel]
+    . exact Ne.symm (ne_of_lt (hfp x hxp))
+  linarith
+
+
+lemma imo_2022_p2_3_2
+  (f : ℝ → ℝ)
+  (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y)
+  (h₂ : ∀ (x : ℝ), 0 < x → x * f x ≤ 1)
+  -- (hc : ∃ x, 0 < x ∧ x * f x < 1)
+  (x z d₁: ℝ)
+  (hxp : 0 < x)
+  (h₃ : x * f x < 1)
+  (hd₀ : d₁ = 1 - x * f x)
+  (hd₁ : x * f x = 1 - d₁)
+  (hz : z = x + d₁ / f x) :
+  False := by
+  have hzp: 0 < z := by
+    rw [hz]
+    refine add_pos hxp ?_
+    refine div_pos ?_ ?_
+    . rw [hd₀]
+      exact sub_pos.mpr h₃
+    . exact hfp x hxp
+  have hxz: ¬ x = z := by
+    by_contra! hcz₀
+    rw [← hcz₀] at hz
+    have hcz₁: 0 < d₁ / f x := by
+      refine div_pos ?_ (hfp x hxp)
+      rw [hd₀]
+      exact sub_pos.mpr h₃
+    linarith
+  have h₄: ¬ (x * f z + z * f x ≤ 2) := by
+    have h₄₁: x * f z + z * f x ≤ 2 → x = z := by
+      exact h₁ x z { left := hxp, right := hzp }
+    exact mt h₄₁ hxz
+  have h₅: x * f z < 1 := by
+    suffices h₅₁: z * f z ≤ 1 by
+      refine lt_of_lt_of_le ?_ h₅₁
+      refine (mul_lt_mul_right (hfp z hzp)).mpr ?_
+      rw [hz]
+      refine lt_add_of_pos_right x ?_
+      refine div_pos ?_ (hfp x hxp)
+      rw [hd₀]
+      exact sub_pos.mpr h₃
+    exact h₂ z hzp
+  have h₆: x * f z + z * f x < 2 := by
+    suffices h₇: z * f x ≤ 1 by
+      linarith
+    rw [hz, add_mul, hd₁]
+    rw [div_mul_comm d₁ (f x) (f x)]
+    rw [div_self]
+    . rw [one_mul, sub_add_cancel]
+    . exact Ne.symm (ne_of_lt (hfp x hxp))
+  linarith
+
+
+lemma imo_2022_p2_3_3
+  (f : ℝ → ℝ)
+  (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y)
+  -- (h₂ : ∀ (x : ℝ), 0 < x → x * f x ≤ 1)
+  -- (hc : ∃ x, 0 < x ∧ x * f x < 1)
+  (x d₁ z : ℝ)
+  (hxp : 0 < x)
+  (h₃ : x * f x < 1)
+  (hd₀ : d₁ = 1 - x * f x)
+  -- (hd₁ : x * f x = 1 - d₁)
+  (hz : z = x + d₁ / f x) :
+  0 < z := by
+  rw [hz]
+  refine add_pos hxp ?_
+  refine div_pos ?_ ?_
+  . rw [hd₀]
+    exact sub_pos.mpr h₃
+  . exact hfp x hxp
+
+
+lemma imo_2022_p2_3_4
+  (f : ℝ → ℝ)
+  (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y)
+  -- (h₂ : ∀ (x : ℝ), 0 < x → x * f x ≤ 1)
+  -- (hc : ∃ x, 0 < x ∧ x * f x < 1)
+  (x d₁ : ℝ)
+  (hxp : 0 < x)
+  (h₃ : x * f x < 1)
+  (hd₀ : d₁ = 1 - x * f x) :
+  -- (hd₁ : x * f x = 1 - d₁)
+  -- (hz : z = x + d₁ / f x) :
+  0 < d₁ / f x := by
+  refine div_pos ?_ ?_
+  . rw [hd₀]
+    exact sub_pos.mpr h₃
+  . exact hfp x hxp
+
+
+lemma imo_2022_p2_3_5
+  (f : ℝ → ℝ)
+  (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y)
+  -- (h₂ : ∀ (x : ℝ), 0 < x → x * f x ≤ 1)
+  -- (hc : ∃ x, 0 < x ∧ x * f x < 1)
+  (x d₁ z: ℝ)
+  (hxp : 0 < x)
+  (h₃ : x * f x < 1)
+  (hd₀ : d₁ = 1 - x * f x)
+  -- (hd₁ : x * f x = 1 - d₁)
+  (hz : z = x + d₁ / f x)
+  (hzp : 0 < z) :
+  ¬x = z := by
+  by_contra! hcz₀
+  rw [← hcz₀] at hz
+  have hcz₁: 0 < d₁ / f x := by
+    refine div_pos ?_ (hfp x hxp)
+    rw [hd₀]
+    exact sub_pos.mpr h₃
+  linarith
+
+
+lemma imo_2022_p2_3_6
+  (f : ℝ → ℝ)
+  (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y)
+  -- (h₂ : ∀ (x : ℝ), 0 < x → x * f x ≤ 1)
+  -- (hc : ∃ x, 0 < x ∧ x * f x < 1)
+  (x d₁ : ℝ)
+  (hxp : 0 < x)
+  (h₃ : x * f x < 1)
+  (hd₀ : d₁ = 1 - x * f x)
+  -- (hd₁ : x * f x = 1 - d₁)
+  (hz : x = x + d₁ / f x) :
+  -- (hzp : 0 < z)
+  -- (hcz₀ : x = z) :
+  False := by
+  have hcz₁: 0 < d₁ / f x := by
+    refine div_pos ?_ (hfp x hxp)
+    rw [hd₀]
+    exact sub_pos.mpr h₃
+  linarith
+
+
+lemma imo_2022_p2_3_7
+  (f : ℝ → ℝ)
+  (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y)
+  (h₂ : ∀ (x : ℝ), 0 < x → x * f x ≤ 1)
+  -- (hc : ∃ x, 0 < x ∧ x * f x < 1)
+  (x z d₁ : ℝ)
+  (hxp : 0 < x)
+  (h₃ : x * f x < 1)
+  (hd₀ : d₁ = 1 - x * f x)
+  (hd₁ : x * f x = 1 - d₁)
+  (hz : z = x + d₁ / f x)
+  (hzp : 0 < z)
+  (hxz : ¬x = z) :
+  ¬x * f z + z * f x ≤ 2 := by
+  have h₄: ¬ (x * f z + z * f x ≤ 2) := by
+    have h₄₁: x * f z + z * f x ≤ 2 → x = z := by
+      exact h₁ x z { left := hxp, right := hzp }
+    exact mt h₄₁ hxz
+  have h₅: x * f z < 1 := by
+    suffices h₅₁: z * f z ≤ 1 by
+      refine lt_of_lt_of_le ?_ h₅₁
+      refine (mul_lt_mul_right (hfp z hzp)).mpr ?_
+      rw [hz]
+      refine lt_add_of_pos_right x ?_
+      refine div_pos ?_ (hfp x hxp)
+      rw [hd₀]
+      exact sub_pos.mpr h₃
+    exact h₂ z hzp
+  have h₆: x * f z + z * f x < 2 := by
+    suffices h₇: z * f x ≤ 1 by
+      linarith
+    rw [hz, add_mul, hd₁]
+    rw [div_mul_comm d₁ (f x) (f x)]
+    rw [div_self]
+    . rw [one_mul, sub_add_cancel]
+    . exact Ne.symm (ne_of_lt (hfp x hxp))
+  linarith
+
+
+lemma imo_2022_p2_3_8
+  (f : ℝ → ℝ)
+  (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y)
+  (h₂ : ∀ (x : ℝ), 0 < x → x * f x ≤ 1)
+  -- (hc : ∃ x, 0 < x ∧ x * f x < 1)
+  (x z d₁ : ℝ)
+  (hxp : 0 < x)
+  (h₃ : x * f x < 1)
+  (hd₀ : d₁ = 1 - x * f x)
+  (hd₁ : x * f x = 1 - d₁)
+  (hz : z = x + d₁ / f x)
+  (hzp : 0 < z)
+  -- (hxz : ¬x = z)
+  (h₄ : ¬x * f z + z * f x ≤ 2) :
+  x * f z < 1 := by
+  have h₅: x * f z < 1 := by
+    suffices h₅₁: z * f z ≤ 1 by
+      refine lt_of_lt_of_le ?_ h₅₁
+      refine (mul_lt_mul_right (hfp z hzp)).mpr ?_
+      rw [hz]
+      refine lt_add_of_pos_right x ?_
+      refine div_pos ?_ (hfp x hxp)
+      rw [hd₀]
+      exact sub_pos.mpr h₃
+    exact h₂ z hzp
+  have h₆: x * f z + z * f x < 2 := by
+    suffices h₇: z * f x ≤ 1 by
+      linarith
+    rw [hz, add_mul, hd₁]
+    rw [div_mul_comm d₁ (f x) (f x)]
+    rw [div_self]
+    . rw [one_mul, sub_add_cancel]
+    . exact Ne.symm (ne_of_lt (hfp x hxp))
+  linarith
+
+
+lemma imo_2022_p2_3_9
+  (f : ℝ → ℝ)
+  (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y)
+  -- (h₂ : ∀ (x : ℝ), 0 < x → x * f x ≤ 1)
+  -- (hc : ∃ x, 0 < x ∧ x * f x < 1)
+  (x z d₁ : ℝ)
+  (hxp : 0 < x)
+  (h₃ : x * f x < 1)
+  (hd₀ : d₁ = 1 - x * f x)
+  -- (hd₁ : x * f x = 1 - d₁)
+  (hz : z = x + d₁ / f x)
+  (hzp : 0 < z)
+  -- (hxz : ¬x = z)
+  -- (h₄ : ¬x * f z + z * f x ≤ 2)
+  (h₅₁ : z * f z ≤ 1) :
+  x * f z < 1 := by
+  refine lt_of_lt_of_le ?_ h₅₁
+  refine (mul_lt_mul_right (hfp z hzp)).mpr ?_
+  rw [hz]
+  refine lt_add_of_pos_right x ?_
+  refine div_pos ?_ (hfp x hxp)
+  rw [hd₀]
+  exact sub_pos.mpr h₃
+
+
+lemma imo_2022_p2_3_10
+  (f : ℝ → ℝ)
+  (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y)
+  -- (h₂ : ∀ (x : ℝ), 0 < x → x * f x ≤ 1)
+  -- (hc : ∃ x, 0 < x ∧ x * f x < 1)
+  (x z d₁ : ℝ)
+  (hxp : 0 < x)
+  -- (h₃ : x * f x < 1)
+  -- (hd₀ : d₁ = 1 - x * f x)
+  (hd₁ : x * f x = 1 - d₁)
+  (hz : z = x + d₁ / f x)
+  -- (hzp : 0 < z)
+  -- (hxz : ¬x = z)
+  (h₄ : ¬x * f z + z * f x ≤ 2)
+  (h₅ : x * f z < 1) :
+  x * f z + z * f x < 2 := by
+  have h₆: x * f z + z * f x < 2 := by
+    suffices h₇: z * f x ≤ 1 by
+      linarith
+    rw [hz, add_mul, hd₁]
+    rw [div_mul_comm d₁ (f x) (f x)]
+    rw [div_self]
+    . rw [one_mul, sub_add_cancel]
+    . exact Ne.symm (ne_of_lt (hfp x hxp))
+  linarith
+
+
+lemma imo_2022_p2_3_11
+  (f : ℝ → ℝ)
+  (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y)
+  -- (h₂ : ∀ (x : ℝ), 0 < x → x * f x ≤ 1)
+  -- (hc : ∃ x, 0 < x ∧ x * f x < 1)
+  (x z d₁ : ℝ)
+  (hxp : 0 < x)
+  -- (h₃ : x * f x < 1)
+  -- (hd₀ : d₁ = 1 - x * f x)
+  (hd₁ : x * f x = 1 - d₁)
+  (hz : z = x + d₁ / f x)
+  -- (hzp : 0 < z)
+  -- (hxz : ¬x = z)
+  (h₄ : ¬x * f z + z * f x ≤ 2)
+  (h₅ : x * f z < 1) :
+  z * f x ≤ 1 := by
+  suffices h₇: z * f x ≤ 1 by
+    linarith
+  rw [hz, add_mul, hd₁]
+  rw [div_mul_comm d₁ (f x) (f x)]
+  rw [div_self]
+  . rw [one_mul, sub_add_cancel]
+  . exact Ne.symm (ne_of_lt (hfp x hxp))
+
+
+lemma imo_2022_p2_3_12
+  (f : ℝ → ℝ)
+  (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y)
+  -- (h₂ : ∀ (x : ℝ), 0 < x → x * f x ≤ 1)
+  -- (hc : ∃ x, 0 < x ∧ x * f x < 1)
+  (x d₁ : ℝ)
+  (hxp : 0 < x) :
+  -- (h₃ : x * f x < 1)
+  -- (hd₀ : d₁ = 1 - x * f x)
+  -- (hd₁ : x * f x = 1 - d₁)
+  -- (hz : z = x + d₁ / f x)
+  -- (hzp : 0 < z)
+  -- (hxz : ¬x = z)
+  -- (h₄ : ¬x * f z + z * f x ≤ 2)
+  -- (h₅ : x * f z < 1) :
+  1 - d₁ + d₁ / f x * f x ≤ 1 := by
+  rw [div_mul_comm d₁ (f x) (f x)]
+  rw [div_self]
+  . rw [one_mul, sub_add_cancel]
+  . exact Ne.symm (ne_of_lt (hfp x hxp))
+
+
+lemma imo_2022_p2_3_13
+  (f : ℝ → ℝ)
+  (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y)
+  -- (h₂ : ∀ (x : ℝ), 0 < x → x * f x ≤ 1)
+  -- (hc : ∃ x, 0 < x ∧ x * f x < 1)
+  (x d₁ : ℝ)
+  (hxp : 0 < x) :
+  -- (h₃ : x * f x < 1) :
+  -- (hd₀ : d₁ = 1 - x * f x)
+  -- (hd₁ : x * f x = 1 - d₁)
+  -- (hz : z = x + d₁ / f x)
+  -- (hzp : 0 < z)
+  -- (hxz : ¬x = z)
+  -- (h₄ : ¬x * f z + z * f x ≤ 2)
+  -- (h₅ : x * f z < 1) :
+  1 - d₁ + f x / f x * d₁ ≤ 1 := by
+  rw [div_self]
+  . rw [one_mul, sub_add_cancel]
+  . exact Ne.symm (ne_of_lt (hfp x hxp))
+
+
+lemma imo_2022_p2_3_14
+  -- (f : ℝ → ℝ)
+  -- (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y)
+  -- (h₂ : ∀ (x : ℝ), 0 < x → x * f x ≤ 1)
+  -- (hc : ∃ x, 0 < x ∧ x * f x < 1)
+  (d₁ : ℝ) :
+  -- (hxp : 0 < x)
+  -- (h₃ : x * f x < 1)
+  -- (hd₀ : d₁ = 1 - x * f x)
+  -- (hd₁ : x * f x = 1 - d₁)
+  -- (hz : z = x + d₁ / f x)
+  -- (hzp : 0 < z)
+  -- (hxz : ¬x = z)
+  -- (h₄ : ¬x * f z + z * f x ≤ 2)
+  -- (h₅ : x * f z < 1) :
+  1 - d₁ + 1 * d₁ ≤ 1 := by
+  rw [one_mul]
+  refine le_of_eq ?_
+  exact sub_add_cancel 1 d₁
+
+
+lemma imo_2022_p2_3_15
+  (f : ℝ → ℝ)
+  (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y)
+  -- (h₂ : ∀ (x : ℝ), 0 < x → x * f x ≤ 1)
+  -- (hc : ∃ x, 0 < x ∧ x * f x < 1)
+  (x : ℝ)
+  (hxp : 0 < x) :
+  -- (h₃ : x * f x < 1)
+  -- (hd₀ : d₁ = 1 - x * f x)
+  -- (hd₁ : x * f x = 1 - d₁)
+  -- (hz : z = x + d₁ / f x)
+  -- (hzp : 0 < z)
+  -- (hxz : ¬x = z)
+  -- (h₄ : ¬x * f z + z * f x ≤ 2)
+  -- (h₅ : x * f z < 1) :
+  f x ≠ 0 := by
+  refine PartialHomeomorph.unitBallBall.proof_2 (f x) ?_
+  exact (hfp x hxp)
+
+
+lemma imo_2022_p2_4
+  (f : ℝ → ℝ)
+  -- (hfp : ∀ (x : ℝ), 0 < x → 0 < f x)
+  -- (h₀ : ∀ (x : ℝ), 0 < x → ∃! y, 0 < y ∧ x * f y + y * f x ≤ 2)
+  -- (h₁ : ∀ (x y : ℝ), 0 < x ∧ 0 < y → x * f y + y * f x ≤ 2 → x = y)
+  (h₂ : ∀ (x : ℝ), 0 < x → x * f x ≤ 1)
+  (h₃ : ∀ (x : ℝ), 0 < x → ¬x * f x < 1) :
+  ∀ (x : ℝ), 0 < x → f x = 1 / x := by
+  intros x hxp
+  have h₄: x * f x ≤ 1 := by exact h₂ x hxp
+  have h₅: ¬ x * f x < 1 := by exact h₃ x hxp
+  refine eq_div_of_mul_eq ?_ ?_
+  . exact ne_of_gt hxp
+  . push_neg at h₅
+    linarith
+
+
+lemma imo_2022_p2_4_1
+  (f : ℝ → ℝ)
+  (x : ℝ)
+  (hxp : 0 < x)
+  (h₄ : x * f x ≤ 1)
+  (h₅ : ¬x * f x < 1) :
+  f x = 1 / x := by
+  refine eq_div_of_mul_eq ?_ ?_
+  . exact ne_of_gt hxp
+  . push_neg at h₅
+    rw [mul_comm]
+    exact le_antisymm h₄ h₅

Lemmas/lake-manifest.json

@@ -0,0 +1,95 @@
+{"version": "1.1.0",
+ "packagesDir": ".lake/packages",
+ "packages":
+ [{"url": "https://github.com/leanprover-community/mathlib4",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover-community",
+   "rev": "d066138f11f7fdf68dcda20d1ed2d296e9d992d7",
+   "name": "mathlib",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "master",
+   "inherited": false,
+   "configFile": "lakefile.lean"},
+  {"url": "https://github.com/leanprover-community/plausible",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover-community",
+   "rev": "59a8514bb0ee5bae2689d8be717b5272c9b3dc1c",
+   "name": "plausible",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "main",
+   "inherited": true,
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/leanprover-community/LeanSearchClient",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover-community",
+   "rev": "0c169a0d55fef3763cfb3099eafd7b884ec7e41d",
+   "name": "LeanSearchClient",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "main",
+   "inherited": true,
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/leanprover-community/import-graph",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover-community",
+   "rev": "461b96f5527089718cb23d3f1fd2960a5d0ff516",
+   "name": "importGraph",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "main",
+   "inherited": true,
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/leanprover-community/ProofWidgets4",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover-community",
+   "rev": "8fff3f074da9237cd4e179fd6dd89be6c4022d41",
+   "name": "proofwidgets",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "v0.0.52-pre",
+   "inherited": true,
+   "configFile": "lakefile.lean"},
+  {"url": "https://github.com/leanprover-community/aesop",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover-community",
+   "rev": "ba9a63be53f16b3b6e4043641c6bad4883e650b4",
+   "name": "aesop",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "master",
+   "inherited": true,
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/leanprover-community/quote4",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover-community",
+   "rev": "7b3b0c8327b3c0214ac49ca6d6922edbb81ab8c9",
+   "name": "Qq",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "master",
+   "inherited": true,
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/leanprover-community/batteries",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover-community",
+   "rev": "b18855cb0f9a19bd4d7e21f3e5525272e377f431",
+   "name": "batteries",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "main",
+   "inherited": true,
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/leanprover/lean4-cli",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover",
+   "rev": "a2eb24a3dbf681f2b655f82ba5ee5b139d4a5abc",
+   "name": "Cli",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "main",
+   "inherited": true,
+   "configFile": "lakefile.toml"}],
+ "name": "imo_steps",
+ "lakeDir": ".lake"}

Lemmas/lakefile.toml

@@ -0,0 +1,16 @@
+name = "imo_steps"
+version = "0.1.0"
+keywords = ["math"]
+defaultTargets = ["ImoSteps"]
+
+[leanOptions]
+pp.unicode.fun = true # pretty-prints `fun a ↦ b`
+autoImplicit = false
+
+[[require]]
+name = "mathlib"
+scope = "leanprover-community"
+rev = "v4.17.0"
+
+[[lean_lib]]
+name = "ImoSteps"

Lemmas/lean-toolchain

@@ -0,0 +1 @@
+leanprover/lean4:v4.17.0

imo_proofs/ImoSteps.lean

@@ -0,0 +1,3 @@
+-- This module serves as the root of the `ImoSteps` library.
+-- Import modules here that should be built as part of the library.
+import ImoSteps.Basic

imo_proofs/imo_1959_p1.lean

@@ -0,0 +1,20 @@
+import Mathlib
+
+
+open Nat
+
+theorem imo_1959_p1
+  (n : ℕ)
+  (h₀ : 0 < n) :
+  Nat.gcd (21*n + 4) (14*n + 3) = 1 := by
+  have h₁: Nat.gcd (21*n + 4) (14*n + 3) = Nat.gcd (7*n + 1) (14*n + 3) := by
+    have g₀: (21 * n + 4) = (7*n + 1) + 1 * (14 * n + 3) := by linarith
+    rw [g₀]
+    exact gcd_add_mul_right_left (7 * n + 1) (14 * n + 3) 1
+  have h₂: Nat.gcd (7*n + 1) (14*n + 3) = Nat.gcd (7*n + 1) (1) := by
+    have g₁: 14 * n + 3 = (7 * n + 1) * 2 + 1 := by linarith
+    rw [g₁]
+    exact gcd_mul_left_add_right (7 * n + 1) 1 2
+  have h₃: Nat.gcd (7*n + 1) (1) = 1 := by
+    exact Nat.gcd_one_right (7 * n + 1)
+  linarith

imo_proofs/imo_1960_p2.lean

@@ -0,0 +1,40 @@
+import Mathlib
+
+open Real
+
+
+theorem imo_1960_p2
+  (x : ℝ)
+  (h₀ : 0 ≤ 1 + 2 * x)
+  (h₁ : (1 - Real.sqrt (1 + 2 * x))^2 ≠ 0)
+  (h₂ : (4 * x^2) / (1 - Real.sqrt (1 + 2*x))^2 < 2*x + 9) :
+  -(1 / 2) ≤ x ∧ x < 45 / 8 := by
+  apply And.intro
+  . linarith
+  . have h₃: 4 * x ^ 2 < (2 * x + 9) * (1 - sqrt (1 + 2 * x) ) ^ 2 := by
+      refine' (div_lt_iff₀ _).mp h₂
+      refine Ne.lt_of_le (id (Ne.symm h₁)) ?_
+      exact sq_nonneg (1 - sqrt (1 + 2 * x))
+    have h₄: (1 - sqrt (1 + 2 * x)) ^ 2 = (2 + 2 * x) - 2 * sqrt (1 + 2 * x) := by
+      ring_nf at *
+      rw [Real.sq_sqrt h₀]
+      ring_nf
+    have h₅: (2 * x + 9) ^ 2 * (sqrt (1 + 2 * x)) ^ 2 < (11 * x + 9) ^ 2 := by
+      rw [← mul_pow]
+      refine' pow_lt_pow_left₀ _  _ (by norm_num)
+      rw [h₄] at h₃
+      simp_all only [ne_eq, zero_lt_two]
+      . linarith
+      . refine' mul_nonneg _ _
+        linarith
+        exact sqrt_nonneg (1 + 2 * x)
+    have h₆: 8 * x^3 < 45 * x^2 := by
+      rw [Real.sq_sqrt h₀] at h₅
+      ring_nf at h₅
+      linarith
+    have h₇₁: 0 ≤ x^2 := by exact sq_nonneg x
+    have h₇: 8 * x < 45 := by
+      refine' lt_of_mul_lt_mul_right ?_ h₇₁
+      ring_nf at *
+      exact h₆
+    linarith

imo_proofs/imo_1962_p2.lean

@@ -0,0 +1,64 @@
+import Mathlib
+
+open Real
+set_option linter.unusedVariables.analyzeTactics true
+
+
+theorem imo_1962_p2
+  (x : ℝ)
+  (h₀ : 0 ≤ 3 - x)
+  (h₁ : 0 ≤ x + 1)
+  (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) :
+  -1 ≤ x ∧ x < 1 - Real.sqrt 31 / 8 := by
+  constructor
+  . exact neg_le_iff_add_nonneg.mpr h₁
+  have h₃: (2 *sqrt (3 - x) * sqrt (x + 1)) ^ 2 < (4 - 1 / 4) ^ 2 := by
+    refine' pow_lt_pow_left₀ _ _ (by norm_num)
+    . refine lt_tsub_iff_left.mpr ?_
+      refine lt_tsub_iff_right.mp ?_
+      suffices g₀: 4 - 2 * sqrt (3 - x) * sqrt (x + 1) = (sqrt (3 - x) - sqrt (x + 1)) ^ 2
+      . rw [g₀]
+        have g₁:  (1:ℝ) / 4 = (1/2)^2 := by norm_num
+        rw [g₁]
+        exact pow_lt_pow_left₀ h₂ (by norm_num) (by norm_num)
+      rw [sub_sq]
+      rw [sq_sqrt h₀, sq_sqrt h₁]
+      ring_nf
+    . refine' mul_nonneg _ _
+      . refine mul_nonneg (by norm_num) ?_
+        exact sqrt_nonneg (3 - x)
+      . exact sqrt_nonneg (x + 1)
+  have h₄: 4 * (x + 1) * (3 - x) < 225 / 16 := by
+    norm_num at h₃
+    suffices g₀: 4 * (x + 1) * (3 - x) = (2 * sqrt (3 - x) * sqrt (x + 1)) ^ 2
+    . exact Eq.trans_lt g₀ h₃
+    . rw [mul_pow, mul_pow, sq_sqrt h₀, sq_sqrt h₁]
+      norm_num
+      exact mul_right_comm 4 (x + 1) (3 - x)
+  have hx1: x < 1 := by
+    suffices g₀: x + 1 < 3 - x
+    . linarith
+    . rw [← sq_sqrt h₀, ← sq_sqrt h₁]
+      refine' pow_lt_pow_left₀ _ _ (by norm_num)
+      . linarith
+      exact sqrt_nonneg (x + 1)
+  have h₅: x < 1 - sqrt 31 / 8 ∨ 1 + sqrt 31 / 8 < x := by
+    ring_nf at h₄
+    have g₀: 0 < x * x + -2 * x + 33 / 64 := by linarith
+    let a:ℝ := sqrt 31 / 8
+    have g₁: x * x + -2 * x + 33 / 64 = (x - (1 + a)) * (x - (1 - a)) := by
+      simp
+      ring_nf
+      norm_num
+      linarith
+    rw [g₁] at g₀
+    by_cases g₂: (x - (1 - a)) < 0
+    . left
+      exact sub_neg.mp g₂
+    push_neg at g₂
+    right
+    have g₃: 0 < (x - (1 + a)) := by exact pos_of_mul_pos_left g₀ g₂
+    exact sub_pos.mp g₃
+  cases h₅ with
+  | inl h₅ => exact h₅
+  | inr h₅ => linarith

imo_proofs/imo_1963_p5.lean

@@ -0,0 +1,53 @@
+import Mathlib
+
+open Real
+set_option linter.unusedVariables.analyzeTactics true
+
+
+lemma sin_mul_cos
+  (x y : ℝ) :
+  Real.sin x * Real.cos y = (sin (x + y) + sin (x - y)) / 2 := by
+    rw [sin_add, sin_sub]
+    simp
+
+theorem imo_1963_p5 :
+  Real.cos (π / 7) - Real.cos (2 * π / 7) + Real.cos (3 * π / 7) = 1 / 2 := by
+  let S:ℝ := Real.cos (π / 7) - Real.cos (2 * π / 7) + Real.cos (3 * π / 7)
+  have h₀: Real.sin (π / 7) * (S * 2) = Real.sin (π / 7) := by
+    ring_nf
+    have h₀₀: sin (π * (1 / 7)) * cos (π * (1 / 7)) * 2 = sin (2 * (π * (1 / 7))) := by
+      rw [Real.sin_two_mul]
+      exact (mul_rotate 2 (sin (π * (1 / 7))) (cos (π * (1 / 7)))).symm
+    rw [h₀₀, sin_mul_cos, sin_mul_cos]
+    rw [← mul_add, ← mul_sub, ← mul_add, ← mul_sub]
+    norm_num
+    ring_nf
+    have h₀₁: -sin (π * (3 / 7)) + sin (π * (4 / 7)) = 0 := by
+      rw [add_comm]
+      refine add_neg_eq_of_eq_add ?_
+      simp
+      refine sin_eq_sin_iff.mpr ?_
+      use 0
+      right
+      ring
+    linarith
+  have h₁: S = 1 / 2 := by
+    refine eq_div_of_mul_eq (by norm_num) ?_
+    nth_rewrite 2 [← mul_one (sin (π / 7))] at h₀
+    refine (mul_right_inj' ?_).mp h₀
+    refine sin_ne_zero_iff.mpr ?_
+    intro n
+    ring_nf
+    rw [mul_comm]
+    simp
+    push_neg
+    constructor
+    . by_contra! hc₀
+      have hc₁: 7 * (↑n:ℝ) = 1 := by
+        rw [mul_comm]
+        exact (mul_eq_one_iff_eq_inv₀ (by norm_num)).mpr hc₀
+      norm_cast at hc₁
+      have g₀: 0 < n := by linarith
+      linarith
+    . exact pi_ne_zero
+  exact h₁

imo_proofs/imo_1964_p2.lean

@@ -0,0 +1,55 @@
+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

imo_proofs/imo_1965_p2.lean

@@ -0,0 +1,198 @@
+import Mathlib
+set_option linter.unusedVariables.analyzeTactics true
+
+
+theorem imo_1965_p2
+  (x y z : ℝ)
+  (a : ℕ → ℝ)
+  (h₀ : 0 < a 0 ∧ 0 < a 4 ∧ 0 < a 8)
+  (h₁ : a 1 < 0 ∧ a 2 < 0)
+  (h₂ : a 3 < 0 ∧ a 5 < 0)
+  (h₃ : a 6 < 0 ∧ a 7 < 0)
+  (h₄ : 0 < a 0 + a 1 + a 2)
+  (h₅ : 0 < a 3 + a 4 + a 5)
+  (h₆ : 0 < a 6 + a 7 + a 8)
+  (h₇ : a 0 * x + a 1 * y + a 2 * z = 0)
+  (h₈ : a 3 * x + a 4 * y + a 5 * z = 0)
+  (h₉ : a 6 * x + a 7 * y + a 8 * z = 0) :
+  x = 0 ∧ y = 0 ∧ z = 0 := by
+  by_cases hx0: x = 0
+  . rw [hx0] at h₇
+    constructor
+    . exact hx0
+    . rw [hx0] at h₈ h₉
+      simp at h₇ h₈ h₉
+      by_cases hy0: y = 0
+      . constructor
+        . exact hy0
+        . rw [hy0] at h₇
+          simp at h₇
+          . cases' h₇ with h₇₀ h₇₁
+            . exfalso
+              linarith
+            . exact h₇₁
+      . by_cases hyn: y < 0
+        . have g1: 0 < a 1 * y := by exact mul_pos_of_neg_of_neg h₁.1 hyn
+          have g2: a 1 * y = -a 2 * z := by linarith
+          rw [g2] at g1
+          have g3: a 2 *z < 0 := by linarith
+          have hzp: 0 < z := by exact pos_of_mul_neg_right g3 (le_of_lt h₁.2)
+          exfalso
+          have g4: a 4 * y < 0 := by exact mul_neg_of_pos_of_neg h₀.2.1 hyn
+          have g5: a 5 * z < 0 := by exact mul_neg_of_neg_of_pos h₂.2 hzp
+          linarith
+        . push_neg at hy0 hyn
+          have hyp: 0 < y := by exact lt_of_le_of_ne hyn hy0.symm
+          exfalso
+          have g1: a 1 * y < 0 := by exact mul_neg_of_neg_of_pos h₁.1 hyp
+          have g2: 0 < z * a 2 := by linarith
+          have hzp: z < 0 := by exact neg_of_mul_pos_left g2 (le_of_lt h₁.2)
+          have g3: 0 < a 4 * y := by exact mul_pos h₀.2.1 hyp
+          have g4: 0 < a 5 * z := by exact mul_pos_of_neg_of_neg h₂.2 hzp
+          linarith
+  . exfalso
+    push_neg at hx0
+    by_cases hxp: 0 < x
+    . by_cases hy0: y = 0
+      . rw [hy0] at h₇ h₈ h₉
+        simp at h₇ h₈ h₉
+        have g1: 0 < a 0 * x := by exact mul_pos h₀.1 hxp
+        have g2: a 2 * z < 0 := by linarith
+        have hzn: 0 < z := by exact pos_of_mul_neg_right g2 (le_of_lt h₁.2)
+        have g3: a 3 * x < 0 := by exact mul_neg_of_neg_of_pos h₂.1 hxp
+        have g4: a 5 * z < 0 := by exact mul_neg_of_neg_of_pos h₂.2 hzn
+        linarith
+      . push_neg at hy0
+        by_cases hyp: 0 < y
+        . have g1: a 6 * x < 0 := by exact mul_neg_of_neg_of_pos h₃.1 hxp
+          have g2: a 7 * y < 0 := by exact mul_neg_of_neg_of_pos h₃.2 hyp
+          have g3: 0 < z * a 8 := by linarith
+          have hzp: 0 < z := by exact pos_of_mul_pos_left g3 (le_of_lt h₀.2.2)
+          ------ here we consider all the possible relationships between x, y, z
+          by_cases rxy: x ≤ y
+          . by_cases ryz: y ≤ z
+              -- x <= y <= z
+            . have g2: 0 < (a 6 + a 7 + a 8) * y := by exact mul_pos h₆ hyp
+              have g3: 0 ≤ a 6 * (x-y) := by
+                exact mul_nonneg_of_nonpos_of_nonpos (le_of_lt h₃.1) (by linarith)-- exact mul_nonneg (le_of_lt h₃.1) (by linarith),},
+              have g4: 0 ≤ a 8 * (z-y) := by exact mul_nonneg (le_of_lt h₀.2.2) (by linarith)
+              linarith
+            push_neg at ryz
+            by_cases rxz: x ≤ z
+              -- x <= z < y
+            . have g2: 0 < (a 3 + a 4 + a 5) * y := by exact mul_pos h₅ hyp
+              have g3: 0 ≤ a 3 * (x-y) := by
+                exact mul_nonneg_of_nonpos_of_nonpos (le_of_lt h₂.1) (by linarith)
+              have g4: 0 < a 5 * (z-y) := by
+                exact mul_pos_of_neg_of_neg h₂.2 (by linarith)
+              linarith
+            push_neg at rxz -- z < x <= y
+            have g2: 0 < (a 3 + a 4 + a 5) * y := by exact mul_pos h₅ hyp
+            have g3: 0 ≤ a 3 * (x-y) := by
+              exact mul_nonneg_of_nonpos_of_nonpos (le_of_lt h₂.1) (by linarith)
+            have g4: 0 < a 5 * (z-y) := by
+              exact mul_pos_of_neg_of_neg h₂.2 (by linarith)
+            linarith
+          push_neg at rxy
+          by_cases rzy: z ≤ y
+            -- z <= y < x
+          . have g2: 0 < (a 0 + a 1 + a 2) * y := by exact mul_pos h₄ hyp
+            have g3: 0 < a 0 * (x-y) := by exact mul_pos h₀.1 (by linarith)
+            have g4: 0 ≤ a 2 * (z-y) := by
+              exact mul_nonneg_of_nonpos_of_nonpos (le_of_lt h₁.2) (by linarith)
+            linarith
+          . push_neg at rzy
+            by_cases rzx: z ≤ x
+              -- y < z <= x
+            . have g2: 0 < (a 0 + a 1 + a 2) * z := by exact mul_pos h₄ hzp
+              have g3: 0 ≤ a 0 * (x-z) := by exact mul_nonneg (le_of_lt h₀.1) (by linarith)
+              have g4: 0 < a 1 * (y-z) := by exact mul_pos_of_neg_of_neg h₁.1 (by linarith)
+              linarith
+            . push_neg at rzx
+                -- y < x < z
+              have g2: 0 < (a 6 + a 7 + a 8) * z := by exact mul_pos h₆ hzp
+              have g3: 0 < a 6 * (x-z) := by exact mul_pos_of_neg_of_neg h₃.1 (by linarith)
+              have g4: 0 < a 7 * (y-z) := by exact mul_pos_of_neg_of_neg h₃.2 (by linarith)
+              linarith
+        -------- new world where y < 0 and 0 < x
+        . push_neg at hyp
+          have hyn: y < 0 :=  by exact lt_of_le_of_ne hyp hy0
+          -- show from a 0 that 0 < z
+          have g1: 0 < a 0 * x := by exact mul_pos h₀.1 hxp
+          have g2: 0 < a 1 * y := by exact mul_pos_of_neg_of_neg h₁.1 hyn
+          have g3: a 2 * z < 0 := by linarith
+          have hzp: 0 < z := by exact pos_of_mul_neg_right g3 (le_of_lt h₁.2)
+          -- then show from a 3 that's not possible
+          have g4: a 3 * x < 0 := by exact mul_neg_of_neg_of_pos h₂.1 hxp
+          have g5: a 4 * y < 0 := by exact mul_neg_of_pos_of_neg h₀.2.1 hyn
+          have g6: a 5 * z < 0 := by exact mul_neg_of_neg_of_pos h₂.2 hzp
+          linarith
+    . push_neg at hxp
+      have hxn: x < 0 := by exact lt_of_le_of_ne hxp hx0
+      by_cases hyp: 0 ≤ y
+      . have g1: a 0 * x < 0 := by exact mul_neg_of_pos_of_neg h₀.1 hxn
+        have g2: a 1 * y ≤ 0 := by
+          refine mul_nonpos_iff.mpr ?_
+          right
+          constructor
+          . exact le_of_lt h₁.1
+          . exact hyp
+        have g3: 0 < z * a 2 := by linarith
+        have hzn: z < 0 := by exact neg_of_mul_pos_left g3 (le_of_lt h₁.2)
+        -- demonstrate the contradiction
+        have g4: 0 < a 3 * x := by exact mul_pos_of_neg_of_neg h₂.1 hxn
+        have g5: 0 ≤ a 4 * y := by exact mul_nonneg (le_of_lt h₀.2.1) hyp
+        have g6: 0 < a 5 * z := by exact mul_pos_of_neg_of_neg h₂.2 hzn
+        linarith
+      . push_neg at hyp
+        -- have hyn: y < 0, {exact lt_of_le_of_ne hyp hy0,},
+        have g1: 0 < a 6 * x := by exact mul_pos_of_neg_of_neg h₃.1 hxn
+        have g2: 0 < a 7 * y := by exact mul_pos_of_neg_of_neg h₃.2 hyp
+        have g3: z * a 8 < 0 := by linarith
+        have hzp: z < 0 := by exact neg_of_mul_neg_left g3 (le_of_lt h₀.2.2)
+          -- we have x,y,z < 0 -- we will examine all the orders they can have
+        by_cases rxy: x ≤ y
+        . by_cases ryz: y ≤ z
+            -- x <= y <= z
+          . have g2: (a 0 + a 1 + a 2) * y < 0 := by exact mul_neg_of_pos_of_neg h₄ hyp
+            have g3: a 0 * (x-y) ≤ 0 := by
+              exact mul_nonpos_of_nonneg_of_nonpos (le_of_lt h₀.1) (by linarith)
+            have g4: a 2 * (z-y) ≤ 0 := by
+              exact mul_nonpos_of_nonpos_of_nonneg (le_of_lt h₁.2) (by linarith)
+            linarith
+          . push_neg at ryz
+            by_cases rxz: x ≤ z
+              -- x <= z < y
+            . have g2: (a 0 + a 1 + a 2) * z < 0 := by exact mul_neg_of_pos_of_neg h₄ hzp
+              have g3: a 0 * (x-z) ≤ 0 := by
+                exact mul_nonpos_of_nonneg_of_nonpos (le_of_lt h₀.1) (by linarith)
+              have g4: a 1 * (y-z) < 0 := by
+                exact mul_neg_of_neg_of_pos h₁.1 (by linarith)
+              linarith
+            . push_neg at rxz -- z < x <= y
+              have g2: (a 6 + a 7 + a 8) * z < 0 := by exact mul_neg_of_pos_of_neg h₆ hzp
+              have g3: a 6 * (x-z) < 0 := by exact mul_neg_of_neg_of_pos h₃.1 (by linarith)
+              have g4: a 7 * (y-z) < 0 := by exact mul_neg_of_neg_of_pos h₃.2 (by linarith)
+              linarith
+        . push_neg at rxy
+          by_cases rzy: z ≤ y
+            -- z <= y < x
+          . have g2: (a 6 + a 7 + a 8) * y < 0 := by exact mul_neg_of_pos_of_neg h₆ hyp
+            have g3: a 6 * (x-y) < 0 := by exact mul_neg_of_neg_of_pos h₃.1 (by linarith)
+            have g4: a 8 * (z-y) ≤ 0 := by
+              exact mul_nonpos_of_nonneg_of_nonpos (le_of_lt h₀.2.2) (by linarith)
+            linarith
+          . push_neg at rzy
+            by_cases rzx: z ≤ x
+              -- y < z <= x
+            . have g2: (a 3 + a 4 + a 5) * z < 0 := by exact mul_neg_of_pos_of_neg h₅ hzp
+              have g3: a 3 * (x-z) ≤ 0 := by
+                exact mul_nonpos_of_nonpos_of_nonneg (le_of_lt h₂.1) (by linarith)
+              have g4: a 4 * (y-z) < 0 := by exact mul_neg_of_pos_of_neg h₀.2.1 (by linarith)
+              linarith
+            . push_neg at rzx
+              -- y < x < z
+              have g2: (a 3 + a 4 + a 5) * y < 0 := by exact mul_neg_of_pos_of_neg h₅ hyp
+              have g3: a 3 * (x-y) < 0 := by exact mul_neg_of_neg_of_pos h₂.1 (by linarith)
+              have g4: a 5 * (z-y) < 0 := by exact mul_neg_of_neg_of_pos h₂.2 (by linarith)
+              linarith

imo_proofs/imo_1968_p5_1.lean

@@ -0,0 +1,37 @@
+import Mathlib
+set_option linter.unusedVariables.analyzeTactics true
+
+open Real
+
+
+theorem imo_1968_p5_1
+  (a : ℝ)
+  (f : ℝ → ℝ)
+  (h₀ : 0 < a)
+  (h₁ : ∀ x, f (x + a) = 1 / 2 + Real.sqrt (f x - (f x)^2))
+  (h₂ : ∀ x, 1 / 2 ≤ f x ∧ f x ≤ 1) :
+  ∃ b > 0, ∀ x, f (x + b) = f x := by
+  use (2 * a)
+  constructor
+  . refine mul_pos (by norm_num) h₀
+  . intro x
+    have h₃: f (x + a) = 1 / 2 + Real.sqrt (f x - (f x)^2) := by
+      exact h₁ x
+    have h₄: f (x + 2 * a) = 1 / 2 + Real.sqrt (f (x + a) - (f (x + a)^2)) := by
+      rw [two_mul, ← add_assoc]
+      exact h₁ (x + a)
+    have h₅: f (x + a) - (f (x + a) ^ 2) = (f x - 1 / 2) ^ 2 := by
+      have h₅₁: 0 ≤ f x - (f x)^2 := by
+        refine sub_nonneg_of_le ?_
+        rw [pow_two]
+        nth_rw 3 [← mul_one (f x)]
+        refine (mul_le_mul_left ?_).mpr ?_
+        . linarith [h₂ x]
+        . exact (h₂ x).2
+      rw [h₃, add_sq, sub_sq, sq_sqrt h₅₁]
+      ring_nf
+    rw [h₅, sqrt_sq ?_] at h₄
+    . linarith
+    . have h₆: 1 / 2 ≤ f x := by
+        exact (h₂ x).1
+      linarith [h₆]

imo_proofs/imo_1969_p2.lean

@@ -0,0 +1,157 @@
+import Mathlib
+set_option linter.unusedVariables.analyzeTactics true
+
+open Real BigOperators
+
+theorem imo_1969_p2
+  (m n : ℝ)
+  (k : ℕ)
+  (a : ℕ → ℝ)
+  (f : ℝ → ℝ)
+  -- (h₀ : 0 < k)
+  -- (h₁ : ∀ x, f x = ∑ i in Finset.range k, ((Real.cos (a i + x)) / (2^i)))
+  (h₁ : ∀ x, f x = Finset.sum (Finset.range k) fun i => ((Real.cos (a i + x)) / (2^i)))
+  (h₂ : f m = 0)
+  (h₃ : f n = 0)
+  (h₄: Finset.sum (Finset.range k) (fun i => (((cos (a i)) / (2 ^ i)))) ≠ 0) :
+  ∃ t : ℤ, m - n = t * π := by
+  let Ccos := Finset.sum (Finset.range k) (fun i => (((cos (a i)) / (2 ^ i))))
+  let Csin := Finset.sum (Finset.range k) (fun i => (((sin (a i)) / (2 ^ i))))
+  have hCcos: Ccos = Finset.sum (Finset.range k) (fun i => (((cos (a i)) / (2 ^ i)))) := by
+    exact rfl
+  have hCsin: Csin = Finset.sum (Finset.range k) (fun i => (((sin (a i)) / (2 ^ i)))) := by
+    exact rfl
+  have h₅: ∀ x, f x = Ccos * cos x - Csin * sin x := by
+    intro x
+    rw [h₁ x]
+    have h₅₀: ∑ i ∈ Finset.range k, (cos (a i + x) / 2 ^ i)
+              = ∑ i ∈ Finset.range k, (((cos (a i) * cos (x) - sin (a i) * sin (x)) / (2^i))) := by
+      refine Finset.sum_congr (by rfl) ?_
+      simp
+      intros i _
+      refine (div_eq_div_iff ?_ ?_).mpr ?_
+      . exact Ne.symm (NeZero.ne' (2 ^ i))
+      . exact Ne.symm (NeZero.ne' (2 ^ i))
+      . refine mul_eq_mul_right_iff.mpr ?_
+        simp
+        exact cos_add (a i) x
+    rw [h₅₀]
+    ring_nf
+    rw [Finset.sum_sub_distrib]
+    have h₅₂: ∑ i ∈ Finset.range k, cos (a i) * cos x * (1 / 2) ^ i
+            = ∑ i ∈ Finset.range k, (cos (a i) * (1 / 2) ^ i) * cos x := by
+      refine Finset.sum_congr (by rfl) ?_
+      simp
+      intro i _
+      linarith
+    have h₅₃: ∑ x_1 ∈ Finset.range k, sin (a x_1) * sin x * (1 / 2) ^ x_1
+            = ∑ x_1 ∈ Finset.range k, ((sin (a x_1) * (1 / 2) ^ x_1) * sin x) := by
+      refine Finset.sum_congr (by rfl) ?_
+      simp
+      intro i _
+      linarith
+    rw [h₅₂, ← Finset.sum_mul _ _ (cos x)]
+    rw [h₅₃, ← Finset.sum_mul _ _ (sin x)]
+    ring_nf at hCcos
+    ring_nf at hCsin
+    rw [hCcos, hCsin]
+  have h₆: (∃ x, (f x = 0 ∧ cos x = 0)) → ∀ y, f y = Ccos * cos y := by
+    intro g₀
+    obtain ⟨x, hx₀, hx₁⟩ := g₀
+    have g₁: Finset.sum (Finset.range k) (fun i => (((sin (a i)) / (2 ^ i)))) = 0 := by
+      rw [h₅ x, hx₁] at hx₀
+      simp at hx₀
+      cases' hx₀ with hx₂ hx₃
+      . exact hx₂
+      . exfalso
+        apply sin_eq_zero_iff_cos_eq.mp at hx₃
+        cases' hx₃ with hx₃ hx₄
+        . linarith
+        . linarith
+    intro y
+    rw [h₅ y]
+    have g₂: Csin = 0 := by
+      linarith
+    rw [g₂, zero_mul]
+    exact sub_zero (Ccos * cos y)
+  by_cases hmn: (cos m = 0) ∨ (cos n = 0)
+  . have h₇: ∀ (x : ℝ), f x = Ccos * cos x := by
+      refine h₆ ?_
+      cases' hmn with hm hn
+      . use m
+      . use n
+    have h₈: ∀ x, f x = 0 → cos x = 0 := by
+      intros x hx₀
+      rw [h₇ x] at hx₀
+      refine eq_zero_of_ne_zero_of_mul_left_eq_zero ?_ hx₀
+      exact h₄
+    have hm₀: ∃ t:ℤ , m = (2 * ↑ t + 1) * π / 2 := by
+      refine cos_eq_zero_iff.mp ?_
+      exact h₈ m h₂
+    have hn₀: ∃ t:ℤ , n = (2 * ↑ t + 1) * π / 2 := by
+      refine cos_eq_zero_iff.mp ?_
+      exact h₈ n h₃
+    obtain ⟨tm, hm₁⟩ := hm₀
+    obtain ⟨tn, hn₁⟩ := hn₀
+    rw [hm₁, hn₁]
+    use (tm - tn)
+    rw [Int.cast_sub]
+    ring_nf
+  . push_neg at hmn
+    have h₇: tan m = tan n := by
+      have h₇₀: ∀ (x:ℝ), (f x = 0 ∧ cos x ≠ 0) → tan x = Ccos / Csin := by
+        intro x hx₀
+        rw [tan_eq_sin_div_cos]
+        symm
+        refine (div_eq_div_iff ?_ ?_).mp ?_
+        . simp
+          exact hx₀.2
+        . simp
+          have hx₁: Ccos * cos x ≠ 0 := by
+            refine mul_ne_zero ?_ hx₀.2
+            exact h₄
+          have hx₂: Ccos * cos x = Csin * sin x := by
+            rw [h₅ x] at hx₀
+            refine eq_of_sub_eq_zero ?_
+            exact hx₀.1
+          have hx₃: Csin * sin x ≠ 0 := by
+            rw [← hx₂]
+            exact hx₁
+          exact left_ne_zero_of_mul hx₃
+        . simp
+          symm
+          refine eq_of_sub_eq_zero ?_
+          rw [h₅ x] at hx₀
+          linarith
+      have h₇₁: tan m = Ccos / Csin := by
+        refine h₇₀ m ?_
+        constructor
+        . exact h₂
+        . exact hmn.1
+      have h₇₂: tan n = Ccos / Csin := by
+        refine h₇₀ n ?_
+        constructor
+        . exact h₃
+        . exact hmn.2
+      rw [h₇₁, h₇₂]
+    have h₈: sin (m - n) = 0 := by
+      have h₈₀: tan m - tan n = 0 := by exact sub_eq_zero_of_eq h₇
+      have h₈₁: (sin m * cos n - cos m * sin n) / (cos m * cos n) = 0 := by
+        rw [← div_sub_div (sin m) (sin n) hmn.1 hmn.2]
+        repeat rw [← tan_eq_sin_div_cos]
+        exact h₈₀
+      have h₈₂: sin (m - n) / (cos m * cos n) = 0 := by
+        rw [sin_sub]
+        exact h₈₁
+      apply div_eq_zero_iff.mp at h₈₂
+      cases' h₈₂ with h₈₂ h₈₃
+      . exact h₈₂
+      . exfalso
+        simp at h₈₃
+        cases' h₈₃ with h₈₄ h₈₅
+        . exact hmn.1 h₈₄
+        . exact hmn.2 h₈₅
+    apply sin_eq_zero_iff.mp at h₈
+    let ⟨t, ht⟩ := h₈
+    use t
+    exact ht.symm

imo_proofs/imo_1974_p3.lean

@@ -0,0 +1,514 @@
+import Mathlib
+set_option linter.unusedVariables.analyzeTactics true
+
+open Nat BigOperators Finset
+
+
+lemma aux_1
+  (a : ℕ) :
+  ¬ a ^ 2 ≡ 2 [MOD 5] := by
+  intro ha₀
+  induction' a with n hn
+  . simp at ha₀
+    have ha₁: ¬ 0 ≡ 2 [MOD 5] := by decide
+    exact ha₁ ha₀
+  . let b:ℕ := n % 5
+    have hb₀: b < 5 := by omega
+    have hb₁: n ≡ b [MOD 5] := by exact Nat.ModEq.symm (Nat.mod_modEq n 5)
+    have hb₂: (n + 1) ≡ (b + 1) [MOD 5] := by
+      exact Nat.ModEq.add_right 1 hb₁
+    have hb₃: (n + 1) ^ 2 ≡ (b + 1) ^ 2 [MOD 5] := by
+      exact Nat.ModEq.pow 2 hb₂
+    interval_cases b
+    . simp at *
+      have g₀: 1 ≡ 2 [MOD 5] := by
+        refine Nat.ModEq.trans hb₃.symm ha₀
+      have g₁: ¬ 1 ≡ 2 [MOD 5] := by decide
+      exact g₁ g₀
+    . simp at hb₃
+      have g₀: 4 ≡ 2 [MOD 5] := by
+        refine Nat.ModEq.trans hb₃.symm ha₀
+      have g₁: ¬ 4 ≡ 2 [MOD 5] := by decide
+      exact g₁ g₀
+    . simp at hb₃
+      have g₀: 9 ≡ 2 [MOD 5] := by
+        refine Nat.ModEq.trans hb₃.symm ha₀
+      have g₁: ¬ 9 ≡ 2 [MOD 5] := by decide
+      exact g₁ g₀
+    . simp at hb₃
+      have g₀: 16 ≡ 2 [MOD 5] := by
+        refine Nat.ModEq.trans hb₃.symm ha₀
+      have g₁: ¬ 16 ≡ 2 [MOD 5] := by decide
+      exact g₁ g₀
+    . simp at hb₃
+      have g₀: 25 ≡ 2 [MOD 5] := by
+        refine Nat.ModEq.trans hb₃.symm ha₀
+      have g₁: ¬ 25 ≡ 2 [MOD 5] := by decide
+      exact g₁ g₀
+
+
+lemma aux_2
+  (a : ℕ) :
+  ¬ a ^ 2 ≡ 3 [MOD 5] := by
+  intro ha₀
+  induction' a with n hn
+  . simp at ha₀
+    have ha₁: ¬ 0 ≡ 3 [MOD 5] := by decide
+    exact ha₁ ha₀
+  . let b:ℕ := n % 5
+    have hb₀: b < 5 := by omega
+    have hb₁: n ≡ b [MOD 5] := by exact Nat.ModEq.symm (Nat.mod_modEq n 5)
+    have hb₂: (n + 1) ≡ (b + 1) [MOD 5] := by
+      exact Nat.ModEq.add_right 1 hb₁
+    have hb₃: (n + 1) ^ 2 ≡ (b + 1) ^ 2 [MOD 5] := by
+      exact Nat.ModEq.pow 2 hb₂
+    interval_cases b
+    . simp at *
+      have g₀: 1 ≡ 3 [MOD 5] := by
+        refine Nat.ModEq.trans hb₃.symm ha₀
+      have g₁: ¬ 1 ≡ 3 [MOD 5] := by decide
+      exact g₁ g₀
+    . simp at hb₃
+      have g₀: 4 ≡ 3 [MOD 5] := by
+        refine Nat.ModEq.trans hb₃.symm ha₀
+      have g₁: ¬ 4 ≡ 3 [MOD 5] := by decide
+      exact g₁ g₀
+    . simp at hb₃
+      have g₀: 9 ≡ 3 [MOD 5] := by
+        refine Nat.ModEq.trans hb₃.symm ha₀
+      have g₁: ¬ 9 ≡ 3 [MOD 5] := by decide
+      exact g₁ g₀
+    . simp at hb₃
+      have g₀: 16 ≡ 3 [MOD 5] := by
+        refine Nat.ModEq.trans hb₃.symm ha₀
+      have g₁: ¬ 16 ≡ 3 [MOD 5] := by decide
+      exact g₁ g₀
+    . simp at hb₃
+      have g₀: 25 ≡ 3 [MOD 5] := by
+        refine Nat.ModEq.trans hb₃.symm ha₀
+      have g₁: ¬ 25 ≡ 3 [MOD 5] := by decide
+      exact g₁ g₀
+
+
+lemma aux_3
+  (n : ℕ) :
+  7 ^ (2 * n + 1) ≡ 2 [MOD 5] ∨ 7 ^ (2 * n + 1) ≡ 3 [MOD 5] := by
+  induction' n with d hd
+  . simp
+    left
+    decide
+  . let b:ℕ := (7 ^ (2 * d + 1)) % 5
+    have hb: b = (7 ^ (2 * d + 1)) % 5 := by rfl
+    have hb₀: b < 5 := by
+      rw [hb]
+      omega
+    have hb₁: (7 ^ (2 * d + 1)) ≡ b [MOD 5] := by
+      exact ModEq.symm (mod_modEq (7 ^ (2 * d + 1)) 5)
+    ring_nf at *
+    have hb₂: 7 ^ (d * 2) * 7 * 49 ≡ b * 49 [MOD 5] := by
+      exact ModEq.mul hb₁ rfl
+    have hb₃: 7 ^ (d * 2) * 7 * 49 ≡ 2 * 49 [MOD 5] ∨ 7 ^ (d * 2) * 7 * 49 ≡ 3 * 49 [MOD 5] := by
+      cases' hd with hd₀ hd₁
+      . left
+        exact ModEq.mul hd₀ rfl
+      . right
+        exact ModEq.mul hd₁ rfl
+    ring_nf at hb₂
+    ring_nf at *
+    cases' hb₃ with hb₄ hb₅
+    . interval_cases b
+      . ring_nf at hb₂
+        have g₀: 0 ≡ 98 [MOD 5] := by
+          refine Nat.ModEq.trans hb₂.symm hb₄
+        have g₁: ¬ 0 ≡ 98 [MOD 5] := by decide
+        exact (g₁ g₀).elim
+      . ring_nf at hb₂
+        have g₀: 49 ≡ 98 [MOD 5] := by
+          refine Nat.ModEq.trans hb₂.symm hb₄
+        have g₁: ¬ 49 ≡ 98 [MOD 5] := by decide
+        exact (g₁ g₀).elim
+      . ring_nf at hb₂
+        have g₀: 98 ≡ 3 [MOD 5] := by decide
+        right
+        refine Nat.ModEq.trans hb₂ g₀
+      . ring_nf at hb₂
+        have g₀: 147 ≡ 98 [MOD 5] := by
+          refine Nat.ModEq.trans hb₂.symm hb₄
+        have g₁: ¬ 147 ≡ 98 [MOD 5] := by decide
+        exact (g₁ g₀).elim
+      . ring_nf at hb₂
+        have g₀: 196 ≡ 98 [MOD 5] := by
+          refine Nat.ModEq.trans hb₂.symm hb₄
+        have g₁: ¬ 196 ≡ 98 [MOD 5] := by decide
+        exact (g₁ g₀).elim
+    . interval_cases b
+      . ring_nf at hb₂
+        have g₀: 0 ≡ 147 [MOD 5] := by
+          refine Nat.ModEq.trans hb₂.symm hb₅
+        have g₁: ¬ 0 ≡ 147 [MOD 5] := by decide
+        exact (g₁ g₀).elim
+      . ring_nf at hb₂
+        have g₀: 49 ≡ 147 [MOD 5] := by
+          refine Nat.ModEq.trans hb₂.symm hb₅
+        have g₁: ¬ 49 ≡ 147 [MOD 5] := by decide
+        exact (g₁ g₀).elim
+      . ring_nf at hb₂
+        have g₀: 98 ≡ 147 [MOD 5] := by
+          refine Nat.ModEq.trans hb₂.symm hb₅
+        have g₁: ¬ 98 ≡ 147 [MOD 5] := by decide
+        exact (g₁ g₀).elim
+      . ring_nf at hb₂
+        exact Or.intro_left (7 ^ (d * 2) * 343 ≡ 3 [MOD 5]) hb₅
+      . ring_nf at hb₂
+        have g₀: 196 ≡ 147 [MOD 5] := by
+          refine Nat.ModEq.trans hb₂.symm hb₅
+        have g₁: ¬ 196 ≡ 147 [MOD 5] := by decide
+        exact (g₁ g₀).elim
+
+
+lemma aux_4
+  (n b a : ℕ)
+  (k : ℝ)
+  -- (hk : k = √8)
+  -- (hb : b = ∑ k ∈ range (n + 1), (2 * n + 1).choose (2 * k + 1) * 2 ^ (3 * k))
+  -- (ha : a = ∑ k ∈ range (n + 1), (2 * n + 1).choose (2 * k) * 2 ^ (3 * k))
+  (hb₁ : ↑b = 1 / k * ∑ x ∈ range (n + 1), ↑((2 * n + 1).choose (2 * x + 1)) * k ^ (2 * x + 1))
+  (ha₁ : ↑a = ∑ x ∈ range (n + 1), ↑((2 * n + 1).choose (2 * x)) * k ^ (2 * x))
+  (hk₀ : k * k⁻¹ = 1) :
+  (1 + k) ^ (2 * n + 1) = ↑a + ↑b * k := by
+  rw [mul_comm _ k, hb₁, ← mul_assoc]
+  rw [← inv_eq_one_div, hk₀, one_mul, ha₁]
+  rw [add_comm, add_pow k 1 (2 * n + 1)]
+  simp
+  clear hb₁ ha₁ b a hk₀
+  let f : ℕ → ℝ := fun i => ↑((2 * n + 1).choose (i)) * k ^ i
+  let fs₂ := Finset.range (2 * n + 2)
+  -- let fs₀ : Finset ℕ := Finset.filter (fun x => Odd x) (Finset.range (2 * n + 2))
+  let fs₀ : Finset ℕ := fs₂.filter (fun x => Odd x)
+  let fs₁ : Finset ℕ := fs₂.filter (fun x => Even x)
+  let fs₃ : Finset ℕ := Finset.range (n + 1)
+  have h₀: ∑ x ∈ range (n + 1), ↑((2 * n + 1).choose (2 * x + 1)) * k ^ (2 * x + 1) =
+    ∑ x ∈ fs₀, ↑((2 * n + 1).choose (x)) * k ^ (x) := by
+    have h₀₁: ∑ x ∈ fs₃, f (2 * x + 1) = ∑ x ∈ (fs₀), f x := by
+      refine sum_bij ?i ?_ ?i_inj ?i_surj ?h
+      . intros a _
+        exact (2 * a + 1)
+      . intros a ha₀
+        have ha₁: a ≤ n := by exact mem_range_succ_iff.mp ha₀
+        have ha₂: 2 * a + 1 ≤ 2 * n + 1 := by linarith
+        have ha₃: (2 * a + 1) ∈ fs₂ := by exact mem_range_succ_iff.mpr ha₂
+        have ha₄: Odd (2 * a + 1) := by exact odd_two_mul_add_one a
+        refine mem_filter.mpr ?_
+        exact And.symm ⟨ha₄, ha₃⟩
+      . intros a _ b _ h₃
+        linarith
+      . intros b hb₀
+        use ((b - 1) / 2)
+        refine exists_prop.mpr ?_
+        have hb₁: b ∈ fs₂ ∧ Odd b := by exact mem_filter.mp hb₀
+        have hb₂: 1 ≤ b := by
+          by_contra! hc₀
+          interval_cases b
+          have hc₁: ¬ Odd 0 := by decide
+          apply hc₁ hb₁.2
+        have hb₃: Even (b - 1) := by
+          refine (Nat.even_sub hb₂).mpr ?_
+          simp only [not_even_one, iff_false, not_even_iff_odd]
+          exact hb₁.2
+        constructor
+        . have hb₄: b < 2 * n + 2 := by exact List.mem_range.mp hb₁.1
+          have hb₅: (b - 1) / 2 < n + 1 := by omega
+          exact mem_range.mpr hb₅
+        . have hb₆: 2 * ((b - 1) / 2) = b - 1 := by exact two_mul_div_two_of_even hb₃
+          rw [hb₆]
+          exact Nat.sub_add_cancel hb₂
+      . exact fun a _ => rfl
+    exact h₀₁
+  have h₁: ∑ x ∈ range (n + 1), ↑((2 * n + 1).choose (2 * x)) * k ^ (2 * x) =
+    ∑ x ∈ fs₁, ↑((2 * n + 1).choose (x)) * k ^ (x) := by
+    have h₁₁: ∑ x ∈ fs₃, f (2 * x) = ∑ x ∈ (fs₁), f x := by
+      refine sum_bij ?_ ?_ ?_ ?_ ?_
+      . intros a _
+        exact (2 * a)
+      . intros a ha₀
+        have ha₁: a < n + 1 := by exact List.mem_range.mp ha₀
+        have ha₂: 2 * a < 2 * n + 2 := by linarith
+        refine mem_filter.mpr ?_
+        constructor
+        . exact mem_range.mpr ha₂
+        . exact even_two_mul a
+      . intros a _ b _ h₃
+        exact Nat.eq_of_mul_eq_mul_left (by norm_num) h₃
+      . intros b hb₀
+        use (b/2)
+        refine exists_prop.mpr ?_
+        have hb₁: b ∈ fs₂ ∧ Even b := by exact mem_filter.mp hb₀
+        constructor
+        . have hb₂: b < 2 * n + 2 := by exact List.mem_range.mp hb₁.1
+          have hb₃: (b / 2) < n + 1 := by exact Nat.div_lt_of_lt_mul hb₂
+          exact mem_range.mpr hb₃
+        . exact two_mul_div_two_of_even hb₁.2
+      . exact fun a _ => rfl
+    exact h₁₁
+  have h₂: ∑ x ∈ range (2 * n + 1 + 1), k ^ x * ↑((2 * n + 1).choose x) =
+    ∑ x ∈ fs₂, ↑((2 * n + 1).choose x) * k ^ x := by
+    refine Finset.sum_congr (rfl) ?_
+    intros x _
+    rw [mul_comm]
+  rw [h₀, h₁, h₂]
+  have h₃: fs₂ = fs₀ ∪ fs₁ := by
+    refine Finset.ext_iff.mpr ?_
+    intro a
+    constructor
+    . intro ha₀
+      refine mem_union.mpr ?mp.a
+      have ha₁: Odd a ∨ Even a := by exact Or.symm (even_or_odd a)
+      cases' ha₁ with ha₂ ha₃
+      . left
+        refine mem_filter.mpr ?mp.a.inl.h.a
+        exact And.symm ⟨ha₂, ha₀⟩
+      . right
+        refine mem_filter.mpr ?mp.a.inl.h.b
+        exact And.symm ⟨ha₃, ha₀⟩
+    . intro ha₀
+      apply mem_union.mp at ha₀
+      cases' ha₀ with ha₁ ha₂
+      . exact mem_of_mem_filter a ha₁
+      . exact mem_of_mem_filter a ha₂
+  have h₄: Disjoint fs₀ fs₁ := by
+    refine disjoint_filter.mpr ?_
+    intros x _ hx₁
+    exact not_even_iff_odd.mpr hx₁
+  nth_rw 2 [add_comm]
+  rw [h₃, Finset.sum_union h₄]
+
+
+lemma aux_5
+  (n b a : ℕ)
+  (k : ℝ)
+  -- (hk : k = √8)
+  -- (hb : b = ∑ k ∈ range (n + 1), (2 * n + 1).choose (2 * k + 1) * 2 ^ (3 * k))
+  -- (ha : a = ∑ k ∈ range (n + 1), (2 * n + 1).choose (2 * k) * 2 ^ (3 * k))
+  (hb₁ : ↑b = 1 / k * ∑ x ∈ range (n + 1), ↑((2 * n + 1).choose (2 * x + 1)) * k ^ (2 * x + 1))
+  (ha₁ : ↑a = ∑ x ∈ range (n + 1), ↑((2 * n + 1).choose (2 * x)) * k ^ (2 * x))
+  (hk₀ : k * k⁻¹ = 1) :
+  (1 - k) ^ (2 * n + 1) = ↑a - ↑b * k := by
+  rw [mul_comm _ k, hb₁, ← mul_assoc]
+  rw [← inv_eq_one_div, hk₀, one_mul, ha₁, sub_eq_add_neg]
+  rw [add_comm 1 _, add_pow (-k) 1 (2 * n + 1)]
+  simp
+  clear hb₁ ha₁ b a hk₀
+  let f₀ : ℕ → ℝ := fun i => ↑((2 * n + 1).choose (i)) * k ^ i
+  let f₁ : ℕ → ℝ := fun i => ↑((2 * n + 1).choose (i)) * (-k) ^ i
+  let fs₂ := Finset.range (2 * n + 2)
+  let fs₀ : Finset ℕ := fs₂.filter (fun x => Odd x)
+  let fs₁ : Finset ℕ := fs₂.filter (fun x => Even x)
+  let fs₃ : Finset ℕ := Finset.range (n + 1)
+  have h₀: ∑ x ∈ range (n + 1), ↑((2 * n + 1).choose (2 * x + 1)) * k ^ (2 * x + 1) =
+    - ∑ x ∈ fs₀, ↑((2 * n + 1).choose (x)) * (-k) ^ (x) := by
+    rw [neg_eq_neg_one_mul, Finset.mul_sum]
+    have h₀₁: ∑ x ∈ fs₃, f₀ (2 * x + 1) = ∑ x ∈ (fs₀), -1 * f₁ x := by
+      refine sum_bij ?i ?_ ?i_inj ?i_surj ?h
+      . intros a _
+        exact (2 * a + 1)
+      . intros a ha₀
+        have ha₁: a ≤ n := by exact mem_range_succ_iff.mp ha₀
+        have ha₂: 2 * a + 1 ≤ 2 * n + 1 := by linarith
+        have ha₃: (2 * a + 1) ∈ fs₂ := by exact mem_range_succ_iff.mpr ha₂
+        have ha₄: Odd (2 * a + 1) := by exact odd_two_mul_add_one a
+        refine mem_filter.mpr ?_
+        exact And.symm ⟨ha₄, ha₃⟩
+      . intros a _ b _ h₃
+        linarith
+      . intros b hb₀
+        use ((b - 1) / 2)
+        refine exists_prop.mpr ?_
+        have hb₁: b ∈ fs₂ ∧ Odd b := by exact mem_filter.mp hb₀
+        have hb₂: 1 ≤ b := by
+          by_contra! hc₀
+          interval_cases b
+          have hc₁: ¬ Odd 0 := by decide
+          apply hc₁ hb₁.2
+        have hb₃: Even (b - 1) := by
+          refine (Nat.even_sub hb₂).mpr ?_
+          simp only [not_even_one, iff_false, not_even_iff_odd]
+          exact hb₁.2
+        constructor
+        . have hb₄: b < 2 * n + 2 := by exact List.mem_range.mp hb₁.1
+          have hb₅: (b - 1) / 2 < n + 1 := by omega
+          exact mem_range.mpr hb₅
+        . have hb₆: 2 * ((b - 1) / 2) = b - 1 := by exact two_mul_div_two_of_even hb₃
+          rw [hb₆]
+          exact Nat.sub_add_cancel hb₂
+      . intros b hb₀
+        ring_nf
+        have hb₁: (-1:ℝ) ^ (b * 2) = 1 := by
+          refine (neg_one_pow_eq_one_iff_even (by norm_num)).mpr ?_
+          rw [mul_comm]
+          exact even_two_mul b
+        rw [hb₁, mul_one]
+    exact h₀₁
+  have h₁: ∑ x ∈ range (n + 1), ↑((2 * n + 1).choose (2 * x)) * k ^ (2 * x) =
+    ∑ x ∈ fs₁, ↑((2 * n + 1).choose (x)) * (-k) ^ (x) := by
+    have h₁₁: ∑ x ∈ fs₃, f₀ (2 * x) = ∑ x ∈ (fs₁), f₁ x := by
+      refine sum_bij ?_ ?_ ?_ ?_ ?_
+      . intros a _
+        exact (2 * a)
+      . intros a ha₀
+        have ha₁: a < n + 1 := by exact List.mem_range.mp ha₀
+        have ha₂: 2 * a < 2 * n + 2 := by linarith
+        refine mem_filter.mpr ?_
+        constructor
+        . exact mem_range.mpr ha₂
+        . exact even_two_mul a
+      . intros a _ b _ h₃
+        exact Nat.eq_of_mul_eq_mul_left (by norm_num) h₃
+      . intros b hb₀
+        use (b/2)
+        refine exists_prop.mpr ?_
+        have hb₁: b ∈ fs₂ ∧ Even b := by exact mem_filter.mp hb₀
+        constructor
+        . have hb₂: b < 2 * n + 2 := by exact List.mem_range.mp hb₁.1
+          have hb₃: (b / 2) < n + 1 := by exact Nat.div_lt_of_lt_mul hb₂
+          exact mem_range.mpr hb₃
+        . exact two_mul_div_two_of_even hb₁.2
+      . intros b hb₀
+        ring_nf
+        have hb₁: (-1:ℝ) ^ (b * 2) = 1 := by
+          refine (neg_one_pow_eq_one_iff_even (by norm_num)).mpr ?_
+          rw [mul_comm]
+          exact even_two_mul b
+        rw [hb₁, mul_one]
+    exact h₁₁
+  have h₂: ∑ x ∈ range (2 * n + 1 + 1), (-k) ^ x * ↑((2 * n + 1).choose x) =
+    ∑ x ∈ fs₂, ↑((2 * n + 1).choose x) * (-k) ^ x := by
+    refine Finset.sum_congr (rfl) ?_
+    intros x _
+    rw [mul_comm]
+  rw [h₀, h₁, h₂, sub_neg_eq_add]
+  have h₃: fs₂ = fs₀ ∪ fs₁ := by
+    refine Finset.ext_iff.mpr ?_
+    intro a
+    constructor
+    . intro ha₀
+      refine mem_union.mpr ?mp.a
+      have ha₁: Odd a ∨ Even a := by exact Or.symm (even_or_odd a)
+      cases' ha₁ with ha₂ ha₃
+      . left
+        refine mem_filter.mpr ?mp.a.inl.h.a
+        exact And.symm ⟨ha₂, ha₀⟩
+      . right
+        refine mem_filter.mpr ?mp.a.inl.h.b
+        exact And.symm ⟨ha₃, ha₀⟩
+    . intro ha₀
+      apply mem_union.mp at ha₀
+      cases' ha₀ with ha₁ ha₂
+      . exact mem_of_mem_filter a ha₁
+      . exact mem_of_mem_filter a ha₂
+  have h₄: Disjoint fs₀ fs₁ := by
+    refine disjoint_filter.mpr ?_
+    intros x _ hx₁
+    exact not_even_iff_odd.mpr hx₁
+  nth_rw 2 [add_comm]
+  rw [h₃, Finset.sum_union h₄]
+
+
+
+
+theorem imo_1974_p3
+  (n : ℕ) :
+  ¬ 5 ∣ ∑ k ∈ Finset.range (n + 1), (Nat.choose (2 * n + 1) (2 * k + 1)) * (2^(3 * k)) := by
+  let k:ℝ := Real.sqrt (8:ℝ)
+  have hk: k = Real.sqrt (8:ℝ) := by rfl
+  let b:ℕ := ∑ k ∈ Finset.range (n + 1), (Nat.choose (2 * n + 1) (2 * k + 1)) * (2^(3 * k))
+  have hb: b = ∑ k ∈ Finset.range (n + 1), (Nat.choose (2 * n + 1) (2 * k + 1)) * (2^(3 * k)) := by rfl
+  rw [← hb]
+  let a:ℕ := ∑ k ∈ Finset.range (n + 1), (Nat.choose (2 * n + 1) (2 * k) * (2 ^ (3 * k)))
+  have ha: a = ∑ k ∈ Finset.range (n + 1), (Nat.choose (2 * n + 1) (2 * k) * (2 ^ (3 * k))) := by rfl
+  have hb₁: b = (1 / k) *
+    ∑ x ∈ Finset.range (n + 1), (Nat.choose (2 * n + 1) (2 * x + 1)) * (k ^ (2 * x + 1)) := by
+    rw [hb, hk]
+    simp
+    rw [Finset.mul_sum]
+    refine Finset.sum_congr (rfl) ?_
+    intros x _
+    rw [mul_comm ((√8)⁻¹), mul_assoc]
+    refine mul_eq_mul_left_iff.mpr ?_
+    left
+    rw [pow_succ, pow_mul, pow_mul, Real.sq_sqrt (by norm_num)]
+    norm_num
+  have ha₁: a = ∑ x ∈ Finset.range (n + 1), (Nat.choose (2 * n + 1) (2 * x) * (k ^ (2 * x))) := by
+    rw [ha, hk]
+    simp
+    refine Finset.sum_congr (rfl) ?_
+    intros x _
+    refine mul_eq_mul_left_iff.mpr ?_
+    left
+    rw [pow_mul, pow_mul, Real.sq_sqrt (by norm_num)]
+    norm_num
+  have hk₀: k * k⁻¹ = 1 := by
+    refine (mul_inv_eq_one₀ ?_).mpr (rfl)
+    rw [hk]
+    norm_num
+  have h₀: (1 + k) ^ (2 * n + 1) = a + b * k := by
+    exact aux_4 n b a k hb₁ ha₁ hk₀
+  have h₁: (1 - k) ^ (2 * n + 1) = a - b * k := by
+    exact aux_5 n b a k hb₁ ha₁ hk₀
+  have h₂: ((1 + k) * (1 - k)) ^ (2 * n + 1) = (a + b * k) * (a - b * k) := by
+    rw [mul_pow, h₀, h₁]
+  rw [← sq_sub_sq 1 k] at h₂
+  rw [← sq_sub_sq (↑a) ((↑b:ℝ) * k)] at h₂
+  rw [mul_pow, hk] at h₂
+  norm_num at h₂
+  have h₃: (7:ℕ) ^ (2 * n + 1) = b ^ 2 * 8 - a ^ 2 := by
+    have h₃₀: Odd (2 * n + 1) := by exact odd_two_mul_add_one n
+    have h₃₁: (-7:ℝ) = (-1:ℝ) * (7:ℕ) := by norm_num
+    have h₃₂: (-1:ℝ) ^ (2 * n + 1) = -1 := by exact Odd.neg_one_pow h₃₀
+    have h₃₃: ↑a ^ 2 - ↑b ^ 2 * 8 = (-1:ℝ) * (↑b ^ 2 * 8 - ↑a ^ 2) := by
+      linarith
+    rw [h₃₁, mul_pow, h₃₂, h₃₃] at h₂
+    simp at h₂
+    have h₃₄: (7:ℝ) ^ (2 * n + 1) = ↑b ^ 2 * 8 - ↑a ^ 2 := by
+      linarith
+    norm_cast at h₃₄
+    rw [Int.subNatNat_eq_coe] at h₃₄
+    rw [← Int.toNat_sub, ← h₃₄]
+    exact rfl
+  have h₄: 7 ^ (2 * n + 1) ≡ 2 [MOD 5] ∨ 7 ^ (2 * n + 1) ≡ 3 [MOD 5] := by
+    refine aux_3 n
+  by_contra! hc₀
+  have hc₁: b^2 * 8 ≡ 0^2 * 8 [MOD 5] := by
+    refine ModEq.mul ?_ rfl
+    refine ModEq.pow 2 ?_
+    exact modEq_zero_iff_dvd.mpr hc₀
+  simp at hc₁
+  have h₅: a ^ 2 < b ^ 2 * 8 := by
+    have h₅₀: 0 < 7 ^ (2 * n + 1) := by
+      exact Nat.pow_pos (by norm_num)
+    rw [h₃] at h₅₀
+    exact Nat.lt_of_sub_pos h₅₀
+  cases' h₄ with h₄₀ h₄₁
+  . rw [h₃] at h₄₀
+    have hc₂: b ^ 2 * 8 - a ^ 2 + a ^ 2 ≡ 2 + a ^ 2 [MOD 5] := by
+      exact ModEq.add_right (a ^ 2) h₄₀
+    rw [Nat.sub_add_cancel (le_of_lt h₅)] at hc₂
+    have hc₃: 3 + (2 + a ^ 2) ≡ 3 [MOD 5] := by
+      apply Nat.ModEq.trans hc₂.symm at hc₁
+      exact ModEq.add_left 3 hc₁
+    have hc₄: a ^ 2 ≡ 3 [MOD 5] := by
+      rw [← add_assoc, ← zero_add 3] at hc₃
+      norm_num at hc₃
+      have hc₄: 5 ≡ 0 [MOD 5] := by decide
+      exact Nat.ModEq.add_left_cancel hc₄ hc₃
+    have hc₅: ¬ a ^ 2 ≡ 3 [MOD 5] := by exact aux_2 a
+    exact hc₅ hc₄
+  . rw [h₃] at h₄₁
+    have hc₂: b ^ 2 * 8 - a ^ 2 + a ^ 2 ≡ 3 + a ^ 2 [MOD 5] := by
+      exact ModEq.add_right (a ^ 2) h₄₁
+    rw [Nat.sub_add_cancel (le_of_lt h₅)] at hc₂
+    apply Nat.ModEq.trans hc₂.symm at hc₁
+    have hc₃: a ^ 2 ≡ 2 [MOD 5] := by
+      refine Nat.ModEq.add_left_cancel'  3 ?_
+      exact hc₁
+    have hc₄: ¬ a ^ 2 ≡ 2 [MOD 5] := by exact aux_1 a
+    exact hc₄ hc₃

imo_proofs/imo_1981_p6.lean

@@ -0,0 +1,44 @@
+import Mathlib
+set_option linter.unusedVariables.analyzeTactics true
+
+
+open Nat
+
+theorem imo_1981_p6
+  (f : ℕ → ℕ → ℕ)
+  (h₀ : ∀ y, f 0 y = y + 1)
+  (h₁ : ∀ x, f (x + 1) 0 = f x 1)
+  (h₂ : ∀ x y, f (x + 1) (y + 1) = f x (f (x + 1) y)) :
+  ∀ y, f 4 (y + 1) = 2 ^ (f 4 y + 3) - 3 := by
+  have h₃: ∀ y, f 1 y = y + 2 := by
+    intro y
+    induction' y with n hn
+    . simp_all only [zero_eq, zero_add]
+    . nth_rw 1 [← zero_add 1]
+      rw [h₂ 0 n, h₀ (f (0 + 1) n), hn]
+  have h₄: ∀ y, f 2 y = 2 * y + 3 := by
+    intro y
+    induction' y with n hn
+    . simp_all only [zero_eq, zero_add, mul_zero]
+    . rw [h₂, h₃, hn, mul_add]
+  have h₅: ∀ y, f 3 y =  2 ^ (y + 3) - 3 := by
+    intro y
+    induction' y with n hn
+    . simp_all only [zero_eq, zero_add, mul_zero]
+      omega
+    . rw [h₂, h₄, hn]
+      rw [Nat.mul_sub_left_distrib]
+      ring_nf
+      by_cases hn₀: 0 < n
+      . rw [← Nat.add_sub_assoc, add_comm]
+        . omega
+        . have hn₂: 2 ^ 1 ≤ 2 ^ n := by exact Nat.pow_le_pow_of_le (by norm_num) hn₀
+          linarith
+      . have hn₁: n = 0 := by linarith
+        rw [hn₁]
+        omega
+  intro y
+  induction' y with n hn
+  . simp
+    rw [h₂, h₁, h₅]
+  . rw [hn, h₂, h₅, h₂, h₅]

imo_proofs/imo_1982_p1.lean

@@ -0,0 +1,78 @@
+import Mathlib
+set_option linter.unusedVariables.analyzeTactics true
+
+
+open Nat
+
+theorem imo_1982_p1
+  (f : ℕ → ℤ)
+  (h₀ : ∀ m n, (0 < m ∧ 0 < n) → f (m + n) - f m - f n = 0 ∨ f (m + n) - f m - f n = 1)
+  (h₁ : f 2 = 0)
+  (h₂ : 0 < f 3)
+  (h₃ : f 9999 = 3333) :
+  f 1982 = 660 := by
+  have h₀₀: ∀ m n, (0 < m ∧ 0 < n) → f m  + f n ≤ f (m + n) := by
+    intros m n hmn
+    have g₀: f (m + n) - f m - f n = 0 ∨ f (m + n) - f m - f n = 1 := by
+      exact h₀ m n hmn
+    omega
+  have h₀₁: ∀ m k, (0 < m ∧ 0 < k) → k * f m ≤ f (k * m) := by
+    intros m k hmk
+    have g₁: 1 ≤ k := by linarith
+    refine Nat.le_induction ?_ ?_ k g₁
+    . simp
+    . intros n hmn  g₂
+      rw [cast_add]
+      rw [add_mul, add_mul, one_mul]
+      simp
+      have g₃: f (n * m) + f (m) ≤ f (n * m + m) := by
+        refine h₀₀ (n * m) m ?_
+        constructor
+        . refine mul_pos ?_ hmk.1
+          exact hmn
+        . exact hmk.1
+      refine le_trans ?_ g₃
+      exact (Int.add_le_add_iff_right (f m)).mpr g₂
+  have h₄: f 3 = 1 := by
+    have g₀ : 3333 * f 3 ≤ f (9999) := by
+      refine h₀₁ 3 3333 ?_
+      omega
+    linarith
+  have h₅: f 1980 = 660 := by
+    have h₅₀: f 1980 ≤ 660 := by
+      have g₀ : f (5 * 1980) + f 99 ≤ f (9999) := by
+        refine h₀₀ (5 * 1980) 99 (by omega)
+      have g₁: 5 * f (1980) ≤ f (5 * 1980) := by
+        exact h₀₁ 1980 5 (by omega)
+      have g₂: 33 * f 3 ≤ f 99 := by
+        exact h₀₁ 3 33 (by omega)
+      rw [h₃] at g₀
+      linarith
+    have h₅₁: 660 ≤ f 1980 := by
+      have g₀ : 660 * f 3 ≤ f (1980) := by
+        refine h₀₁ 3 660 ?_
+        omega
+      rw [h₄] at g₀
+      exact g₀
+    exact le_antisymm h₅₀ h₅₁
+  have h₆: f 1982 - f 1980 - f 2 = 0 ∨ f 1982 - f 1980 - f 2 = 1 := by
+    refine h₀ 1980 2 ?_
+    omega
+  cases' h₆ with h₆₀ h₆₁
+  . linarith
+  . exfalso
+    rw [h₅, h₁] at h₆₁
+    have h₆₂: f 1982 = 661 := by
+      linarith
+    have h₆₃: 5 * f 1982 + 29 ≤ 3333 := by
+      have g₀ : f (5 * 1982) + f 89 ≤ f 9999 := by
+        refine h₀₀ (5 * 1982) 89 (by omega)
+      have g₁: f (29 * 3) + f 2 ≤ f 89 := by
+        refine h₀₀ (29 * 3) 2 (by omega)
+      have g₂: 5 * f (1982) ≤ f (5 * 1982) := by
+        exact h₀₁ 1982 5 (by omega)
+      have g₃: 29 * f 3 ≤ f (87) := by
+        exact h₀₁ 3 29 (by omega)
+      linarith
+    rw [h₆₂] at h₆₃
+    linarith

imo_proofs/imo_1983_p6.lean

@@ -0,0 +1,181 @@
+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

imo_proofs/imo_1984_p6.lean

@@ -0,0 +1,436 @@
+import Mathlib
+set_option linter.unusedVariables.analyzeTactics true
+
+open Nat
+
+
+lemma mylemma_sub_sq
+  (a b : ℕ)
+  (h₀: b < a) :
+  ((a - b) ^ 2 = a ^ 2 + b ^ 2 - 2 * a * b) := by
+  have h₁: b^2 ≤ a * b := by
+    rw [pow_two]
+    refine Nat.mul_le_mul_right ?_ ?_
+    exact Nat.le_of_lt h₀
+  have h₂: a * b ≤ a ^ 2 := by
+    rw [pow_two]
+    refine Nat.mul_le_mul_left ?_ ?_
+    exact Nat.le_of_lt h₀
+  repeat rw [pow_two]
+  repeat rw [Nat.mul_sub_left_distrib]
+  repeat rw [Nat.mul_sub_right_distrib a b a]
+  rw [Nat.sub_right_comm]
+  repeat rw [Nat.mul_sub_right_distrib a b b]
+  ring_nf
+  have h₃: a ^ 2 - (a * b - b ^ 2) = a ^ 2 - a * b + b ^ 2 := by
+    refine tsub_tsub_assoc ?h₁ h₁
+    exact h₂
+  rw [h₃]
+  rw [← Nat.sub_add_comm h₂]
+  . rw [← Nat.sub_add_eq, ← mul_two]
+
+
+lemma mylemma_k_le_m_alt
+  (a b c d k m : ℕ)
+  (h₂ : a < b ∧ b < c ∧ c < d)
+  (h₃ : a * d = b * c)
+  (h₄ : a + d = 2 ^ k)
+  (h₅ : b + c = 2 ^ m)
+  (hkm : k ≤ m) :
+  False := by
+  have h₆: (a + d) ^ 2 ≤ (b + c) ^ 2 := by
+    refine Nat.pow_le_pow_of_le_left ?_ 2
+    rw [h₄,h₅]
+    exact pow_le_pow_right₀ (by norm_num) hkm
+  rw [add_sq, add_sq, mul_assoc, h₃, mul_assoc] at h₆
+  have h₇: (d - a) ^ 2 ≤ (c - b) ^ 2 := by
+    have hda: a < d := by
+      refine lt_trans h₂.1 ?_
+      exact lt_trans h₂.2.1 h₂.2.2
+    rw [mylemma_sub_sq d a hda]
+    rw [mylemma_sub_sq c b h₂.2.1]
+    rw [mul_assoc, mul_assoc]
+    rw [mul_comm d a, mul_comm c b]
+    rw [h₃]
+    refine Nat.sub_le_sub_right ?_ (2 * (b * c))
+    linarith
+  have h₈: (c - b) ^ 2 < (d - a) ^ 2 := by
+    refine Nat.pow_lt_pow_left ?_ (by norm_num)
+    have h₈₀: c - a < d - a := by
+      have g₀: c - a + a < d - a + a := by
+        rw [Nat.sub_add_cancel ?_]
+        rw [Nat.sub_add_cancel ?_]
+        . exact h₂.2.2
+        . linarith
+        . linarith
+      exact Nat.lt_of_add_lt_add_right g₀
+    refine lt_trans ?_ h₈₀
+    refine Nat.sub_lt_sub_left ?_ h₂.1
+    exact lt_trans h₂.1 h₂.2.1
+  have h₉: (d - a) ^ 2 ≠ (d - a) ^ 2 := by
+    refine Nat.ne_of_lt ?_
+    exact lt_of_le_of_lt h₇ h₈
+  refine false_of_ne h₉
+
+
+
+
+lemma mylemma_k_le_m
+  (a b c d k m : ℕ)
+  (h₂ : a < b ∧ b < c ∧ c < d)
+  (h₃ : a * d = b * c)
+  (h₄ : a + d = 2 ^ k)
+  (h₅ : b + c = 2 ^ m) :
+  (m < k) := by
+  have h₆: (c - b) ^ 2 < (d - a) ^ 2 := by
+    refine Nat.pow_lt_pow_left ?_ (by norm_num)
+    have h₈₀: c - a < d - a := by
+      have g₀: c - a + a < d - a + a := by
+        rw [Nat.sub_add_cancel ?_]
+        rw [Nat.sub_add_cancel ?_]
+        . exact h₂.2.2
+        . linarith
+        . linarith
+      exact Nat.lt_of_add_lt_add_right g₀
+    refine lt_trans ?_ h₈₀
+    refine Nat.sub_lt_sub_left ?_ h₂.1
+    exact lt_trans h₂.1 h₂.2.1
+  have h₇: (b + c) ^ 2 < (a + d) ^ 2 := by
+    rw [add_sq b c, add_sq a d]
+    have hda: a < d := by
+      refine lt_trans h₂.1 ?_
+      exact lt_trans h₂.2.1 h₂.2.2
+    rw [mylemma_sub_sq d a hda] at h₆
+    rw [mylemma_sub_sq c b h₂.2.1] at h₆
+    rw [mul_assoc 2 b c, ← h₃, ← mul_assoc]
+    rw [mul_assoc 2 c b, mul_comm c b, ← h₃, ← mul_assoc] at h₆
+    rw [add_assoc, add_comm _ (c ^ 2), ← add_assoc]
+    rw [add_assoc (a ^ 2), add_comm _ (d ^ 2), ← add_assoc]
+    rw [mul_assoc 2 d a, mul_comm d a, ← mul_assoc] at h₆
+    rw [add_comm (d ^ 2) (a ^ 2)] at h₆
+    rw [add_comm (c ^ 2) (b ^ 2)] at h₆
+    have g₀: 2 * a * d ≤ 4 * a * d := by
+      ring_nf
+      exact Nat.mul_le_mul_left (a * d) (by norm_num)
+    have g₁: 2 * a * d = 4 * a * d - 2 * a * d := by
+      ring_nf
+      rw [← Nat.mul_sub_left_distrib]
+      norm_num
+    have g₂: 2 * a * d ≤ b ^ 2 + c ^ 2 := by
+      rw [mul_assoc, h₃, ← mul_assoc]
+      exact two_mul_le_add_sq b c
+    have g₃: 2 * a * d ≤ a ^ 2 + d ^ 2 := by
+      exact two_mul_le_add_sq a d
+    rw [g₁, ← Nat.add_sub_assoc (g₀) (b ^ 2 + c ^ 2)]
+    rw [← Nat.add_sub_assoc (g₀) (a ^ 2 + d ^ 2)]
+    rw [Nat.sub_add_comm g₂, Nat.sub_add_comm g₃]
+    exact (Nat.add_lt_add_iff_right).mpr h₆
+  have h2 : 1 < 2 := by norm_num
+  refine (Nat.pow_lt_pow_iff_right h2).mp ?_
+  rw [← h₄, ← h₅]
+  exact (Nat.pow_lt_pow_iff_left (by norm_num) ).mp h₇
+
+
+
+lemma mylemma_h8
+  (a b c d k m : ℕ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  (h₁ : Odd a ∧ Odd b ∧ Odd c ∧ Odd d)
+  (h₂ : a < b ∧ b < c ∧ c < d)
+  (h₅ : b + c = 2 ^ m)
+  (hkm : m < k)
+  (h₆ : b * 2 ^ m - a * 2 ^ k = (b - a) * (b + a))
+  (h₇ : 2 ^ m ∣ (b - a) * (b + a)) :
+  (b + a = 2 ^ (m - 1)) := by
+  have h₇₁: ∃ y z, y ∣ b - a ∧ z ∣ b + a ∧ y * z = 2 ^ m := by
+    exact Nat.dvd_mul.mp h₇
+  let ⟨p, q, hpd⟩ := h₇₁
+  cases' hpd with hpd hqd
+  cases' hqd with hqd hpq
+  have hm1: 1 ≤ m := by
+    by_contra! hc
+    interval_cases m
+    linarith
+  have h₈₀: b - a < 2 ^ (m - 1) := by
+    have g₀: b < (b + c) / 2 := by
+      refine (Nat.lt_div_iff_mul_lt' ?_ b).mpr ?_
+      . refine even_iff_two_dvd.mp ?_
+        exact Odd.add_odd h₁.2.1 h₁.2.2.1
+      . linarith
+    have g₁: (b + c) / 2 = 2 ^ (m-1) := by
+      rw [h₅]
+      rw [← Nat.pow_sub_mul_pow 2 hm1]
+      simp
+    rw [← g₁]
+    refine lt_trans ?_ g₀
+    exact Nat.sub_lt h₀.2.1 h₀.1
+  have hp: p = 2 := by
+    have hp₀: 2 * b < 2 ^ m := by
+      rw [← h₅, two_mul]
+      exact Nat.add_lt_add_left h₂.2.1 b
+    have hp₁: b + a < 2 ^ (m) := by
+      have g₀: b + a < b + b := by
+        exact Nat.add_lt_add_left h₂.1 b
+      refine Nat.lt_trans g₀ ?_
+      rw [← two_mul]
+      exact hp₀
+    have hp₂: q < 2 ^ m := by
+      refine Nat.lt_of_le_of_lt (Nat.le_of_dvd ?_ hqd) hp₁
+      exact Nat.add_pos_right b h₀.1
+    have hp₃: 1 < p := by
+      rw [← hpq] at hp₂
+      exact one_lt_of_lt_mul_left hp₂
+    have h2prime: Nat.Prime 2 := by exact prime_two
+    have hp₅: ∀ i j:ℕ , 2 ^ i ∣ (b - a) ∧ 2 ^ j ∣ (b + a) → (i < 2 ∨ j < 2) := by
+      by_contra! hc
+      let ⟨i, j, hi⟩ := hc
+      have hti: 2 ^ 2 ∣ 2 ^ i := by exact Nat.pow_dvd_pow 2 hi.2.1
+      have htj: 2 ^ 2 ∣ 2 ^ j := by exact Nat.pow_dvd_pow 2 hi.2.2
+      norm_num at hti htj
+      have hi₄: 4 ∣ b - a := by exact Nat.dvd_trans hti hi.1.1
+      have hi₅: 4 ∣ b + a := by exact Nat.dvd_trans htj hi.1.2
+      have hi₆: 4 ∣ (b - a) + (b + a) := by exact Nat.dvd_add hi₄ hi₅
+      have hi₇: 2 ∣ b := by
+        have g₀: 0 < 2 := by norm_num
+        refine Nat.dvd_of_mul_dvd_mul_left g₀ ?_
+        rw [← Nat.add_sub_cancel (2 * b) a, Nat.two_mul b]
+        rw [add_assoc, Nat.sub_add_comm (le_of_lt h₂.1)]
+        exact hi₆
+      have hi₈: Even b := by
+        exact even_iff_two_dvd.mpr hi₇
+      apply Nat.not_odd_iff_even.mpr hi₈
+      exact h₁.2.1
+    have hp₆: ∀ i j:ℕ , i + j = m ∧ 2 ^ i ∣ (b - a) ∧ 2 ^ j ∣ (b + a) → (¬ j < 2) := by
+      by_contra! hc
+      let ⟨i, j, hi⟩ := hc
+      have hi₀: m - 1 ≤ i := by
+        rw [← hi.1.1]
+        simp
+        exact Nat.le_pred_of_lt hi.2
+      have hi₁: 2 ^ (m - 1) ≤ 2 ^ i := by exact Nat.pow_le_pow_right (by norm_num) hi₀
+      have hi₂: 2 ^ i < 2 ^ (m - 1) := by
+        refine lt_of_le_of_lt ?_ h₈₀
+        refine Nat.le_of_dvd ?_ hi.1.2.1
+        exact Nat.sub_pos_of_lt h₂.1
+      linarith [hi₁, hi₂]
+    have hi₀: ∃ i ≤ m, p = 2 ^ i := by
+      have g₀: p ∣ 2 ^ m := by
+        rw [← hpq]
+        exact Nat.dvd_mul_right p q
+      exact (Nat.dvd_prime_pow h2prime).mp g₀
+    let ⟨i, hp⟩ := hi₀
+    cases' hp with him hp
+    let j:ℕ := m - i
+    have hj₀: j = m - i := by linarith
+    have hj₁: i + j = m := by
+      rw [add_comm, ← Nat.sub_add_cancel him]
+    have hq: q = 2 ^ j := by
+      rw [hp] at hpq
+      rw [hj₀, ← Nat.pow_div him (by norm_num)]
+      refine Nat.eq_div_of_mul_eq_right ?_ hpq
+      refine Nat.ne_of_gt ?_
+      rw [← hp]
+      linarith [hp₃]
+    rw [hp] at hpd
+    rw [hq] at hqd
+    have hj₃: ¬ j < 2 := by
+      exact hp₆ i j {left:= hj₁ , right:= { left := hpd , right:= hqd} }
+    have hi₂: i < 2 := by
+      have g₀: i < 2 ∨ j < 2 := by
+        exact hp₅ i j { left := hpd , right:= hqd }
+      omega
+    have hi₃: 0 < i := by
+      rw [hp] at hp₃
+      refine Nat.zero_lt_of_ne_zero ?_
+      exact (Nat.one_lt_two_pow_iff).mp hp₃
+    have hi₄: i = 1 := by
+      interval_cases i
+      rfl
+    rw [hi₄] at hp
+    exact hp
+  have hq: q = 2 ^ (m - 1) := by
+    rw [hp, ← Nat.pow_sub_mul_pow 2 hm1, pow_one, mul_comm] at hpq
+    exact Nat.mul_right_cancel (by norm_num) hpq
+  rw [hq] at hqd
+  have h₈₂: ∃ c, (b + a) = c * 2 ^ (m - 1) := by
+    exact exists_eq_mul_left_of_dvd hqd
+  let ⟨f, hf⟩ := h₈₂
+  have hfeq1: f = 1 := by
+    have hf₀: f * 2 ^ (m - 1) < 2 * 2 ^ (m - 1) := by
+      rw [← hf, ← Nat.pow_succ', ← Nat.succ_sub hm1]
+      rw [Nat.succ_sub_one, ← h₅]
+      refine Nat.add_lt_add_left ?_ b
+      exact lt_trans h₂.1 h₂.2.1
+    have hf₁: f < 2 := by
+      exact Nat.lt_of_mul_lt_mul_right hf₀
+    interval_cases f
+    . simp at hf
+      exfalso
+      linarith [hf]
+    . linarith
+  rw [hfeq1, one_mul] at hf
+  exact hf
+
+
+
+
+theorem imo_1984_p6
+  (a b c d k m : ℕ)
+  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
+  (h₁ : Odd a ∧ Odd b ∧ Odd c ∧ Odd d)
+  (h₂ : a < b ∧ b < c ∧ c < d)
+  (h₃ : a * d = b * c)
+  (h₄ : a + d = (2:ℕ)^k)
+  (h₅ : b + c = 2^m) :
+  a = 1 := by
+  by_cases hkm: k ≤ m
+  . exfalso
+    apply Nat.not_lt_of_le at hkm
+    rw [← not_true_eq_false]
+    refine (not_congr ?_).mp hkm
+    refine iff_true_intro ?_
+    exact mylemma_k_le_m a b c d k m h₂ h₃ h₄ h₅
+  . push_neg at hkm
+    have h₆: b * 2 ^ m - a * 2 ^ k = (b - a) * (b + a) := by
+      have h₆₀: c = 2 ^ m - b := by exact (tsub_eq_of_eq_add_rev (id h₅.symm)).symm
+      have h₆₁: d = 2 ^ k - a := by exact (tsub_eq_of_eq_add_rev (id h₄.symm)).symm
+      rw [h₆₀, h₆₁] at h₃
+      repeat rw [Nat.mul_sub_left_distrib, ← pow_two] at h₃
+      have h₆₂: b * 2 ^ m - a * 2 ^ k =  b ^ 2 - a ^ 2 := by
+        symm at h₃
+        refine Nat.sub_eq_of_eq_add ?_
+        rw [add_comm, ← Nat.add_sub_assoc]
+        . rw [Nat.sub_add_comm]
+          . refine Nat.eq_add_of_sub_eq ?_ h₃
+            rw [pow_two]
+            refine le_of_lt ?_
+            refine mul_lt_mul' (by linarith) ?_ (le_of_lt h₀.2.1) h₀.2.1
+            linarith
+          . rw [pow_two]
+            refine le_of_lt ?_
+            refine mul_lt_mul' (by linarith) ?_ (le_of_lt h₀.1) h₀.1
+            linarith
+        . refine le_of_lt ?_
+          rw [pow_two, pow_two]
+          exact mul_lt_mul h₂.1 (le_of_lt h₂.1) h₀.1 (le_of_lt h₀.2.1)
+      rw [Nat.sq_sub_sq b a] at h₆₂
+      linarith
+    have h₇: 2 ^ m ∣ (b - a) * (b + a) := by
+      have h₇₀: k = (k - m) + m := by exact (Nat.sub_add_cancel (le_of_lt hkm)).symm
+      rw [h₇₀, pow_add] at h₆
+      have h₇₁: (b - a * 2 ^ (k - m)) * (2 ^ m) = (b - a) * (b + a) := by
+        rw [Nat.mul_sub_right_distrib]
+        rw [mul_assoc a _ _]
+        exact h₆
+      exact Dvd.intro_left (b - a * 2 ^ (k - m)) h₇₁
+    have h₈: b + a = 2 ^ (m - 1) := by
+      exact mylemma_h8 a b c d k m h₀ h₁ h₂ h₅ hkm h₆ h₇
+    have h₉: a = 2 ^ (2 * m - 2) / 2 ^ k := by
+      have ga: 1 ≤ a := by exact Nat.succ_le_of_lt h₀.1
+      have gb: 3 ≤ b := by
+        by_contra! hc
+        interval_cases b
+        . linarith
+        . linarith [ga, h₂.1]
+        . have g₀: ¬ Odd 2 := by decide
+          exact g₀ h₁.2.1
+      have gm: 3 ≤ m := by
+        have gm₀: 2 ^ 2 ≤ 2 ^ (m - 1) := by
+          norm_num
+          rw [← h₈]
+          linarith
+        have gm₁: 2 ≤ m - 1 := by
+          exact (Nat.pow_le_pow_iff_right (by norm_num)).mp gm₀
+        omega
+      have g₀: a < 2 ^ (m - 2) := by
+        have g₀₀: a + a < b + a := by simp [h₂.1]
+        rw [h₈, ← mul_two a] at g₀₀
+        have g₀₁: m - 1 = Nat.succ (m - 2) := by
+          rw [← Nat.succ_sub ?_]
+          . rw [succ_eq_add_one]
+            omega
+          . linarith
+        rw [g₀₁, Nat.pow_succ 2 _] at g₀₀
+        exact Nat.lt_of_mul_lt_mul_right g₀₀
+      have h₉₀: b = 2 ^ (m - 1) - a := by
+        symm
+        exact Nat.sub_eq_of_eq_add h₈.symm
+      rw [h₈, h₉₀] at h₆
+      repeat rw [Nat.mul_sub_right_distrib] at h₆
+      repeat rw [← Nat.pow_add] at h₆
+      have hm1: 1 ≤ m := by
+        linarith
+      repeat rw [← Nat.sub_add_comm hm1] at h₆
+      repeat rw [← Nat.add_sub_assoc hm1] at h₆
+      ring_nf at h₆
+      rw [← Nat.sub_add_eq _ 1 1] at h₆
+      norm_num at h₆
+      rw [← Nat.sub_add_eq _ (a * 2 ^ (m - 1)) (a * 2 ^ (m - 1))] at h₆
+      rw [← two_mul (a * 2 ^ (m - 1))] at h₆
+      rw [mul_comm 2 _] at h₆
+      rw [mul_assoc a (2 ^ (m - 1)) 2] at h₆
+      rw [← Nat.pow_succ, succ_eq_add_one] at h₆
+      rw [Nat.sub_add_cancel hm1] at h₆
+      rw [← Nat.sub_add_eq ] at h₆
+      have h₉₁: 2 ^ (m * 2 - 1) = 2 ^ (m * 2 - 2) - a * 2 ^ m + (a * 2 ^ m + a * 2 ^ k) := by
+        refine Nat.eq_add_of_sub_eq ?_ h₆
+        by_contra! hc
+        have g₁: 2 ^ (m * 2 - 1) - (a * 2 ^ m + a * 2 ^ k) = 0 := by
+          exact Nat.sub_eq_zero_of_le (le_of_lt hc)
+        rw [g₁] at h₆
+        have g₂: 2 ^ (m * 2 - 2) ≤ a * 2 ^ m := by exact Nat.le_of_sub_eq_zero h₆.symm
+        have g₃: 2 ^ (m - 2) ≤ a := by
+          rw [mul_two, Nat.add_sub_assoc (by linarith) m] at g₂
+          rw [Nat.pow_add, mul_comm] at g₂
+          refine Nat.le_of_mul_le_mul_right g₂ ?_
+          exact Nat.two_pow_pos m
+        linarith [g₀, g₃]
+      rw [← Nat.add_assoc] at h₉₁
+      have h₉₂: a * 2 ^ k = 2 * 2 ^ (2 * m - 2) - 2 ^ (2 * m - 2) := by
+        rw [Nat.sub_add_cancel ?_] at h₉₁
+        . rw [add_comm] at h₉₁
+          symm
+          rw [← Nat.pow_succ', succ_eq_add_one]
+          rw [← Nat.sub_add_comm ?_]
+          . refine Nat.sub_eq_of_eq_add ?_
+            rw [mul_comm 2 m, ← h₉₁]
+            exact rfl
+          . linarith [hm1]
+        . refine le_of_lt ?_
+          rw [mul_two, Nat.add_sub_assoc, Nat.pow_add, mul_comm (2 ^ m) _]
+          refine (Nat.mul_lt_mul_right (by linarith)).mpr g₀
+          linarith
+      nth_rewrite 2 [← Nat.one_mul (2 ^ (2 * m - 2))] at h₉₂
+      rw [← Nat.mul_sub_right_distrib 2 1 (2 ^ (2 * m - 2))] at h₉₂
+      norm_num at h₉₂
+      refine Nat.eq_div_of_mul_eq_left ?_ h₉₂
+      exact Ne.symm (NeZero.ne' (2 ^ k))
+    by_cases hk2m: k ≤ 2 * m - 2
+    . rw [Nat.pow_div hk2m (by norm_num)] at h₉
+      rw [Nat.sub_right_comm (2*m) 2 k] at h₉
+      by_contra! hc
+      cases' (lt_or_gt_of_ne hc) with hc₀ hc₁
+      . interval_cases a
+        linarith
+      . have hc₂: ¬ Odd a := by
+          refine (not_odd_iff_even).mpr ?_
+          have hc₃: 1 ≤ 2 * m - k - 2 := by
+            by_contra! hc₄
+            interval_cases (2 * m - k - 2)
+            simp at h₉
+            rw [h₉] at hc₁
+            contradiction
+          have hc₄: 2 * m - k - 2 = succ (2 * m - k - 3) := by
+            rw [succ_eq_add_one]
+            exact Nat.eq_add_of_sub_eq hc₃ rfl
+          rw [h₉, hc₄, Nat.pow_succ']
+          exact even_two_mul (2 ^ (2 * m - k - 3))
+        exact hc₂ h₁.1
+    . push_neg at hk2m
+      exfalso
+      have ha: a = 0 := by
+        rw [h₉]
+        refine (Nat.div_eq_zero_iff).mpr ?_
+        right
+        exact Nat.pow_lt_pow_right (by norm_num) hk2m
+      linarith [ha, h₀.1]

imo_proofs/imo_1985_p6.lean

@@ -0,0 +1,1318 @@
+import Mathlib
+
+set_option linter.unusedVariables.analyzeTactics true
+
+lemma aux_1
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n)) :
+  ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x := by
+  intros n x hp
+  have hz₇: n ≤ 7 ∨ 7 < n := by
+    exact le_or_lt n 7
+  cases' hp with hn₀ hx₀
+  by_cases hn₁: 1 < n
+  . refine Nat.le_induction ?_ ?_ n hn₁
+    . rw [h₁ 1 x (by norm_num)]
+      rw [h₀ x]
+      refine mul_pos hx₀ ?_
+      refine add_pos hx₀ (by norm_num)
+    . intros m hm₀ hm₁
+      rw [h₁ m x (by linarith)]
+      refine mul_pos hm₁ ?_
+      refine add_pos hm₁ ?_
+      refine one_div_pos.mpr ?_
+      norm_cast
+      exact Nat.zero_lt_of_lt hm₀
+  . interval_cases n
+    rw [h₀ x]
+    exact hx₀
+
+
+lemma aux_2
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x) :
+  ∀ (n : ℕ) (x y : NNReal), 0 < n → x < y → f n x < f n y := by
+  intros n x y hn hxy
+  by_cases hn₁: 1 < n
+  . refine Nat.le_induction ?_ ?_ n hn₁
+    . rw [h₁ 1 x (by norm_num)]
+      rw [h₁ 1 y (by norm_num)]
+      norm_num
+      refine mul_lt_mul ?_ ?_ ?_ ?_
+      . rw [h₀ x, h₀ y]
+        exact hxy
+      . refine _root_.add_le_add ?_ (by norm_num)
+        rw [h₀ x, h₀ y]
+        exact le_of_lt hxy
+      . refine add_pos_of_nonneg_of_pos ?_ (by linarith)
+        rw [h₀ x]
+        exact NNReal.zero_le_coe
+      . refine le_of_lt ?_
+        refine h₂ 1 y ?_
+        norm_num
+        exact pos_of_gt hxy
+    . intros m hm₀ hm₁
+      rw [h₁ m x (by linarith)]
+      rw [h₁ m y (by linarith)]
+      refine mul_lt_mul hm₁ ?_ ?_ ?_
+      . refine _root_.add_le_add ?_ (by norm_num)
+        exact le_of_lt hm₁
+      . refine add_pos_of_nonneg_of_pos ?_ ?_
+        . exact h₃ m x (by linarith)
+        . refine one_div_pos.mpr ?_
+          norm_cast
+          exact Nat.zero_lt_of_lt hm₀
+      . refine le_of_lt ?_
+        refine h₂ m y ?_
+        constructor
+        . exact Nat.zero_lt_of_lt hm₀
+        . exact pos_of_gt hxy
+  . interval_cases n
+    rw [h₀ x, h₀ y]
+    exact hxy
+
+
+lemma aux_3
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₄ : ∀ (n : ℕ) (x y : NNReal), 0 < n → x < y → f n x < f n y) :
+  ∀ (n : ℕ) (x : NNReal), 1 < n ∧ 1 ≤ x → 1 < f n x := by
+  intros n x hx₀
+  cases' hx₀ with hn₀ hx₁
+  have g₂₀: f n 1 ≤ f n x := by
+    by_cases hx₂: 1 < x
+    . refine le_of_lt ?_
+      refine h₄ n 1 x ?_ hx₂
+      exact Nat.zero_lt_of_lt hn₀
+    . push_neg at hx₂
+      have hx₃: x = 1 := by exact le_antisymm hx₂ hx₁
+      rw [hx₃]
+  have g₂₁: f 1 1 < f n 1 := by
+    rw [h₀]
+    refine Nat.le_induction ?_ ?_ n hn₀
+    . rw [h₁ 1 1 (by norm_num), h₀]
+      norm_num
+    . intros m hm₀ hm₁
+      rw [h₁ m 1 (by linarith)]
+      refine one_lt_mul_of_lt_of_le hm₁ ?_
+      nth_rw 1 [← add_zero 1]
+      refine add_le_add ?_ ?_
+      . exact le_of_lt hm₁
+      . refine one_div_nonneg.mpr ?_
+        exact Nat.cast_nonneg' m
+  refine lt_of_lt_of_le ?_ g₂₀
+  exact (lt_iff_lt_of_cmp_eq_cmp (congrFun (congrArg cmp (h₀ 1)) (f n 1))).mp g₂₁
+
+
+lemma aux_4
+  (f : ℕ → NNReal → ℝ)
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (h₄ : ∀ (n : ℕ) (x y : NNReal), 0 < n → x < y → f n x < f n y)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₀ : f₀ = fun n x => (f n x).toNNReal) :
+  ∀ (n : ℕ), 0 < n → StrictMono (f₀ n) := by
+  intros n hn₀
+  refine Monotone.strictMono_of_injective ?_ ?_
+  . refine monotone_iff_forall_lt.mpr ?_
+    intros a b hab
+    refine le_of_lt ?_
+    rw [hf₀]
+    exact (Real.toNNReal_lt_toNNReal_iff_of_nonneg (h₃ n a hn₀)).mpr (h₄ n a b hn₀ hab)
+  . intros p q hpq
+    contrapose! hpq
+    apply lt_or_gt_of_ne at hpq
+    cases' hpq with hpq hpq
+    . refine ne_of_lt ?_
+      rw [hf₀]
+      exact (Real.toNNReal_lt_toNNReal_iff_of_nonneg (h₃ n p hn₀)).mpr (h₄ n p q hn₀ hpq)
+    . symm
+      refine ne_of_lt ?_
+      rw [hf₀]
+      exact (Real.toNNReal_lt_toNNReal_iff_of_nonneg (h₃ n q hn₀)).mpr (h₄ n q p hn₀ hpq)
+
+
+lemma aux_5
+  (f : ℕ → NNReal → ℝ)
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hfi : fi = fun n => Function.invFun (f₀ n)):
+  ∀ (n : ℕ) (x y : NNReal), 0 < n → f₀ n x = y → fi n y = x := by
+  intros n x y hn₀ hn₁
+  have hf₃: ∀ n y, fi n y = Function.invFun (f₀ n) y := by
+    exact fun n y => congrFun (congrFun hfi n) y
+  rw [← hn₁, hf₃]
+  have hmo₃: ∀ n, 0 < n → Function.Injective (f₀ n) := by
+    exact fun n a => StrictMono.injective (hmo₂ n a)
+  have hn₂: (Function.invFun (f₀ n)) ∘ (f₀ n) = id := by exact Function.invFun_comp (hmo₃ n hn₀)
+  rw [Function.comp_def (Function.invFun (f₀ n)) (f₀ n)] at hn₂
+  have hn₃: (fun x => Function.invFun (f₀ n) (f₀ n x)) x = id x := by exact Eq.symm (NNReal.eq (congrArg NNReal.toReal (congrFun (id (Eq.symm hn₂)) x)))
+  exact hmo₁ n hn₀ (congrArg (f n) hn₃)
+
+
+lemma aux_6
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₀ : f₀ = fun n x => (f n x).toNNReal) :
+  ∀ (n : ℕ), 0 < n → Continuous (f₀ n) := by
+  intros n hn₀
+  rw [hf₀]
+  refine Continuous.comp' ?_ ?_
+  . exact continuous_real_toNNReal
+  . refine Nat.le_induction ?_ ?_ n hn₀
+    . have hn₁: f 1 = fun (x:NNReal) => (x:ℝ) := by exact (Set.eqOn_univ (f 1) fun x => ↑x).mp fun ⦃x⦄ _ => h₀ x
+      rw [hn₁]
+      exact NNReal.continuous_coe
+    . intros d hd₀ hd₁
+      have hd₂: f (d + 1) = fun x => f d x * (f d x + 1 / ↑d) := by
+        exact (Set.eqOn_univ (f (d + 1)) fun x => f d x * (f d x + 1 / ↑d)).mp fun ⦃x⦄ _ => h₁ d x hd₀
+      rw [hd₂]
+      refine Continuous.mul hd₁ ?_
+      refine Continuous.add hd₁ ?_
+      exact continuous_const
+
+
+lemma aux_7
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₃ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 ≤ f n x)
+  (h₅ : ∀ (n : ℕ) (x : NNReal), 1 < n ∧ 1 ≤ x → 1 < f n x)
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (hmo₄ : ∀ (n : ℕ), 0 < n → Continuous (f₀ n)) :
+  ∀ (n : ℕ), 0 < n → Function.Surjective (f₀ n) := by
+  intros n hn₀
+  refine Continuous.surjective (hmo₄ n hn₀) ?_ ?_
+  . refine Monotone.tendsto_atTop_atTop ?_ ?_
+    . exact StrictMono.monotone (hmo₂ n hn₀)
+    . intro b
+      use (b + 1)
+      refine Nat.le_induction ?_ ?_ n hn₀
+      . rw [hf₂ 1 (b + 1) (by linarith), h₀]
+        simp
+      . intros d hd₀ hd₁
+        rw [hf₂ (d + 1) (b + 1) (by linarith), h₁ d (b + 1) (by linarith)]
+        have hd₂: b ≤ f d (b + 1) := by
+          rw [hf₂ d (b + 1) (by linarith)] at hd₁
+          exact (Real.le_toNNReal_iff_coe_le (h₃ d (b + 1) hd₀)).mp hd₁
+        have hd₃: 1 < (f d (b + 1) + 1 / ↑d) := by
+          by_cases hd₄: 1 < d
+          . refine lt_add_of_lt_of_pos ?_ ?_
+            . refine h₅ d (b + 1) ?_
+              constructor
+              . exact hd₄
+              . exact le_add_self
+            . refine div_pos (by linarith) ?_
+              exact Nat.cast_pos'.mpr hd₀
+          . have hd₅: d = 1 := by linarith
+            rw [hd₅, h₀]
+            simp
+            norm_cast
+            refine add_pos_of_nonneg_of_pos ?_ ?_
+            . exact _root_.zero_le b
+            . exact zero_lt_one' NNReal
+        refine NNReal.le_toNNReal_of_coe_le ?_
+        nth_rw 1 [← mul_one (↑b:ℝ)]
+        refine mul_le_mul hd₂ (le_of_lt hd₃) (by linarith) ?_
+        exact h₃ d (b + 1) hd₀
+  . refine Filter.tendsto_atBot_atBot.mpr ?_
+    intro b
+    use 0
+    intro a ha₀
+    have ha₁: a = 0 := by exact nonpos_iff_eq_zero.mp ha₀
+    have ha₂: f₀ n 0 = 0 := by
+      refine Nat.le_induction ?_ ?_ n hn₀
+      . rw [hf₂ 1 0 (by linarith), h₀]
+        exact Real.toNNReal_coe
+      . intros d hd₀ hd₁
+        rw [hf₂ (d + 1) 0 (by linarith), h₁ d 0 (by linarith)]
+        have hd₂: 0 ≤ f d 0 := by exact h₃ d 0 hd₀
+        have hd₃: f d 0 = 0 := by
+          rw [hf₂ d 0 (by linarith)] at hd₁
+          apply Real.toNNReal_eq_zero.mp at hd₁
+          exact eq_of_le_of_le hd₁ hd₂
+        rw [hd₃, zero_mul]
+        exact Real.toNNReal_zero
+    rw [ha₁, ha₂]
+    exact _root_.zero_le b
+
+
+lemma aux_8
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (hmo₁ : ∀ (n : ℕ), 0 < n → Function.Injective (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (sn : Set ℕ)
+  (fb : ↑sn → NNReal)
+  (hsn₁ : ∀ (n : ↑sn), ↑n ∈ sn ∧ 0 < n.1)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n) :
+  ∀ (n : ↑sn), fb n < 1 := by
+  intros n
+  have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+  let z := fb n
+  have hz₀: z = fb n := by rfl
+  rw [← hz₀]
+  by_contra! hc₀
+  have hc₁: 1 ≤ f n z := by
+    by_cases hn₁: 1 < (n:ℕ)
+    . refine le_of_lt ?_
+      refine aux_3 f h₀ h₁ ?_ (↑n) z ?_
+      . exact fun n x y a a_1 => hmo₀ n a a_1
+      . exact ⟨hn₁, hc₀⟩
+    . have hn₂: (n:ℕ) = 1 := by linarith
+      rw [hn₂, h₀]
+      exact hc₀
+  have hz₁: f₀ n z = 1 - 1 / n := by
+    exact hfb₁ n
+  have hz₃: f n z = 1 - 1 / n := by
+    rw [hf₂ n z hn₀] at hz₁
+    by_cases hn₁: 1 < (n:ℕ)
+    . have hz₂: 1 - 1 / (n:NNReal) ≠ 0 := by
+        have g₀: (n:NNReal) ≠ 0 := by
+          norm_cast
+          linarith
+        nth_rw 1 [← div_self g₀, ← NNReal.sub_div]
+        refine div_ne_zero ?_ g₀
+        norm_cast
+        exact Nat.sub_ne_zero_iff_lt.mpr hn₁
+      apply (Real.toNNReal_eq_iff_eq_coe hz₂).mp at hz₁
+      rw [hz₁]
+      exact Eq.symm ((fun {r} {p:NNReal} hp => (Real.toNNReal_eq_iff_eq_coe hp).mp) hz₂ (hmo₁ n hn₀ rfl))
+    . have hn₂: (n:ℕ) = 1 := by linarith
+      rw [hn₂, h₀] at hz₁
+      simp at hz₁
+      rw [hn₂, h₀, hz₁]
+      simp
+  rw [hz₃] at hc₁
+  have hz₄: 0 < 1 / (n:ℝ) := by
+    refine div_pos (by linarith) ?_
+    exact Nat.cast_pos'.mpr hn₀
+  linarith
+
+
+lemma aux_9
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (hmo₂ : ∀ (n : ℕ), 0 < n → StrictMono (f₀ n))
+  (fi : ℕ → NNReal → NNReal)
+  (hf₅ : ∀ (x : NNReal), fi 1 x = x)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (hf₇ : ∀ (n : ℕ) (x y : NNReal), 0 < n → (f₀ n x = y ↔ fi n y = x))
+  (fb : ℕ → NNReal)
+  (hfb₀ : fb = fun n => fi n (1 - 1 / ↑n))
+  (sn : Set ℕ)
+  (hsn : sn = Set.Ici 1) :
+  StrictMonoOn fb sn := by
+  rw [hsn]
+  refine strictMonoOn_Ici_of_pred_lt ?hψ
+  intros m hm₀
+  rw [hfb₀]
+  refine Nat.le_induction ?_ ?_ m hm₀
+  . have g₁: fi 1 0 = 0 := by exact hf₅ 0
+    have g₂: (2:NNReal).IsConjExponent (2:NNReal) := by
+      refine (NNReal.isConjExponent_iff_eq_conjExponent ?_).mpr ?_
+      . exact one_lt_two
+      . norm_cast
+        simp
+    simp
+    norm_cast
+    rw [g₁, NNReal.IsConjExponent.one_sub_inv g₂]
+    let x := fi 2 2⁻¹
+    have hx₀: x = fi 2 2⁻¹ := by rfl
+    have hx₁: f₀ 2 x = 2⁻¹ := by
+      rw [hx₀]
+      have g₃: Function.RightInverse (fi 2) (f₀ 2) := by exact hmo₇ 2 (by linarith)
+      exact g₃ 2⁻¹
+    rw [← hx₀]
+    contrapose! hx₁
+    have hc₁: x = 0 := by exact nonpos_iff_eq_zero.mp hx₁
+    have hc₃: f₀ 2 x = 0 := by
+      rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+      norm_cast
+      rw [zero_mul]
+      exact Real.toNNReal_zero
+    rw [hc₃]
+    exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₂)
+  . simp
+    intros n hn₀ _
+    let i := fi n (1 - (↑n)⁻¹)
+    let j := fi (n + 1) (1 - ((↑n:NNReal) + 1)⁻¹)
+    have hi₀: i = fi n (1 - (↑n)⁻¹) := by rfl
+    have hj₀: j = fi (n + 1) (1 - ((↑n:NNReal) + 1)⁻¹) := by rfl
+    have hi₁: f₀ n i = (1 - (↑n)⁻¹) := by exact (hf₇ n i (1 - (↑n:NNReal)⁻¹) (by linarith)).mpr hi₀.symm
+    have hj₁: f₀ (n + 1) j = (1 - ((↑n:NNReal) + 1)⁻¹) := by
+      exact (hf₇ (n + 1) j _ (by linarith)).mpr hj₀.symm
+    have hj₂: (1 - ((↑n:NNReal) + 1)⁻¹) = (1 - ((n:ℝ) + 1)⁻¹).toNNReal := by
+      exact rfl
+    have hn₂: f₀ (n + 1) i < f₀ (n + 1) j := by
+      rw [hj₁, hj₂, hf₂ (n + 1) _ (by linarith), h₁ n i (by linarith)]
+      rw [hf₁ n i (by linarith), hi₁]
+      refine (Real.toNNReal_lt_toNNReal_iff ?_).mpr ?_
+      . refine sub_pos.mpr ?_
+        refine inv_lt_one_of_one_lt₀ ?_
+        norm_cast
+        exact Nat.lt_add_right 1 hn₀
+      . have g₀: (↑n:NNReal)⁻¹ ≤ 1 := by exact Nat.cast_inv_le_one n
+        rw [NNReal.coe_sub g₀, NNReal.coe_inv]
+        simp
+        refine inv_strictAnti₀ ?_ ?_
+        . norm_cast
+          exact Nat.zero_lt_of_lt hn₀
+        . norm_cast
+          exact lt_add_one n
+    refine (StrictMono.lt_iff_lt ?_).mp hn₂
+    exact hmo₂ (n + 1) (by linarith)
+
+
+lemma aux_10
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ (x : NNReal), f 1 x = ↑x)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₂ : ∀ (n : ℕ) (x : NNReal), 0 < n → f₀ n x = (f n x).toNNReal)
+  (fi : ℕ → NNReal → NNReal)
+  (hmo₇ : ∀ (n : ℕ), 0 < n → Function.RightInverse (fi n) (f₀ n))
+  (sn : Set ℕ)
+  (sb : Set NNReal)
+  (fb : ↑sn → NNReal)
+  (hsn₀ : sn = Set.Ici 1)
+  (hfb₀ : fb = fun n:↑sn => fi (↑n) (1 - 1 / ↑��n))
+  (hsb₀ : sb = Set.range fb)
+  (fr : NNReal → ℝ)
+  (hfr: fr = fun x => ↑x)
+  (sbr : Set ℝ)
+  (hsbr: sbr = fr '' sb)
+  (br: ℝ)
+  (hbr₀ : IsLUB sbr br) :
+  0 < br := by
+  have hnb₀: 2 ∈ sn := by
+    rw [hsn₀]
+    decide
+  let nb : ↑sn := ⟨2, hnb₀⟩
+  have g₀: 0 < fb nb := by
+    have g₁: (2:NNReal).IsConjExponent (2:NNReal) := by
+      refine (NNReal.isConjExponent_iff_eq_conjExponent ?_).mpr ?_
+      . exact one_lt_two
+      . norm_cast
+        simp
+    rw [hfb₀]
+    simp
+    have hnb₁: nb.val = 2 := by exact rfl
+    rw [hnb₁]
+    norm_cast
+    rw [NNReal.IsConjExponent.one_sub_inv g₁]
+    let x := fi 2 2⁻¹
+    have hx₀: x = fi 2 2⁻¹ := by rfl
+    have hx₁: f₀ 2 x = 2⁻¹ := by
+      rw [hx₀]
+      have g₃: Function.RightInverse (fi 2) (f₀ 2) := by exact hmo₇ 2 (by linarith)
+      exact g₃ 2⁻¹
+    rw [← hx₀]
+    contrapose! hx₁
+    have hc₁: x = 0 := by exact nonpos_iff_eq_zero.mp hx₁
+    have hc₃: f₀ 2 x = 0 := by
+      rw [hc₁, hf₂ 2 0 (by linarith), h₁ 1 0 (by linarith), h₀ 0]
+      norm_cast
+      rw [zero_mul]
+      exact Real.toNNReal_zero
+    rw [hc₃]
+    exact Ne.symm (NNReal.IsConjExponent.inv_ne_zero g₁)
+  have g₁: ∃ x, 0 < x ∧ x ∈ sbr := by
+    use (fb nb).toReal
+    constructor
+    . exact g₀
+    . rw [hsbr]
+      simp
+      use fb ↑nb
+      constructor
+      . rw [hsb₀]
+        exact Set.mem_range_self nb
+      . exact congrFun hfr (fb ↑nb)
+  obtain ⟨x, hx₀, hx₁⟩ := g₁
+  have hx₂: br ∈ upperBounds sbr := by
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  exact gt_of_ge_of_gt (hx₂ hx₁) hx₀
+
+
+lemma aux_11
+  (sn : Set ℕ)
+  (fb fc : ↑sn → NNReal)
+  (hfc₂ : ∀ (n : ↑sn), fb n < fc n)
+  (hfb₃ : StrictMono fb)
+  (hfc₃ : StrictAnti fc)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x ↦ ↑x)
+  (sbr scr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (hscr : scr = fr '' sc)
+  (br cr : ℝ)
+  (hbr₀ : IsLUB sbr br)
+  (hcr₀ : IsGLB scr cr)
+  (hfb₄ : ∀ (n : ↑sn), 0 ≤ fb n) :
+  br ≤ cr := by
+  have hfc₄: ∀ nb nc, fb nb < fc nc := by
+    intros nb nc
+    cases' (lt_or_le nb nc) with hn₀ hn₀
+    . refine lt_trans ?_ (hfc₂ nc)
+      exact hfb₃ hn₀
+    cases' lt_or_eq_of_le hn₀ with hn₁ hn₁
+    . refine lt_trans (hfc₂ nb) ?_
+      exact hfc₃ hn₁
+    . rw [hn₁]
+      exact hfc₂ nb
+  by_contra! hc₀
+  have hc₁: ∃ x ∈ sbr, cr < x ∧ x ≤ br := by exact IsLUB.exists_between hbr₀ hc₀
+  let ⟨x, hx₀, hx₁, _⟩ := hc₁
+  have hc₂: ∃ y ∈ scr, cr ≤ y ∧ y < x := by exact IsGLB.exists_between hcr₀ hx₁
+  let ⟨y, hy₀, _, hy₂⟩ := hc₂
+  have hc₃: x < y := by
+    have hx₃: x.toNNReal ∈ sb := by
+      rw [hsbr] at hx₀
+      apply (Set.mem_image fr sb x).mp at hx₀
+      obtain ⟨z, hz₀, hz₁⟩ := hx₀
+      rw [← hz₁, hfr, Real.toNNReal_coe]
+      exact hz₀
+    have hy₃: y.toNNReal ∈ sc := by
+      rw [hscr] at hy₀
+      apply (Set.mem_image fr sc y).mp at hy₀
+      obtain ⟨z, hz₀, hz₁⟩ := hy₀
+      rw [← hz₁, hfr, Real.toNNReal_coe]
+      exact hz₀
+    rw [hsb₀] at hx₃
+    rw [hsc₀] at hy₃
+    apply Set.mem_range.mp at hx₃
+    apply Set.mem_range.mp at hy₃
+    let ⟨nx, hnx₀⟩ := hx₃
+    let ⟨ny, hny₀⟩ := hy₃
+    have hy₄: 0 < y := by
+      contrapose! hy₃
+      have hy₅: y.toNNReal = 0 := by exact Real.toNNReal_of_nonpos hy₃
+      intro z
+      rw [hy₅]
+      refine ne_of_gt ?_
+      refine lt_of_le_of_lt ?_ (hfc₂ z)
+      exact hfb₄ z
+    refine (Real.toNNReal_lt_toNNReal_iff hy₄).mp ?_
+    rw [← hnx₀, ← hny₀]
+    exact hfc₄ nx ny
+  refine (lt_self_iff_false x).mp ?_
+  exact lt_trans hc₃ hy₂
+
+
+lemma aux_exists
+  (f : ℕ → NNReal → ℝ)
+  (h₂ : ∀ (n : ℕ) (x : NNReal), 0 < n ∧ 0 < x → 0 < f n x)
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (f₀ : ℕ → NNReal → NNReal)
+  (hf₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x = ↑(f₀ n x))
+  (sn : Set ℕ)
+  (hsn₀ : sn = Set.Ici 1)
+  (fb fc : ↑sn → NNReal)
+  (hfb₁ : ∀ (n : ↑sn), f₀ (↑n) (fb n) = 1 - 1 / ↑↑n)
+  (hfc₁ : ∀ (n : ↑sn), f₀ (↑n) (fc n) = 1)
+  (hfb₃ : StrictMono fb)
+  (hfc₃ : StrictAnti fc)
+  (sb sc : Set NNReal)
+  (hsb₀ : sb = Set.range fb)
+  (hsc₀ : sc = Set.range fc)
+  (fr : NNReal → ℝ)
+  (hfr : fr = fun x => ↑x)
+  (sbr scr : Set ℝ)
+  (hsbr : sbr = fr '' sb)
+  (hscr : scr = fr '' sc)
+  (br cr : ℝ)
+  (h₈ : ∀ (n : ℕ) (x : NNReal), 0 < n → 0 < x → 1 - 1 / ↑n < f n x → f n x < f (n + 1) x)
+  (hbr₁ : 0 < br)
+  (hu₅ : br ≤ cr)
+  (hbr₃ : ∀ x ∈ sbr, x ≤ br)
+  (hcr₃ : ∀ x ∈ scr, cr ≤ x) :
+  ∃ x, ∀ (n : ℕ), 0 < n → 0 < f n x ∧ f n x < f (n + 1) x ∧ f (n + 1) x < 1 := by
+  cases' lt_or_eq_of_le hu₅ with hu₆ hu₆
+  . apply exists_between at hu₆
+    let ⟨a, ha₀, ha₁⟩ := hu₆
+    have ha₂: 0 < a := by exact gt_trans ha₀ hbr₁
+    have ha₃: 0 < a.toNNReal := by exact Real.toNNReal_pos.mpr ha₂
+    use a.toNNReal
+    intros n hn₀
+    have hn₁: n ∈ sn := by
+      rw [hsn₀]
+      exact hn₀
+    constructor
+    . exact h₂ n a.toNNReal ⟨hn₀, ha₃⟩
+    constructor
+    . refine h₈ n a.toNNReal hn₀ ?_ ?_
+      . exact Real.toNNReal_pos.mpr ha₂
+      . let nn : ↑sn := ⟨n, hn₁⟩
+        have hn₂: f n (fb nn) = 1 - 1 / n := by
+          rw [hf₁ n _ hn₀, hfb₁ nn]
+          refine NNReal.coe_sub ?_
+          refine div_le_self ?_ ?_
+          . exact zero_le_one' NNReal
+          . exact Nat.one_le_cast.mpr hn₀
+        rw [← hn₂]
+        refine hmo₀ n hn₀ ?_
+        refine Real.lt_toNNReal_iff_coe_lt.mpr ?_
+        refine lt_of_le_of_lt ?_ ha₀
+        refine hbr₃ _ ?_
+        rw [hsbr]
+        refine (Set.mem_image fr sb _).mpr ?_
+        use (fb nn)
+        rw [hfr, hsb₀]
+        refine ⟨?_, rfl⟩
+        exact Set.mem_range_self nn
+    . have hn₂: n + 1 ∈ sn := by
+        rw [hsn₀]
+        exact Set.mem_Ici.mpr (by linarith)
+      let nn : ↑sn := ⟨n + 1, hn₂⟩
+      have hn₃: f (n + 1) (fc (nn)) = 1 := by
+        rw [hf₁ (n + 1) _ (by linarith), hfc₁ nn]
+        exact rfl
+      rw [← hn₃]
+      refine hmo₀ (n + 1) (by linarith) ?_
+      refine (Real.toNNReal_lt_iff_lt_coe (le_of_lt ha₂)).mpr ?_
+      refine lt_of_lt_of_le ha₁ ?_
+      refine hcr₃ _ ?_
+      rw [hscr]
+      refine (Set.mem_image fr sc _).mpr ?_
+      use (fc nn)
+      rw [hfr, hsc₀]
+      refine ⟨?_, rfl⟩
+      exact Set.mem_range_self nn
+  . use br.toNNReal
+    intros n hn₀
+    have hn₁: n ∈ sn := by
+      rw [hsn₀]
+      exact hn₀
+    constructor
+    . refine h₂ n br.toNNReal ⟨hn₀, ?_⟩
+      exact Real.toNNReal_pos.mpr hbr₁
+    constructor
+    . refine h₈ n br.toNNReal hn₀ ?_ ?_
+      . exact Real.toNNReal_pos.mpr hbr₁
+      . let nn : ↑sn := ⟨n, hn₁⟩
+        have hn₂: fb nn < br := by
+          by_contra! hc₀
+          have hbr₅: (fb nn) = br := by
+            refine eq_of_le_of_le ?_ hc₀
+            refine hbr₃ _ ?_
+            rw [hsbr]
+            refine (Set.mem_image fr sb _).mpr ?_
+            use (fb nn)
+            rw [hfr, hsb₀]
+            constructor
+            . exact Set.mem_range_self nn
+            . exact rfl
+          have hn₂: n + 1 ∈ sn := by
+            rw [hsn₀]
+            refine Set.mem_Ici.mpr ?_
+            exact Nat.le_add_right_of_le hn₀
+          let ns : ↑sn := ⟨n + 1, hn₂⟩
+          have hc₁: fb nn < fb ns := by
+            refine hfb₃ ?_
+            refine Subtype.mk_lt_mk.mpr ?_
+            exact lt_add_one n
+          have hbr₆: fb ns ≤ fb nn := by
+            refine NNReal.coe_le_coe.mp ?_
+            rw [hbr₅]
+            refine hbr₃ _ ?_
+            rw [hsbr]
+            refine (Set.mem_image fr sb _).mpr ?_
+            use (fb ns)
+            rw [hfr, hsb₀]
+            refine ⟨?_, rfl⟩
+            exact Set.mem_range_self ns
+          refine (lt_self_iff_false (fb nn)).mp ?_
+          exact lt_of_lt_of_le hc₁ hbr₆
+        have hn₃: f n (fb nn) = 1 - 1 / n := by
+          rw [hf₁ n _ hn₀, hfb₁ nn]
+          refine NNReal.coe_sub ?_
+          refine div_le_self ?_ ?_
+          . exact zero_le_one' NNReal
+          . exact Nat.one_le_cast.mpr hn₀
+        rw [← hn₃]
+        refine hmo₀ n hn₀ ?_
+        exact Real.lt_toNNReal_iff_coe_lt.mpr hn₂
+    . have hn₂: n + 1 ∈ sn := by
+        rw [hsn₀]
+        exact Set.mem_Ici.mpr (by linarith)
+      let nn : ↑sn := ⟨n + 1, hn₂⟩
+      have hcr₁: 0 < cr := by exact gt_of_ge_of_gt hu₅ hbr₁
+      have hn₃: f (n + 1) (fc (nn)) = 1 := by
+        rw [hf₁ (n + 1) _ (by linarith), hfc₁ nn]
+        exact rfl
+      rw [← hn₃, hu₆]
+      refine hmo₀ (n + 1) (by linarith) ?_
+      refine (Real.toNNReal_lt_iff_lt_coe (le_of_lt hcr₁)).mpr ?_
+      by_contra! hc₀
+      have hc₁: fc nn = cr := by
+        refine eq_of_le_of_le hc₀ ?_
+        refine hcr₃ _ ?_
+        rw [hscr]
+        refine (Set.mem_image fr sc _).mpr ?_
+        use (fc nn)
+        rw [hfr, hsc₀]
+        refine ⟨?_, rfl⟩
+        exact Set.mem_range_self nn
+      have hn₄: n + 2 ∈ sn := by
+        rw [hsn₀]
+        refine Set.mem_Ici.mpr ?_
+        exact Nat.le_add_right_of_le hn₀
+      let ns : ↑sn := ⟨n + 2, hn₄⟩
+      have hn₅: fc ns < fc nn := by
+        refine hfc₃ ?_
+        refine Subtype.mk_lt_mk.mpr ?_
+        exact Nat.lt_add_one (n + 1)
+      have hc₂: fc nn ≤ fc ns := by
+        refine NNReal.coe_le_coe.mp ?_
+        rw [hc₁]
+        refine hcr₃ _ ?_
+        rw [hscr]
+        refine (Set.mem_image fr sc _).mpr ?_
+        use (fc ns)
+        rw [hfr, hsc₀]
+        refine ⟨?_, rfl⟩
+        exact Set.mem_range_self ns
+      refine (lt_self_iff_false (fc ns)).mp ?_
+      exact lt_of_lt_of_le hn₅ hc₂
+
+
+
+
+lemma aux_unique_top_ind
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n)
+  (a b : NNReal)
+  (ha₀ : a < b)
+  (hd₃: ∀ (nd : ↑sd), nd.1 + (1:ℕ) ∈ sd)
+  (hd₂ : ∀ (nd : ↑sd), fd a b nd * (2 - 1 / ↑↑nd) ≤ fd a b ⟨nd.1 + 1, hd₃ nd⟩)
+  (hi₀ : 2 ∈ sd)
+  (i : ↑sd)
+  (hi₁ : i = ⟨2, hi₀⟩) :
+  ∀ (nd : ↑sd), fd a b i * (3 / 2) ^ (nd.1 - 2) ≤ fd a b nd := by
+  intro nd
+  rw [hfd₁ a b nd]
+  have hnd₀: 2 ≤ nd.1 := by
+    refine Set.mem_Ici.mp ?_
+    rw [← hsd]
+    exact nd.2
+  refine Nat.le_induction ?_ ?_ nd.1 hnd₀
+  . have hi₂: i.val = (2:ℕ) := by
+      simp_all only [Subtype.forall]
+    rw [hfd₁ a b i, hi₂]
+    simp
+  . simp
+    intros n hn₀ hn₁
+    have hn₂: n - 1 = n - 2 + 1 := by
+      simp
+      exact (Nat.sub_eq_iff_eq_add hn₀).mp rfl
+    have hn₃: n ∈ sd := by
+      rw [hsd]
+      exact hn₀
+    let nn : ↑sd := ⟨n, hn₃⟩
+    have hnn: nn.1 = n := by exact rfl
+    have hn₄: nn.1 + 1 ∈ sd := by
+      rw [hnn, hsd]
+      refine Set.mem_Ici.mpr ?_
+      exact Nat.le_add_right_of_le hn₀
+    have hn₅: fd a b nn * (2 - 1 / ↑n) ≤ fd a b ⟨nn.1 + 1, hn₄⟩ := by exact hd₂ nn
+    rw [hfd₁ a b ⟨nn.1 + 1, hn₄⟩] at hn₅
+    have hn₆: f (↑nn + 1) b - f (↑nn + 1) a = f (n + 1) b - f (n + 1) a := by exact rfl
+    rw [hn₆] at hn₅
+    refine le_trans ?_ hn₅
+    rw [hn₂, pow_succ (3/2) (n - 2), ← mul_assoc (fd a b i)]
+    refine mul_le_mul ?_ ?_ (by linarith) ?_
+    . refine le_of_le_of_eq hn₁ ?_
+      rw [hfd₁]
+    . refine (div_le_iff₀ (two_pos)).mpr ?_
+      rw [sub_mul, one_div_mul_eq_div _ 2]
+      refine le_sub_iff_add_le.mpr ?_
+      refine le_sub_iff_add_le'.mp ?_
+      refine (div_le_iff₀ ?_).mpr ?_
+      . refine Nat.cast_pos.mpr ?_
+        exact lt_of_lt_of_le (two_pos) hn₀
+      . ring_nf
+        exact Nat.ofNat_le_cast.mpr hn₀
+    . exact le_of_lt (hd₁ nn a b ha₀)
+
+
+
+lemma aux_unique_top
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n) :
+  ∀ (a b : NNReal),
+    a < b →
+      (∀ (n : ↑sd), f (↑n) a < f (↑n + 1) a ∧ f (↑n) b < f (↑n + 1) b)
+      → Filter.Tendsto (fd a b) Filter.atTop Filter.atTop := by
+  intros a b ha₀ ha₁
+  have hd₀: ∀ (nd:↑sd), (nd.1 + 1) ∈ sd := by
+    intro nd
+    let t : ℕ := nd.1
+    have ht: t = nd.1 := by rfl
+    rw [← ht, hsd]
+    refine Set.mem_Ici.mpr ?_
+    refine Nat.le_add_right_of_le ?_
+    refine Set.mem_Ici.mp ?_
+    rw [ht, ← hsd]
+    exact nd.2
+  have hd₂: ∀ nd, fd a b nd  * (2 - 1 / nd.1) ≤ fd a b ⟨nd.1 + 1, hd₀ nd⟩ := by
+    intro nd
+    have hnd₀: 0 < nd.1 := by
+      have g₀: 2 ≤ nd.1 := by
+        refine Set.mem_Ici.mp ?_
+        rw [← hsd]
+        exact nd.2
+      exact Nat.zero_lt_of_lt g₀
+    rw [hfd₁, hfd₁, h₁ nd.1 _ hnd₀, h₁ nd.1 _ hnd₀]
+    have hnd₁: f (↑nd) b * (f (↑nd) b + 1 / ↑↑nd) - f (↑nd) a * (f (↑nd) a + 1 / ↑↑nd) =
+      (f (↑nd) b - f (↑nd) a) * (f (↑nd) b + f (↑nd) a + 1 / nd.1) := by
+      ring_nf
+    rw [hnd₁]
+    refine (mul_le_mul_left ?_).mpr ?_
+    . rw [← hfd₁]
+      exact hd₁ nd a b ha₀
+    . refine le_sub_iff_add_le.mp ?_
+      rw [sub_neg_eq_add]
+      have hnd₂: 1 - 1 / nd.1 < f (↑nd) b := by
+        exact h₇ nd.1 b hnd₀ (ha₁ nd).2
+      have hnd₃: 1 - 1 / nd.1 < f (↑nd) a := by
+        exact h₇ nd.1 a hnd₀ (ha₁ nd).1
+      linarith
+  have hi: 2 ∈ sd := by
+    rw [hsd]
+    decide
+  let i : ↑sd := ⟨(2:ℕ), hi⟩
+  have hd₃: ∀ nd, fd a b i * (3 / 2) ^ (nd.1 - 2) ≤ fd a b nd := by
+    intro nd
+    exact aux_unique_top_ind f sd hsd fd hfd₁ hd₁ a b ha₀ hd₀ hd₂ hi i rfl nd
+  have hsd₁: Nonempty ↑sd := by
+    refine Set.Nonempty.to_subtype ?_
+    exact Set.nonempty_of_mem (hd₀ i)
+  refine Filter.tendsto_atTop_atTop.mpr ?_
+  intro z
+  by_cases hz₀: z ≤ fd a b i
+  . use i
+    intros j _
+    refine le_trans hz₀ ?_
+    refine le_trans ?_ (hd₃ j)
+    refine le_mul_of_one_le_right ?_ ?_
+    . refine le_of_lt ?_
+      exact hd₁ i a b ha₀
+    . refine one_le_pow₀ ?_
+      linarith
+  . push_neg at hz₀
+    have hz₁: 0 < fd a b i := by exact hd₁ i a b ha₀
+    have hz₂: 0 < Real.log (z / fd a b i) := by
+      refine Real.log_pos ?_
+      exact (one_lt_div hz₁).mpr hz₀
+    let j : ℕ := Nat.ceil (2 + Real.log (z / fd a b i) / Real.log (3 / 2))
+    have hj₀: 2 < j := by
+      refine Nat.lt_ceil.mpr ?_
+      norm_cast
+      refine lt_add_of_pos_right 2 ?_
+      refine div_pos ?_ ?_
+      . exact hz₂
+      . refine Real.log_pos ?_
+        linarith
+    have hj₁: j ∈ sd := by
+      rw [hsd]
+      exact Set.mem_Ici_of_Ioi hj₀
+    use ⟨j, hj₁⟩
+    intro k hk₀
+    have hk₁: fd a b i * (3 / 2) ^ (k.1 - 2) ≤ fd a b k := by
+      exact hd₃ k
+    have hk₂: i < k := by
+      refine lt_of_lt_of_le ?_ hk₀
+      refine Subtype.mk_lt_mk.mpr ?_
+      refine Nat.lt_ceil.mpr ?_
+      norm_cast
+      refine lt_add_of_pos_right 2 ?_
+      refine div_pos ?_ ?_
+      . exact hz₂
+      . refine Real.log_pos ?_
+        linarith
+    refine le_trans ?_ hk₁
+    refine (div_le_iff₀' ?_).mp ?_
+    . exact hz₁
+    . refine Real.le_pow_of_log_le (by linarith) ?_
+      refine (div_le_iff₀ ?_).mp ?_
+      . refine Real.log_pos ?_
+        linarith
+      . rw [Nat.cast_sub ?_]
+        . rw [Nat.cast_two]
+          refine le_sub_iff_add_le'.mpr ?_
+          exact Nat.le_of_ceil_le hk₀
+        . exact Nat.le_of_succ_le hk₂
+
+
+
+lemma aux_unique_nhds
+  (f : ℕ → NNReal → ℝ)
+  (sd : Set ℕ)
+  (hsd : sd = Set.Ici 2)
+  (fd : NNReal → NNReal → ↑sd → ℝ)
+  (hfd₁ : ∀ (y₁ y₂ : NNReal) (n : ↑sd), fd y₁ y₂ n = f (↑n) y₂ - f (↑n) y₁)
+  (hd₁ : ∀ (n : ↑sd) (a b : NNReal), a < b → 0 < fd a b n) :
+  ∀ (a b : NNReal),
+    a < b →
+      (∀ (n : ↑sd), (1 - 1 / ↑↑n < f (↑n) a ∧ 1 - 1 / ↑↑n < f (↑n) b) ∧ f (↑n) a < 1 ∧ f (↑n) b < 1) →
+        Filter.Tendsto (fd a b) Filter.atTop (nhds 0) := by
+  intros a b ha₀ ha₁
+  have hsd₁: Nonempty ↑sd := by
+    rw [hsd]
+    refine Set.Nonempty.to_subtype ?_
+    exact Set.nonempty_Ici
+  refine tendsto_atTop_nhds.mpr ?_
+  intros U hU₀ hU₁
+  have hU₂: U ∈ nhds 0 := by exact IsOpen.mem_nhds hU₁ hU₀
+  apply mem_nhds_iff_exists_Ioo_subset.mp at hU₂
+  obtain ⟨l, u, hl₀, hl₁⟩ := hU₂
+  have hl₂: 0 < u := by exact (Set.mem_Ioo.mpr hl₀).2
+  let nd := 2 + Nat.ceil (1/u)
+  have hnd₀: nd ∈ sd := by
+    rw [hsd]
+    refine Set.mem_Ici.mpr ?_
+    exact Nat.le_add_right 2 ⌈1 / u⌉₊
+  use ⟨nd, hnd₀⟩
+  intros n hn₀
+  refine (IsOpen.mem_nhds_iff hU₁).mp ?_
+  refine mem_nhds_iff.mpr ?_
+  use Set.Ioo l u
+  constructor
+  . exact hl₁
+  constructor
+  . exact isOpen_Ioo
+  . refine Set.mem_Ioo.mpr ?_
+    constructor
+    . refine lt_trans ?_ (hd₁ n a b ha₀)
+      exact (Set.mem_Ioo.mp hl₀).1
+    . have hn₁: fd a b n < 1 / n := by
+        rw [hfd₁]
+        have ha₂: 1 - 1 / n < f n a := by exact (ha₁ n).1.1
+        have hb₁: f n b < 1 := by exact (ha₁ n).2.2
+        refine sub_lt_iff_lt_add.mpr ?_
+        refine lt_trans hb₁ ?_
+        exact sub_lt_iff_lt_add'.mp ha₂
+      have hn₂: (1:ℝ) / n ≤ 1 / nd := by
+        refine one_div_le_one_div_of_le ?_ ?_
+        . refine Nat.cast_pos.mpr ?_
+          rw [hsd] at hnd₀
+          exact lt_of_lt_of_le (Nat.zero_lt_two) hnd₀
+        . exact Nat.cast_le.mpr hn₀
+      refine lt_of_lt_of_le hn₁ ?_
+      refine le_trans hn₂ ?_
+      refine div_le_of_le_mul₀ ?_ ?_ ?_
+      . exact Nat.cast_nonneg' nd
+      . exact le_of_lt hl₂
+      . have hl₃: u * (2 + 1 / u) ≤ u * ↑((2:ℕ) + ⌈(1:ℝ) / u⌉₊) := by
+          refine (mul_le_mul_left hl₂).mpr ?_
+          rw [Nat.cast_add 2 _, Nat.cast_two]
+          refine add_le_add_left ?_ 2
+          exact Nat.le_ceil (1 / u)
+        refine le_trans ?_ hl₃
+        rw [mul_add, mul_one_div u u, div_self (ne_of_gt hl₂)]
+        refine le_of_lt ?_
+        refine sub_lt_iff_lt_add.mp ?_
+        rw [sub_self 1]
+        exact mul_pos hl₂ (two_pos)
+
+
+
+
+lemma aux_unique
+  (f : ℕ → NNReal → ℝ)
+  (h₁ : ∀ (n : ℕ) (x : NNReal), 0 < n → f (n + 1) x = f n x * (f n x + 1 / ↑n))
+  (hmo₀ : ∀ (n : ℕ), 0 < n → StrictMono (f n))
+  (h₇ : ∀ (n : ℕ) (x : NNReal), 0 < n → f n x < f (n + 1) x → 1 - 1 / ↑n < f n x) :
+  ∀ (y₁ y₂ : NNReal),
+    (∀ (n : ℕ), 0 < n → 0 < f n y₁ ∧ f n y₁ < f (n + 1) y₁ ∧ f (n + 1) y₁ < 1) →
+      (∀ (n : ℕ), 0 < n → 0 < f n y₂ ∧ f n y₂ < f (n + 1) y₂ ∧ f (n + 1) y₂ < 1) → y₁ = y₂ := by
+  intros x y hx₀ hy₀
+  let sd : Set ℕ := Set.Ici 2
+  let fd : NNReal → NNReal → ↑sd → ℝ := fun y₁ y₂ n => (f n.1 y₂ - f n.1 y₁)
+  have hfd₁: ∀ y₁ y₂ n, fd y₁ y₂ n = f n.1 y₂ - f n.1 y₁ := by exact fun y₁ y₂ n => rfl
+  have hd₁: ∀ n a b, a < b → 0 < fd a b n := by
+    intros nd a b hnd₀
+    rw [hfd₁]
+    refine sub_pos.mpr ?_
+    refine hmo₀ nd.1 ?_ hnd₀
+    exact lt_of_lt_of_le (Nat.zero_lt_two) nd.2
+  have hfd₂: ∀ a b, a < b → (∀ n:↑sd, f n.1 a < f (n.1 + 1) a ∧ f n.1 b < f (n.1 + 1) b)
+      → Filter.Tendsto (fd a b) Filter.atTop Filter.atTop := by
+    intros a b ha₀ ha₁
+    exact aux_unique_top f h₁ h₇ sd rfl fd hfd₁ hd₁ a b ha₀ ha₁
+  have hfd₃: ∀ a b, a < b → (∀ (n:↑sd), (1 - 1 / n.1 < f n.1 a ∧ 1 - 1 / n.1 < f n.1 b) ∧ (f n.1 a < 1 ∧ f n.1 b < 1))
+      → Filter.Tendsto (fd a b) Filter.atTop (nhds 0) := by
+    intros a b ha₀ ha₁
+    exact aux_unique_nhds f sd rfl fd hfd₁ hd₁ a b ha₀ ha₁
+  by_contra! hc₀
+  by_cases hy₁: x < y
+  . have hy₂: Filter.Tendsto (fd x y) Filter.atTop Filter.atTop := by
+      refine hfd₂ x y hy₁ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by exact lt_of_lt_of_le (two_pos) nd.2
+      constructor
+      . exact (hx₀ nd.1 hnd₀).2.1
+      . exact (hy₀ nd.1 hnd₀).2.1
+    have hy₃: Filter.Tendsto (fd x y) Filter.atTop (nhds 0) := by
+      refine hfd₃ x y hy₁ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by
+        refine lt_of_lt_of_le ?_ nd.2
+        exact Nat.zero_lt_two
+      have hnd₁: nd.1 - 1 + 1 = nd.1 := by exact Nat.sub_add_cancel hnd₀
+      have hnd₂: 0 < nd.1 - 1 := by
+        refine Nat.sub_pos_of_lt ?_
+        refine lt_of_lt_of_le ?_ nd.2
+        exact Nat.one_lt_two
+      constructor
+      . constructor
+        . refine h₇ nd.1 x hnd₀ ?_
+          exact (hx₀ (nd.1) hnd₀).2.1
+        . refine h₇ nd.1 y hnd₀ ?_
+          exact (hy₀ (nd.1) hnd₀).2.1
+      . constructor
+        . rw [← hnd₁]
+          exact (hx₀ (nd.1 - 1) hnd₂).2.2
+        . rw [← hnd₁]
+          exact (hy₀ (nd.1 - 1) hnd₂).2.2
+    apply Filter.tendsto_atTop_atTop.mp at hy₂
+    apply tendsto_atTop_nhds.mp at hy₃
+    contrapose! hy₃
+    clear hy₃
+    let sx : Set ℝ := Set.Ioo (-1) 1
+    use sx
+    constructor
+    . refine Set.mem_Ioo.mpr ?_
+      simp
+    constructor
+    . exact isOpen_Ioo
+    . intro N
+      have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd x y a := by exact hy₂ (N + 3)
+      obtain ⟨i, hi₀⟩ := hy₅
+      have hi₁: (N.1 + i.1) ∈ sd := by
+        refine Set.mem_Ici.mpr ?_
+        rw [← add_zero 2]
+        refine Nat.add_le_add ?_ ?_
+        . exact N.2
+        . refine le_trans ?_ i.2
+          exact Nat.zero_le 2
+      let a : ↑sd := ⟨N + i, hi₁⟩
+      use a
+      constructor
+      . refine Subtype.mk_le_mk.mpr ?_
+        exact Nat.le_add_right ↑N ↑i
+      . refine Set.not_mem_Ioo_of_ge ?_
+        have hi₂: ↑↑N + 3 ≤ fd x y a := by
+          refine hi₀ a ?_
+          refine Subtype.mk_le_mk.mpr ?_
+          exact Nat.le_add_left ↑i ↑N
+        refine le_trans ?_ hi₂
+        norm_cast
+        exact Nat.le_add_left 1 (↑N + 2)
+  . have hy₂: y < x := by
+      push_neg at hy₁
+      exact lt_of_le_of_ne hy₁ hc₀.symm
+    have hy₃: Filter.Tendsto (fd y x) Filter.atTop Filter.atTop := by
+      refine hfd₂ y x hy₂ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by exact lt_of_lt_of_le (two_pos) nd.2
+      constructor
+      . exact (hy₀ nd.1 hnd₀).2.1
+      . exact (hx₀ nd.1 hnd₀).2.1
+    have hy₄: Filter.Tendsto (fd y x) Filter.atTop (nhds 0) := by
+      refine hfd₃ y x hy₂ ?_
+      intro nd
+      have hnd₀: 0 < nd.1 := by exact lt_of_lt_of_le (Nat.zero_lt_two) nd.2
+      have hnd₁: nd.1 - 1 + 1 = nd.1 := by exact Nat.sub_add_cancel hnd₀
+      have hnd₂: 0 < nd.1 - 1 := by
+        refine Nat.sub_pos_of_lt ?_
+        exact lt_of_lt_of_le (Nat.one_lt_two) nd.2
+      constructor
+      . constructor
+        . refine h₇ nd.1 y hnd₀ ?_
+          exact (hy₀ (nd.1) hnd₀).2.1
+        . refine h₇ nd.1 x hnd₀ ?_
+          exact (hx₀ (nd.1) hnd₀).2.1
+      . constructor
+        . rw [← hnd₁]
+          exact (hy₀ (nd.1 - 1) hnd₂).2.2
+        . rw [← hnd₁]
+          exact (hx₀ (nd.1 - 1) hnd₂).2.2
+    apply Filter.tendsto_atTop_atTop.mp at hy₃
+    apply tendsto_atTop_nhds.mp at hy₄
+    contrapose! hy₄
+    clear hy₄
+    let sx : Set ℝ := Set.Ioo (-1) 1
+    use sx
+    constructor
+    . refine Set.mem_Ioo.mpr ?_
+      simp
+    constructor
+    . exact isOpen_Ioo
+    . intro N
+      have hy₅: ∃ i, ∀ (a : ↑sd), i ≤ a → N + 3 ≤ fd y x a := by exact hy₃ (N + 3)
+      obtain ⟨i, hi₀⟩ := hy₅
+      have hi₁: (N.1 + i.1) ∈ sd := by
+        refine Set.mem_Ici.mpr ?_
+        rw [← add_zero 2]
+        refine Nat.add_le_add ?_ ?_
+        . exact N.2
+        . refine le_trans ?_ i.2
+          exact Nat.zero_le 2
+      let a : ↑sd := ⟨N + i, hi₁⟩
+      use a
+      constructor
+      . refine Subtype.mk_le_mk.mpr ?_
+        exact Nat.le_add_right ↑N ↑i
+      . refine Set.not_mem_Ioo_of_ge ?_
+        have hi₂: ↑↑N + 3 ≤ fd y x a := by
+          refine hi₀ a ?_
+          refine Subtype.mk_le_mk.mpr ?_
+          exact Nat.le_add_left ↑i ↑N
+        refine le_trans ?_ hi₂
+        norm_cast
+        exact Nat.le_add_left 1 (↑N + 2)
+
+
+
+
+theorem imo_1985_p6
+  (f : ℕ → NNReal → ℝ)
+  (h₀ : ∀ x, f 1 x = x)
+  (h₁ : ∀ n x, 0 < n → f (n + 1) x = f n x * (f n x + 1 / n)) :
+  ∃! a, ∀ n, 0 < n → 0 < f n a ∧ f n a < f (n + 1) a ∧ f (n + 1) a < 1 := by
+  have h₂: ∀ n x, 0 < n ∧ 0 < x → 0 < f n x := by
+    exact fun n x a => aux_1 f h₀ h₁ n x a
+  have h₃: ∀ n x, 0 < n → 0 ≤ f n x := by
+    intros n x hn
+    refine Nat.le_induction ?_ ?_ n hn
+    . rw [h₀ x]
+      exact NNReal.zero_le_coe
+    . intros d hd₀ hd₁
+      rw [h₁ d x hd₀]
+      refine mul_nonneg hd₁ ?_
+      refine add_nonneg hd₁ ?_
+      refine div_nonneg (by linarith) ?_
+      exact Nat.cast_nonneg' d
+  have hmo₀: ∀ n, 0 < n → StrictMono (f n) := by
+    intros n hn₀
+    refine Monotone.strictMono_of_injective ?h₁ ?h₂
+    . refine monotone_iff_forall_lt.mpr ?h₁.a
+      intros a b hab
+      refine le_of_lt ?_
+      exact aux_2 f h₀ h₁ h₂ h₃ n a b hn₀ hab
+    . intros p q hpq
+      contrapose! hpq
+      apply lt_or_gt_of_ne at hpq
+      cases' hpq with hpq hpq
+      . refine ne_of_lt ?_
+        exact aux_2 f h₀ h₁ h₂ h₃ n p q hn₀ hpq
+      . symm
+        refine ne_of_lt ?_
+        exact aux_2 f h₀ h₁ h₂ h₃ n q p hn₀ hpq
+  have hmo₁: ∀ n, 0 < n → Function.Injective (f n) := by exact fun n a => StrictMono.injective (hmo₀ n a)
+  let f₀: ℕ → NNReal → NNReal := fun n x => (f n x).toNNReal
+  have hf₀: f₀ = fun n x => (f n x).toNNReal := by rfl
+  have hf₁: ∀ n x, 0 < n → f n x = f₀ n x := by
+    intros n x hn₀
+    rw [hf₀]
+    simp
+    exact h₃ n x hn₀
+  have hf₂: ∀ n x, 0 < n → f₀ n x = (f n x).toNNReal := by
+    intros n x _
+    rw [hf₀]
+  have hmo₂: ∀ n, 0 < n → StrictMono (f₀ n) := by
+    intros n hn₀
+    refine aux_4 f h₃ ?_ f₀ hf₀ n hn₀
+    exact fun n x y a a_1 => hmo₀ n a a_1
+  let fi : ℕ → NNReal → NNReal := fun n => Function.invFun (f₀ n)
+  have hmo₇: ∀ n, 0 < n → Function.RightInverse (fi n) (f₀ n) := by
+    intros n hn₀
+    refine Function.rightInverse_invFun ?_
+    have h₄: ∀ n x y, 0 < n → x < y → f n x < f n y := by
+      exact fun n x y a a_1 => aux_2 f h₀ h₁ h₂ h₃ n x y a a_1
+    refine aux_7 f h₀ h₁ h₃ ?_ f₀ hf₂ hmo₂ ?_ n hn₀
+    . exact fun n x a => aux_3 f h₀ h₁ h₄ n x a
+    . intros m hm₀
+      exact aux_6 f h₀ h₁ f₀ hf₀ m hm₀
+  have hf₇: ∀ n x y, 0 < n → (f₀ n x = y ↔ fi n y = x) := by
+    intros n x y hn₀
+    constructor
+    . intro hn₁
+      rw [← hn₁, hf₀]
+      have hn₂: (Function.invFun (f n)) ∘ (f n) = id := by exact Function.invFun_comp (hmo₁ n hn₀)
+      rw [Function.comp_def (Function.invFun (f n)) (f n)] at hn₂
+      exact aux_5 f hmo₁ f₀ hmo₂ fi rfl n x ((fun n x => (f n x).toNNReal) n x) hn₀ (hf₂ n x hn₀)
+    . intro hn₁
+      rw [← hn₁]
+      exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ y))
+  let sn : Set ℕ := Set.Ici 1
+  let fb : ↑sn → NNReal := sn.restrict (fun (n:ℕ) => fi n (1 - 1 / (n:NNReal)))
+  let fc : ↑sn → NNReal := sn.restrict (fun (n:ℕ) => fi n 1)
+  have hsn₁: ∀ n:↑sn, ↑n ∈ sn ∧ 0 < (↑n:ℕ) := by
+    intro n
+    have hn₀: ↑n ∈ sn := by exact Subtype.coe_prop n
+    constructor
+    . exact Subtype.coe_prop n
+    . exact hn₀
+  have hfb₀: fb = fun (n:↑sn) => fi n (1 - 1 / (n:NNReal)) := by rfl
+  have hfc₀: fc = fun (n:↑sn) => fi n 1 := by rfl
+  have hfb₁: ∀ n:↑sn, f₀ n (fb n) = 1 - 1 / (n:NNReal) := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    rw [hfb₀]
+    exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ (1 - 1 / (n:NNReal))))
+  have hfc₁: ∀ n:↑sn, f₀ n (fc n) = 1 := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    rw [hfc₀]
+    exact hmo₁ n hn₀ (congrArg (f n) (hmo₇ n hn₀ 1))
+  have hu₁: ∀ n:↑sn, fb n < 1 := by
+    exact aux_8 f h₀ h₁ hmo₀ hmo₁ f₀ hf₂ sn fb hsn₁ hfb₁
+  have hfc₂: ∀ n:↑sn, fb n < fc n := by
+    intros n
+    have hn₀: 0 < (n:ℕ) := by exact (hsn₁ n).2
+    have g₀: f₀ n (fb n) < f₀ n (fc n) := by
+      rw [hfb₁ n, hfc₁ n]
+      simp
+      exact (hsn₁ n).2
+    exact (StrictMono.lt_iff_lt (hmo₂ n hn₀)).mp g₀
+  have hfb₃: StrictMono fb := by
+    refine StrictMonoOn.restrict ?_
+    refine aux_9 f h₀ h₁ f₀ hf₁ hf₂ hmo₂ fi ?_ hmo₇ hf₇ _ (by rfl) sn (by rfl)
+    intro x
+    refine (hf₇ 1 x x (by linarith)).mp ?_
+    rw [hf₂ 1 x (by linarith), h₀]
+    exact Real.toNNReal_coe
+  have hfc₃: StrictAnti fc := by
+    have g₀: StrictAntiOn (fun n => fi n 1) sn := by
+      refine strictAntiOn_Ici_of_lt_pred ?_
+      intros m hm₀
+      have hm₁: 0 < m - 1 := by exact Nat.zero_lt_sub_of_lt hm₀
+      have hm₂: m = m - 1 + 1 := by rw [Nat.sub_add_cancel (le_of_lt hm₀)]
+      have hm₃: 0 < m := by exact Nat.zero_lt_of_lt hm₀
+      simp
+      let x := fi m 1
+      let y := fi (m - 1) 1
+      have hx₀: x = fi m 1 := by rfl
+      have hy₀: y = fi (m - 1) 1 := by rfl
+      have hx₁: f₀ m x = 1 := by exact (hf₇ m x 1 (by linarith)).mpr hx₀.symm
+      have hy₁: f₀ (m - 1) y = 1 := by
+        exact (hf₇ (m - 1) y 1 hm₁).mpr hy₀.symm
+      have hy₂: f (m - 1) y = 1 := by
+        rw [hf₁ (m - 1) y hm₁, hy₁]
+        exact rfl
+      have hf: StrictMono (f m) := by exact hmo₀ m hm₃
+      refine (StrictMono.lt_iff_lt hf).mp ?_
+      rw [← hx₀, ← hy₀]
+      rw [hf₁ m x hm₃, hf₁ m y hm₃]
+      refine NNReal.coe_lt_coe.mpr ?_
+      rw [hx₁, hf₂ m y hm₃, hm₂, h₁ (m - 1) y hm₁, hy₂]
+      simp
+      exact hm₀
+    intros m n hmn
+    rw [hfc₀]
+    simp
+    let mn : ℕ := ↑m
+    let nn : ℕ := ↑n
+    have hm₀: mn ∈ sn := by exact Subtype.coe_prop m
+    have hn₀: nn ∈ sn := by exact Subtype.coe_prop n
+    exact g₀ hm₀ hn₀ hmn
+  let sb := Set.range fb
+  let sc := Set.range fc
+  have hsb₀: sb = Set.range fb := by rfl
+  have hsc₀: sc = Set.range fc := by rfl
+  let fr : NNReal → ℝ := fun x => x.toReal
+  let sbr := Set.image fr sb
+  let scr := Set.image fr sc
+  have hu₃: ∃ br, IsLUB sbr br := by
+    refine Real.exists_isLUB ?_ ?_
+    . exact Set.Nonempty.of_subtype
+    . refine NNReal.bddAbove_coe.mpr ?_
+      refine (bddAbove_iff_exists_ge 1).mpr ?_
+      use 1
+      constructor
+      . exact Preorder.le_refl 1
+      . intros y hy₀
+        apply Set.mem_range.mp at hy₀
+        obtain ⟨na, hna₀⟩ := hy₀
+        refine le_of_lt ?_
+        rw [← hna₀]
+        exact hu₁ na
+  have hu₄: ∃ cr, IsGLB scr cr := by
+    refine Real.exists_isGLB ?_ ?_
+    . refine Set.Nonempty.image fr ?_
+      exact Set.range_nonempty fc
+    . exact NNReal.bddBelow_coe sc
+  obtain ⟨br, hbr₀⟩ := hu₃
+  obtain ⟨cr, hcr₀⟩ := hu₄
+  have h₇: ∀ n x, 0 < n → (f n x < f (n + 1) x → 1 - 1 / n < f n x) := by
+    intros n x hn₀ hn₁
+    rw [h₁ n x hn₀] at hn₁
+    nth_rw 1 [← mul_one (f n x)] at hn₁
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . exact sub_right_lt_of_lt_add g₀
+    . refine lt_of_mul_lt_mul_left hn₁ ?_
+      exact h₃ n x hn₀
+  have h₈: ∀ n x, 0 < n → 0 < x → 1 - 1 / n < f n x → f n x < f (n + 1) x := by
+    intros n x hn₀ hx₀ hn₁
+    rw [h₁ n x hn₀]
+    suffices g₀: 1 < f n x + 1 / ↑n
+    . nth_rw 1 [← mul_one (f n x)]
+      refine mul_lt_mul' ?_ g₀ ?_ ?_
+      . exact Preorder.le_refl (f n x)
+      . exact zero_le_one' ℝ
+      . exact gt_of_gt_of_ge (hmo₀ n hn₀ hx₀) (h₃ n 0 hn₀)
+    . exact lt_add_of_tsub_lt_right hn₁
+  have hbr₁: 0 < br := by
+    exact aux_10 f h₀ h₁ f₀ hf₂ fi hmo₇ sn sb fb (by rfl) hfb₀ hsb₀ fr (by rfl) sbr (by rfl) br hbr₀
+  have hfb₄: ∀ n, 0 ≤ fb n := by
+    intro n
+    have hfb₂: fb = fun (n:↑sn) => Function.invFun (f₀ n) (1 - 1 / ↑n) := by exact hfb₀
+    rw [hfb₂]
+    simp
+  have hu₅: br ≤ cr := by
+    exact aux_11 sn fb fc hfc₂ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr (by rfl) sbr scr (by rfl) (by rfl) br cr hbr₀ hcr₀ hfb₄
+  have hbr₃: ∀ x ∈ sbr, x ≤ br := by
+    refine mem_upperBounds.mp ?_
+    refine (isLUB_le_iff hbr₀).mp ?_
+    exact Preorder.le_refl br
+  have hcr₃: ∀ x ∈ scr, cr ≤ x := by
+      refine mem_lowerBounds.mp ?_
+      refine (le_isGLB_iff hcr₀).mp ?_
+      exact Preorder.le_refl cr
+  refine existsUnique_of_exists_of_unique ?_ ?_
+  . exact aux_exists f h₂ hmo₀ f₀ hf₁ sn (by rfl) fb fc hfb₁ hfc₁ hfb₃ hfc₃ sb sc hsb₀ hsc₀ fr (by rfl) sbr scr (by rfl) (by rfl) br cr h₈ hbr₁ hu₅ hbr₃ hcr₃
+  . intros x y hx₀ hy₀
+    exact aux_unique f h₁ hmo₀ h₇ x y hx₀ hy₀

imo_proofs/imo_1992_p1.lean

@@ -0,0 +1,484 @@
+import Mathlib
+set_option linter.unusedVariables.analyzeTactics true
+
+open Int Rat
+
+
+lemma mylemma_main_lt2
+  (p q r: ℤ)
+  (hpl: 4 ≤ p)
+  (hql: 5 ≤ q)
+  (hrl: 6 ≤ r) :
+  (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ ↑2 := by
+  have h₁: (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ)
+    = (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)) := by
+    norm_cast
+    simp
+  have hp: (↑p/↑(p-1):ℚ) ≤ ((4/3):ℚ) := by
+    have g₁: 0 < (↑(p - 1):ℚ) := by
+      norm_cast
+      linarith [hpl]
+    have g₂: ↑p * ↑(3:ℚ) ≤ ↑(4:ℚ) * (↑(p - 1):ℚ) := by
+      norm_cast
+      linarith
+    refine (div_le_iff₀ g₁).mpr ?_
+    rw [div_mul_eq_mul_div]
+    refine (le_div_iff₀ ?_).mpr g₂
+    norm_num
+  have hq: (↑q/↑(q-1)) ≤ ((5/4):ℚ) := by
+    have g₁: 0 < (↑(q - 1):ℚ) := by
+      norm_cast
+      linarith[hql]
+    have g₂: ↑q * ↑(4:ℚ) ≤ ↑(5:ℚ) * (↑(q - 1):ℚ) := by
+      norm_cast
+      linarith
+    refine (div_le_iff₀ g₁).mpr ?_
+    rw [div_mul_eq_mul_div]
+    refine (le_div_iff₀ ?_).mpr g₂
+    norm_num
+  have hr: (↑r/↑(r-1)) ≤ ((6/5):ℚ) := by
+    have g₁: 0 < (↑(r - 1):ℚ) := by
+      norm_cast
+      linarith[hql]
+    have g₂: ↑r * ↑(5:ℚ) ≤ ↑(6:ℚ) * (↑(r - 1):ℚ) := by
+      norm_cast
+      linarith
+    refine (div_le_iff₀ g₁).mpr ?_
+    rw [div_mul_eq_mul_div]
+    refine (le_div_iff₀ ?_).mpr g₂
+    norm_num
+  have hub: (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)) ≤ (4/3:ℚ) * ((5/4):ℚ) * ((6/5):ℚ) := by
+    have hq_nonneg: 0 ≤ (↑q:ℚ) := by
+      norm_cast
+      linarith
+    have hq_1_nonneg: 0 ≤ (↑(q - 1):ℚ) := by
+      norm_cast
+      linarith
+    have h₂: 0 ≤ (((q:ℚ) / ↑(q - 1)):ℚ) := by
+      exact div_nonneg hq_nonneg hq_1_nonneg
+    have hub1: (↑p/↑(p-1)) * (↑q/↑(q-1)) ≤ ((4/3):ℚ) * ((5/4):ℚ) := by
+      exact mul_le_mul hp hq h₂ (by norm_num)
+    have hr_nonneg: 0 ≤ (↑r:ℚ) := by
+      norm_cast
+      linarith
+    have hr_1_nonneg: 0 ≤ (↑(r - 1):ℚ) := by
+      norm_cast
+      linarith
+    have h₃: 0 ≤ (((r:ℚ) / ↑(r - 1)):ℚ) := by
+      exact div_nonneg hr_nonneg hr_1_nonneg
+    exact mul_le_mul hub1 hr h₃ (by norm_num)
+  norm_num at hub
+  rw [h₁]
+  norm_num
+  exact hub
+
+
+lemma mylemma_k_lt_2
+  (p q r k: ℤ)
+  (hk: p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  (hpl: 4 ≤ p)
+  (hql: 5 ≤ q)
+  (hrl: 6 ≤ r)
+  (hden: 0 < (p - 1) * (q - 1) * (r - 1) ) :
+  (k < 2) := by
+  have h₁: (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ ↑2 := by
+    exact mylemma_main_lt2 p q r hpl hql hrl
+  have h₂: ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+    have g₁: ↑(p * q * r - 1) = ↑k * (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+      norm_cast
+      linarith
+    symm
+    have g₂: (↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≠ 0 := by
+      norm_cast
+      linarith[hden]
+    exact (div_eq_iff g₂).mpr g₁
+  have h₃: ↑k < (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+    rw [h₂]
+    have g₁: (↑(p * q * r - 1):ℚ) < (↑(p * q * r):ℚ) := by
+      norm_cast
+      exact sub_one_lt (p * q * r)
+    have g₂: 0 < (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+      norm_cast
+    exact div_lt_div_of_pos_right g₁ g₂
+  have h₄: (↑k:ℚ) < ↑2 := by
+    exact lt_of_lt_of_le h₃ h₁
+  norm_cast at h₄
+
+
+lemma mylemma_main_lt4
+  (p q r: ℤ)
+  (hpl: 2 ≤ p)
+  (hql: 3 ≤ q)
+  (hrl: 4 ≤ r) :
+  (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ ↑4 := by
+  have h₁: (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ)
+      = (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)) := by
+    norm_cast
+    simp
+  have hp: (↑p/↑(p-1):ℚ) ≤ ↑(2:ℚ) := by
+    have g₁: 0 < (↑(p - 1):ℚ) := by
+      norm_cast
+      linarith[hpl]
+    have g₂: ↑p ≤ ↑(2:ℚ) * (↑(p - 1):ℚ) := by
+      norm_cast
+      linarith
+    exact (div_le_iff₀ g₁).mpr g₂
+  have hq: (↑q/↑(q-1)) ≤ ((3/2):ℚ) := by
+    have g₁: 0 < (↑(q - 1):ℚ) := by
+      norm_cast
+      linarith[hql]
+    have g₂: ↑q * ↑(2:ℚ) ≤ ↑(3:ℚ) * (↑(q - 1):ℚ) := by
+      norm_cast
+      linarith
+    refine (div_le_iff₀ g₁).mpr ?_
+    rw [div_mul_eq_mul_div]
+    refine (le_div_iff₀ ?_).mpr g₂
+    norm_num
+  have hr: (↑r/↑(r-1)) ≤ ((4/3):ℚ) := by
+    have g₁: 0 < (↑(r - 1):ℚ) := by
+      norm_cast
+      linarith[hql]
+    have g₂: ↑r * ↑(3:ℚ) ≤ ↑(4:ℚ) * (↑(r - 1):ℚ) := by
+      norm_cast
+      linarith
+    refine (div_le_iff₀ g₁).mpr ?_
+    rw [div_mul_eq_mul_div]
+    refine (le_div_iff₀ ?_).mpr g₂
+    norm_num
+  have hub: (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)) ≤ (2:ℚ) * ((3/2):ℚ) * ((4/3):ℚ) := by
+    have hq_nonneg: 0 ≤ (↑q:ℚ) := by
+      norm_cast
+      linarith
+    have hq_1_nonneg: 0 ≤ (↑(q - 1):ℚ) := by
+      norm_cast
+      linarith
+    have h₂: 0 ≤ (((q:ℚ) / ↑(q - 1)):ℚ) := by
+      exact div_nonneg hq_nonneg hq_1_nonneg
+    have hub1: (↑p/↑(p-1)) * (↑q/↑(q-1)) ≤ (2:ℚ) * ((3/2):ℚ) := by
+      exact mul_le_mul hp hq h₂ (by norm_num)
+    have hr_nonneg: 0 ≤ (↑r:ℚ) := by
+      norm_cast
+      linarith
+    have hr_1_nonneg: 0 ≤ (↑(r - 1):ℚ) := by
+      norm_cast
+      linarith
+    have h₃: 0 ≤ (((r:ℚ) / ↑(r - 1)):ℚ) := by
+      exact div_nonneg hr_nonneg hr_1_nonneg
+    exact mul_le_mul hub1 hr h₃ (by norm_num)
+  norm_num at hub
+  rw [h₁]
+  norm_num
+  exact hub
+
+
+
+lemma mylemma_k_lt_4
+  (p q r k: ℤ)
+  (hk: p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  (hpl: 2 ≤ p)
+  (hql: 3 ≤ q)
+  (hrl: 4 ≤ r)
+  (hden: 0 < (p - 1) * (q - 1) * (r - 1) ) :
+  (k < 4) := by
+  have h₁: (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ ↑4 := by
+    exact mylemma_main_lt4 p q r hpl hql hrl
+  have h₂: ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+    have g₁: ↑(p * q * r - 1) = ↑k * (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+      norm_cast
+      linarith
+    symm
+    have g₂: (↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≠ 0 := by
+      norm_cast
+      linarith [hden]
+    exact (div_eq_iff g₂).mpr g₁
+  have h₃: ↑k < (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+    rw [h₂]
+    have g₁: (↑(p * q * r - 1):ℚ) < (↑(p * q * r):ℚ) := by
+      norm_cast
+      exact sub_one_lt (p * q * r)
+    have g₂: 0 < (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+      norm_cast
+    exact div_lt_div_of_pos_right g₁ g₂
+  have h₄: (↑k:ℚ) < ↑4 := by
+    exact lt_of_lt_of_le h₃ h₁
+  norm_cast at h₄
+
+
+
+lemma mylemma_k_gt_1
+  (p q r k: ℤ)
+  (hk: p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  (h₁: ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ))
+  (hpl: 2 ≤ p)
+  (hql: 3 ≤ q)
+  (hrl: 4 ≤ r)
+  (hden: 0 < (p - 1) * (q - 1) * (r - 1) ) :
+  (1 < k) := by
+  have hk0: 0 < (↑k:ℚ) := by
+    have g₁: 2*3*4 ≤ p * q * r := by
+      have g₂: 2*3 ≤ p * q := by
+        exact mul_le_mul hpl hql (by norm_num) (by linarith[hpl])
+      exact mul_le_mul g₂ hrl (by norm_num) (by linarith[g₂])
+    have g₂: 0 < (↑(p * q * r - 1):ℚ) := by
+      norm_cast
+      linarith[g₁]
+    have g₃: 0 < (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+      norm_cast
+    rw [h₁]
+    exact div_pos g₂ g₃
+  norm_cast at hk0
+  by_contra hc
+  push_neg at hc
+  interval_cases k
+  simp at hk
+  exfalso
+  have g₁: p*q + q*r + r*p = p+q+r := by linarith
+  have g₂: p < p*q := by exact lt_mul_right (by linarith) (by linarith)
+  have g₃: q < q*r := by exact lt_mul_right (by linarith) (by linarith)
+  have g₄: r < r*p := by exact lt_mul_right (by linarith) (by linarith)
+  have g₅: p+q+r < p*q + q*r + r*p := by linarith[g₂,g₃,g₄]
+  linarith [g₁,g₅]
+
+
+
+lemma mylemma_p_lt_4
+  (p q r k: ℤ)
+  (h₀ : 1 < p ∧ p < q ∧ q < r)
+  (hk: p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k)
+  (h₁: ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ))
+  (hpl: 2 ≤ p)
+  (hql: 3 ≤ q)
+  (hrl: 4 ≤ r)
+  (hden: 0 < (p - 1) * (q - 1) * (r - 1) ) :
+  (p < 4) := by
+  by_contra hcp
+  push_neg at hcp
+  have hcq: 5 ≤ q := by linarith
+  have hcr: 6 ≤ r := by linarith
+  have h₃: k < 2 := by exact mylemma_k_lt_2 p q r k hk hcp hcq hcr hden
+  have h₄: 1 < k := by exact mylemma_k_gt_1 p q r k hk h₁ hpl hql hrl hden
+  linarith
+
+
+lemma q_r_divisor_of_prime
+  (q r : ℤ)
+  (p: ℕ)
+  (h₀ : q * r = ↑p)
+  (h₁: Nat.Prime p) :
+  q = -1 ∨ q = 1 ∨ q = -p ∨ q = p := by
+  have hq : q ≠ 0 := by
+    intro h
+    rw [h] at h₀
+    simp at h₀
+    symm at h₀
+    norm_cast at h₀
+    rw [h₀] at h₁
+    exact Nat.not_prime_zero h₁
+  have hr : r ≠ 0 := by
+    intro h
+    rw [h] at h₀
+    simp at h₀
+    norm_cast at h₀
+    rw [← h₀] at h₁
+    exact Nat.not_prime_zero h₁
+  have hqr : abs q * abs r = p := by
+    have h₃: abs q = q.natAbs := by exact abs_eq_natAbs q
+    have h₄: abs r = r.natAbs := by exact abs_eq_natAbs r
+    rw [h₃,h₄]
+    norm_cast
+    exact Int.natAbs_mul_natAbs_eq h₀
+  have h_abs: abs (↑(q.natAbs):ℤ) = 1 ∨ abs q = p := by
+    cases' Int.natAbs_eq q with h_1 h_2
+    . rw [h_1] at hqr
+      have h₂: abs (↑(q.natAbs):ℤ) ∣ p := by exact Dvd.intro (abs r) hqr
+      have h₃: (↑(q.natAbs):ℕ) ∣ p := by
+        norm_cast at *
+      have h₄: (↑(q.natAbs):ℕ) = 1 ∨ (↑(q.natAbs):ℕ) = p := by
+        exact Nat.Prime.eq_one_or_self_of_dvd h₁ (↑(q.natAbs):ℕ) h₃
+      cases' h₄ with h₄₀ h₄₁
+      . left
+        norm_cast at *
+      have h₅: abs q = q.natAbs := by exact abs_eq_natAbs q
+      right
+      rw [h₅]
+      norm_cast at *
+    . rw [h_2] at hqr
+      rw [abs_neg _] at hqr
+      have h₂: abs (↑(q.natAbs):ℤ) ∣ p := by exact Dvd.intro (abs r) hqr
+      have h₃: (↑(q.natAbs):ℕ) ∣ p := by
+        norm_cast at *
+      have h₄: (↑(q.natAbs):ℕ) = 1 ∨ (↑(q.natAbs):ℕ) = p := by
+        exact Nat.Prime.eq_one_or_self_of_dvd h₁ (↑(q.natAbs):ℕ) h₃
+      cases' h₄ with h₄₀ h₄₁
+      . left
+        norm_cast at *
+      . have h₅: abs q = q.natAbs := by exact abs_eq_natAbs q
+        right
+        rw [h₅]
+        norm_cast
+  cases' h_abs with hq_abs hq_abs
+  . norm_cast at *
+    have h₄: q = ↑(q.natAbs) ∨ q = -↑(q.natAbs) := by
+      exact Int.natAbs_eq q
+    rw [hq_abs] at h₄
+    norm_cast at h₄
+    cases' h₄ with h₄₀ h₄₁
+    . right
+      left
+      exact h₄₀
+    . left
+      exact h₄₁
+  . right
+    right
+    have h₂: abs q = q.natAbs := by exact abs_eq_natAbs q
+    rw [h₂] at hq_abs
+    norm_cast at hq_abs
+    refine or_comm.mp ?_
+    refine (Int.natAbs_eq_natAbs_iff).mp ?_
+    norm_cast
+
+
+lemma mylemma_qr_11
+  (q r: ℤ)
+  (h₀: (4 - q) * (4 - r) = 11) :
+  (4 - q = -1 ∨ 4 - q = 1 ∨ 4 - q = -11 ∨ 4 - q = 11) := by
+  have h₁: Nat.Prime (11) := by decide
+  exact q_r_divisor_of_prime (4-q) (4-r) 11 h₀ h₁
+
+
+lemma mylemma_qr_5
+  (q r: ℤ)
+  (h₀: (q - 3) * (r - 3) = 5) :
+  (q - 3 = -1 ∨ q - 3 = 1 ∨ q - 3 = -5 ∨ q - 3 = 5) := by
+  have h₁: Nat.Prime (5) := by decide
+  exact q_r_divisor_of_prime (q - 3) (r - 3) 5 h₀ h₁
+
+
+lemma mylemma_63qr_5
+  (q r: ℤ)
+  (h₀: (6 - 3*q) * (2 - r) = 5) :
+  (6 - 3*q = -1 ∨ 6 - 3*q = 1 ∨ 6 - 3*q = -5 ∨ 6 - 3*q = 5) := by
+  have h₁: Nat.Prime (5) := by decide
+  exact q_r_divisor_of_prime (6 - 3*q) (2 - r) 5 h₀ h₁
+
+
+lemma mylemma_case_k_2
+  (p q r: ℤ)
+  (h₀: 1 < p ∧ p < q ∧ q < r)
+  (hpl: 2 ≤ p)
+  (hql: 3 ≤ q)
+  (hrl: 4 ≤ r)
+  (hpu: p < 4)
+  (hk: p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * 2) :
+  (p, q, r) = (2, 4, 8) ∨ (p, q, r) = (3, 5, 15) := by
+  interval_cases p
+  . exfalso
+    norm_num at *
+    have g₁: 2*q + 2*r = 3 := by linarith
+    linarith [g₁,hql,hrl]
+  . right
+    norm_num at *
+    -- have g₁: q*r - 4*q - 4*r + 5 = 0 := by linarith
+    have g₂: (4-q)*(4-r) = 11 := by linarith
+    have g₃: (4-q) = -1 ∨ (4-q) = 1 ∨ (4-q) = -11 ∨ (4-q) = 11 := by
+      exact mylemma_qr_11 q r g₂
+    cases' g₃ with g₃₁ g₃₂
+    . have hq: q = 5 := by linarith
+      constructor
+      . exact hq
+      . rw [hq] at g₂
+        linarith[g₂]
+    . exfalso
+      cases' g₃₂ with g₃₂ g₃₃
+      . have hq: q = 3 := by linarith[g₃₂]
+        rw [hq] at g₂
+        have hr: r = -7 := by linarith[g₂]
+        linarith[hrl,hr]
+      . cases' g₃₃ with g₃₃ g₃₄
+        . have hq: q = 15 := by linarith[g₃₃]
+          rw [hq] at g₂
+          have hr: r = 5 := by linarith[g₂]
+          linarith[hq,hr,h₀.2]
+        . have hq: q = -7 := by linarith[g₃₄]
+          linarith[hq,hql]
+
+
+lemma mylemma_case_k_3
+  (p q r: ℤ)
+  (h₀: 1 < p ∧ p < q ∧ q < r)
+  (hpl: 2 ≤ p)
+  (hql: 3 ≤ q)
+  (hrl: 4 ≤ r)
+  (hpu: p < 4)
+  (hk: p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * 3) :
+  (p, q, r) = (2, 4, 8) ∨ (p, q, r) = (3, 5, 15) := by
+  interval_cases p
+  -- p = 2
+  . norm_num at *
+    -- have g₁: q*r - 3*q - 3*r + 4 = 0 := by linarith
+    have g₂: (q-3)*(r-3) = 5 := by linarith
+    have g₃: (q-3) = -1 ∨ (q-3) = 1 ∨ (q-3) = -5 ∨ (q-3) = 5 := by
+      exact mylemma_qr_5 q r g₂
+    cases' g₃ with g₃₁ g₃₂
+    . exfalso
+      linarith [hql,g₃₁]
+    . cases' g₃₂ with g₃₂ g₃₃
+      . have hq: q = 4 := by linarith
+        rw [hq] at g₂
+        have hr: r = 8 := by linarith[g₂]
+        exact { left := hq, right := hr }
+      . exfalso
+        cases' g₃₃ with g₃₃ g₃₄
+        . linarith[hql,g₃₃]
+        . have hq: q = 8 := by linarith
+          rw [hq] at g₂
+          norm_num at g₂
+          have hr: r = 4 := by linarith
+          linarith[hrl,hr]
+  -- p = 3
+  . right
+    norm_num at *
+    -- have g₁: 3 * q * r - 6 * q - 6 * r + 7 = 0 := by linarith
+    have g₂: (6 - 3*q) * (2 - r) = 5 := by linarith
+    have g₃: (6 - 3*q) = -1 ∨ (6 - 3*q) = 1 ∨ (6 - 3*q) = -5 ∨ (6 - 3*q) = 5 := by
+      exact mylemma_63qr_5 q r g₂
+    exfalso
+    cases' g₃ with g₃₁ g₃₂
+    . linarith[g₃₁,q]
+    . cases' g₃₂ with g₃₂ g₃₃
+      . linarith[g₃₂,q]
+      . cases' g₃₃ with g₃₃ g₃₄
+        . linarith[g₃₃,q]
+        . linarith[g₃₄,q]
+
+
+
+theorem imo_1992_p1
+  (p q r : ℤ)
+  (h₀ : 1 < p ∧ p < q ∧ q < r)
+  (h₁ : (p - 1) * (q - 1) * (r - 1)∣(p * q * r - 1)) :
+  (p, q, r) = (2, 4, 8) ∨ (p, q, r) = (3, 5, 15) := by
+  cases' h₁ with k hk
+  have hpl: 2 ≤ p := by linarith
+  have hql: 3 ≤ q := by linarith
+  have hrl: 4 ≤ r := by linarith
+  have hden: 0 < (((p - 1) * (q - 1)) * (r - 1)) := by
+    have gp: 0 < (p - 1) := by linarith
+    have gq: 0 < (q - 1) := by linarith
+    have gr: 0 < (r - 1) := by linarith
+    exact mul_pos (mul_pos gp gq) gr
+  have h₁: ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+    have g₁: ↑(p * q * r - 1) = ↑k * (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by
+      norm_cast
+      linarith
+    symm
+    have g₂: (↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≠ 0 := by
+      norm_cast
+      linarith[hden]
+    exact (div_eq_iff g₂).mpr g₁
+  have hk4: k < 4 := by exact mylemma_k_lt_4 p q r k hk hpl hql hrl hden
+  have hk1: 1 < k := by exact mylemma_k_gt_1 p q r k hk h₁ hpl hql hrl hden
+  have hpu: p < 4 := by exact mylemma_p_lt_4 p q r k h₀ hk h₁ hpl hql hrl hden
+  interval_cases k
+  . exact mylemma_case_k_2 p q r h₀ hpl hql hrl hpu hk
+  . exact mylemma_case_k_3 p q r h₀ hpl hql hrl hpu hk

imo_proofs/imo_1997_p5.lean

@@ -0,0 +1,402 @@
+import Mathlib
+set_option linter.unusedVariables.analyzeTactics true
+
+open Nat Real
+
+
+lemma mylemma_xy_le_y
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  -- (g : x ^ y ^ 2 = (x ^ y) ^ y)
+  (hxy : x ≤ y)
+  (h₁ : (x ^ y) ^ y = y ^ x) :
+  x ^ y ≤ y := by
+  by_contra hc
+  push_neg at hc
+  have h₂: y^x ≤ y^y := by
+    { exact Nat.pow_le_pow_of_le_right h₀.2 hxy }
+  have h₃: y^y < (x^y)^y := by
+    refine Nat.pow_lt_pow_left hc ?_
+    refine Nat.pos_iff_ne_zero.mp h₀.2
+  rw [h₁] at h₃
+  linarith [h₂, h₃]
+
+
+lemma four_times_k_less_than_two_pow_k
+  (k : ℕ)
+  (hk : 5 ≤ k) :
+  4 * k < 2 ^ k := by
+  -- Proceed by induction on k
+  induction' k using Nat.case_strong_induction_on with n ih
+  -- Base case: k = 0 is not possible since 5 ≤ k
+  -- so we start directly with k = 5 as the base case
+  . norm_num
+  by_cases h₀ : n < 5
+  . have hn: n = 4 := by linarith
+    rw [hn]
+    norm_num
+  . push_neg at h₀
+    have ih₁ : 4 * n < 2 ^ n := ih n (le_refl n) h₀
+    rw [mul_add, pow_add, mul_one, pow_one, mul_two]
+    refine Nat.add_lt_add ih₁ ?_
+    refine lt_trans ?_ ih₁
+    refine (Nat.lt_mul_iff_one_lt_right (by norm_num)).mpr ?_
+    refine Nat.lt_of_lt_of_le ?_ h₀
+    norm_num
+
+
+lemma mylemma_case_xley
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x^(y^2) = y^x)
+  (g₁ : x^(y^2) = (x^y)^y)
+  (hxy : x ≤ y) :
+  (x, y) = (1, 1) ∨ (x, y) = (16, 2) ∨ (x, y) = (27, 3) := by
+  rw [g₁] at h₁
+  have g2: x^y ≤ y := by
+    exact mylemma_xy_le_y x y h₀ hxy h₁
+  have g3: x = 1 := by
+    by_contra hc
+    have g3: 2 ≤ x := by
+      by_contra gc
+      push_neg at gc
+      interval_cases x
+      . linarith
+      . omega
+    have g4: 2^y ≤ x^y := by { exact Nat.pow_le_pow_of_le_left g3 y }
+    have g5: y < 2^y := by exact Nat.lt_two_pow_self
+    linarith
+  rw [g3] at h₁
+  simp at h₁
+  left
+  norm_num
+  exact { left := g3, right := id h₁.symm }
+
+
+lemma mylemma_exp_log
+  (x: ℕ)
+  (h₀: 0 < x):
+  (↑x = Real.exp (Real.log ↑x)):= by
+  have hx_pos : 0 < (↑x : ℝ) := by exact Nat.cast_pos.mpr h₀
+  symm
+  exact Real.exp_log hx_pos
+
+
+
+lemma mylemma_y2_lt_x
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (hxy : y < x) :
+  y ^ 2 < x := by
+  by_cases hy: 1 < y
+  . have hx: 2 ≤ x := by linarith
+    have h₂: y ^ x < x ^ x := by
+      refine Nat.pow_lt_pow_left hxy ?_
+      exact Nat.ne_of_lt' h₀.1
+    rw [← h₁] at h₂
+    exact (Nat.pow_lt_pow_iff_right hx).mp h₂
+  . push_neg at hy
+    interval_cases y
+    . simp
+      exact h₀.1
+    . simp at *
+      assumption
+
+
+
+lemma mylemma_5
+  (x y: ℕ)
+  (h₀: 0 < x ∧ 0 < y)
+  (h₁: x ^ y ^ 2 = y ^ x) :
+  (↑x / ↑y^2) ^ y ^ 2 = (↑y:ℝ)^ ((↑x:ℝ) - 2 * ↑y ^ 2) := by
+  have g₁: (↑x:ℝ) ^ (↑y:ℝ) ^ 2 = (↑y:ℝ) ^ (↑x:ℝ) := by
+    norm_cast
+  have g₂: 0 < ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2)) := by
+    norm_cast
+    exact pow_pos h₀.2 _
+  have g₃: ((↑x:ℝ) ^ (↑y:ℝ) ^ 2) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2))
+        = ((↑y:ℝ) ^ (↑x:ℝ)) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2)) := by
+    refine (div_left_inj' ?_).mpr g₁
+    norm_cast
+    refine pow_ne_zero _ ?_
+    linarith [h₀.2]
+  have gy: 0 < (↑y:ℝ) := by
+    norm_cast
+    exact h₀.2
+  rw [← Real.rpow_sub gy (↑x) (2 * ↑y ^ 2)] at g₃
+  have g₄: ((↑x:ℝ) ^ (↑y:ℝ) ^ 2) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2))
+        = (↑x / ↑y^2) ^ y ^ 2 := by
+    have g₅: (↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2) = ((↑y:ℝ) ^ 2) ^ ((↑y:ℝ) ^ 2) := by
+      norm_cast
+      refine pow_mul y 2 (y^2)
+    rw [g₅]
+    symm
+    norm_cast
+    have g₆: ((↑x:ℝ) / ↑y ^ 2) ^ y ^ 2 = ↑x ^ y ^ 2 / (↑y ^ 2) ^ y ^ 2 := by
+      refine div_pow (↑x:ℝ) ((↑y:ℝ) ^ 2) (y^2)
+    norm_cast at *
+  rw [g₄] at g₃
+  norm_cast at *
+
+
+
+
+lemma mylemma_2y2_lt_x
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (hxy : y < x) :
+  2 * y ^ 2 < x := by
+  by_cases hy1: y = 1
+  . rw [hy1]
+    norm_num
+    by_contra hc
+    push_neg at hc
+    interval_cases x
+    . linarith
+    . linarith
+    . rw [hy1] at h₁
+      simp at h₁
+  . have hy: 1 < y := by
+      contrapose! hy1
+      linarith
+    clear hy1
+    have h₂: (↑y:ℝ) ^ 2 < ↑x := by
+      norm_cast
+      exact mylemma_y2_lt_x x y h₀ h₁ hxy
+    have h₃: 1 < ↑x / (↑y:ℝ) ^ 2 := by
+      refine (one_lt_div ?_).mpr h₂
+      norm_cast
+      exact pow_pos h₀.2 2  -- rw ← one_mul ((↑y:ℝ)^2) at h₂, refine lt_div_iff_mul_lt.mpr h₂, },
+    have h₄: 1 < (↑x / (↑y:ℝ)^2)^(y^2) := by
+      refine one_lt_pow₀ h₃ ?_
+      refine Nat.ne_of_gt ?_
+      refine sq_pos_of_pos ?_
+      exact lt_of_succ_lt hy
+    have h₅: (↑x/ (↑y:ℝ)^2)^(y^2) = (↑y:ℝ)^((↑x:ℝ) - 2*(↑y:ℝ)^2) := by
+      exact mylemma_5 x y h₀ h₁
+    rw [h₅] at h₄
+    have h₆: 0 < (↑x:ℝ) - 2 * (↑y:ℝ) ^ 2 := by
+      by_contra hc
+      push_neg at hc
+      cases' lt_or_eq_of_le hc with hlt heq
+      . have gy: 1 < (↑y:ℝ) := by
+          norm_cast
+        have glt: (↑x:ℝ) - 2*(↑y:ℝ)^2 < (↑0:ℝ) := by
+          norm_cast at *
+        have g₁: (↑y:ℝ) ^ ((↑x:ℝ) - 2*(↑y:ℝ)^2) < (↑y:ℝ) ^ (↑0:ℝ) := by
+          exact Real.rpow_lt_rpow_of_exponent_lt gy glt
+        simp at g₁
+        linarith[ h₄,g₁]
+      . rw [heq] at h₄
+        simp at h₄
+    simp at h₆
+    norm_cast at h₆
+
+
+lemma mylemma_castdvd
+  (x y: ℕ)
+  (h₀: 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (hyx: y < x) :
+  (y^2 ∣ x) := by
+  have h₂: (x ^ y ^ 2).factorization = (y^x).factorization := by
+    exact congr_arg Nat.factorization h₁
+  simp at h₂
+  symm at h₂
+  have hxy1: 2 * y^2 ≤ x := by exact le_of_lt (mylemma_2y2_lt_x x y h₀ h₁ hyx)
+  have hxy: 2 • y^2 ≤ x := by exact hxy1
+  have h₃: 2 • y^2 • x.factorization ≤ x • x.factorization := by
+    rw [← smul_assoc]
+    refine nsmul_le_nsmul_left ?_ hxy
+    norm_num
+  rw [← h₂] at h₃
+  have h₄: 2 • x • y.factorization = x • (2 • y.factorization) := by
+    rw [← smul_assoc, ← smul_assoc]
+    have g₄: 2 • x = x • 2 := by
+      simp
+      exact mul_comm 2 x
+    rw [g₄]
+  rw [h₄] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  rw [← Nat.factorization_pow] at h₃
+  have h₅: (y ^ 2) ^ x ∣ x^x := by
+    have g₁: (y ^ 2) ^ x ≠ 0 := by
+      refine pow_ne_zero x ?_
+      refine pow_ne_zero 2 ?_
+      linarith
+    have g₂: x ^ x ≠ 0 := by
+      refine pow_ne_zero x ?_
+      linarith
+    exact (Nat.factorization_le_iff_dvd g₁ g₂).mp h₃
+  refine (Nat.pow_dvd_pow_iff ?_).mp h₅
+  exact Nat.ne_of_gt h₀.1
+
+
+
+
+lemma mylemma_xsuby_eq_2xy2_help
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (h₂ : Real.log (↑x:ℝ)  = Real.log ↑y * ↑x / (↑(y ^ 2:ℕ ):ℝ) )
+  (hxy : y < x) :
+  x = y ^ (x / y ^ 2) := by
+  have h_exp : Real.exp (Real.log ↑x)
+            = Real.exp (Real.log ↑y * (↑x:ℝ)  / ((↑y:ℝ)) ^ 2) := by
+    rw [h₂]
+    norm_cast
+  rw [← mylemma_exp_log x h₀.1] at h_exp
+  rw [← mul_div] at h_exp
+  rw [Real.exp_mul] at h_exp
+  rw [← mylemma_exp_log y h₀.2] at h_exp
+  have h₃: (↑x:ℝ) / ((↑y:ℝ)^2) = (↑(x / y^2:ℕ):ℝ) := by
+    norm_cast
+    symm
+    have g₂: y^2 ∣ x := by
+      exact mylemma_castdvd x y h₀ h₁ hxy
+    have h₃: (↑(y^(2:ℕ)):ℝ) ≠ 0 := by
+      norm_cast
+      exact pow_ne_zero 2 ( by linarith)
+    exact Nat.cast_div g₂ h₃
+  have h₄ : (↑(y ^ (x / y ^ (2:ℕ))):ℝ) = (↑y:ℝ)^((↑x:ℝ) / ((↑y:ℝ)^2)) := by
+    rw [Nat.cast_pow, h₃]
+    norm_cast
+  rw [←h₄] at h_exp
+  exact Nat.cast_inj.mp h_exp
+
+
+theorem mylemma_xsuby_eq_2xy2
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x ^ y ^ 2 = y ^ x)
+  (hxy : y < x) :
+  x = y ^ (x / y ^ 2) := by
+  -- sketch: y^2 * log x = x * log y
+  have h₃: Real.log (x^(y^2)) = Real.log (y^x) := by
+    norm_cast
+    rw [h₁]
+  have h₄: (↑(y ^ (2:ℕ)):ℝ)  * Real.log x = ↑x * Real.log y := by
+    have h41: Real.log (y^x) = (↑x:ℝ) * Real.log (y) := by
+      exact Real.log_pow y x
+    have h42: Real.log (x^(y^2)) = (↑(y ^ (2:ℕ)):ℝ) * Real.log x := by
+      exact Real.log_pow x (y^2)
+    rw [h41,h42] at h₃
+    exact h₃
+  ring_nf at h₄
+  have h₅: Real.log ↑x = Real.log ↑y * ↑x / (↑(y ^ (2:ℕ)):ℝ) := by
+    by_contra hc
+    rw [mul_comm (Real.log ↑y) (↑x)] at hc
+    rw [← h₄, mul_comm, ← mul_div] at hc
+    rw [div_self, mul_one] at hc
+    . apply hc
+      norm_cast
+    . norm_cast
+      push_neg
+      refine pow_ne_zero 2 ?_
+      exact Nat.ne_of_gt h₀.2
+  have h₆: x = y ^ (x / y ^ 2) := by
+    exact mylemma_xsuby_eq_2xy2_help x y h₀ h₁ h₅ hxy
+  exact h₆
+
+
+
+theorem imo_1997_p5
+  (x y : ℕ)
+  (h₀ : 0 < x ∧ 0 < y)
+  (h₁ : x^(y^2) = y^x) :
+  (x, y) = (1, 1) ∨ (x, y) = (16, 2) ∨ (x, y) = (27, 3) := by
+  have g₁: x^(y^2) = (x^y)^y := by
+    rw [Nat.pow_two]
+    exact Nat.pow_mul x y y
+  by_cases hxy: x ≤ y
+  . exact mylemma_case_xley x y h₀ h₁ g₁ hxy
+  . push_neg at hxy
+    right
+    have h₃: x = y ^ (x / y ^ 2) := by
+      exact mylemma_xsuby_eq_2xy2 x y h₀ h₁ hxy
+    let k:ℕ  := x / y^2 -- { admit },
+    have hk_def: k = x / y^2 := by exact rfl
+    by_cases hk: k < 2
+    . rw [← hk_def] at h₃
+      interval_cases k
+      . exfalso
+        simp at h₃
+        linarith
+      . exfalso
+        simp at *
+        linarith [hxy,h₃] --simp at h₃, rw h₃ at hxy, linarith[hxy], },
+    . push_neg at hk
+      rw [← hk_def] at h₃
+      have h₅: k = y^(k-2) := by
+        rw [h₃] at hk_def
+        nth_rewrite 1 [hk_def]
+        exact Nat.pow_div hk h₀.2
+      by_cases hk5: k < 5
+      . interval_cases k
+        . exfalso
+          simp at h₅
+        . right
+          norm_num
+          simp at h₅
+          symm at h₅
+          rw [h₅] at h₃
+          norm_num at h₃
+          exact { left := h₃, right := h₅ }
+        . simp at h₅
+          symm at h₅
+          have g₂: y^4 = y^2 * y^2 := by ring_nf
+          rw [g₂, h₅] at h₃
+          norm_num at h₃
+          left
+          norm_num
+          constructor
+          . exact h₃
+          . have h₆ : y ^ 2 = 2 ^ 2 := by
+              norm_num
+              exact h₅
+            have h₇: 0 ≤ y := by
+              linarith
+            exact (sq_eq_sq₀ h₇ (by linarith)).mp (h₆)
+      push_neg at hk5
+      by_cases hy: 2 ≤ y
+      . have h₅₁: k < y^(k-2) := by
+          have h₆: 2^(k-2) ≤ y^(k-2) := by
+            have hk1: 3 ≤ k - 2 := by exact Nat.sub_le_sub_right hk5 2
+            exact (Nat.pow_le_pow_iff_left (by linarith)).mpr hy
+          have h₇: 4*k < 2^k := by
+            exact four_times_k_less_than_two_pow_k k hk5
+          have h₇: k < 2^(k-2) := by
+            have h₈ : k < 2 ^ k / 4 := by
+              have h81: 4 ∣ 2^k := by
+                have h82: 2^k = 4*2^(k-2) := by
+                  have h83: k = 2 + (k -2) := by
+                    ring_nf
+                    exact (add_sub_of_le hk).symm
+                  nth_rewrite 1 [h83]
+                  rw [pow_add]
+                  norm_num
+                rw [h82]
+                exact Nat.dvd_mul_right 4 (2^(k-2))
+              exact (Nat.lt_div_iff_mul_lt' h81 k).mpr h₇
+            have h₉: 2 ^ k / 4 = 2 ^ (k-2) := by
+              have g2: k = k - 2 + 2 := by
+                exact (Nat.sub_eq_iff_eq_add hk).mp rfl
+              have h1: 2^k = 2^(k - 2 + 2) := by
+                exact congrArg (HPow.hPow 2) g2
+              have h2: 2 ^ (k - 2 + 2) = 2 ^ (k - 2) * 2 ^ 2 := by rw [pow_add]
+              rw [h1, h2]
+              ring_nf
+              simp
+            linarith
+          linarith
+        exfalso
+        linarith
+      . push_neg at hy
+        interval_cases y
+        . linarith
+        . simp at h₅
+          simp at h₃
+          linarith

imo_proofs/imo_2007_p6.lean

@@ -0,0 +1,571 @@
+import Mathlib
+set_option linter.unusedVariables.analyzeTactics true
+
+open NNReal Nat BigOperators Finset
+
+-- imo-official.org/problems/IMO2007SL.pdf
+
+
+lemma aux1
+  (a : ℕ → NNReal)
+  (m : ℕ)
+  (hm₀ : Nat.succ 4 ≤ m) :
+  a (m - 1) ^ 2 + a m ^ 2 + a 1 ^ 2 + a 2 ^ 2 ≤ ∑ x ∈ Finset.range m, a (x + 1) ^ 2 := by
+  let fs: Finset ℕ := {0, 1, m-2, m-1}
+  have h₀: fs = {0, 1, m-2, m-1} := by rfl
+  have h₁: fs ⊆ Finset.range m := by
+    refine insert_subset ?_ ?_
+    . refine mem_range.mpr ?_
+      exact zero_lt_of_lt hm₀
+    . refine insert_subset ?_ ?_
+      . refine mem_range.mpr ?_
+        linarith
+      . refine insert_subset ?_ ?_
+        . refine mem_range.mpr ?_
+          refine sub_lt ?_ (by norm_num)
+          exact zero_lt_of_lt hm₀
+        . refine singleton_subset_iff.mpr ?_
+          refine mem_range.mpr ?_
+          exact sub_one_lt_of_lt hm₀
+  rw [← Finset.sum_sdiff h₁]
+  have h₂: ∑ x ∈ fs, a (x + 1) ^ 2 = a (m - 1) ^ 2 + a m ^ 2 + a 1 ^ 2 + a 2 ^ 2 := by
+    rw [h₀]
+    have g₀: 0 ∈ fs := by exact mem_insert_self 0 {1, m - 2, m - 1}
+    rw [← Finset.add_sum_erase fs _ g₀]
+    simp
+    have g₁: 4 ≤ m - 1 := by exact Nat.le_sub_one_of_lt hm₀
+    have g₂: 3 ≤ m - 2 := by exact le_sub_of_add_le hm₀
+    have g₃: fs.erase 0 = ({1, m - 2, m - 1}:(Finset ℕ)) := by
+      rw [h₀]
+      refine erase_insert ?h
+      refine forall_mem_not_eq'.mp ?_
+      intros b hb₀ hb₁
+      rw [hb₁] at hb₀
+      norm_num at hb₀
+      cases' hb₀ with hb₀ hb₀
+      . rw [← hb₀] at g₂
+        linarith
+      . rw [← hb₀] at g₁
+        linarith
+    rw [g₃]
+    have g₄: (1:ℕ) ∈ ({1, m - 2, m - 1}:(Finset ℕ)) := by
+      exact mem_insert_self 1 {m - 2, m - 1}
+    rw [← Finset.add_sum_erase _ _ g₄]
+    simp
+    rw [Finset.erase_eq_self.mpr ?_]
+    . have g₅: (m - 2) ∈ ({m - 2, m - 1}:(Finset ℕ)) := by
+        exact mem_insert_self (m - 2) {m - 1}
+      rw [← Finset.add_sum_erase _ _ g₅]
+      simp
+      rw [Finset.erase_eq_self.mpr ?_]
+      . rw [Finset.sum_singleton, Nat.sub_add_cancel (by linarith)]
+        rw [← Nat.sub_add_comm (by linarith)]
+        simp
+        ring_nf
+      . refine Finset.not_mem_singleton.mpr ?_
+        omega
+    . refine forall_mem_not_eq'.mp ?_
+      intros b hb₀ hb₁
+      rw [hb₁] at hb₀
+      simp at hb₀
+      cases' hb₀ with hb₀ hb₀
+      . rw [← hb₀] at g₂
+        linarith
+      . rw [← hb₀] at g₁
+        linarith
+  rw [add_comm _ (∑ x ∈ fs, a (x + 1) ^ 2), h₂]
+  exact le_self_add
+
+
+
+lemma aux2
+  (a : ℕ → NNReal) :
+  ∀ (n : ℕ),
+    4 < n ∧ n < 101 →
+      (∀ (x y : ℕ), x % n = y % n → a (x + 1) = a (y + 1)) →
+        ∑ x ∈ range n, (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) ≤
+          (∑ x ∈ range n, a (x + 1) ^ 2) ^ 2 := by
+  intro n hn₀ hn₂
+  cases' hn₀ with hn₀ hn₁
+  have hn₃: n = (n - 2) + 1 + 1 := by omega
+  nth_rw 1 [hn₃,]
+  rw [Finset.sum_range_succ, sum_range_succ]
+  have hn₄: a (n - 2 + 1) = a (n - 1) := by
+    refine congrArg a (by omega)
+  have hn₅: a (n - 2 + 3) = a 1 := by
+    refine hn₂ (n - 2 + 2) 0 ?_
+    rw [Nat.zero_mod, Nat.sub_add_cancel ?_]
+    . rw [Nat.mod_self n]
+    . linarith
+  have hn₆: a (n - 2 + 1 + 3) = a 2 := by
+    refine hn₂ (n - 2 + 3) 1 ?_
+    symm
+    rw [Nat.mod_eq_of_lt (by linarith)]
+    have g₀: n - 2 + 3 = n + 1 := by linarith
+    rw [g₀]
+    refine Eq.symm (mod_eq_of_modEq ?_ (by linarith))
+    exact Nat.add_modEq_left
+  rw [← hn₃, hn₄, hn₅, hn₆]
+  refine le_induction ?_ ?_ n hn₀
+  . repeat rw [Finset.sum_range_succ]
+    simp
+    ring_nf
+    repeat refine add_le_add_right ?_ _
+    refine le_of_eq ?_
+    rfl
+  . intros m hm₀ hm₁
+    have hm₂: m + 1 - 2 = m - 2 + 1 := by
+      rw [add_comm, add_comm _ 1, Nat.add_sub_assoc ?_ 1]
+      omega
+    rw [hm₂, Finset.sum_range_succ, sum_range_succ]
+    have hm₃: m - 2 + 1 = m - 1 := by exact id (Eq.symm hm₂)
+    have hm₄: m - 2 + 2 = m := by exact Eq.symm ((fun {m n} => pred_eq_succ_iff.mp) hm₂)
+    have hm₅: m - 2 + 3 = m + 1 := by omega
+    have hm₆: m + 1 - 1 = m := by exact rfl
+    rw [hm₃, hm₄, hm₅, hm₆]
+    clear hm₃ hm₄ hm₅ hm₆
+    rw [add_sq, add_assoc ((∑ x ∈ Finset.range m, a (x + 1) ^ 2) ^ 2)]
+    have h₅₀: 2 * a (m - 1) ^ 2 * a (m + 1) ^ 2 + 2 * a m ^ 2 * a (m + 1) ^ 2
+      + 2 * a (m + 1) ^ 2 * a 1 ^ 2 + 2 * a (m + 1) ^ 2 * a 2 ^ 2 + a (m + 1) ^ 4 ≤
+      (2 * ∑ x ∈ Finset.range m, a (x + 1) ^ 2) * a (m + 1) ^ 2 + (a (m + 1) ^ 2) ^ 2 := by
+      rw [← pow_mul]
+      simp
+      have h₅₁: 2 * a (m - 1) ^ 2 * a (m + 1) ^ 2 + 2 * a m ^ 2 * a (m + 1) ^ 2 + 2 * a (m + 1) ^ 2 * a 1 ^ 2 +
+        2 * a (m + 1) ^ 2 * a 2 ^ 2 =
+        2 * a (m + 1) ^ 2 * (a (m - 1) ^ 2 + a m ^ 2 + a 1 ^ 2 + a 2 ^ 2) := by
+        ring_nf
+      rw [h₅₁, mul_assoc 2 _ (a (m + 1) ^ 2), mul_comm (∑ x ∈ Finset.range m, a (x + 1) ^ 2), ← mul_assoc 2]
+      have h₅₂: a (m - 1) ^ 2 + a m ^ 2 + a 1 ^ 2 + a 2 ^ 2 ≤ ∑ x ∈ Finset.range m, a (x + 1) ^ 2 := by
+        exact aux1 a m hm₀
+      refine mul_le_mul ?_ ?_ ?_ ?_
+      . exact le_of_eq (by rfl)
+      . exact h₅₂
+      . exact _root_.zero_le (a (m - 1) ^ 2 + a m ^ 2 + a 1 ^ 2 + a 2 ^ 2)
+      . exact _root_.zero_le (2 * a (m + 1) ^ 2)
+    have h₅₃: ∑ x ∈ Finset.range (m - 2), (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) +
+        a (m - 1) ^ 4 + 2 * a (m - 1) ^ 2 * a m ^ 2 + a m ^ 4 + 2 * a m ^ 2 * a 1 ^ 2
+      ≤ (∑ x ∈ Finset.range m, a (x + 1) ^ 2) ^ 2 := by
+      have h₅₄: ∑ x ∈ Finset.range (m - 2), (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) +
+          a (m - 1) ^ 4 + 2 * a (m - 1) ^ 2 * a m ^ 2 + a m ^ 4 + 2 * a m ^ 2 * a 1 ^ 2
+        ≤ ∑ x ∈ Finset.range (m - 2), (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) +
+          (a (m - 1) ^ 4 + 2 * a (m - 1) ^ 2 * a m ^ 2 + 2 * a (m - 1) ^ 2 * a 1 ^ 2) +
+          (a m ^ 4 + 2 * a m ^ 2 * a 1 ^ 2 + 2 * a m ^ 2 * a 2 ^ 2) := by
+        repeat rw [add_assoc]
+        repeat refine add_le_add_left ?_ _
+        have h₅₅: 2 * a (m - 1) ^ 2 * a 1 ^ 2 + (a m ^ 4 + (2 * a m ^ 2 * a 1 ^ 2 + 2 * a m ^ 2 * a 2 ^ 2)) =
+          (a m ^ 4 + 2 * a m ^ 2 * a 1 ^ 2) + (2 * a (m - 1) ^ 2 * a 1 ^ 2 + 2 * a m ^ 2 * a 2 ^ 2) := by
+          ring_nf
+        rw [h₅₅]
+        exact le_self_add
+      exact le_trans h₅₄ hm₁
+    apply add_le_add h₅₃ at h₅₀
+    have h₅₆: ∑ x ∈ Finset.range (m - 2), (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2)
+        + a (m - 1) ^ 4 + 2 * a (m - 1) ^ 2 * a m ^ 2 + a m ^ 4 + 2 * a m ^ 2 * a 1 ^ 2
+        + (2 * a (m - 1) ^ 2 * a (m + 1) ^ 2 + 2 * a m ^ 2 * a (m + 1) ^ 2 + 2 * a (m + 1) ^ 2 * a 1 ^ 2
+        + 2 * a (m + 1) ^ 2 * a 2 ^ 2 + a (m + 1) ^ 4)
+      = ∑ x ∈ Finset.range (m - 2), (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) +
+        (a (m - 1) ^ 4 + 2 * a (m - 1) ^ 2 * a m ^ 2 + 2 * a (m - 1) ^ 2 * a (m + 1) ^ 2) +
+      (a m ^ 4 + 2 * a m ^ 2 * a (m + 1) ^ 2 + 2 * a m ^ 2 * a 1 ^ 2) +
+    (a (m + 1) ^ 4 + 2 * a (m + 1) ^ 2 * a 1 ^ 2 + 2 * a (m + 1) ^ 2 * a 2 ^ 2) := by
+      repeat rw [add_assoc]
+      simp
+      ring_nf
+    rw [← h₅₆]
+    exact h₅₀
+
+
+theorem imo_2007_p6
+  (a : ℕ → NNReal)
+  (h₀ : ∑ x ∈ Finset.range 100, ((a (x + 1)) ^ 2) = 1)
+  (h₁ : ∀ x y, x % 100 = y % 100 → a (x + 1) = a (y + 1)) :
+  ∑ x ∈ Finset.range (99), ((a (x + 1)) ^ 2 * a (x + 2)) + (a 100) ^ 2 * a 1 < (12:NNReal) / (25:NNReal) := by
+  have h₂: ∀ x, 2 * a x ^ 2 * a (x + 1) * a (x + 2) ≤
+    (a x * a (x + 1)) ^ 2 + (a x * a (x + 2)) ^ 2 := by
+    intro x
+    have h₂₀: 2 * (a x * a (x + 1)) * (a x * a (x + 2)) ≤
+      (a x * a (x + 1)) ^ 2 + (a x * a (x + 2)) ^ 2 := by
+      exact two_mul_le_add_sq (a x * a (x + 1)) (a x * a (x + 2))
+    have h₂₁: 2 * (a x * a (x + 1)) * (a x * a (x + 2)) = 2 * a x ^ 2 * a (x + 1) * a (x + 2) := by
+      rw [pow_two]
+      ring_nf
+    exact le_of_eq_of_le (id (Eq.symm h₂₁)) h₂₀
+  have h₃: ∀ x ∈ Finset.range 100, a (x + 1) ≤ 1 := by
+    intros x hx₀
+    by_contra hx₁
+    push_neg at hx₁
+    let fsx : Finset ℕ := {x}
+    have hx₂: 1 < ∑ x ∈ range 100, a (x + 1) ^ 2 := by
+      have hx₃: 0 ≤ ∑ x ∈ (range 100 \ fsx), a (x + 1) ^ 2 := by
+        exact _root_.zero_le (∑ x ∈ range 100 \ fsx, a (x + 1) ^ 2)
+      have hx₄: 1 < ∑ x ∈ (fsx), a (x + 1) ^ 2 := by
+        rw [Finset.sum_singleton]
+        refine one_lt_pow₀ hx₁ ?_
+        norm_num
+      have hx₅: ∑ x ∈ (range 100 \ fsx), a (x + 1) ^ 2 + ∑ x ∈ (fsx), a (x + 1) ^ 2 =
+        ∑ x ∈ range 100, a (x + 1) ^ 2 := by
+        rw [← Finset.sum_union ?_]
+        . rw [Finset.sdiff_union_self_eq_union]
+          have hx₆: range 100 ∪ fsx = range 100 := by
+            refine Finset.union_eq_left.mpr ?_
+            exact singleton_subset_iff.mpr hx₀
+          rw [hx₆]
+        . exact sdiff_disjoint
+      rw [← hx₅]
+      exact lt_add_of_nonneg_of_lt hx₃ hx₄
+    simp_all only [mem_range, lt_self_iff_false]
+  have h₄: (∑ x ∈ Finset.range 100, (a (x + 2) * (a (x + 1) ^ 2  + 2 * a (x + 2) * a (x + 3)))) ^ 2 ≤
+    ∑ x ∈ Finset.range 100, (a (x + 1) ^ 4 + 6 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) := by
+    have h₄₀: (∑ x ∈ Finset.range 100, (a (x + 2) * (a (x + 1) ^ 2 + 2 * a (x + 2) * a (x + 3)))) ^ 2 ≤
+      (∑ x ∈ Finset.range 100, (a (x + 2) ^ 2)) *
+      (∑ x ∈ Finset.range 100, ((a (x + 1) ^ 2 + 2 * a (x + 2) * a (x + 3))) ^ 2) := by
+      refine sum_mul_sq_le_sq_mul_sq (range 100) (fun i => a (i + 2)) _
+    have h₄₁: ∑ x ∈ Finset.range 100, (a (x + 2) ^ 2) = 1 := by
+      rw [Finset.sum_range_succ'] at h₀
+      simp at h₀
+      rw [Finset.sum_range_succ]
+      have h₄₁₁: a 1 = a 101 := by exact h₁ 0 100 rfl
+      rw [← h₄₁₁]
+      exact h₀
+    have h₄₂: ∑ x ∈ Finset.range 100, ((a (x + 1) ^ 2  + 2 * a (x + 2) * a (x + 3))) ^ 2 =
+      ∑ x ∈ Finset.range 100, ((a (x + 1) ^ 4 + 4 * a (x + 1) ^ 2 * a (x + 2) * a (x + 3)
+        + 4 * a (x + 2) ^ 2 * a (x + 3) ^ 2)) := by
+      refine Finset.sum_congr (rfl) ?_
+      intros x _
+      rw [add_sq]
+      ring_nf
+    rw [h₄₁, one_mul, h₄₂] at h₄₀
+    have h₄₃: ∑ x ∈ Finset.range 100, ((a (x + 1) ^ 4 + 4 * a (x + 1) ^ 2 * a (x + 2) * a (x + 3)
+        + 4 * a (x + 2) ^ 2 * a (x + 3) ^ 2)) ≤
+      ∑ x ∈ Finset.range 100, ((a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * (a (x + 2) ^ 2 + a (x + 3) ^ 2)
+        + 4 * a (x + 2) ^ 2 * a (x + 3) ^ 2)) := by
+      refine Finset.sum_le_sum ?_
+      intros x _
+      rw [add_comm (a (x + 1) ^ 4) _, add_comm (a (x + 1) ^ 4) _]
+      rw [add_assoc, add_assoc]
+      refine add_le_add ?_ ?_
+      . have hx₁: 2 * a (x + 1) ^ 2 * a (x + 1 + 1) * a (x + 1 + 2) ≤
+          (a (x + 1) * a (x + 1 + 1)) ^ 2 + (a (x + 1) * a (x + 1 + 2)) ^ 2 := by
+          exact h₂ (x + 1)
+        have hx₂: 2 * a (x + 1) ^ 2 * a (x + 2) * a (x + 3) ≤
+          a (x + 1) ^ 2 * (a (x + 2) ^ 2 + a (x + 3) ^ 2) := by
+          rw [mul_add]
+          refine le_of_le_of_eq hx₁ ?_
+          ring_nf
+        have hx₃: (4:NNReal) = 2 * 2 := by norm_num
+        rw [hx₃]
+        repeat rw [mul_assoc]
+        have hx₄: 0 < (2:NNReal) := by norm_num
+        refine (mul_le_mul_left hx₄).mpr ?_
+        ring_nf
+        ring_nf at hx₂
+        exact hx₂
+      . exact Preorder.le_refl (a (x + 1) ^ 4 + 4 * a (x + 2) ^ 2 * a (x + 3) ^ 2)
+    have h₄₄: ∑ x ∈ Finset.range 100, ((a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * (a (x + 2) ^ 2 + a (x + 3) ^ 2)
+        + 4 * a (x + 2) ^ 2 * a (x + 3) ^ 2)) =
+      ∑ x ∈ Finset.range 100, (a (x + 1) ^ 4 + 6 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2
+        * a (x + 1) ^ 2 * a (x + 3) ^ 2) := by
+      rw [Finset.sum_add_distrib]
+      have h₄₄₁: ∑ x ∈ range 100, 4 * a (x + 2) ^ 2 * a (x + 3) ^ 2 =
+        ∑ x ∈ range 100, 4 * a (x + 1) ^ 2 * a (x + 2) ^ 2 := by
+        rw [Finset.sum_range_succ _ 99, sum_range_succ' _ 99]
+        have g₀: a 101 = a 1 := by exact h₁ 100 0 rfl
+        have g₁: a 102 = a 2 := by exact h₁ 101 1 rfl
+        rw [g₀, g₁]
+      rw [h₄₄₁, ← Finset.sum_add_distrib]
+      refine Finset.sum_congr (rfl) ?_
+      intros x _
+      rw [mul_add]
+      ring_nf
+    rw [h₄₄] at h₄₃
+    exact le_trans h₄₀ h₄₃
+  have h₆: ∑ x ∈ range 100, 4 * a (x + 1) ^ 2 * a (x + 2) ^ 2 ≤ 1 := by
+    have h₆₀: ∑ x ∈ range 100, 4 * a (x + 1) ^ 2 * a (x + 2) ^ 2 =
+      ∑ x ∈ range 100, 4 * (a (x + 1) ^ 2 * a (x + 2) ^ 2) := by
+      refine Finset.sum_congr rfl ?_
+      intros x _
+      ring_nf
+    rw [h₆₀, ← Finset.mul_sum]
+    let fs₂ := Finset.range (100)
+    let fs₀ : Finset ℕ := fs₂.filter (fun x => Odd x)
+    let fs₁ : Finset ℕ := fs₂.filter (fun x => Even x)
+    have h₆₁ : Disjoint fs₀ fs₁ := by
+      refine Finset.sdiff_eq_self_iff_disjoint.mp (by rfl)
+    have h₆₂ : fs₀ ∪ fs₁ = fs₂ := by
+      symm
+      refine Finset.ext_iff.mpr ?_
+      intro a
+      constructor
+      . intro ha₀
+        refine mem_union.mpr ?mp.a
+        have ha₁: Odd a ∨ Even a := by exact Or.symm (even_or_odd a)
+        cases' ha₁ with ha₂ ha₃
+        . left
+          refine mem_filter.mpr ?mp.a.inl.h.a
+          exact And.symm ⟨ha₂, ha₀⟩
+        . right
+          refine mem_filter.mpr ?mp.a.inl.h.b
+          exact And.symm ⟨ha₃, ha₀⟩
+      . intro ha₀
+        apply mem_union.mp at ha₀
+        cases' ha₀ with ha₁ ha₂
+        . exact mem_of_mem_filter a ha₁
+        . exact mem_of_mem_filter a ha₂
+    have h₆₃: 4 * ∑ i ∈ fs₂, a (i + 1) ^ 2 * a (i + 2) ^ 2 ≤
+      4 * ((∑ i ∈ fs₀, (a (i + 1) ^ 2)) * (∑ i ∈ fs₁, (a (i + 1) ^ 2))) := by
+      refine mul_le_mul (by norm_num) ?_ ?_ (by norm_num)
+      . rw [← h₆₂, Finset.sum_union h₆₁]
+        have g₀: ∑ i ∈ fs₁, a (i + 1) ^ 2 = ∑ i ∈ fs₀, (a i) ^ 2 := by
+          refine sum_bij ?_ ?h.b2 ?h.b3 ?h.b4 ?h.b5
+          . intros b _
+            exact (b + 1)
+          . intros b hb₀
+            apply mem_filter.mp at hb₀
+            cases' hb₀ with hb₀ hb₁
+            have hb₂: Odd (b + 1) := by exact Even.add_one hb₁
+            have hb₃: b ≤ 98 := by
+              by_contra hc₀
+              apply mem_range.mp at hb₀
+              interval_cases b
+              have hc₁: ¬ Even 99 := by decide
+              exact hc₁ hb₁
+            have hb₄: b + 1 < 100 := by linarith
+            have hb₅: (b + 1) ∈ fs₂ := by exact mem_range.mpr hb₄
+            refine mem_filter.mpr ?_
+            exact And.symm ⟨hb₂, hb₅⟩
+          . intros b _ c _ hb₂
+            linarith
+          . intros b hb₀
+            use (b - 1)
+            refine exists_prop.mpr ?h.a
+            have hb₁: b ∈ fs₂ ∧ Odd b := by exact mem_filter.mp hb₀
+            have hb₂: 1 ≤ b := by
+              by_contra hc
+              interval_cases b
+              have hb₃: ¬ Odd 0 := by decide
+              exact hb₃ hb₁.2
+            constructor
+            . cases' hb₁ with hb₁ hb₃
+              have hb₄: Even (b - 1) := by exact Nat.Odd.sub_odd hb₃ (by decide)
+              have hb₅: (b - 1) ∈ fs₂ := by
+                refine mem_range.mpr ?_
+                have hb₆: b < 100 := by exact List.mem_range.mp hb₁
+                omega
+              refine mem_filter.mpr ?_
+              exact And.symm ⟨hb₄, hb₅⟩
+            . exact Nat.sub_add_cancel hb₂
+          . exact fun a_1 _ => rfl
+        have g₁: ∑ x ∈ fs₁, a (x + 1) ^ 2 * a (x + 2) ^ 2 =
+          ∑ x ∈ fs₀, a (x) ^ 2 * a (x + 1) ^ 2 := by
+          refine sum_bij ?_ ?_ ?_ ?_ ?_
+          . intros b _
+            exact (b + 1)
+          . intros b hb₀
+            apply mem_filter.mp at hb₀
+            cases' hb₀ with hb₀ hb₁
+            have hb₂: Odd (b + 1) := by exact Even.add_one hb₁
+            have hb₃: b ≤ 98 := by
+              by_contra hc₀
+              apply mem_range.mp at hb₀
+              interval_cases b
+              have hc₁: ¬ Even 99 := by decide
+              exact hc₁ hb₁
+            have hb₄: b + 1 < 100 := by linarith
+            have hb₅: (b + 1) ∈ fs₂ := by exact mem_range.mpr hb₄
+            refine mem_filter.mpr ?_
+            exact And.symm ⟨hb₂, hb₅⟩
+          . intros b _ c _ hb₂
+            linarith
+          . intros b hb₀
+            use (b - 1)
+            refine exists_prop.mpr ?h.b
+            have hb₁: b ∈ fs₂ ∧ Odd b := by exact mem_filter.mp hb₀
+            have hb₂: 1 ≤ b := by
+              by_contra hc
+              interval_cases b
+              have hb₃: ¬ Odd 0 := by decide
+              exact hb₃ hb₁.2
+            constructor
+            . cases' hb₁ with hb₁ hb₃
+              have hb₄: Even (b - 1) := by exact Nat.Odd.sub_odd hb₃ (by decide)
+              have hb₅: (b - 1) ∈ fs₂ := by
+                refine mem_range.mpr ?_
+                have hb₆: b < 100 := by exact List.mem_range.mp hb₁
+                omega
+              refine mem_filter.mpr ?_
+              exact And.symm ⟨hb₄, hb₅⟩
+            . exact Nat.sub_add_cancel hb₂
+          . exact fun a_1 _ => rfl
+        rw [g₀, g₁, Finset.sum_mul_sum, add_comm, ← sum_add_distrib]
+        refine sum_le_sum ?_
+        intros x hx₀
+        apply mem_filter.mp at hx₀
+        cases' hx₀ with hx₀ hx₁
+        apply mem_range.mp at hx₀
+        by_cases hx₃: x < 99
+        . clear h₀ h₁ h₂ h₃ h₄ h₆₀ g₀ g₁
+          let fs₃ : Finset ℕ := {x, (x + 2)}
+          have hx₄: fs₃ ⊆ fs₀ := by
+            intros b hb₀
+            have hb₁: b = x ∨ b = x + 2 := by
+              have g₀: fs₃ = {x, x + 2} := by rfl
+              simp_all only [mem_insert, mem_singleton]
+            cases' hb₁ with hb₁ hb₁
+            . rw [hb₁]
+              refine mem_filter.mpr ?_
+              apply mem_range.mpr at hx₀
+              exact And.symm ⟨hx₁, hx₀⟩
+            . rw [hb₁]
+              refine mem_filter.mpr ?_
+              constructor
+              . have hx₄: x < 98 := by
+                  by_contra hc
+                  interval_cases x
+                  have hx₅: ¬ Odd 98 := by decide
+                  apply hx₅ hx₁
+                refine mem_range.mpr ?_
+                linarith
+              . refine Odd.add_even hx₁ ?_
+                decide
+          have hx₅: ∑ j ∈ fs₃, a (x + 1) ^ 2 * a j ^ 2 = a (x + 1) ^ 2 * a x ^ 2 + a (x + 1) ^ 2 * a (x + 2) ^ 2 := by
+            have hx₆: fs₃ = {x, x + 2} := by rfl
+            refine Finset.sum_eq_add_of_mem (x) (x + 2) ?_ ?_ (by norm_num) ?_
+            . rw [hx₆]
+              exact mem_insert_self x {x + 2}
+            . rw [hx₆]
+              simp
+            . intros c hc₀ hc₁
+              exfalso
+              rw [hx₆] at hc₀
+              simp only [mem_insert, mem_singleton] at hc₀
+              have hc₃: ¬ (c ≠ x ∧ c ≠ x + 2) := by
+                omega
+              exact hc₃ hc₁
+          rw [← Finset.sum_sdiff hx₄, hx₅]
+          refine le_add_left ?_
+          refine le_of_eq ?_
+          rw [mul_comm (a x ^ 2) (a (x + 1) ^ 2)]
+        . interval_cases x
+          norm_num
+          have hx₄: a 101 = a 1 := by exact h₁ 100 0 rfl
+          let fs₃: Finset ℕ := {1, 99}
+          have hx₅: fs₃ ⊆ fs₀ := by
+            refine Finset.subset_iff.mpr ?_
+            intros b hb₀
+            have hb₁: b = 1 ∨ b = 99 := by exact List.mem_pair.mp hb₀
+            cases' hb₁ with hb₂ hb₂
+            . refine mem_filter.mpr ?_
+              rw [hb₂]
+              constructor
+              . refine mem_range.mpr (by decide)
+              . decide
+            . rw [hb₂]
+              refine mem_filter.mpr ?_
+              constructor
+              . exact self_mem_range_succ 99
+              . decide
+          have hx₆: ∑ x ∈ fs₃, a 100 ^ 2 * a x ^ 2 = a 100 ^ 2 * a 99 ^ 2 + a 100 ^ 2 * a 1 ^ 2 := by
+            clear h₀ h₁ h₂ h₃ h₄ h₆₀
+            have hx₇: fs₃ = {1, 99} := by rfl
+            refine Finset.sum_eq_add_of_mem (99:ℕ) (1:ℕ) ?_ ?_ (by norm_num) ?_
+            . rw [hx₇]
+              decide
+            . rw [hx₇]
+              decide
+            . intros c hc₀ hc₁
+              exfalso
+              have hc₂: c = 99 ∨ c = 1 := by
+                refine Or.symm ?_
+                exact List.mem_pair.mp hc₀
+              have hc₃: ¬ (c ≠ 99 ∧ c ≠ 1) := by omega
+              exact hc₃ hc₁
+          rw [← Finset.sum_sdiff hx₅, hx₄, hx₆]
+          refine le_add_left ?_
+          refine le_of_eq ?_
+          rw [mul_comm (a 99 ^ 2) (a 100 ^ 2)]
+      . exact _root_.zero_le (∑ i ∈ range 100, a (i + 1) ^ 2 * a (i + 2) ^ 2)
+    have h₆₄: 4 * ((∑ i ∈ fs₀, (a (i + 1) ^ 2)) * (∑ i ∈ fs₁, (a (i + 1) ^ 2))) ≤
+      (∑ i ∈ fs₀, (a (i + 1) ^ 2) + ∑ i ∈ fs₁, (a (i + 1) ^ 2)) ^ 2 := by
+      have g₀: ∀ x y : ℝ, 4 * x * y ≤ (x + y) ^ 2 := by
+        intros x y
+        rw [add_sq]
+        have g₁: 2 * x * y ≤ x ^ 2 + y ^ 2 := by exact two_mul_le_add_sq x y
+        linarith
+      rw [← mul_assoc]
+      let x := (∑ i ∈ fs₀, a (i + 1) ^ 2)
+      let y := (∑ i ∈ fs₁, a (i + 1) ^ 2)
+      refine g₀ x y
+    have h₆₅: (∑ i ∈ fs₀, (a (i + 1) ^ 2) + ∑ i ∈ fs₁, (a (i + 1) ^ 2)) ^ 2 = 1 := by
+      rw [← Finset.sum_union h₆₁, h₆₂, h₀]
+      exact one_pow 2
+    refine le_trans h₆₃ ?_
+    refine le_trans h₆₄ ?_
+    rw [h₆₅]
+  let S : NNReal := ∑ x ∈ Finset.range 99, ((a (x + 1)) ^ 2 * a (x + 2)) + (a 100) ^ 2 * a 1
+  have hS : S = ∑ x ∈ Finset.range 99, ((a (x + 1)) ^ 2 * a (x + 2)) + (a 100) ^ 2 * a 1 := by rfl
+  rw [← hS]
+  have hS₁ : S = ∑ x ∈ Finset.range 100, ((a (x + 1)) ^ 2 * a (x + 2)) := by
+    rw [Finset.sum_range_succ]
+    norm_num
+    have g₀: a 101 = a 1 := by exact h₁ 100 0 rfl
+    rw [g₀]
+  have h₇: (3 * S) ^ 2 ≤ 2 := by
+    have h₇₀: 3 * S = ∑ x ∈ Finset.range 100, (a (x + 2) * (a (x + 1) ^ 2  + 2 * a (x + 2) * a (x + 3))) := by
+      have g₀: ∑ x ∈ Finset.range 100, (a (x + 2) * (a (x + 1) ^ 2  + 2 * a (x + 2) * a (x + 3))) =
+        ∑ x ∈ Finset.range 100, (a (x + 1) ^ 2 * a (x + 2) + 2 * a (x + 2) ^ 2 * a (x + 3)) := by
+        refine Finset.sum_congr rfl ?_
+        intros x _
+        ring_nf
+      have g₁: (3:NNReal) = 1 + 2 := by norm_num
+      rw [g₀, Finset.sum_add_distrib]
+      rw [g₁, hS₁, add_mul, one_mul, Finset.mul_sum]
+      simp
+      rw [Finset.sum_range_succ' _ 99, sum_range_succ _ 99]
+      norm_num
+      have g₂: a 101 = a 1 := by exact h₁ 100 0 rfl
+      have g₃: a 102 = a 2 := by exact h₁ 101 1 rfl
+      rw [g₂, g₃, ← mul_assoc 2]
+      simp
+      refine Finset.sum_congr rfl ?_
+      intros x _
+      ring_nf
+    rw [← h₇₀] at h₄
+    refine le_trans h₄ ?_
+    have h₇₁: ∑ x ∈ range 100, (a (x + 1) ^ 4 + 6 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) =
+      ∑ x ∈ range 100, (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) +
+      ∑ x ∈ range 100, 4 * a (x + 1) ^ 2 * a (x + 2) ^ 2 := by
+      rw [← Finset.sum_add_distrib]
+      refine Finset.sum_congr rfl ?_
+      intros x _
+      ring_nf
+    have h₇₂: ∑ x ∈ range 100, (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) ≤ 1 := by
+      refine le_trans (aux2 a 100 ?_ h₁) ?_
+      . omega
+      . refine (sq_le_one_iff₀ ?_).mpr ?_
+        . exact _root_.zero_le (∑ x ∈ range 100, a (x + 1) ^ 2)
+        . rw [← h₀]
+    rw [h₇₁, ← one_add_one_eq_two]
+    refine add_le_add ?_ h₆
+    norm_num
+    exact h₇₂
+  have h₈ : S ≤ (NNReal.sqrt 2) / (3:NNReal) := by
+    have h₆₀: NNReal.sqrt (((3:NNReal) * S) ^ 2) ≤ NNReal.sqrt 2 := by
+      exact NNReal.sqrt_le_sqrt.mpr h₇
+    rw [sqrt_sq, mul_comm] at h₆₀
+    refine (le_div_iff₀ (by norm_num)).mpr h₆₀
+  have h₉: (NNReal.sqrt 2) / (3:NNReal) < (12:NNReal) / (25:NNReal)  := by
+    have h₇₁: 2 < 144 / (625:NNReal) * 9 := by
+      refine (one_lt_div (by norm_num)).mp ?_
+      rw [mul_comm_div, ← mul_div_assoc, div_div]
+      norm_num
+      refine (one_lt_div (by norm_num)).mpr ?_
+      norm_num
+    have h₇₂: (NNReal.sqrt 2 / 3:NNReal) ^ 2 < (12 / 25:NNReal) ^ 2 := by
+      rw [div_pow, div_pow]
+      norm_num
+      refine (div_lt_iff₀ ?_).mpr h₇₁
+      exact ofNat_pos'
+    have h₇₃: NNReal.sqrt ((NNReal.sqrt 2 / 3) ^ 2) < NNReal.sqrt ((12 / 25) ^ 2) := by
+      exact sqrt_lt_sqrt.mpr h₇₂
+    rw [sqrt_sq, sqrt_sq] at h₇₃
+    exact h₇₃
+  exact lt_of_le_of_lt h₈ h₉

imo_proofs/imo_2022_p2.lean

@@ -0,0 +1,256 @@
+import Mathlib
+set_option linter.unusedVariables.analyzeTactics true
+
+
+theorem imo_2022_p2_simple
+  (g: ℝ → ℝ)
+  (h₀: ∀ x, 0 < x → ∃ y:ℝ , (0 < y ∧ g (x) + g (y) ≤ 2 * x * y
+        ∧ (∀ z:ℝ, (0 < z ∧ z ≠ y) →  ¬ g (x) + g (z) ≤ 2 * x * z) )) :
+  (∀ x:ℝ , 0 < x → g x = x^2) := by
+  have h₁: ∀ x y:ℝ , 0 < x ∧ 0 < y → g x + g y ≤ 2 * x * y → x = y := by
+    intros x y hp h₁
+    by_contra! hc
+    have g₁: 2 * x * x < g x + g x := by
+      let ⟨p,h₁₁⟩ := h₀ x hp.1
+      cases' h₁₁ with h₁₁ h₁₂
+      cases' h₁₂ with h₁₂ h₁₃
+      by_cases hxp: x ≠ p
+      . have h₁₄: ¬ g x + g x ≤ 2 * x * x := by
+          refine h₁₃ x ?_
+          constructor
+          . exact hp.1
+          . exact hxp
+        exact not_le.mp h₁₄
+      . push_neg at hxp
+        exfalso
+        have hpy: y ≠ p := by exact Ne.trans_eq (id (Ne.symm hc)) hxp
+        have hcy: ¬g x + g y ≤ 2 * x * y := by
+          refine h₁₃ y ?_
+          constructor
+          . exact hp.2
+          . exact hpy
+        linarith
+    have g₂: 2 * y * y < g y + g y := by
+      let ⟨p,h₁₁⟩ := h₀ y hp.2
+      cases' h₁₁ with h₁₁ h₁₂
+      cases' h₁₂ with h₁₂ h₁₃
+      by_cases hyp: y ≠ p
+      . have h₁₄: ¬ g y + g y ≤ 2 * y * y := by
+          refine h₁₃ y ?_
+          constructor
+          . exact hp.2
+          . exact hyp
+        exact not_le.mp h₁₄
+      . push_neg at hyp
+        exfalso
+        have hpx: x ≠ p := by exact Ne.trans_eq hc hyp
+        have hcy: ¬g x + g y ≤ 2 * x * y := by
+          rw [add_comm, mul_right_comm]
+          refine h₁₃ x ?_
+          constructor
+          . exact hp.1
+          . exact hpx
+        linarith
+    ring_nf at g₁ g₂
+    simp at g₁ g₂
+    have g₃: x ^ 2 + y ^ 2 < g x + g y := by exact add_lt_add g₁ g₂
+    have g₄: x ^ 2 + y ^ 2 < 2 * x * y := by exact LT.lt.trans_le g₃ h₁
+    have g₅: (x - y) ^ 2 < 0 := by
+      rw [sub_sq, sub_add_eq_add_sub]
+      exact sub_neg.mpr g₄
+    have g₆: (x - y) ≠ 0 := by exact sub_ne_zero.mpr hc
+    have g₇: 0 < (x - y) ^ 2 := by exact (sq_pos_iff).mpr g₆
+    have g₈: (0:ℝ) ≠ 0 := by
+      refine ne_of_lt ?_
+      exact lt_trans g₇ g₅
+    refine false_of_ne g₈
+  have h₂: ∀ x:ℝ , 0 < x → g x ≤ x ^ 2 := by
+    intros x hxp
+    let ⟨y,h₁₁⟩ := h₀ x hxp
+    cases' h₁₁ with h₁₁ h₁₂
+    cases' h₁₂ with h₁₂ h₁₃
+    have hxy: x = y := by
+      apply h₁ x y
+      . exact { left := hxp, right := h₁₁ }
+      . exact h₁₂
+    rw [← hxy] at h₁₂
+    linarith
+  have h₃: ∀ x:ℝ , 0 < x → ¬ g x < x ^ 2 := by
+    simp
+    by_contra! hc
+    let ⟨x,hxp⟩ := hc
+    cases' hxp with hxp h₃
+    let d₁:ℝ := x ^ 2 - g x
+    have hd₁ : g x = x ^ 2 - d₁ := by exact (sub_sub_self (x ^ 2) (g x)).symm
+    let z:ℝ := x + Real.sqrt d₁
+    have hz: z = x + Real.sqrt d₁ := by exact rfl
+    have hzp: 0 < z := by
+      refine add_pos hxp ?_
+      refine Real.sqrt_pos_of_pos ?_
+      exact sub_pos.mpr h₃
+    have hxz: z ≠ x := by
+      rw [hz]
+      simp
+      push_neg
+      refine Real.sqrt_ne_zero'.mpr ?_
+      exact sub_pos.mpr h₃
+    have h₅: g x + g z ≤ 2 * x * z := by
+      rw [hd₁]
+      have h₅₁: - d₁ + Real.sqrt (x ^ 2 - (x ^ 2 - d₁)) ^ 2 ≤ 0 := by
+        simp
+        rw [Real.sq_sqrt]
+        exact sub_nonneg_of_le (h₂ x hxp)
+      have h₅₂: x ^ 2 - d₁ + z ^ 2 ≤ 2 * x * z := by
+        rw [hz, mul_add, add_sq]
+        ring_nf
+        repeat rw [add_assoc]
+        refine add_le_add_left ?_ (x * Real.sqrt (x ^ 2 - g x) * 2)
+        rw [hd₁]
+        linarith
+      exact add_le_of_add_le_left h₅₂ (h₂ z hzp)
+    let ⟨y,hyp⟩ := h₀ x hxp
+    cases' hyp with hyp h₆
+    cases' h₆ with h₆ h₇
+    have hxy: x = y := by
+      apply h₁
+      . exact { left := hxp, right := hyp }
+      . exact h₆
+    have h₈: ¬g x + g z ≤ 2 * x * z := by
+        refine h₇ z ?_
+        constructor
+        . exact hzp
+        . exact Ne.trans_eq hxz hxy
+    linarith[h₅,h₈]
+  intros x hxp
+  have g₂: g x ≤ x ^ 2 := by exact h₂ x hxp
+  have g₃: ¬ g x < x ^ 2 := by exact h₃ x hxp
+  linarith
+
+
+
+
+
+theorem imo_2022_p2
+  (f: ℝ → ℝ)
+  (hfp: ∀ x:ℝ, 0 < x → 0 < f x)
+  (h₀: ∀ x:ℝ , 0 < x → ∃! y:ℝ , 0 < y ∧ (x * f (y) + y * f (x) ≤ 2) ):
+  ∀ x:ℝ , 0 < x → f (x) = 1 / x := by
+  have h₁: ∀ x y:ℝ , (0 < x ∧ 0 < y) → (x * f (y) + y * f (x) ≤ 2) → x = y := by
+    intros x y hp h₁
+    by_contra! hc
+    have h₁₀: x * f x + x * f x > 2 := by
+      let ⟨z,h₁₁⟩ := h₀ x hp.1
+      cases' h₁₁ with h₁₁ h₁₂
+      have h₁₄: y = z := by
+        apply h₁₂ y
+        constructor
+        . exact hp.2
+        . exact h₁
+      have hxz: ¬ x = z := by exact Ne.trans_eq hc h₁₄
+      have h₁₆: ¬ (fun y => 0 < y ∧ x * f y + y * f x ≤ 2) x := by
+        exact mt (h₁₂ x) hxz
+      have h₁₇: ¬ (0 < x ∧ x * f x + x * f x ≤ 2) := by exact h₁₆
+      push_neg at h₁₇
+      exact h₁₇ hp.1
+    have h₁₁: y * f y + y * f y > 2 := by
+      let ⟨z,h₁₁⟩ := h₀ y hp.2
+      cases' h₁₁ with h₁₁ h₁₂
+      have h₁₄: x = z := by
+        apply h₁₂ x
+        constructor
+        . exact hp.1
+        . rw [add_comm]
+          exact h₁
+      have hxz: ¬ y = z := by exact Ne.trans_eq (id (Ne.symm hc)) h₁₄
+      have h₁₆: ¬ (fun y_2 => 0 < y_2 ∧ y * f y_2 + y_2 * f y ≤ 2) y := by
+        exact mt (h₁₂ y) hxz
+      have h₁₇: ¬ (0 < y ∧ y * f y + y * f y ≤ 2) := by exact h₁₆
+      push_neg at h₁₇
+      exact h₁₇ hp.2
+    ring_nf at h₁₀ h₁₁
+    simp at h₁₀ h₁₁
+    have h₁₅: 1 / x < f x := by exact (div_lt_iff₀' hp.1).mpr (h₁₀)
+    have h₁₆: 1 / y < f y := by exact (div_lt_iff₀' hp.2).mpr (h₁₁)
+    have h₁₂: x / y + y / x < 2 := by
+      refine lt_of_le_of_lt' h₁ ?_
+      refine add_lt_add ?_ ?_
+      . rw [← mul_one_div]
+        exact (mul_lt_mul_left hp.1).mpr h₁₆
+      . rw [← mul_one_div]
+        exact (mul_lt_mul_left hp.2).mpr h₁₅
+    have h₁₃: 2 < x / y + y / x := by
+      refine lt_of_mul_lt_mul_right ?_ (le_of_lt hp.1)
+      refine lt_of_mul_lt_mul_right ?_ (le_of_lt hp.2)
+      repeat rw [add_mul, mul_assoc]
+      rw [mul_comm x y, ← mul_assoc (x/y)]
+      rw [div_mul_comm x y y, div_mul_comm y x x, div_self, div_self]
+      . ring_nf
+        refine lt_of_sub_pos ?_
+        rw [mul_comm _ 2, ← mul_assoc]
+        rw [← sub_sq']
+        refine sq_pos_of_ne_zero ?_
+        exact sub_ne_zero.mpr hc.symm
+      . exact ne_of_gt hp.1
+      . exact ne_of_gt hp.2
+    linarith
+  have h₂: ∀ x:ℝ , 0 < x → x * f x ≤ 1 := by
+    intros x hxp
+    let ⟨y,h₂₁⟩ := h₀ x hxp
+    cases' h₂₁ with h₂₁ h₂₂
+    have hxy: x = y := by
+      apply h₁ x y
+      . constructor
+        . exact hxp
+        . exact h₂₁.1
+      . exact h₂₁.2
+    rw [← hxy] at h₂₁
+    linarith
+  have h₃: ∀ x:ℝ , 0 < x → ¬ x * f x < 1 := by
+    by_contra! hc
+    let ⟨x,hxp⟩ := hc
+    cases' hxp with hxp h₃
+    let d₁:ℝ := 1 - x * f x
+    have hd₁ : x * f x = 1 - d₁ := by exact (sub_sub_self 1 (x * f x)).symm
+    let z:ℝ := x + d₁ / f x
+    have hz: z = x + d₁ / f x := by exact rfl
+    have hzp: 0 < z := by
+      refine add_pos hxp ?_
+      refine div_pos ?_ ?_
+      . exact sub_pos.mpr h₃
+      . exact hfp x hxp
+    have hxz: ¬ x = z := by
+      by_contra! hcz₀
+      rw [← hcz₀] at hz
+      have hcz₁: 0 < d₁ / f x := by
+        refine div_pos ?_ (hfp x hxp)
+        exact sub_pos.mpr h₃
+      linarith
+    have h₄: ¬ (x * f z + z * f x ≤ 2) := by
+      have h₄₁: x * f z + z * f x ≤ 2 → x = z := by
+        exact h₁ x z { left := hxp, right := hzp }
+      exact mt h₄₁ hxz
+    have h₅: x * f z < 1 := by
+      suffices h₅₁: z * f z ≤ 1 by
+        refine lt_of_lt_of_le ?_ h₅₁
+        refine (mul_lt_mul_right (hfp z hzp)).mpr ?_
+        rw [hz]
+        refine lt_add_of_pos_right x ?_
+        refine div_pos ?_ (hfp x hxp)
+        exact sub_pos.mpr h₃
+      exact h₂ z hzp
+    have h₆: x * f z + z * f x < 2 := by
+      suffices h₇: z * f x ≤ 1 by
+        linarith
+      rw [hz, add_mul, hd₁]
+      rw [div_mul_comm d₁ (f x) (f x)]
+      rw [div_self]
+      . rw [one_mul, sub_add_cancel]
+      . exact Ne.symm (ne_of_lt (hfp x hxp))
+    linarith
+  intros x hxp
+  have h₄: x * f x ≤ 1 := by exact h₂ x hxp
+  have h₅: ¬ x * f x < 1 := by exact h₃ x hxp
+  refine eq_div_of_mul_eq ?_ ?_
+  . exact ne_of_gt hxp
+  . push_neg at h₅
+    linarith

imo_proofs/imo_2022_p5.lean

@@ -0,0 +1,587 @@
+import Mathlib
+set_option linter.unusedVariables.analyzeTactics true
+
+open Nat
+
+
+lemma mylemma_1
+  (b p: ℕ)
+  (h₀: 0 < b)
+  (hbp: b < p) :
+  (1 + (b * p + b ^ p) ≤ (1 + b) ^ p) := by
+  refine Nat.le_induction ?_ ?_ p hbp
+  . rw [add_pow 1 b b.succ]
+    rw [Finset.sum_range_succ _ b.succ]
+    simp
+    rw [Finset.sum_range_succ _ b]
+    simp
+    rw [add_comm _ (b * (b + 1))]
+    have gb: b = b - 1 + 1 := by exact (Nat.sub_eq_iff_eq_add h₀).mp rfl
+    nth_rewrite 7 [gb]
+    rw [Finset.sum_range_succ' _ (b-1)]
+    simp
+    omega
+  . intros n _ h₂
+    nth_rewrite 2 [pow_add]
+    rw [pow_one]
+    have h₃: (1 + (b * n + b ^ n)) * (1 + b) ≤ ((1 + b) ^ n) * (1 + b) := by
+      exact mul_le_mul_right' h₂ (1 + b)
+    have h₄: 1 + (b * (n + 1) + b ^ (n + 1)) ≤ (1 + (b * n + b ^ n)) * (1 + b) := by
+      ring_nf
+      rw [Nat.add_assoc _ (b ^ 2 * n) (b ^ n)]
+      exact Nat.le_add_right (1 + b + b * b ^ n + b * n) (b ^ 2 * n + b ^ n)
+    exact le_trans h₄ h₃
+
+
+lemma mylemma_2
+  (b: ℕ) :
+  (b.factorial ≤ b ^ b) := by
+  -- exact factorial_le_pow b
+  -- lean 4 has the lemma factorial_le_pow
+  induction' b with n hi
+  . norm_num
+  . by_cases hnp: 0 < n
+    . rw [ factorial_succ, pow_add, pow_one, mul_comm ]
+      refine mul_le_mul_right (n + 1) ?_
+      have h₂: n^ n ≤ (n + 1)^n := by
+        refine (Nat.pow_le_pow_iff_left ?_).mpr ?_
+        . linarith
+        . linarith
+      exact le_trans hi h₂
+    . push_neg at hnp
+      interval_cases n
+      simp
+
+
+lemma mylemma_3
+  (a b p: ℕ)
+  (hp: Nat.Prime p)
+  (h₁: a ^ p = b.factorial + p)
+  (hbp: p ≤ b) :
+  (p ∣ a) := by
+  have h₂: p ∣ b.factorial := by exact Nat.dvd_factorial (Nat.Prime.pos hp) hbp
+  have h₃: p ∣ b.factorial + p := by exact Nat.dvd_add_self_right.mpr h₂
+  have h₄: p ∣ a^p := by
+    rw [h₁]
+    exact h₃
+  exact Nat.Prime.dvd_of_dvd_pow hp h₄
+
+
+lemma mylemma_42
+  (a b : ℕ)
+  (h₀: 2 ≤ a)
+  (h₁: a < b) :
+  (a + b < a * b ) := by
+  have h₂: a + b < b + b := by exact add_lt_add_right h₁ b
+  have h₃: b + b ≤ a * b := by
+    rw [← two_mul]
+    exact mul_le_mul_right' h₀ b
+  exact gt_of_ge_of_gt h₃ h₂
+
+
+lemma mylemma_43
+  (p: ℕ) :
+  (Finset.Ico p (2 * p - 1)).prod (fun x => x + 1)
+    = (Finset.range (p - 1)).prod (fun x => p + (x + 1)) := by
+  rw [Finset.prod_Ico_eq_prod_range _ (p) (2 * p - 1)]
+  have h₀: 2 * p - 1 - p = p - 1 := by omega
+  rw [h₀]
+  exact rfl
+
+
+lemma mylemma_44
+  (p: ℕ)
+  (hp: 2 ≤ p) :
+  (Finset.range (p - 1)).prod (fun (x : ℕ) => x + 1)
+      = (Finset.range (p - 1)).prod (fun (x : ℕ) => p - (x + 1)) := by
+  refine Nat.le_induction ?_ ?_ p hp
+  . norm_num
+  . intros n hn2 h₀
+    simp at *
+    have hn: 0 < n := by exact lt_of_succ_lt hn2
+    rw [← Nat.mul_factorial_pred hn, h₀]
+    let f: (ℕ → ℕ) := fun (x : ℕ) => n - x
+    have h₁: (Finset.range n).prod f =
+        (Finset.range 1).prod f * (Finset.Ico 1 n).prod f := by
+      exact (Finset.prod_range_mul_prod_Ico (fun k => n - k) hn).symm
+    rw [h₁]
+    have h₂: (Finset.range 1).prod f = n := by
+      exact Finset.prod_range_one fun k => n - k
+    rw [h₂]
+    simp
+    left
+    rw [Finset.prod_Ico_eq_prod_range f 1 n]
+    ring_nf
+    exact rfl
+
+
+lemma mylemma_41
+  (b p: ℕ)
+  -- (h₀: 0 < b)
+  (hp: Nat.Prime p)
+  (hb2p: b < 2 * p) :
+  b.factorial + p < p ^ (2 * p) := by
+  have h₁: b.factorial ≤ (2*p - 1).factorial := by
+    refine factorial_le ?_
+    exact le_pred_of_lt hb2p
+  have gp: 2 ≤ p := by exact Prime.two_le hp
+  have gp1: (p - 1) + 1 = p := by
+    refine Nat.sub_add_cancel ?_
+    exact one_le_of_lt gp
+  let f: (ℕ → ℕ) := (fun (x : ℕ) => x + 1)
+  have h₂: (Finset.range (2 * p - 1)).prod f =
+      (Finset.range (p - 1)).prod (fun (x : ℕ) => p^2 - (x+1)^2) * p := by
+    -- break the prod into three segments rang(p-1) + p + (p+1) until 2p-1
+    have g₀: (Finset.range (2 * p - 1)).prod f = (Finset.range ((p - 1) + 1)).prod f
+           * (Finset.Ico ((p - 1) + 1) (2 * p - 1)).prod f := by
+      symm
+      refine Finset.prod_range_mul_prod_Ico f ?_
+      rw [gp1]
+      have gg₀: p + 2 - 1 ≤ 2 * p - 1 := by
+        refine Nat.sub_le_sub_right ?_ 1
+        rw [add_comm]
+        exact add_le_mul (by norm_num) gp
+      exact le_of_lt gg₀
+    have g₁: (Finset.range ((p - 1) + 1)).prod (fun (x : ℕ) => x + 1) =
+       (Finset.range (p - 1)).prod (fun (x : ℕ) => x + 1) * ((p - 1) + 1) := by
+      exact Finset.prod_range_succ _ (p - 1)
+    rw [g₁] at g₀
+    nth_rewrite 2 [mul_comm] at g₀
+    rw [← mul_assoc] at g₀
+    rw [gp1] at g₀ g₁
+    have g₂: (Finset.Ico ((p - 1) + 1) (2 * p - 1)).prod (fun (x : ℕ) => x + 1)
+              = (Finset.range (p - 1)).prod (fun (x : ℕ) => p + (x+1)) := by
+      rw [gp1]
+      exact mylemma_43 p
+    have g₃: (Finset.range (p - 1)).prod (fun (x : ℕ) => x + 1)
+              = (Finset.range (p - 1)).prod (fun (x : ℕ) => p - (x+1)) := by
+      exact mylemma_44 p gp
+    rw [gp1] at g₂
+    rw [g₂,g₃] at g₀
+    have g₄: (Finset.range (p - 1)).prod (fun (x : ℕ) => p + (x+1))
+      * (Finset.range (p - 1)).prod (fun (x : ℕ) => p - (x+1))
+            = (Finset.range (p - 1)).prod (fun (x : ℕ) => (p + (x+1)) * (p - (x+1))) := by
+      symm
+      exact Finset.prod_mul_distrib
+    have g₅: (fun (x : ℕ) => p ^ 2 - (x+1) ^ 2) = (fun (x : ℕ) => (p + (x+1)) * (p - (x+1))) := by
+      ext1 x
+      exact Nat.sq_sub_sq p (x + 1)
+    rw [g₄,← g₅] at g₀
+    exact g₀
+  have h₃: (Finset.range (p - 1)).prod (fun (x : ℕ) => p^2 - (x+1)^2) * p
+      ≤ (p^2)^(Finset.range (p - 1)).card * p := by
+    refine Nat.mul_le_mul_right ?_ ?_
+    refine Finset.prod_le_pow_card (Finset.range (p - 1)) ?_ (p^2) ?_
+    intros x _
+    exact (p ^ 2).sub_le ((x + 1) ^ 2)
+  simp at *
+  have h₄: b.factorial + p ≤ (p ^ 2) ^ (p - 1) * p + p := by
+    refine add_le_add_right ?_ p
+    refine le_trans ?_ h₃
+    rw [← h₂]
+    rw [Finset.prod_range_add_one_eq_factorial]
+    exact h₁
+  have h₅: b.factorial + p < (p ^ 2) ^ (p - 1) * p * p := by
+    refine lt_of_le_of_lt h₄ ?_
+    rw [add_comm]
+    nth_rewrite 2 [mul_comm]
+    refine mylemma_42 p ((p ^ 2) ^ (p - 1) * p) gp ?_
+    refine lt_mul_left (by linarith) ?_
+    rw [← pow_mul]
+    refine Nat.one_lt_pow ?_ (Nat.lt_of_succ_le gp)
+    refine Nat.mul_ne_zero (by norm_num) ?_
+    exact Nat.sub_ne_zero_iff_lt.mpr gp
+  rw [mul_assoc _ p p, ← pow_two p] at h₅
+  rw [← Nat.pow_succ, succ_eq_add_one, gp1] at h₅
+  rw [Nat.pow_mul]
+  exact h₅
+
+
+lemma mylemma_4
+  (a b p: ℕ)
+  (h₀: 0 < a ∧ 0 < b)
+  (hp: Nat.Prime p)
+  (h₁: a ^ p = b.factorial + p)
+  (hbp: p ≤ b)
+  (h₂: p ∣ a)
+  (hb2p: b < 2 * p) :
+  (a = p) := by
+  have gp: p ≤ a := by exact Nat.le_of_dvd h₀.1 h₂
+  cases' lt_or_eq_of_le gp with h₃ h₃
+  . exfalso
+    cases' h₂ with c h₂
+    have gc: 0 < c := by
+      by_contra! hc0
+      interval_cases c
+      simp at *
+      linarith
+    by_cases hc: c < p
+    . have g₁: c ∣ c^p := by exact dvd_pow_self c (by linarith)
+      have h₄: c ∣ a^p := by
+        rw [h₂, mul_pow]
+        exact dvd_mul_of_dvd_right g₁ (p ^ p)
+      have h₅: c ∣ b.factorial := by exact Nat.dvd_factorial gc (by linarith)
+      have g₂: p = a ^ p - b.factorial := by
+        symm
+        rw [add_comm] at h₁
+        refine (Nat.sub_eq_iff_eq_add ?_).mpr h₁
+        rw [add_comm] at h₁
+        exact le.intro (h₁.symm)
+      have h₆: c ∣ p := by
+        rw [g₂]
+        exact dvd_sub' h₄ h₅
+      have h₇: c = 1 ∨ c = p := by exact (dvd_prime hp).mp h₆
+      cases' h₇ with h₇₀ h₇₁
+      . rw [h₇₀, mul_one] at h₂
+        rw [h₂] at h₃
+        linarith [h₃]
+      . rw [h₇₁] at hc
+        simp at hc
+    . push_neg at hc
+      have g₃: p^2 ≤ a := by
+        rw [h₂, pow_two]
+        exact mul_le_mul_left' hc p
+      have h₃: p^(2*p) ≤ a^p := by
+        rw [pow_mul]
+        exact pow_left_mono p g₃
+      have h₇: b.factorial + p < p^(2*p) := by exact mylemma_41 b p hp hb2p
+      rw [←h₁] at h₇
+      linarith
+  exact h₃.symm
+
+
+lemma mylemma_53
+  (p: ℕ)
+  (hp5: 5 ≤ p) :
+  ((↑p:ℤ) ^ p ≡ ↑p * (↑p + 1) * (-1) ^ (p - 1) + (-1) ^ p [ZMOD (↑p + 1) ^ 2]) := by
+  -- have h₁: ↑p ^ p = Finset.range -- binomial expansion
+  -- take the first two elements out
+  -- show that all the other elements are individually divisible by (p+1)^2
+  -- conclude that their sum is divisible by (p+1)^2
+  -- summation ≡ 0 [ZMOD (↑p + 1) ^ 2]
+  -- now show that Nat.modeq.add
+  have h₀: (↑p:ℤ) = (↑p + 1) - 1 := by simp
+  have h₁: ↑p ^ p ≡ ((↑p + 1) - 1) ^ p [ZMOD (↑p + 1) ^ 2] := by rw [← h₀]
+  have h₂: (((↑p:ℤ) + 1) - 1) ^ p = (↑p * (↑p + 1) * (-1:ℤ) ^ (p - 1) + (-1) ^ p)
+           + (Finset.Ico 2 (p + 1)).sum (fun (k : ℕ) =>
+           (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(p.choose k)) := by
+    rw [sub_eq_add_neg]
+    rw [add_pow ((↑p:ℤ) + 1) (-1:ℤ)]
+    have g₀: 2 ≤ p + 1 := by
+      have gg₀: 5 + 1 ≤ p + 1 := by exact add_le_add_right hp5 1
+      refine le_trans ?_ gg₀
+      norm_num
+    have g₁: 1 ≤ 2 := by norm_num
+    rw [← Finset.sum_range_add_sum_Ico _ g₀]
+    rw [← Finset.sum_range_add_sum_Ico _ g₁]
+    simp
+    rw [add_comm]
+    simp
+    rw [mul_comm]
+    rw [mul_assoc]
+  have h₃: 0 ≡ (Finset.Ico 2 (p + 1)).sum (fun (k : ℕ) => (↑p + 1) ^ k * (-1) ^ (p - k) * ↑(p.choose k))
+                [ZMOD (↑p + 1) ^ 2] := by
+    refine Int.modEq_of_dvd ?_
+    simp
+    refine Finset.dvd_sum ?_
+    intros x g₀
+    have gx: 2 ≤ x := by exact (Finset.mem_Ico.mp g₀).left
+    rw [mul_assoc]
+    refine dvd_mul_of_dvd_left ?_ ((-1:ℤ) ^ (p - x) * ↑(p.choose x))
+    refine pow_dvd_pow ((↑p:ℤ) + 1) gx
+  rw [h₂] at h₁
+  rw [← add_zero ((↑p:ℤ) ^ p)] at h₁
+  exact Int.ModEq.add_right_cancel h₃ h₁
+
+
+lemma mylemma_52
+  (p: ℕ)
+  (hp: Nat.Prime p)
+  (hp5: 5 ≤ p)
+  (h₀: (p + 1) ^ 2 ∣ p ^ p - p) :
+  False := by
+  have h₁: ((↑p^p - ↑p):ℤ) ≡ (↑(p.choose 1) * ↑(p + 1) * (-1:ℤ)^(p-1) + (-1:ℤ)^p) - ↑p
+      [ZMOD ↑(p+1)^2] := by
+    refine Int.ModEq.sub_right (↑p) ?_
+    simp
+    exact mylemma_53 p hp5
+  have gpo: Odd p := by
+    refine Nat.Prime.odd_of_ne_two hp ?_
+    linarith [hp5]
+  have gpe: Even (p - 1) := by
+    refine hp.even_sub_one ?_
+    linarith [hp5]
+  have g₁: (-1:ℤ) ^ (p - 1) = 1 := by exact Even.neg_one_pow gpe
+  have g₂: (-1:ℤ) ^ (p) = -1 := by exact Odd.neg_one_pow gpo
+  rw [g₁,g₂] at h₁
+  simp at h₁
+  have h₂: (p ^ p - p) ≡ (p * (p + 1)) - 1 - p [MOD ((p + 1) ^ 2)] := by
+    refine Int.natCast_modEq_iff.mp ?_
+    have g₃: p ≤ p^p := by
+      refine Nat.le_self_pow (by linarith) _
+    rw [Nat.cast_sub g₃]
+    have g₄: p ≤ p * (p + 1) - 1 := by
+      rw [mul_add]
+      simp
+      rw [add_comm, Nat.add_sub_assoc]
+      . simp
+      . rw [← pow_two]
+        refine Nat.one_le_pow 2 p (by linarith)
+    rw [Nat.cast_sub g₄]
+    have g₅: 1 ≤ p * (p + 1) := by
+      rw [← mul_one (p * (p + 1))]
+      refine Nat.le_mul_of_pos_left ?_ ?_
+      refine Nat.mul_pos (by linarith) (by linarith)
+    rw [Nat.cast_sub g₅]
+    rw [← sub_eq_add_neg] at h₁
+    norm_cast
+    norm_cast at h₁
+  have h₃: p * (p + 1) - 1 - p = p^2 - 1 := by
+    rw [Nat.sub_sub, mul_add]
+    simp
+    rw [← pow_two]
+    exact Nat.add_sub_add_right (p^2) p 1
+  rw [h₃] at h₂
+  clear h₃ gpo gpe g₁ g₂
+  -- now derive a line of contradictions from h₀
+  have hc₁: (p ^ p - p) ≡ 0 [MOD (p+1)^2] := by exact modEq_zero_iff_dvd.mpr h₀
+  -- mix the contradiction with what we had in h₂
+  have h₄: p ^ 2 - 1 ≡ 0 [MOD (p+1)^2] := by
+    apply Nat.ModEq.symm at h₂
+    exact Nat.ModEq.trans h₂ hc₁
+  have h₅: p - 1 ≡ 0 [MOD (p+1)] := by
+    rw [pow_two] at h₄
+    have g₀: p^2 - 1^2 = (p-1) * (p+1) := by
+      rw [mul_comm]
+      exact Nat.sq_sub_sq p 1
+    simp at g₀
+    rw [g₀] at h₄
+    have g₁: p + 1 ≠ 0 := by linarith
+    refine Nat.ModEq.mul_right_cancel' g₁ ?_
+    rw [zero_mul]
+    exact h₄
+  have h₆: p - 1 ≤ 0 := by
+    refine Nat.ModEq.le_of_lt_add h₅ ?_
+    simp
+    rw [← succ_eq_add_one]
+    refine Nat.sub_lt_succ p 1
+  have h₇: 0 < p - 1 := by
+    simp
+    linarith
+  linarith [h₆,h₇]
+
+
+lemma mylemma_51
+  (p: ℕ)
+  (hpl: 5 ≤ p) :
+  (p + p.factorial < p ^ p) := by
+  -- we use induction
+  refine Nat.le_induction ?_ ?_ p (hpl)
+  . exact Nat.lt_of_sub_eq_succ rfl
+  . intros n hn h₁
+    have h₂: n + 1 + (n + 1).factorial = (n.factorial + 1) * (n + 1) := by
+      rw[add_mul, one_mul, Nat.factorial_succ]
+      rw [add_comm (n + 1)]
+      rw [mul_comm (n + 1)]
+    rw [h₂, pow_add, pow_one ]
+    refine Nat.mul_lt_mul_of_pos_right ?_ (by linarith)
+    have h₅: n ^ n < (n + 1) ^ n := by
+      refine Nat.pow_lt_pow_left ?_ ?_
+      . exact lt_add_one n
+      . refine Nat.ne_of_gt ?_
+        linarith
+    linarith
+
+
+lemma mylemma_5
+  (b p: ℕ)
+  (hp: Nat.Prime p)
+  (hbp: p ≤ b)
+  (h₁: p ^ p = b.factorial + p)
+  (hp5: 5 ≤ p) :
+  (False) := by
+  -- first prove that b = p cannot be
+  by_cases h₄: b = p
+  . have h₅: p + p.factorial < p^p := by exact mylemma_51 p hp5
+    rw [h₄] at h₁
+    linarith
+  . have hpb: p < b := by exact lt_of_le_of_ne' hbp h₄
+    clear hbp h₄
+    have h₂: (p + 1) ^ 2 ∣ b.factorial := by
+      have g₁: p + 1 ≤ b := by exact succ_le_iff.mpr hpb
+      have g₂: 2 ∣ (p + 1) := by
+        have gg₁: Odd p := by
+          refine hp.odd_of_ne_two ?_
+          linarith
+        have gg₂: Even (p + 1) := by
+          refine gg₁.add_odd ?_
+          norm_num
+        exact even_iff_two_dvd.mp gg₂
+      have g₃: 2 * ((p+1)/2) * (p + 1) ∣ b.factorial := by
+        have gg₁: (p + 1).factorial ∣ b.factorial := by exact Nat.factorial_dvd_factorial g₁
+        have gg₂: (p + 1).factorial = (p + 1) * p.factorial := by exact factorial_succ p
+        rw [mul_comm] at gg₂
+        have gg₃: 6/2 ≤ (p + 1)/2 := by
+          refine Nat.div_le_div_right ?_
+          linarith
+        norm_num at gg₃
+        have gg₄: 2 + (p+1)/2 ≤ p := by -- strong induction
+          refine Nat.le_induction ?_ ?_ p (hp5)
+          . norm_num
+          . intros n _ h₂
+            ring_nf
+            have ggg₁: (n / 2).succ ≤ (n + 1) / 2 + 1 := by
+              rw [← succ_eq_add_one]
+              refine Nat.succ_le_succ ?_
+              refine Nat.div_le_div_right ?_
+              linarith
+            simp
+            nth_rewrite 1 [← mul_one 2]
+            rw [Nat.two_mul 1, add_assoc]
+            refine Nat.add_le_add_left ?_ 1
+            refine le_trans ?_ h₂
+            rw [add_comm 2 _]
+            nth_rewrite 3 [← mul_one 2]
+            rw [Nat.two_mul 1, ← add_assoc, add_comm 1]
+            exact Nat.add_le_add_right ggg₁ 1
+        have gg₅: (2+(p+1)/2).factorial ∣ p.factorial := by
+          exact factorial_dvd_factorial gg₄
+        have gg₆: (2:ℕ).factorial * ((p+1)/2).factorial ∣ p.factorial := by
+          refine dvd_trans ?_ gg₅
+          exact (2:ℕ).factorial_mul_factorial_dvd_factorial_add ((p + 1) / 2)
+        have gg₇: 2 * ((p+1)/2) ∣ p.factorial := by
+          refine dvd_trans ?_ gg₆
+          simp
+          refine mul_dvd_mul_left 2 ?_
+          refine Nat.dvd_factorial (by linarith[gg₃]) (by linarith)
+        have gg₈: 2 * ((p+1)/2) * (p + 1) ∣ p.factorial * (p + 1) := by
+          refine mul_dvd_mul_right ?_ (p + 1)
+          exact gg₇
+        rw [gg₂] at gg₁
+        exact dvd_trans gg₈ gg₁
+      have g₄: 2 * ((p+1)/2) = (p + 1) := by
+        exact Nat.mul_div_cancel' g₂
+      rw [g₄] at g₃
+      ring_nf at *
+      exact g₃
+    have h₃: b.factorial = p ^ p - p := by exact eq_tsub_of_add_eq (h₁.symm)
+    rw [h₃] at h₂
+    exact mylemma_52 p hp hp5 h₂
+
+
+lemma mylemma_6
+  (a b p: ℕ)
+  (hp: Nat.Prime p)
+  (h₂: p ∣ a)
+  (hb2p: 2 * p ≤ b) :
+  (p ^ 2 ∣ a ^ p - b.factorial) := by
+  have g₁: p^p ∣ a^p := by exact pow_dvd_pow_of_dvd h₂ p
+  have g₂: 2 ≤ p := by exact Prime.two_le hp
+  have h₃: p^2 ∣ a^p := by exact pow_dvd_of_le_of_pow_dvd g₂ g₁
+  have g₃: (2*p).factorial ∣ b.factorial := by exact factorial_dvd_factorial hb2p
+  have g₄: p.factorial * p.factorial ∣ (p+p).factorial := by
+    exact factorial_mul_factorial_dvd_factorial_add p p
+  rw [← pow_two, ← two_mul] at g₄
+  have g₅: p ∣ p.factorial := by exact Nat.dvd_factorial (by linarith) (by linarith)
+  have h₄: p ^ 2 ∣ p.factorial ^ 2 := by exact pow_dvd_pow_of_dvd g₅ 2
+  have g₆: p ^ 2 ∣ (2 * p).factorial := by exact dvd_trans h₄ g₄
+  have h₅: p^2 ∣ b.factorial := by exact dvd_trans g₆ g₃
+  exact dvd_sub' h₃ h₅
+
+
+theorem imo_2022_p5
+  (a b p : ℕ)
+  (h₀: 0 < a ∧ 0 < b)
+  (hp: Nat.Prime p)
+  (h₁: a^p = Nat.factorial b + p) :
+  (a, b, p) = (2, 2, 2) ∨ (a, b, p) = (3, 4, 3) := by
+  by_cases hbp: b < p -- no solution
+  . exfalso
+    by_cases hab: a ≤ b
+    . have h₂: a ∣ b.factorial := by exact Nat.dvd_factorial h₀.1 hab
+      have g₃: a ∣ b.factorial + p := by
+        rw [← h₁]
+        refine dvd_pow_self a ?_
+        exact Nat.Prime.ne_zero hp
+      have h₃: a ∣ p := by exact (Nat.dvd_add_right h₂).mp g₃
+      have h₄: a = 1 := by
+        have g₄: a = 1 ∨ a = p := by
+          exact (Nat.dvd_prime hp).mp h₃
+        cases' g₄ with g₄₀ g₄₁
+        . exact g₄₀
+        . exfalso
+          rw [← g₄₁] at hbp
+          linarith[hbp,hab]
+      rw [h₄] at h₁
+      simp at h₁
+      have h₅: 2 ≤ p := by exact Nat.Prime.two_le hp
+      have g₆: 0 < b.factorial := by exact Nat.factorial_pos b
+      have h₇: 1+2 ≤ b.factorial + p := by exact Nat.add_le_add g₆ h₅
+      rw [← h₁] at h₇
+      linarith
+    . push_neg at hab
+      have h₂: (b+1)^p ≤ a^p := by
+        refine (Nat.pow_le_pow_iff_left ?_).mpr hab
+        exact Nat.Prime.ne_zero hp
+      have h₃: b^p + p*b + 1 ≤ (b+1)^p := by
+        ring_nf
+        rw [add_assoc]
+        exact mylemma_1 b p h₀.2 hbp
+      have g₄: p * 1 ≤ p * b := by
+        refine mul_le_mul ?_ ?_ ?_ ?_
+        . exact rfl.ge
+        . exact h₀.2
+        . norm_num
+        . exact Nat.zero_le p
+      have g₄: b.factorial ≤ b^b := by exact Nat.factorial_le_pow b
+      have g₅: b^b ≤ b^p := by
+        refine Nat.pow_le_pow_of_le_right h₀.2 ?_
+        exact le_of_lt hbp
+      linarith
+  . push_neg at hbp
+    have h₂: p ∣ a := by exact mylemma_3 a b p hp h₁ hbp
+    by_cases hb2p: b < 2*p
+    . have h₃: a = p := by exact mylemma_4 a b p h₀ hp h₁ hbp h₂ hb2p
+      rw [h₃] at h₁
+      by_cases hp5: p < 5
+      . have h₄: 2 ≤ p := by exact Prime.two_le hp
+        interval_cases p
+        . left
+          norm_num at h₁
+          have h₄: b.factorial = 2 := by linarith
+          have g₅: (2:ℕ).factorial = 2 := by norm_num
+          rw [← g₅] at h₄
+          have h₅: b = 2 := by
+            refine (Nat.factorial_inj ?_).mp h₄
+            linarith
+          rw [h₃,h₅]
+        . right
+          norm_num at h₁
+          rw [h₃]
+          have h₄: b.factorial = 24 := by linarith
+          have g₅: (4:ℕ).factorial = 24 := by exact rfl
+          rw [← g₅] at h₄
+          have h₅: b = 4 := by
+            refine (Nat.factorial_inj ?_).mp h₄
+            linarith
+          rw [h₅]
+        . exfalso
+          contradiction
+      . push_neg at hp5
+        exfalso -- lifting the exponent
+        exact mylemma_5 b p hp hbp h₁ hp5
+    . push_neg at hb2p
+      exfalso
+      have h₃: p^2 ∣ a^p - b.factorial := by exact mylemma_6 a b p hp h₂ hb2p
+      have g₃: b.factorial ≤ a^p := by exact le.intro (h₁.symm)
+      have g₄: a^p - b.factorial = p := by
+        rw [add_comm] at h₁
+        exact (Nat.sub_eq_iff_eq_add g₃).mpr h₁
+      have h₄: p^2 ∣ p := by
+        rw [g₄] at h₃
+        exact h₃
+      have gp: 0 < p := by exact Prime.pos hp
+      have h₅: p^2 ≤ p := by exact Nat.le_of_dvd gp h₄
+      have g₆: 1 < p := by exact Prime.one_lt hp
+      have h₆: p^1 < p^2 := by exact Nat.pow_lt_pow_succ g₆
+      linarith

imo_proofs/imo_2023_p4.lean

@@ -0,0 +1,453 @@
+import Mathlib
+import Mathlib.Analysis.SpecialFunctions.Pow.Real
+
+
+set_option linter.unusedVariables.analyzeTactics true
+
+open Real Set
+
+lemma mylemma_1
+  (x a: ℕ → ℝ)
+  (hxp: ∀ (i : ℕ), 0 < x i)
+  (h₀: ∀ (n : ℕ),
+    1 ≤ n ∧ n ≤ 2023 →
+      a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k)
+      * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) :
+  ∀ (n : ℕ), (1 ≤ n ∧ n ≤ 2022) → a (n) < a (n + 1) := by
+  intros n hn
+  have h₂: a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k)
+              * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by
+    refine h₀ n ?_
+    constructor
+    . exact hn.1
+    linarith
+  have h₃: a (n + 1) = sqrt ((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k)
+              * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k) := by
+    refine h₀ (n + 1) ?_
+    constructor
+    . linarith
+    linarith
+  rw [h₂,h₃]
+  refine sqrt_lt_sqrt ?_ ?_
+  . refine le_of_lt ?_
+    refine mul_pos ?_ ?_
+    . refine Finset.sum_pos ?_ ?_
+      . exact fun i _ => hxp i
+      . simp
+        linarith
+    . refine Finset.sum_pos ?_ ?_
+      intros i _
+      exact one_div_pos.mpr (hxp i)
+      . simp
+        linarith
+  have g₀: 1 ≤ n + 1 := by linarith
+  rw [Finset.sum_Ico_succ_top g₀ _, Finset.sum_Ico_succ_top g₀ _]
+  repeat rw [add_mul, mul_add]
+  have h₄: 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) +
+    x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by
+      refine add_pos ?_ ?_
+      . refine mul_pos ?_ ?_
+        . refine Finset.sum_pos ?_ ?_
+          . exact fun i _ => hxp i
+          . simp
+            linarith
+        . exact one_div_pos.mpr (hxp (n + 1))
+      . refine mul_pos ?_ ?_
+        . exact hxp (n + 1)
+        . refine add_pos ?_ ?_
+          . refine Finset.sum_pos ?_ ?_
+            . intros i _
+              exact one_div_pos.mpr (hxp i)
+            . simp
+              linarith
+          exact one_div_pos.mpr (hxp (n + 1))
+  linarith
+
+
+lemma mylemma_amgm
+  (b1 b2 b3 b4 :ℝ)
+  (hb1: 0 ≤ b1)
+  (hb2: 0 ≤ b2)
+  (hb3: 0 ≤ b3)
+  (hb4: 0 ≤ b4) :
+  (4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + b3 + b4) := by
+  let w1 : ℝ := (4:ℝ)⁻¹
+  let w2 : ℝ := w1
+  let w3 : ℝ := w2
+  let w4 : ℝ := w3
+  rw [mul_comm]
+  refine mul_le_of_le_div₀ ?_ (by norm_num) ?_
+  . refine add_nonneg ?_ hb4
+    refine add_nonneg ?_ hb3
+    exact add_nonneg hb1 hb2
+  have h₀: (b1^w1 * b2^w2 * b3^w3 * b4^w4) ≤ w1 * b1 + w2 * b2 + w3 * b3 + w4 * b4 := by
+    refine geom_mean_le_arith_mean4_weighted (by norm_num) ?_ ?_ ?_ hb1 hb2 hb3 hb4 ?_
+    . norm_num
+    . norm_num
+    . norm_num
+    . norm_num
+  repeat rw [mul_rpow _]
+  ring_nf at *
+  linarith
+  repeat { assumption }
+  . exact mul_nonneg hb1 hb2
+  . exact hb4
+  . refine mul_nonneg ?_ hb3
+    exact mul_nonneg hb1 hb2
+
+
+
+lemma mylemma_2
+  (x a: ℕ → ℝ)
+  (hxp: ∀ (i : ℕ), 0 < x i)
+  (h₀: ∀ (n : ℕ),
+    1 ≤ n ∧ n ≤ 2023 →
+      a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k)
+            * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)))
+  (n: ℕ)
+  (hn: 1 ≤ n ∧ n ≤ 2021) :
+  (4 * a n ≤
+    (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) +
+      (x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by
+      repeat rw [mul_add, add_mul]
+      have g₁₁: 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1 := by
+        refine le_of_lt ?_
+        refine Finset.sum_pos ?_ ?_
+        . exact fun i _ => hxp i
+        . simp
+          linarith
+      have g₁₂: 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹ := by
+        refine le_of_lt ?_
+        refine Finset.sum_pos ?_ ?_
+        . intros i _
+          exact inv_pos.mpr (hxp i)
+        . simp
+          linarith
+      have h₃₂: 4 * ( ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) *
+          ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) *
+          ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) *
+          (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹
+        ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) +
+          (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) +
+          ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) +
+          x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by
+            let b1:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))
+            let b2:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))
+            let b3:ℝ := (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)
+            let b4:ℝ := x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)
+            have hb1: b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) := by
+              exact rfl
+            have hb2: b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) := by
+              exact rfl
+            have hb3: b3 = (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by
+              exact rfl
+            have hb4: b4 = x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by
+              exact rfl
+            rw [← hb1, ← hb2, ← hb3, ← hb4]
+            have g₀: 4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + b3 + b4 := by
+              have b1p: 0 ≤ b1 := by
+                rw [hb1]
+                refine mul_nonneg ?_ ?_
+                . ring_nf
+                  exact g₁₁
+                . refine le_of_lt ?_
+                  exact one_div_pos.mpr (hxp (n + 1))
+              have b2p: 0 ≤ b2 := by
+                rw [hb2]
+                refine mul_nonneg ?_ ?_
+                . ring_nf
+                  exact g₁₁
+                . refine le_of_lt ?_
+                  exact one_div_pos.mpr (hxp (n + 2))
+              have b3p: 0 ≤ b3 := by
+                rw [hb3]
+                refine mul_nonneg ?_ ?_
+                . exact LT.lt.le (hxp (n + 1))
+                . ring_nf
+                  exact g₁₂
+              have b4p: 0 ≤ b4 := by
+                rw [hb4]
+                refine mul_nonneg ?_ ?_
+                . exact LT.lt.le (hxp (n + 2))
+                . ring_nf
+                  exact g₁₂
+              exact mylemma_amgm b1 b2 b3 b4 b1p b2p b3p b4p
+            linarith
+      have h₃₃: 4 * a (n) = 4 * (((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) *
+          ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) *
+          ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) *
+          (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ := by
+        simp
+        ring_nf
+        have g₀: (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2
+            * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ *
+            (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2
+          = x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ *
+            (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 *
+            (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 := by
+            linarith
+        have g₁: x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by
+          rw [mul_assoc]
+          have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by
+            refine div_self ?_
+            exact ne_of_gt (hxp (1 + n))
+          have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by
+            refine div_self ?_
+            exact ne_of_gt (hxp (2 + n))
+          rw [gg₁, gg₂]
+          norm_num
+        rw [g₁] at g₀
+        rw [g₀]
+        simp
+        repeat rw [mul_rpow]
+        have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹
+            = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by
+              rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹]
+              norm_num
+        have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹
+            = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by
+              rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹]
+              norm_num
+        -- rw [g₂, ← sqrt_eq_rpow (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1)]
+        -- rw [g₃, ← sqrt_eq_rpow (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹)]
+        have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k)
+                  * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by
+          refine h₀ n ?_
+          constructor
+          . exact hn.1
+          . linarith
+        norm_cast at *
+        rw [g₂, g₃, ← mul_rpow]
+        rw [← sqrt_eq_rpow]
+        ring_nf at g₄
+        exact g₄
+        . exact g₁₁
+        . exact g₁₂
+        . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1)
+        . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹)
+      exact Eq.trans_le h₃₃ h₃₂
+
+
+lemma mylemma_3
+  (x a: ℕ → ℝ)
+  (hxp: ∀ (i : ℕ), 0 < x i)
+  (hx: ∀ (i j : ℕ), i ≠ j → x i ≠ x j)
+  (h₀: ∀ (n : ℕ),
+    1 ≤ n ∧ n ≤ 2023 →
+      a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k)
+            * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)))
+  (h₀₁: ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) :
+  (∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) := by
+    intros n hn
+    have g₀: 0 ≤ a n + 2 := by
+      refine le_of_lt ?_
+      refine add_pos ?_ (by norm_num)
+      refine h₀₁ n ?_
+      constructor
+      . exact hn.1
+      . linarith
+    have g₁: 0 ≤ a (n + 2) := by
+      refine le_of_lt ?_
+      refine h₀₁ (n + 2) ?_
+      constructor
+      . linarith
+      . linarith
+    rw [← sqrt_sq g₀, ← sqrt_sq g₁]
+    have g₂: 0 ≤ (a n + 2) ^ 2 := by exact sq_nonneg (a n + 2)
+    -- simp
+    refine Real.sqrt_lt_sqrt g₂ ?_
+    have g₃: 0 ≤ ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k)
+        * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by
+      refine le_of_lt ?_
+      refine mul_pos ?_ ?_
+      . refine Finset.sum_pos ?_ ?_
+        . exact fun i _ => hxp i
+        . simp
+          linarith
+      . refine Finset.sum_pos ?_ ?_
+        . intros i _
+          exact one_div_pos.mpr (hxp i)
+        . simp
+          linarith
+    have gn₀: a (n) ^ 2 = ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k)
+        * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by
+      rw [← sq_sqrt g₃]
+      have g₄: 0 ≤ a n := by
+        refine le_of_lt ?_
+        refine h₀₁ n ?_
+        constructor
+        . exact hn.1
+        . linarith
+      refine (sq_eq_sq₀ g₄ ?_).mpr ?_
+      . exact
+        sqrt_nonneg
+          ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) *
+            Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)
+      . refine h₀ (n) ?_
+        constructor
+        . exact hn.1
+        . linarith
+    have gn₁: a (n + 2) = sqrt ((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k)
+        * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k) := by
+      refine h₀ (n + 2) ?_
+      constructor
+      . linarith
+      . linarith
+    rw [add_sq, gn₁, sq_sqrt]
+    . have ga₀: 1 ≤ n + 2 := by linarith
+      rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _]
+      have ga₁: 1 ≤ n + 1 := by linarith
+      rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _]
+      rw [add_assoc, add_assoc, add_assoc]
+      rw [add_mul, mul_add]
+      rw [← gn₀]
+      repeat rw [add_assoc]
+      refine add_lt_add_left ?_ (a (n) ^ 2)
+      rw [mul_add (x (n + 1) + x (n + 2))]
+      have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by
+        repeat rw [add_mul, mul_add, mul_add]
+        repeat rw [mul_div_left_comm _ 1 _, one_mul]
+        repeat rw [div_self ?_]
+        . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2))
+            = x (n + 1) * x (n + 1) := by
+            rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_]
+            simp
+            exact ne_of_gt (hxp (n + 2))
+          have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1))
+            = x (n + 2) * x (n + 2) := by
+            rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_]
+            simp
+            exact ne_of_gt (hxp (n + 1))
+          have h₂₀: 0 < x (n + 1) * x (n + 2) := by
+            refine mul_pos ?_ ?_
+            . exact hxp (n + 1)
+            . exact hxp (n + 2)
+          have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by
+            refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀)
+            rw [mul_add, hc₂, hc₃, ← sq, ← sq]
+            refine lt_of_sub_pos ?_
+            have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2
+                      = (x (n + 1) - x (n + 2)) ^ 2 := by
+              rw [sub_sq]
+              linarith
+            rw [gh₂₁]
+            refine (sq_pos_iff).mpr ?_
+            refine sub_ne_zero.mpr ?_
+            exact hx (n+1) (n+2) (by linarith)
+          linarith
+        . exact ne_of_gt (hxp (n + 2))
+        . exact ne_of_gt (hxp (n + 1))
+      clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁
+      have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k)
+          * (1 / x (n + 1) + 1 / x (n + 2)) +
+          ((x (n + 1) + x (n + 2))
+          * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by
+          exact mylemma_2 (fun k => x k) a hxp h₀ n hn
+      linarith
+    . refine mul_nonneg ?_ ?_
+      . refine Finset.sum_nonneg ?_
+        intros i _
+        exact LT.lt.le (hxp i)
+      . refine Finset.sum_nonneg ?_
+        intros i _
+        simp
+        exact LT.lt.le (hxp i)
+
+
+theorem imo_2023_p4
+  (x : ℕ → ℝ)
+  (a : ℕ → ℝ)
+  (hxp: ∀ (i: ℕ), (0 < x i) )
+  (hx: ∀ (i j: ℕ), (i ≠ j) → (x i ≠ x j) )
+  (h₀: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2023) →
+        a n = Real.sqrt ( (Finset.sum (Finset.Ico 1 (n + 1)) fun (k : ℕ) => (x k))
+            * (Finset.sum (Finset.Ico 1 (n + 1)) fun (k : ℕ) => 1 / (x k)) ) )
+  (h₁: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2023) → ∃ (kz:ℤ), (a n = ↑kz )) :
+  (3034 ≤ a 2023) := by
+  have ha1: a 1 = 1 := by
+    have g₀: sqrt ((Finset.sum (Finset.Ico 1 (1 + 1)) fun k => x k)
+            * Finset.sum (Finset.Ico 1 (1 + 1)) fun k => 1 / x k) = 1 := by
+      norm_num
+      refine div_self ?_
+      exact ne_of_gt (hxp 1)
+    rw [← g₀]
+    exact h₀ (1) (by norm_num)
+  have h₀₁: ∀ (n : ℕ), (1 ≤ n ∧ n ≤ 2023) → 0 < a n := by
+    intros n hn
+    have ha: a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k)
+       * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by
+        exact h₀ (n) (hn)
+    rw [ha]
+    refine Real.sqrt_pos.mpr ?_
+    refine mul_pos ?_ ?_
+    . refine Finset.sum_pos ?_ ?_
+      . intros i _
+        exact hxp i
+      simp
+      linarith
+    . refine Finset.sum_pos ?_ ?_
+      . intros i _
+        exact one_div_pos.mpr (hxp i)
+      simp
+      linarith
+  have h₁₁: ∀ (n : ℕ), (1 ≤ n ∧ n ≤ 2023) → ∃ (kn:ℕ), a n = ↑kn := by
+    intros n hn
+    have g₁₁: 0 < a n := by
+      exact h₀₁ n hn
+    let ⟨p, gp⟩ := h₁ n hn
+    let q:ℕ := Int.toNat p
+    have g₁₂: ↑q = p := by
+      refine Int.toNat_of_nonneg ?_
+      rw [gp] at g₁₁
+      norm_cast at g₁₁
+      exact Int.le_of_lt g₁₁
+    use q
+    rw [gp]
+    norm_cast
+    exact id g₁₂.symm
+  have h₂₁: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2021) → a n + 2 < a (n+2) := by
+    exact fun n a_1 => mylemma_3 (fun i => x i) a hxp hx h₀ h₀₁ n a_1
+  have h₂: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2021) → a n + 3 ≤ a (n+2) := by
+    intros n hn
+    have g₀: a n + 2 < a (n + 2) := by exact h₂₁ n hn
+    have g₀₁: ∃ (p:ℕ), a n = ↑p := by
+      apply h₁₁
+      constructor
+      . linarith
+      . linarith
+    have g₀₂: ∃ (q:ℕ), a (n + 2) = ↑q := by
+      apply h₁₁
+      constructor
+      . linarith
+      . linarith
+    let ⟨p, _⟩ := g₀₁
+    let ⟨q, _⟩ := g₀₂
+    have g₁: p + 2 < q := by
+      suffices g₁₁: ↑p + (2:ℝ) < ↑q
+      . norm_cast at g₁₁
+      . linarith
+    have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast
+    linarith
+  have h₃: ∀ (n:ℕ), (0 ≤ n ∧ n ≤ 1010) → a 1 + 3 * (↑(n)  + 1) ≤ a (3 + 2 * n) := by
+    intros n hn
+    induction' n with d hd
+    · simp
+      exact h₂ (1) (by norm_num)
+    · rw [mul_add]
+      simp
+      have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by
+        refine h₂ (3 + 2 * d) ?_
+        constructor
+        . linarith
+        . linarith
+      have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by
+        refine add_le_add_right ?_ (3)
+        apply hd
+        constructor
+        . linarith
+        . linarith
+      refine le_trans (by linarith[g₁]) g₀
+  rw [ha1] at h₃
+  have h₄: (3034:ℝ)  = 1 + 3 * (↑1010 + 1) := by norm_num
+  rw [h₄]
+  exact h₃ (1010) (by norm_num)

imo_proofs/lake-manifest.json

@@ -0,0 +1,95 @@
+{"version": "1.1.0",
+ "packagesDir": ".lake/packages",
+ "packages":
+ [{"url": "https://github.com/leanprover-community/mathlib4",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover-community",
+   "rev": "d066138f11f7fdf68dcda20d1ed2d296e9d992d7",
+   "name": "mathlib",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "master",
+   "inherited": false,
+   "configFile": "lakefile.lean"},
+  {"url": "https://github.com/leanprover-community/plausible",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover-community",
+   "rev": "59a8514bb0ee5bae2689d8be717b5272c9b3dc1c",
+   "name": "plausible",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "main",
+   "inherited": true,
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/leanprover-community/LeanSearchClient",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover-community",
+   "rev": "0c169a0d55fef3763cfb3099eafd7b884ec7e41d",
+   "name": "LeanSearchClient",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "main",
+   "inherited": true,
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/leanprover-community/import-graph",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover-community",
+   "rev": "461b96f5527089718cb23d3f1fd2960a5d0ff516",
+   "name": "importGraph",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "main",
+   "inherited": true,
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/leanprover-community/ProofWidgets4",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover-community",
+   "rev": "8fff3f074da9237cd4e179fd6dd89be6c4022d41",
+   "name": "proofwidgets",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "v0.0.52-pre",
+   "inherited": true,
+   "configFile": "lakefile.lean"},
+  {"url": "https://github.com/leanprover-community/aesop",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover-community",
+   "rev": "ba9a63be53f16b3b6e4043641c6bad4883e650b4",
+   "name": "aesop",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "master",
+   "inherited": true,
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/leanprover-community/quote4",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover-community",
+   "rev": "7b3b0c8327b3c0214ac49ca6d6922edbb81ab8c9",
+   "name": "Qq",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "master",
+   "inherited": true,
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/leanprover-community/batteries",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover-community",
+   "rev": "b18855cb0f9a19bd4d7e21f3e5525272e377f431",
+   "name": "batteries",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "main",
+   "inherited": true,
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/leanprover/lean4-cli",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover",
+   "rev": "a2eb24a3dbf681f2b655f82ba5ee5b139d4a5abc",
+   "name": "Cli",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "main",
+   "inherited": true,
+   "configFile": "lakefile.toml"}],
+ "name": "imo_steps",
+ "lakeDir": ".lake"}

imo_proofs/lakefile.toml

@@ -0,0 +1,16 @@
+name = "imo_steps"
+version = "0.1.0"
+keywords = ["math"]
+defaultTargets = ["ImoSteps"]
+
+[leanOptions]
+pp.unicode.fun = true # pretty-prints `fun a ↦ b`
+autoImplicit = false
+
+[[require]]
+name = "mathlib"
+scope = "leanprover-community"
+rev = "v4.17.0"
+
+[[lean_lib]]
+name = "ImoSteps"

imo_proofs/lean-toolchain

@@ -0,0 +1 @@
+leanprover/lean4:v4.17.0