l3lab/ntp-mathlib huggingface dataset:
/-- Given any triangle in an orthocentric system, the fourth point is
its orthocenter. -/
theorem OrthocentricSystem.eq_insert_orthocenter {s : Set P} (ho : OrthocentricSystem s)
    {t : Triangle ℝ P} (ht : Set.range t.points ⊆ s) :
    s = insert t.orthocenter (Set.range t.points) := by
  rcases ho with ⟨t₀, ht₀o, ht₀s⟩
  rw [ht₀s] at ht
  rcases exists_of_range_subset_orthocentricSystem ht₀o ht t.independent.injective with
    (⟨i₁, i₂, i₃, j₂, j₃, h₁₂, h₁₃, h₂₃, h₁₂₃, h₁, hj₂₃, h₂, h₃⟩ | hs)
  · obtain ⟨j₁, hj₁₂, hj₁₃, hj₁₂₃⟩ :
        ∃ j₁ : Fin 3, j₁ ≠ j₂ ∧ j₁ ≠ j₃ ∧ ∀ j : Fin 3, j = j₁ ∨ j = j₂ ∨ j = j₃ := by
      clear h₂ h₃
      -- porting note: was `decide!`
      fin_cases j₂ <;> fin_cases j₃ <;> 

Proof state:
case 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
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert (Triangle.orthocenter t₀) (Set.range t₀.points)
ht₀o : Triangle.orthocenter t₀ ∉ Set.range t₀.points
ht₀s : s = insert (Triangle.orthocenter t₀) (Set.range t₀.points)
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : Simplex.points t i₁ = Triangle.orthocenter t₀
hj₂₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
⊢ ∃ j₁,
    j₁ ≠ { val := 0, isLt := (_ : 0 < 3) } ∧
      j₁ ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∧
        ∀ (j : Fin 3),
          j = j₁ ∨
            j = { val := 0, isLt := (_ : 0 < 3) } ∨ j = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }

Next tactic:
simp (config := {decide := true}) at hj₂₃ ⊢
/-- Given any triangle in an orthocentric system, the fourth point is
its orthocenter. -/
theorem OrthocentricSystem.eq_insert_orthocenter {s : Set P} (ho : OrthocentricSystem s)
    {t : Triangle ℝ P} (ht : Set.range t.points ⊆ s) :
    s = insert t.orthocenter (Set.range t.points) := by
  rcases ho with ⟨t₀, ht₀o, ht₀s⟩
  rw [ht₀s] at ht
  rcases exists_of_range_subset_orthocentricSystem ht₀o ht t.independent.injective with
    (⟨i₁, i₂, i₃, j₂, j₃, h₁₂, h₁₃, h₂₃, h₁₂₃, h₁, hj₂₃, h₂, h₃⟩ | hs)
  · obtain ⟨j₁, hj₁₂, hj₁₃, hj₁₂₃⟩ :
        ∃ j₁ : Fin 3, j₁ ≠ j₂ ∧ j₁ ≠ j₃ ∧ ∀ j : Fin 3, j = j₁ ∨ j = j₂ ∨ j = j₃ := by
      clear h₂ h₃
      -- porting note: was `decide!`
      fin_cases j₂ <;> fin_cases j₃ <;> 

Proof state:
case 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
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert (Triangle.orthocenter t₀) (Set.range t₀.points)
ht₀o : Triangle.orthocenter t₀ ∉ Set.range t₀.points
ht₀s : s = insert (Triangle.orthocenter t₀) (Set.range t₀.points)
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : Simplex.points t i₁ = Triangle.orthocenter t₀
hj₂₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 0, isLt := (_ : 0 < 3) }
⊢ ∃ j₁,
    j₁ ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∧
      j₁ ≠ { val := 0, isLt := (_ : 0 < 3) } ∧
        ∀ (j : Fin 3),
          j = j₁ ∨ j = { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∨ j = { val := 0, isLt := (_ : 0 < 3) }

Next tactic:
simp (config := {decide := true}) at hj₂₃ ⊢
/-- Given any triangle in an orthocentric system, the fourth point is
its orthocenter. -/
theorem OrthocentricSystem.eq_insert_orthocenter {s : Set P} (ho : OrthocentricSystem s)
    {t : Triangle ℝ P} (ht : Set.range t.points ⊆ s) :
    s = insert t.orthocenter (Set.range t.points) := by
  rcases ho with ⟨t₀, ht₀o, ht₀s⟩
  rw [ht₀s] at ht
  rcases exists_of_range_subset_orthocentricSystem ht₀o ht t.independent.injective with
    (⟨i₁, i₂, i₃, j₂, j₃, h₁₂, h₁₃, h₂₃, h₁₂₃, h₁, hj₂₃, h₂, h₃⟩ | hs)
  · obtain ⟨j₁, hj₁₂, hj₁₃, hj₁₂₃⟩ :
        ∃ j₁ : Fin 3, j₁ ≠ j₂ ∧ j₁ ≠ j₃ ∧ ∀ j : Fin 3, j = j₁ ∨ j = j₂ ∨ j = j₃ := by
      clear h₂ h₃
      -- porting note: was `decide!`
      fin_cases j₂ <;> fin_cases j₃ <;> 

Proof state:
case 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
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert (Triangle.orthocenter t₀) (Set.range t₀.points)
ht₀o : Triangle.orthocenter t₀ ∉ Set.range t₀.points
ht₀s : s = insert (Triangle.orthocenter t₀) (Set.range t₀.points)
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : Simplex.points t i₁ = Triangle.orthocenter t₀
hj₂₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
⊢ ∃ j₁,
    j₁ ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∧
      j₁ ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∧
        ∀ (j : Fin 3),
          j = j₁ ∨
            j = { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∨ j = { val := 1, isLt := (_ : (fun a => a < 3) 1) }

Next tactic:
simp (config := {decide := true}) at hj₂₃ ⊢
/-- Given any triangle in an orthocentric system, the fourth point is
its orthocenter. -/
theorem OrthocentricSystem.eq_insert_orthocenter {s : Set P} (ho : OrthocentricSystem s)
    {t : Triangle ℝ P} (ht : Set.range t.points ⊆ s) :
    s = insert t.orthocenter (Set.range t.points) := by
  rcases ho with ⟨t₀, ht₀o, ht₀s⟩
  rw [ht₀s] at ht
  rcases exists_of_range_subset_orthocentricSystem ht₀o ht t.independent.injective with
    (⟨i₁, i₂, i₃, j₂, j₃, h₁₂, h₁₃, h₂₃, h₁₂₃, h₁, hj₂₃, h₂, h₃⟩ | hs)
  · obtain ⟨j₁, hj₁₂, hj₁₃, hj₁₂₃⟩ :
        ∃ j₁ : Fin 3, j₁ ≠ j₂ ∧ j₁ ≠ j₃ ∧ ∀ j : Fin 3, j = j₁ ∨ j = j₂ ∨ j = j₃ := by
      clear h₂ h₃
      -- porting note: was `decide!`
      fin_cases j₂ <;> fin_cases j₃ <;> 

Proof state:
case 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
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert (Triangle.orthocenter t₀) (Set.range t₀.points)
ht₀o : Triangle.orthocenter t₀ ∉ Set.range t₀.points
ht₀s : s = insert (Triangle.orthocenter t₀) (Set.range t₀.points)
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : Simplex.points t i₁ = Triangle.orthocenter t₀
hj₂₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
⊢ ∃ j₁,
    j₁ ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∧
      j₁ ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∧
        ∀ (j : Fin 3),
          j = j₁ ∨
            j = { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∨
              j = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }

Next tactic:
simp (config := {decide := true}) at hj₂₃ ⊢
/-- Given any triangle in an orthocentric system, the fourth point is
its orthocenter. -/
theorem OrthocentricSystem.eq_insert_orthocenter {s : Set P} (ho : OrthocentricSystem s)
    {t : Triangle ℝ P} (ht : Set.range t.points ⊆ s) :
    s = insert t.orthocenter (Set.range t.points) := by
  rcases ho with ⟨t₀, ht₀o, ht₀s⟩
  rw [ht₀s] at ht
  rcases exists_of_range_subset_orthocentricSystem ht₀o ht t.independent.injective with
    (⟨i₁, i₂, i₃, j₂, j₃, h₁₂, h₁₃, h₂₃, h₁₂₃, h₁, hj₂₃, h₂, h₃⟩ | hs)
  · obtain ⟨j₁, hj₁₂, hj₁₃, hj₁₂₃⟩ :
        ∃ j₁ : Fin 3, j₁ ≠ j₂ ∧ j₁ ≠ j₃ ∧ ∀ j : Fin 3, j = j₁ ∨ j = j₂ ∨ j = j₃ := by
      clear h₂ h₃
      -- porting note: was `decide!`
      fin_cases j₂ <;> fin_cases j₃ <;> 

Proof state:
case 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
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert (Triangle.orthocenter t₀) (Set.range t₀.points)
ht₀o : Triangle.orthocenter t₀ ∉ Set.range t₀.points
ht₀s : s = insert (Triangle.orthocenter t₀) (Set.range t₀.points)
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : Simplex.points t i₁ = Triangle.orthocenter t₀
hj₂₃ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 0, isLt := (_ : 0 < 3) }
⊢ ∃ j₁,
    j₁ ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∧
      j₁ ≠ { val := 0, isLt := (_ : 0 < 3) } ∧
        ∀ (j : Fin 3),
          j = j₁ ∨
            j = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∨ j = { val := 0, isLt := (_ : 0 < 3) }

Next tactic:
simp (config := {decide := true}) at hj₂₃ ⊢
/-- Given any triangle in an orthocentric system, the fourth point is
its orthocenter. -/
theorem OrthocentricSystem.eq_insert_orthocenter {s : Set P} (ho : OrthocentricSystem s)
    {t : Triangle ℝ P} (ht : Set.range t.points ⊆ s) :
    s = insert t.orthocenter (Set.range t.points) := by
  rcases ho with ⟨t₀, ht₀o, ht₀s⟩
  rw [ht₀s] at ht
  rcases exists_of_range_subset_orthocentricSystem ht₀o ht t.independent.injective with
    (⟨i₁, i₂, i₃, j₂, j₃, h₁₂, h₁₃, h₂₃, h₁₂₃, h₁, hj₂₃, h₂, h₃⟩ | hs)
  · obtain ⟨j₁, hj₁₂, hj₁₃, hj₁₂₃⟩ :
        ∃ j₁ : Fin 3, j₁ ≠ j₂ ∧ j₁ ≠ j₃ ∧ ∀ j : Fin 3, j = j₁ ∨ j = j₂ ∨ j = j₃ := by
      clear h₂ h₃
      -- porting note: was `decide!`
      fin_cases j₂ <;> fin_cases j₃ <;> 

Proof state:
case 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
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert (Triangle.orthocenter t₀) (Set.range t₀.points)
ht₀o : Triangle.orthocenter t₀ ∉ Set.range t₀.points
ht₀s : s = insert (Triangle.orthocenter t₀) (Set.range t₀.points)
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : Simplex.points t i₁ = Triangle.orthocenter t₀
hj₂₃ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
⊢ ∃ j₁,
    j₁ ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∧
      j₁ ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∧
        ∀ (j : Fin 3),
          j = j₁ ∨
            j = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∨
              j = { val := 1, isLt := (_ : (fun a => a < 3) 1) }

Next tactic:
simp (config := {decide := true}) at hj₂₃ ⊢
/-- Given any triangle in an orthocentric system, the fourth point is
its orthocenter. -/
theorem OrthocentricSystem.eq_insert_orthocenter {s : Set P} (ho : OrthocentricSystem s)
    {t : Triangle ℝ P} (ht : Set.range t.points ⊆ s) :
    s = insert t.orthocenter (Set.range t.points) := by
  rcases ho with ⟨t₀, ht₀o, ht₀s⟩
  rw [ht₀s] at ht
  rcases exists_of_range_subset_orthocentricSystem ht₀o ht t.independent.injective with
    (⟨i₁, i₂, i₃, j₂, j₃, h₁₂, h₁₃, h₂₃, h₁₂₃, h₁, hj₂₃, h₂, h₃⟩ | hs)
  · obtain ⟨j₁, hj₁₂, hj₁₃, hj₁₂₃⟩ :
        ∃ j₁ : Fin 3, j₁ ≠ j₂ ∧ j₁ ≠ j₃ ∧ ∀ j : Fin 3, j = j₁ ∨ j = j₂ ∨ j = j₃ := by
      clear h₂ h₃
      -- porting note: was `decide!`
      fin_cases j₂ <;> fin_cases j₃ <;> 

Proof state:
case 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
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert (Triangle.orthocenter t₀) (Set.range t₀.points)
ht₀o : Triangle.orthocenter t₀ ∉ Set.range t₀.points
ht₀s : s = insert (Triangle.orthocenter t₀) (Set.range t₀.points)
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : Simplex.points t i₁ = Triangle.orthocenter t₀
hj₂₃ :
  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
⊢ ∃ j₁,
    j₁ ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∧
      j₁ ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∧
        ∀ (j : Fin 3),
          j = j₁ ∨
            j = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∨
              j = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }

Next tactic:
simp (config := {decide := true}) at hj₂₃ ⊢
/-- Given any triangle in an orthocentric system, the fourth point is
its orthocenter. -/
theorem OrthocentricSystem.eq_insert_orthocenter {s : Set P} (ho : OrthocentricSystem s)
    {t : Triangle ℝ P} (ht : Set.range t.points ⊆ s) :
    s = insert t.orthocenter (Set.range t.points) := by
  rcases ho with ⟨t₀, ht₀o, ht₀s⟩
  rw [ht₀s] at ht
  rcases exists_of_range_subset_orthocentricSystem ht₀o ht t.independent.injective with
    (⟨i₁, i₂, i₃, j₂, j₃, h₁₂, h₁₃, h₂₃, h₁₂₃, h₁, hj₂₃, h₂, h₃⟩ | hs)
  · obtain ⟨j₁, hj₁₂, hj₁₃, hj₁₂₃⟩ :
        ∃ j₁ : Fin 3, j₁ ≠ j₂ ∧ j₁ ≠ j₃ ∧ ∀ j : Fin 3, j = j₁ ∨ j = j₂ ∨ j = j₃ := by
      clear h₂ h₃
      -- porting note: was `decide!`
      fin_cases j₂ <;> fin_cases j₃ <;> simp (config := {decide := true}) at hj₂₃ ⊢
    

Proof state:
case intro.intro.inl.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
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
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert (Triangle.orthocenter t₀) (Set.range t₀.points)
ht₀o : Triangle.orthocenter t₀ ∉ Set.range t₀.points
ht₀s : s = insert (Triangle.orthocenter t₀) (Set.range t₀.points)
i₁ i₂ i₃ j₂ j₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : Simplex.points t i₁ = Triangle.orthocenter t₀
hj₂₃ : j₂ ≠ j₃
h₂ : Simplex.points t₀ j₂ = Simplex.points t i₂
h₃ : Simplex.points t₀ j₃ = Simplex.points t i₃
j₁ : Fin 3
hj₁₂ : j₁ ≠ j₂
hj₁₃ : j₁ ≠ j₃
hj₁₂₃ : ∀ (j : Fin 3), j = j₁ ∨ j = j₂ ∨ j = j₃
⊢ s = insert (Triangle.orthocenter t) (Set.range t.points)

Next tactic:
suffices h : t₀.points j₁ = t.orthocenter
/-- Given any triangle in an orthocentric system, the fourth point is
its orthocenter. -/
theorem OrthocentricSystem.eq_insert_orthocenter {s : Set P} (ho : OrthocentricSystem s)
    {t : Triangle ℝ P} (ht : Set.range t.points ⊆ s) :
    s = insert t.orthocenter (Set.range t.points) := by
  rcases ho with ⟨t₀, ht₀o, ht₀s⟩
  rw [ht₀s] at ht
  rcases exists_of_range_subset_orthocentricSystem ht₀o ht t.independent.injective with
    (⟨i₁, i₂, i₃, j₂, j₃, h₁₂, h₁₃, h₂₃, h₁₂₃, h₁, hj₂₃, h₂, h₃⟩ | hs)
  · obtain ⟨j₁, hj₁₂, hj₁₃, hj₁₂₃⟩ :
        ∃ j₁ : Fin 3, j₁ ≠ j₂ ∧ j₁ ≠ j₃ ∧ ∀ j : Fin 3, j = j₁ ∨ j = j₂ ∨ j = j₃ := by
      clear h₂ h₃
      -- porting note: was `decide!`
      fin_cases j₂ <;> fin_cases j₃ <;> simp (config := {decide := true}) at hj₂₃ ⊢
    suffices h : t₀.points j₁ = t.orthocenter
    · 

Proof state:
case intro.intro.inl.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
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
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert (Triangle.orthocenter t₀) (Set.range t₀.points)
ht₀o : Triangle.orthocenter t₀ ∉ Set.range t₀.points
ht₀s : s = insert (Triangle.orthocenter t₀) (Set.range t₀.points)
i₁ i₂ i₃ j₂ j₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : Simplex.points t i₁ = Triangle.orthocenter t₀
hj₂₃ : j₂ ≠ j₃
h₂ : Simplex.points t₀ j₂ = Simplex.points t i₂
h₃ : Simplex.points t₀ j₃ = Simplex.points t i₃
j₁ : Fin 3
hj₁₂ : j₁ ≠ j₂
hj₁₃ : j₁ ≠ j₃
hj₁₂₃ : ∀ (j : Fin 3), j = j₁ ∨ j = j₂ ∨ j = j₃
h : Simplex.points t₀ j₁ = Triangle.orthocenter t
⊢ s = insert (Triangle.orthocenter t) (Set.range t.points)

Next tactic:
have hui : (Set.univ : Set (Fin 3)) = {i₁, i₂, i₃} := by ext x; simpa using h₁₂₃ x
/-- Given any triangle in an orthocentric system, the fourth point is
its orthocenter. -/
theorem OrthocentricSystem.eq_insert_orthocenter {s : Set P} (ho : OrthocentricSystem s)
    {t : Triangle ℝ P} (ht : Set.range t.points ⊆ s) :
    s = insert t.orthocenter (Set.range t.points) := by
  rcases ho with ⟨t₀, ht₀o, ht₀s⟩
  rw [ht₀s] at ht
  rcases exists_of_range_subset_orthocentricSystem ht₀o ht t.independent.injective with
    (⟨i₁, i₂, i₃, j₂, j₃, h₁₂, h₁₃, h₂₃, h₁₂₃, h₁, hj₂₃, h₂, h₃⟩ | hs)
  · obtain ⟨j₁, hj₁₂, hj₁₃, hj₁₂₃⟩ :
        ∃ j₁ : Fin 3, j₁ ≠ j₂ ∧ j₁ ≠ j₃ ∧ ∀ j : Fin 3, j = j₁ ∨ j = j₂ ∨ j = j₃ := by
      clear h₂ h₃
      -- porting note: was `decide!`
      fin_cases j₂ <;> fin_cases j₃ <;> simp (config := {decide := true}) at hj₂₃ ⊢
    suffices h : t₀.points j₁ = t.orthocenter
    · have hui : (Set.univ : Set (Fin 3)) = {i₁, i₂, i₃} := by 

Proof state:
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
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert (Triangle.orthocenter t₀) (Set.range t₀.points)
ht₀o : Triangle.orthocenter t₀ ∉ Set.range t₀.points
ht₀s : s = insert (Triangle.orthocenter t₀) (Set.range t₀.points)
i₁ i₂ i₃ j₂ j₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : Simplex.points t i₁ = Triangle.orthocenter t₀
hj₂₃ : j₂ ≠ j₃
h₂ : Simplex.points t₀ j₂ = Simplex.points t i₂
h₃ : Simplex.points t₀ j₃ = Simplex.points t i₃
j₁ : Fin 3
hj₁₂ : j₁ ≠ j₂
hj₁₃ : j₁ ≠ j₃
hj₁₂₃ : ∀ (j : Fin 3), j = j₁ ∨ j = j₂ ∨ j = j₃
h : Simplex.points t₀ j₁ = Triangle.orthocenter t
⊢ Set.univ = {i₁, i₂, i₃}

Next tactic:
ext x
/-- Given any triangle in an orthocentric system, the fourth point is
its orthocenter. -/
theorem OrthocentricSystem.eq_insert_orthocenter {s : Set P} (ho : OrthocentricSystem s)
    {t : Triangle ℝ P} (ht : Set.range t.points ⊆ s) :
    s = insert t.orthocenter (Set.range t.points) := by
  rcases ho with ⟨t₀, ht₀o, ht₀s⟩
  rw [ht₀s] at ht
  rcases exists_of_range_subset_orthocentricSystem ht₀o ht t.independent.injective with
    (⟨i₁, i₂, i₃, j₂, j₃, h₁₂, h₁₃, h₂₃, h₁₂₃, h₁, hj₂₃, h₂, h₃⟩ | hs)
  · obtain ⟨j₁, hj₁₂, hj₁₃, hj₁₂₃⟩ :
        ∃ j₁ : Fin 3, j₁ ≠ j₂ ∧ j₁ ≠ j₃ ∧ ∀ j : Fin 3, j = j₁ ∨ j = j₂ ∨ j = j₃ := by
      clear h₂ h₃
      -- porting note: was `decide!`
      fin_cases j₂ <;> fin_cases j₃ <;> simp (config := {decide := true}) at hj₂₃ ⊢
    suffices h : t₀.points j₁ = t.orthocenter
    · have hui : (Set.univ : Set (Fin 3)) = {i₁, i₂, i₃} := by ext x; 

Proof state:
case h
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
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert (Triangle.orthocenter t₀) (Set.range t₀.points)
ht₀o : Triangle.orthocenter t₀ ∉ Set.range t₀.points
ht₀s : s = insert (Triangle.orthocenter t₀) (Set.range t₀.points)
i₁ i₂ i₃ j₂ j₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : Simplex.points t i₁ = Triangle.orthocenter t₀
hj₂₃ : j₂ ≠ j₃
h₂ : Simplex.points t₀ j₂ = Simplex.points t i₂
h₃ : Simplex.points t₀ j₃ = Simplex.points t i₃
j₁ : Fin 3
hj₁₂ : j₁ ≠ j₂
hj₁₃ : j₁ ≠ j₃
hj₁₂₃ : ∀ (j : Fin 3), j = j₁ ∨ j = j₂ ∨ j = j₃
h : Simplex.points t₀ j₁ = Triangle.orthocenter t
x : Fin 3
⊢ x ∈ Set.univ ↔ x ∈ {i₁, i₂, i₃}

Next tactic:
simpa using h₁₂₃ x
/-- Given any triangle in an orthocentric system, the fourth point is
its orthocenter. -/
theorem OrthocentricSystem.eq_insert_orthocenter {s : Set P} (ho : OrthocentricSystem s)
    {t : Triangle ℝ P} (ht : Set.range t.points ⊆ s) :
    s = insert t.orthocenter (Set.range t.points) := by
  rcases ho with ⟨t₀, ht₀o, ht₀s⟩
  rw [ht₀s] at ht
  rcases exists_of_range_subset_orthocentricSystem ht₀o ht t.independent.injective with
    (⟨i₁, i₂, i₃, j₂, j₃, h₁₂, h₁₃, h₂₃, h₁₂₃, h₁, hj₂₃, h₂, h₃⟩ | hs)
  · obtain ⟨j₁, hj₁₂, hj₁₃, hj₁₂₃⟩ :
        ∃ j₁ : Fin 3, j₁ ≠ j₂ ∧ j₁ ≠ j₃ ∧ ∀ j : Fin 3, j = j₁ ∨ j = j₂ ∨ j = j₃ := by
      clear h₂ h₃
      -- porting note: was `decide!`
      fin_cases j₂ <;> fin_cases j₃ <;> simp (config := {decide := true}) at hj₂₃ ⊢
    suffices h : t₀.points j₁ = t.orthocenter
    · have hui : (Set.univ : Set (Fin 3)) = {i₁, i₂, i₃} := by ext x; simpa using h₁₂₃ x
      

Proof state:
case intro.intro.inl.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
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
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert (Triangle.orthocenter t₀) (Set.range t₀.points)
ht₀o : Triangle.orthocenter t₀ ∉ Set.range t₀.points
ht₀s : s = insert (Triangle.orthocenter t₀) (Set.range t₀.points)
i₁ i₂ i₃ j₂ j₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : Simplex.points t i₁ = Triangle.orthocenter t₀
hj₂₃ : j₂ ≠ j₃
h₂ : Simplex.points t₀ j₂ = Simplex.points t i₂
h₃ : Simplex.points t₀ j₃ = Simplex.points t i₃
j₁ : Fin 3
hj₁₂ : j₁ ≠ j₂
hj₁₃ : j₁ ≠ j₃
hj₁₂₃ : ∀ (j : Fin 3), j = j₁ ∨ j = j₂ ∨ j = j₃
h : Simplex.points t₀ j₁ = Triangle.orthocenter t
hui : Set.univ = {i₁, i₂, i₃}
⊢ s = insert (Triangle.orthocenter t) (Set.range t.points)

Next tactic:
have huj : (Set.univ : Set (Fin 3)) = {j₁, j₂, j₃} := by ext x; simpa using hj₁₂₃ x
/-- Given any triangle in an orthocentric system, the fourth point is
its orthocenter. -/
theorem OrthocentricSystem.eq_insert_orthocenter {s : Set P} (ho : OrthocentricSystem s)
    {t : Triangle ℝ P} (ht : Set.range t.points ⊆ s) :
    s = insert t.orthocenter (Set.range t.points) := by
  rcases ho with ⟨t₀, ht₀o, ht₀s⟩
  rw [ht₀s] at ht
  rcases exists_of_range_subset_orthocentricSystem ht₀o ht t.independent.injective with
    (⟨i₁, i₂, i₃, j₂, j₃, h₁₂, h₁₃, h₂₃, h₁₂₃, h₁, hj₂₃, h₂, h₃⟩ | hs)
  · obtain ⟨j₁, hj₁₂, hj₁₃, hj₁₂₃⟩ :
        ∃ j₁ : Fin 3, j₁ ≠ j₂ ∧ j₁ ≠ j₃ ∧ ∀ j : Fin 3, j = j₁ ∨ j = j₂ ∨ j = j₃ := by
      clear h₂ h₃
      -- porting note: was `decide!`
      fin_cases j₂ <;> fin_cases j₃ <;> simp (config := {decide := true}) at hj₂₃ ⊢
    suffices h : t₀.points j₁ = t.orthocenter
    · have hui : (Set.univ : Set (Fin 3)) = {i₁, i₂, i₃} := by ext x; simpa using h₁₂₃ x
      have huj : (Set.univ : Set (Fin 3)) = {j₁, j₂, j₃} := by 

Proof state:
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
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert (Triangle.orthocenter t₀) (Set.range t₀.points)
ht₀o : Triangle.orthocenter t₀ ∉ Set.range t₀.points
ht₀s : s = insert (Triangle.orthocenter t₀) (Set.range t₀.points)
i₁ i₂ i₃ j₂ j₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : Simplex.points t i₁ = Triangle.orthocenter t₀
hj₂₃ : j₂ ≠ j₃
h₂ : Simplex.points t₀ j₂ = Simplex.points t i₂
h₃ : Simplex.points t₀ j₃ = Simplex.points t i₃
j₁ : Fin 3
hj₁₂ : j₁ ≠ j₂
hj₁₃ : j₁ ≠ j₃
hj₁₂₃ : ∀ (j : Fin 3), j = j₁ ∨ j = j₂ ∨ j = j₃
h : Simplex.points t₀ j₁ = Triangle.orthocenter t
hui : Set.univ = {i₁, i₂, i₃}
⊢ Set.univ = {j₁, j₂, j₃}

Next tactic:
ext x