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]