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