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