imo_proofs/imo_1969_p2.lean

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