l3lab/ntp-mathlib huggingface dataset:
/-- The orthocenter is the only point lying in any two of the
altitudes. -/
theorem eq_orthocenter_of_forall_mem_altitude {t : Triangle ℝ P} {i₁ i₂ : Fin 3} {p : P}
    (h₁₂ : i₁ ≠ i₂) (h₁ : p ∈ t.altitude i₁) (h₂ : p ∈ t.altitude i₂) : p = t.orthocenter := by
  obtain ⟨i₃, h₂₃, h₁₃⟩ : ∃ i₃, i₂ ≠ i₃ ∧ i₁ ≠ i₃ := by
    clear h₁ h₂
    -- porting note: was `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> decide
  rw [t.altitude_eq_mongePlane h₁₃ h₁₂ h₂₃.symm] at h₁
  rw [t.altitude_eq_mongePlane h₂₃ h₁₂.symm h₁₃.symm] at h₂
  rw [orthocenter_eq_mongePoint]
  have ha : ∀ i, i₃ ≠ i → p ∈ t.mongePlane i₃ i := by
    intro i hi
    have hi₁₂ : i₁ = i ∨ i₂ = i := by
      clear h₁ h₂
      -- porting note: was `decide!`
      fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> fin_cases i <;> 

Proof state:
case head.tail.head.tail.head.tail.head
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
p : P
h₁₂ h₁₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
h₂₃ hi : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
⊢ { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∨
    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ hi ⊢
/-- The orthocenter is the only point lying in any two of the
altitudes. -/
theorem eq_orthocenter_of_forall_mem_altitude {t : Triangle ℝ P} {i₁ i₂ : Fin 3} {p : P}
    (h₁₂ : i₁ ≠ i₂) (h₁ : p ∈ t.altitude i₁) (h₂ : p ∈ t.altitude i₂) : p = t.orthocenter := by
  obtain ⟨i₃, h₂₃, h₁₃⟩ : ∃ i₃, i₂ ≠ i₃ ∧ i₁ ≠ i₃ := by
    clear h₁ h₂
    -- porting note: was `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> decide
  rw [t.altitude_eq_mongePlane h₁₃ h₁₂ h₂₃.symm] at h₁
  rw [t.altitude_eq_mongePlane h₂₃ h₁₂.symm h₁₃.symm] at h₂
  rw [orthocenter_eq_mongePoint]
  have ha : ∀ i, i₃ ≠ i → p ∈ t.mongePlane i₃ i := by
    intro i hi
    have hi₁₂ : i₁ = i ∨ i₂ = i := by
      clear h₁ h₂
      -- porting note: was `decide!`
      fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> fin_cases i <;> 

Proof state:
case head.tail.head.tail.head.tail.tail.head
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
p : P
h₁₂ h₁₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
h₂₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
hi : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
⊢ { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∨
    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ hi ⊢
/-- The orthocenter is the only point lying in any two of the
altitudes. -/
theorem eq_orthocenter_of_forall_mem_altitude {t : Triangle ℝ P} {i₁ i₂ : Fin 3} {p : P}
    (h₁₂ : i₁ ≠ i₂) (h₁ : p ∈ t.altitude i₁) (h₂ : p ∈ t.altitude i₂) : p = t.orthocenter := by
  obtain ⟨i₃, h₂₃, h₁₃⟩ : ∃ i₃, i₂ ≠ i₃ ∧ i₁ ≠ i₃ := by
    clear h₁ h₂
    -- porting note: was `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> decide
  rw [t.altitude_eq_mongePlane h₁₃ h₁₂ h₂₃.symm] at h₁
  rw [t.altitude_eq_mongePlane h₂₃ h₁₂.symm h₁₃.symm] at h₂
  rw [orthocenter_eq_mongePoint]
  have ha : ∀ i, i₃ ≠ i → p ∈ t.mongePlane i₃ i := by
    intro i hi
    have hi₁₂ : i₁ = i ∨ i₂ = i := by
      clear h₁ h₂
      -- porting note: was `decide!`
      fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> fin_cases i <;> 

Proof state:
case head.tail.head.tail.tail.head.head
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
p : P
h₁₂ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
h₁₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₂₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
hi : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 0, isLt := (_ : 0 < 3) }
⊢ { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } ∨
    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ hi ⊢
/-- The orthocenter is the only point lying in any two of the
altitudes. -/
theorem eq_orthocenter_of_forall_mem_altitude {t : Triangle ℝ P} {i₁ i₂ : Fin 3} {p : P}
    (h₁₂ : i₁ ≠ i₂) (h₁ : p ∈ t.altitude i₁) (h₂ : p ∈ t.altitude i₂) : p = t.orthocenter := by
  obtain ⟨i₃, h₂₃, h₁₃⟩ : ∃ i₃, i₂ ≠ i₃ ∧ i₁ ≠ i₃ := by
    clear h₁ h₂
    -- porting note: was `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> decide
  rw [t.altitude_eq_mongePlane h₁₃ h₁₂ h₂₃.symm] at h₁
  rw [t.altitude_eq_mongePlane h₂₃ h₁₂.symm h₁₃.symm] at h₂
  rw [orthocenter_eq_mongePoint]
  have ha : ∀ i, i₃ ≠ i → p ∈ t.mongePlane i₃ i := by
    intro i hi
    have hi₁₂ : i₁ = i ∨ i₂ = i := by
      clear h₁ h₂
      -- porting note: was `decide!`
      fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> fin_cases i <;> 

Proof state:
case head.tail.head.tail.tail.head.tail.head
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
p : P
h₁₂ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
h₁₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₂₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
hi : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
⊢ { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∨
    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ hi ⊢
/-- The orthocenter is the only point lying in any two of the
altitudes. -/
theorem eq_orthocenter_of_forall_mem_altitude {t : Triangle ℝ P} {i₁ i₂ : Fin 3} {p : P}
    (h₁₂ : i₁ ≠ i₂) (h₁ : p ∈ t.altitude i₁) (h₂ : p ∈ t.altitude i₂) : p = t.orthocenter := by
  obtain ⟨i₃, h₂₃, h₁₃⟩ : ∃ i₃, i₂ ≠ i₃ ∧ i₁ ≠ i₃ := by
    clear h₁ h₂
    -- porting note: was `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> decide
  rw [t.altitude_eq_mongePlane h₁₃ h₁₂ h₂₃.symm] at h₁
  rw [t.altitude_eq_mongePlane h₂₃ h₁₂.symm h₁₃.symm] at h₂
  rw [orthocenter_eq_mongePoint]
  have ha : ∀ i, i₃ ≠ i → p ∈ t.mongePlane i₃ i := by
    intro i hi
    have hi₁₂ : i₁ = i ∨ i₂ = i := by
      clear h₁ h₂
      -- porting note: was `decide!`
      fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> fin_cases i <;> 

Proof state:
case head.tail.head.tail.tail.head.tail.tail.head
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
p : P
h₁₂ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
h₁₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₂₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
hi :
  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
⊢ { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∨
    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ hi ⊢
/-- The orthocenter is the only point lying in any two of the
altitudes. -/
theorem eq_orthocenter_of_forall_mem_altitude {t : Triangle ℝ P} {i₁ i₂ : Fin 3} {p : P}
    (h₁₂ : i₁ ≠ i₂) (h₁ : p ∈ t.altitude i₁) (h₂ : p ∈ t.altitude i₂) : p = t.orthocenter := by
  obtain ⟨i₃, h₂₃, h₁₃⟩ : ∃ i₃, i₂ ≠ i₃ ∧ i₁ ≠ i₃ := by
    clear h₁ h₂
    -- porting note: was `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> decide
  rw [t.altitude_eq_mongePlane h₁₃ h₁₂ h₂₃.symm] at h₁
  rw [t.altitude_eq_mongePlane h₂₃ h₁₂.symm h₁₃.symm] at h₂
  rw [orthocenter_eq_mongePoint]
  have ha : ∀ i, i₃ ≠ i → p ∈ t.mongePlane i₃ i := by
    intro i hi
    have hi₁₂ : i₁ = i ∨ i₂ = i := by
      clear h₁ h₂
      -- porting note: was `decide!`
      fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> fin_cases i <;> 

Proof state:
case head.tail.tail.head.head.head
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
p : P
h₁₂ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₁₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 0, isLt := (_ : 0 < 3) }
h₂₃ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 0, isLt := (_ : 0 < 3) }
hi : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 0, isLt := (_ : 0 < 3) }
⊢ { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ hi ⊢
/-- The orthocenter is the only point lying in any two of the
altitudes. -/
theorem eq_orthocenter_of_forall_mem_altitude {t : Triangle ℝ P} {i₁ i₂ : Fin 3} {p : P}
    (h₁₂ : i₁ ≠ i₂) (h₁ : p ∈ t.altitude i₁) (h₂ : p ∈ t.altitude i₂) : p = t.orthocenter := by
  obtain ⟨i₃, h₂₃, h₁₃⟩ : ∃ i₃, i₂ ≠ i₃ ∧ i₁ ≠ i₃ := by
    clear h₁ h₂
    -- porting note: was `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> decide
  rw [t.altitude_eq_mongePlane h₁₃ h₁₂ h₂₃.symm] at h₁
  rw [t.altitude_eq_mongePlane h₂₃ h₁₂.symm h₁₃.symm] at h₂
  rw [orthocenter_eq_mongePoint]
  have ha : ∀ i, i₃ ≠ i → p ∈ t.mongePlane i₃ i := by
    intro i hi
    have hi₁₂ : i₁ = i ∨ i₂ = i := by
      clear h₁ h₂
      -- porting note: was `decide!`
      fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> fin_cases i <;> 

Proof state:
case head.tail.tail.head.head.tail.head
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
p : P
h₁₂ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₁₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 0, isLt := (_ : 0 < 3) }
h₂₃ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 0, isLt := (_ : 0 < 3) }
hi : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
⊢ { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ hi ⊢
/-- The orthocenter is the only point lying in any two of the
altitudes. -/
theorem eq_orthocenter_of_forall_mem_altitude {t : Triangle ℝ P} {i₁ i₂ : Fin 3} {p : P}
    (h₁₂ : i₁ ≠ i₂) (h₁ : p ∈ t.altitude i₁) (h₂ : p ∈ t.altitude i₂) : p = t.orthocenter := by
  obtain ⟨i₃, h₂₃, h₁₃⟩ : ∃ i₃, i₂ ≠ i₃ ∧ i₁ ≠ i₃ := by
    clear h₁ h₂
    -- porting note: was `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> decide
  rw [t.altitude_eq_mongePlane h₁₃ h₁₂ h₂₃.symm] at h₁
  rw [t.altitude_eq_mongePlane h₂₃ h₁₂.symm h₁₃.symm] at h₂
  rw [orthocenter_eq_mongePoint]
  have ha : ∀ i, i₃ ≠ i → p ∈ t.mongePlane i₃ i := by
    intro i hi
    have hi₁₂ : i₁ = i ∨ i₂ = i := by
      clear h₁ h₂
      -- porting note: was `decide!`
      fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> fin_cases i <;> 

Proof state:
case head.tail.tail.head.head.tail.tail.head
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
p : P
h₁₂ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₁₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 0, isLt := (_ : 0 < 3) }
h₂₃ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 0, isLt := (_ : 0 < 3) }
hi : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
⊢ { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =
      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ hi ⊢
/-- The orthocenter is the only point lying in any two of the
altitudes. -/
theorem eq_orthocenter_of_forall_mem_altitude {t : Triangle ℝ P} {i₁ i₂ : Fin 3} {p : P}
    (h₁₂ : i₁ ≠ i₂) (h₁ : p ∈ t.altitude i₁) (h₂ : p ∈ t.altitude i₂) : p = t.orthocenter := by
  obtain ⟨i₃, h₂₃, h₁₃⟩ : ∃ i₃, i₂ ≠ i₃ ∧ i₁ ≠ i₃ := by
    clear h₁ h₂
    -- porting note: was `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> decide
  rw [t.altitude_eq_mongePlane h₁₃ h₁₂ h₂₃.symm] at h₁
  rw [t.altitude_eq_mongePlane h₂₃ h₁₂.symm h₁₃.symm] at h₂
  rw [orthocenter_eq_mongePoint]
  have ha : ∀ i, i₃ ≠ i → p ∈ t.mongePlane i₃ i := by
    intro i hi
    have hi₁₂ : i₁ = i ∨ i₂ = i := by
      clear h₁ h₂
      -- porting note: was `decide!`
      fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> fin_cases i <;> 

Proof state:
case head.tail.tail.head.tail.head.head
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
p : P
h₁₂ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₁₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
h₂₃ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
hi : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 0, isLt := (_ : 0 < 3) }
⊢ { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ hi ⊢
/-- The orthocenter is the only point lying in any two of the
altitudes. -/
theorem eq_orthocenter_of_forall_mem_altitude {t : Triangle ℝ P} {i₁ i₂ : Fin 3} {p : P}
    (h₁₂ : i₁ ≠ i₂) (h₁ : p ∈ t.altitude i₁) (h₂ : p ∈ t.altitude i₂) : p = t.orthocenter := by
  obtain ⟨i₃, h₂₃, h₁₃⟩ : ∃ i₃, i₂ ≠ i₃ ∧ i₁ ≠ i₃ := by
    clear h₁ h₂
    -- porting note: was `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> decide
  rw [t.altitude_eq_mongePlane h₁₃ h₁₂ h₂₃.symm] at h₁
  rw [t.altitude_eq_mongePlane h₂₃ h₁₂.symm h₁₃.symm] at h₂
  rw [orthocenter_eq_mongePoint]
  have ha : ∀ i, i₃ ≠ i → p ∈ t.mongePlane i₃ i := by
    intro i hi
    have hi₁₂ : i₁ = i ∨ i₂ = i := by
      clear h₁ h₂
      -- porting note: was `decide!`
      fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> fin_cases i <;> 

Proof state:
case head.tail.tail.head.tail.head.tail.head
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
p : P
h₁₂ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₁₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
h₂₃ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
hi : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
⊢ { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ hi ⊢
/-- The orthocenter is the only point lying in any two of the
altitudes. -/
theorem eq_orthocenter_of_forall_mem_altitude {t : Triangle ℝ P} {i₁ i₂ : Fin 3} {p : P}
    (h₁₂ : i₁ ≠ i₂) (h₁ : p ∈ t.altitude i₁) (h₂ : p ∈ t.altitude i₂) : p = t.orthocenter := by
  obtain ⟨i₃, h₂₃, h₁₃⟩ : ∃ i₃, i₂ ≠ i₃ ∧ i₁ ≠ i₃ := by
    clear h₁ h₂
    -- porting note: was `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> decide
  rw [t.altitude_eq_mongePlane h₁₃ h₁₂ h₂₃.symm] at h₁
  rw [t.altitude_eq_mongePlane h₂₃ h₁₂.symm h₁₃.symm] at h₂
  rw [orthocenter_eq_mongePoint]
  have ha : ∀ i, i₃ ≠ i → p ∈ t.mongePlane i₃ i := by
    intro i hi
    have hi₁₂ : i₁ = i ∨ i₂ = i := by
      clear h₁ h₂
      -- porting note: was `decide!`
      fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> fin_cases i <;> 

Proof state:
case head.tail.tail.head.tail.head.tail.tail.head
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
p : P
h₁₂ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₁₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
h₂₃ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
hi : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
⊢ { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =
      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ hi ⊢
/-- The orthocenter is the only point lying in any two of the
altitudes. -/
theorem eq_orthocenter_of_forall_mem_altitude {t : Triangle ℝ P} {i₁ i₂ : Fin 3} {p : P}
    (h₁₂ : i₁ ≠ i₂) (h₁ : p ∈ t.altitude i₁) (h₂ : p ∈ t.altitude i₂) : p = t.orthocenter := by
  obtain ⟨i₃, h₂₃, h₁₃⟩ : ∃ i₃, i₂ ≠ i₃ ∧ i₁ ≠ i₃ := by
    clear h₁ h₂
    -- porting note: was `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> decide
  rw [t.altitude_eq_mongePlane h₁₃ h₁₂ h₂₃.symm] at h₁
  rw [t.altitude_eq_mongePlane h₂₃ h₁₂.symm h₁₃.symm] at h₂
  rw [orthocenter_eq_mongePoint]
  have ha : ∀ i, i₃ ≠ i → p ∈ t.mongePlane i₃ i := by
    intro i hi
    have hi₁₂ : i₁ = i ∨ i₂ = i := by
      clear h₁ h₂
      -- porting note: was `decide!`
      fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> fin_cases i <;> 

Proof state:
case head.tail.tail.head.tail.tail.head.head
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
p : P
h₁₂ h₁₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₂₃ :
  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
hi : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 0, isLt := (_ : 0 < 3) }
⊢ { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ hi ⊢
/-- The orthocenter is the only point lying in any two of the
altitudes. -/
theorem eq_orthocenter_of_forall_mem_altitude {t : Triangle ℝ P} {i₁ i₂ : Fin 3} {p : P}
    (h₁₂ : i₁ ≠ i₂) (h₁ : p ∈ t.altitude i₁) (h₂ : p ∈ t.altitude i₂) : p = t.orthocenter := by
  obtain ⟨i₃, h₂₃, h₁₃⟩ : ∃ i₃, i₂ ≠ i₃ ∧ i₁ ≠ i₃ := by
    clear h₁ h₂
    -- porting note: was `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> decide
  rw [t.altitude_eq_mongePlane h₁₃ h₁₂ h₂₃.symm] at h₁
  rw [t.altitude_eq_mongePlane h₂₃ h₁₂.symm h₁₃.symm] at h₂
  rw [orthocenter_eq_mongePoint]
  have ha : ∀ i, i₃ ≠ i → p ∈ t.mongePlane i₃ i := by
    intro i hi
    have hi₁₂ : i₁ = i ∨ i₂ = i := by
      clear h₁ h₂
      -- porting note: was `decide!`
      fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> fin_cases i <;> 

Proof state:
case head.tail.tail.head.tail.tail.head.tail.head
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
p : P
h₁₂ h₁₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₂₃ :
  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
hi : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
⊢ { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ hi ⊢
/-- The orthocenter is the only point lying in any two of the
altitudes. -/
theorem eq_orthocenter_of_forall_mem_altitude {t : Triangle ℝ P} {i₁ i₂ : Fin 3} {p : P}
    (h₁₂ : i₁ ≠ i₂) (h₁ : p ∈ t.altitude i₁) (h₂ : p ∈ t.altitude i₂) : p = t.orthocenter := by
  obtain ⟨i₃, h₂₃, h₁₃⟩ : ∃ i₃, i₂ ≠ i₃ ∧ i₁ ≠ i₃ := by
    clear h₁ h₂
    -- porting note: was `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> decide
  rw [t.altitude_eq_mongePlane h₁₃ h₁₂ h₂₃.symm] at h₁
  rw [t.altitude_eq_mongePlane h₂₃ h₁₂.symm h₁₃.symm] at h₂
  rw [orthocenter_eq_mongePoint]
  have ha : ∀ i, i₃ ≠ i → p ∈ t.mongePlane i₃ i := by
    intro i hi
    have hi₁₂ : i₁ = i ∨ i₂ = i := by
      clear h₁ h₂
      -- porting note: was `decide!`
      fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> fin_cases i <;> 

Proof state:
case head.tail.tail.head.tail.tail.head.tail.tail.head
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
p : P
h₁₂ h₁₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₂₃ hi :
  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
⊢ { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =
      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ hi ⊢
/-- The orthocenter is the only point lying in any two of the
altitudes. -/
theorem eq_orthocenter_of_forall_mem_altitude {t : Triangle ℝ P} {i₁ i₂ : Fin 3} {p : P}
    (h₁₂ : i₁ ≠ i₂) (h₁ : p ∈ t.altitude i₁) (h₂ : p ∈ t.altitude i₂) : p = t.orthocenter := by
  obtain ⟨i₃, h₂₃, h₁₃⟩ : ∃ i₃, i₂ ≠ i₃ ∧ i₁ ≠ i₃ := by
    clear h₁ h₂
    -- porting note: was `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> decide
  rw [t.altitude_eq_mongePlane h₁₃ h₁₂ h₂₃.symm] at h₁
  rw [t.altitude_eq_mongePlane h₂₃ h₁₂.symm h₁₃.symm] at h₂
  rw [orthocenter_eq_mongePoint]
  have ha : ∀ i, i₃ ≠ i → p ∈ t.mongePlane i₃ i := by
    intro i hi
    have hi₁₂ : i₁ = i ∨ i₂ = i := by
      clear h₁ h₂
      -- porting note: was `decide!`
      fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> fin_cases i <;> 

Proof state:
case tail.head.head.head.head
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
p : P
h₁₂ h₁₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 0, isLt := (_ : 0 < 3) }
h₂₃ hi : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 0, isLt := (_ : 0 < 3) }
⊢ { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } ∨
    { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ hi ⊢