l3lab/ntp-mathlib huggingface dataset:
/-- This is an auxiliary lemma giving information about the relation
of two triangles in an orthocentric system; it abstracts some
reasoning, with no geometric content, that is common to some other
lemmas.  Suppose the orthocentric system is generated by triangle `t`,
and we are given three points `p` in the orthocentric system.  Then
either we can find indices `i₁`, `i₂` and `i₃` for `p` such that `p
i₁` is the orthocenter of `t` and `p i₂` and `p i₃` are points `j₂`
and `j₃` of `t`, or `p` has the same points as `t`. -/
theorem exists_of_range_subset_orthocentricSystem {t : Triangle ℝ P}
    (ho : t.orthocenter ∉ Set.range t.points) {p : Fin 3 → P}
    (hps : Set.range p ⊆ insert t.orthocenter (Set.range t.points)) (hpi : Function.Injective p) :
    (∃ i₁ i₂ i₃ j₂ j₃ : Fin 3,
      i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ (∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃) ∧
        p i₁ = t.orthocenter ∧ j₂ ≠ j₃ ∧ t.points j₂ = p i₂ ∧ t.points j₃ = p i₃) ∨
      Set.range p = Set.range t.points := by
  by_cases h : t.orthocenter ∈ Set.range p
  · left
    rcases h with ⟨i₁, h₁⟩
    obtain ⟨i₂, i₃, h₁₂, h₁₃, h₂₃, h₁₂₃⟩ :
        ∃ i₂ i₃ : Fin 3, i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
      clear h₁
      fin_cases i₁ <;> decide
    have h : ∀ i, i₁ ≠ i → ∃ j : Fin 3, t.points j = p i := by
      intro i hi
      replace hps := Set.mem_of_mem_insert_of_ne
        (Set.mem_of_mem_of_subset (Set.mem_range_self i) hps) (h₁ ▸ hpi.ne hi.symm)
      

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
t : Triangle ℝ P
ho : Triangle.orthocenter t ∉ Set.range t.points
p : Fin 3 → P
hpi : Function.Injective p
i₁ : Fin 3
h₁ : p i₁ = Triangle.orthocenter t
i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
i : Fin 3
hi : i₁ ≠ i
hps : p i ∈ Set.range t.points
⊢ ∃ j, Simplex.points t j = p i

Next tactic:
exact hps
/-- This is an auxiliary lemma giving information about the relation
of two triangles in an orthocentric system; it abstracts some
reasoning, with no geometric content, that is common to some other
lemmas.  Suppose the orthocentric system is generated by triangle `t`,
and we are given three points `p` in the orthocentric system.  Then
either we can find indices `i₁`, `i₂` and `i₃` for `p` such that `p
i₁` is the orthocenter of `t` and `p i₂` and `p i₃` are points `j₂`
and `j₃` of `t`, or `p` has the same points as `t`. -/
theorem exists_of_range_subset_orthocentricSystem {t : Triangle ℝ P}
    (ho : t.orthocenter ∉ Set.range t.points) {p : Fin 3 → P}
    (hps : Set.range p ⊆ insert t.orthocenter (Set.range t.points)) (hpi : Function.Injective p) :
    (∃ i₁ i₂ i₃ j₂ j₃ : Fin 3,
      i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ (∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃) ∧
        p i₁ = t.orthocenter ∧ j₂ ≠ j₃ ∧ t.points j₂ = p i₂ ∧ t.points j₃ = p i₃) ∨
      Set.range p = Set.range t.points := by
  by_cases h : t.orthocenter ∈ Set.range p
  · left
    rcases h with ⟨i₁, h₁⟩
    obtain ⟨i₂, i₃, h₁₂, h₁₃, h₂₃, h₁₂₃⟩ :
        ∃ i₂ i₃ : Fin 3, i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
      clear h₁
      fin_cases i₁ <;> decide
    have h : ∀ i, i₁ ≠ i → ∃ j : Fin 3, t.points j = p i := by
      intro i hi
      replace hps := Set.mem_of_mem_insert_of_ne
        (Set.mem_of_mem_of_subset (Set.mem_range_self i) hps) (h₁ ▸ hpi.ne hi.symm)
      exact hps
    

Proof state:
case pos.h.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
t : Triangle ℝ P
ho : Triangle.orthocenter t ∉ Set.range t.points
p : Fin 3 → P
hps : Set.range p ⊆ insert (Triangle.orthocenter t) (Set.range t.points)
hpi : Function.Injective p
i₁ : Fin 3
h₁ : p i₁ = Triangle.orthocenter t
i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h : ∀ (i : Fin 3), i₁ ≠ i → ∃ j, Simplex.points t j = p i
⊢ ∃ i₁ i₂ i₃ j₂ j₃,
    i₁ ≠ i₂ ∧
      i₁ ≠ i₃ ∧
        i₂ ≠ i₃ ∧
          (∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃) ∧
            p i₁ = Triangle.orthocenter t ∧ j₂ ≠ j₃ ∧ Simplex.points t j₂ = p i₂ ∧ Simplex.points t j₃ = p i₃

Next tactic:
rcases h i₂ h₁₂ with ⟨j₂, h₂⟩
/-- This is an auxiliary lemma giving information about the relation
of two triangles in an orthocentric system; it abstracts some
reasoning, with no geometric content, that is common to some other
lemmas.  Suppose the orthocentric system is generated by triangle `t`,
and we are given three points `p` in the orthocentric system.  Then
either we can find indices `i₁`, `i₂` and `i₃` for `p` such that `p
i₁` is the orthocenter of `t` and `p i₂` and `p i₃` are points `j₂`
and `j₃` of `t`, or `p` has the same points as `t`. -/
theorem exists_of_range_subset_orthocentricSystem {t : Triangle ℝ P}
    (ho : t.orthocenter ∉ Set.range t.points) {p : Fin 3 → P}
    (hps : Set.range p ⊆ insert t.orthocenter (Set.range t.points)) (hpi : Function.Injective p) :
    (∃ i₁ i₂ i₃ j₂ j₃ : Fin 3,
      i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ (∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃) ∧
        p i₁ = t.orthocenter ∧ j₂ ≠ j₃ ∧ t.points j₂ = p i₂ ∧ t.points j₃ = p i₃) ∨
      Set.range p = Set.range t.points := by
  by_cases h : t.orthocenter ∈ Set.range p
  · left
    rcases h with ⟨i₁, h₁⟩
    obtain ⟨i₂, i₃, h₁₂, h₁₃, h₂₃, h₁₂₃⟩ :
        ∃ i₂ i₃ : Fin 3, i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
      clear h₁
      fin_cases i₁ <;> decide
    have h : ∀ i, i₁ ≠ i → ∃ j : Fin 3, t.points j = p i := by
      intro i hi
      replace hps := Set.mem_of_mem_insert_of_ne
        (Set.mem_of_mem_of_subset (Set.mem_range_self i) hps) (h₁ ▸ hpi.ne hi.symm)
      exact hps
    rcases h i₂ h₁₂ with ⟨j₂, h₂⟩
    

Proof state:
case pos.h.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
t : Triangle ℝ P
ho : Triangle.orthocenter t ∉ Set.range t.points
p : Fin 3 → P
hps : Set.range p ⊆ insert (Triangle.orthocenter t) (Set.range t.points)
hpi : Function.Injective p
i₁ : Fin 3
h₁ : p i₁ = Triangle.orthocenter t
i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h : ∀ (i : Fin 3), i₁ ≠ i → ∃ j, Simplex.points t j = p i
j₂ : Fin 3
h₂ : Simplex.points t j₂ = p i₂
⊢ ∃ i₁ i₂ i₃ j₂ j₃,
    i₁ ≠ i₂ ∧
      i₁ ≠ i₃ ∧
        i₂ ≠ i₃ ∧
          (∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃) ∧
            p i₁ = Triangle.orthocenter t ∧ j₂ ≠ j₃ ∧ Simplex.points t j₂ = p i₂ ∧ Simplex.points t j₃ = p i₃

Next tactic:
rcases h i₃ h₁₃ with ⟨j₃, h₃⟩
/-- This is an auxiliary lemma giving information about the relation
of two triangles in an orthocentric system; it abstracts some
reasoning, with no geometric content, that is common to some other
lemmas.  Suppose the orthocentric system is generated by triangle `t`,
and we are given three points `p` in the orthocentric system.  Then
either we can find indices `i₁`, `i₂` and `i₃` for `p` such that `p
i₁` is the orthocenter of `t` and `p i₂` and `p i₃` are points `j₂`
and `j₃` of `t`, or `p` has the same points as `t`. -/
theorem exists_of_range_subset_orthocentricSystem {t : Triangle ℝ P}
    (ho : t.orthocenter ∉ Set.range t.points) {p : Fin 3 → P}
    (hps : Set.range p ⊆ insert t.orthocenter (Set.range t.points)) (hpi : Function.Injective p) :
    (∃ i₁ i₂ i₃ j₂ j₃ : Fin 3,
      i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ (∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃) ∧
        p i₁ = t.orthocenter ∧ j₂ ≠ j₃ ∧ t.points j₂ = p i₂ ∧ t.points j₃ = p i₃) ∨
      Set.range p = Set.range t.points := by
  by_cases h : t.orthocenter ∈ Set.range p
  · left
    rcases h with ⟨i₁, h₁⟩
    obtain ⟨i₂, i₃, h₁₂, h₁₃, h₂₃, h₁₂₃⟩ :
        ∃ i₂ i₃ : Fin 3, i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
      clear h₁
      fin_cases i₁ <;> decide
    have h : ∀ i, i₁ ≠ i → ∃ j : Fin 3, t.points j = p i := by
      intro i hi
      replace hps := Set.mem_of_mem_insert_of_ne
        (Set.mem_of_mem_of_subset (Set.mem_range_self i) hps) (h₁ ▸ hpi.ne hi.symm)
      exact hps
    rcases h i₂ h₁₂ with ⟨j₂, h₂⟩
    rcases h i₃ h₁₃ with ⟨j₃, h₃⟩
    

Proof state:
case pos.h.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
t : Triangle ℝ P
ho : Triangle.orthocenter t ∉ Set.range t.points
p : Fin 3 → P
hps : Set.range p ⊆ insert (Triangle.orthocenter t) (Set.range t.points)
hpi : Function.Injective p
i₁ : Fin 3
h₁ : p i₁ = Triangle.orthocenter t
i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h : ∀ (i : Fin 3), i₁ ≠ i → ∃ j, Simplex.points t j = p i
j₂ : Fin 3
h₂ : Simplex.points t j₂ = p i₂
j₃ : Fin 3
h₃ : Simplex.points t j₃ = p i₃
⊢ ∃ i₁ i₂ i₃ j₂ j₃,
    i₁ ≠ i₂ ∧
      i₁ ≠ i₃ ∧
        i₂ ≠ i₃ ∧
          (∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃) ∧
            p i₁ = Triangle.orthocenter t ∧ j₂ ≠ j₃ ∧ Simplex.points t j₂ = p i₂ ∧ Simplex.points t j₃ = p i₃

Next tactic:
have hj₂₃ : j₂ ≠ j₃ := by
      intro he
      rw [he, h₃] at h₂
      exact h₂₃.symm (hpi h₂)
/-- This is an auxiliary lemma giving information about the relation
of two triangles in an orthocentric system; it abstracts some
reasoning, with no geometric content, that is common to some other
lemmas.  Suppose the orthocentric system is generated by triangle `t`,
and we are given three points `p` in the orthocentric system.  Then
either we can find indices `i₁`, `i₂` and `i₃` for `p` such that `p
i₁` is the orthocenter of `t` and `p i₂` and `p i₃` are points `j₂`
and `j₃` of `t`, or `p` has the same points as `t`. -/
theorem exists_of_range_subset_orthocentricSystem {t : Triangle ℝ P}
    (ho : t.orthocenter ∉ Set.range t.points) {p : Fin 3 → P}
    (hps : Set.range p ⊆ insert t.orthocenter (Set.range t.points)) (hpi : Function.Injective p) :
    (∃ i₁ i₂ i₃ j₂ j₃ : Fin 3,
      i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ (∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃) ∧
        p i₁ = t.orthocenter ∧ j₂ ≠ j₃ ∧ t.points j₂ = p i₂ ∧ t.points j₃ = p i₃) ∨
      Set.range p = Set.range t.points := by
  by_cases h : t.orthocenter ∈ Set.range p
  · left
    rcases h with ⟨i₁, h₁⟩
    obtain ⟨i₂, i₃, h₁₂, h₁₃, h₂₃, h₁₂₃⟩ :
        ∃ i₂ i₃ : Fin 3, i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
      clear h₁
      fin_cases i₁ <;> decide
    have h : ∀ i, i₁ ≠ i → ∃ j : Fin 3, t.points j = p i := by
      intro i hi
      replace hps := Set.mem_of_mem_insert_of_ne
        (Set.mem_of_mem_of_subset (Set.mem_range_self i) hps) (h₁ ▸ hpi.ne hi.symm)
      exact hps
    rcases h i₂ h₁₂ with ⟨j₂, h₂⟩
    rcases h i₃ h₁₃ with ⟨j₃, h₃⟩
    have hj₂₃ : 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
t : Triangle ℝ P
ho : Triangle.orthocenter t ∉ Set.range t.points
p : Fin 3 → P
hps : Set.range p ⊆ insert (Triangle.orthocenter t) (Set.range t.points)
hpi : Function.Injective p
i₁ : Fin 3
h₁ : p i₁ = Triangle.orthocenter t
i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h : ∀ (i : Fin 3), i₁ ≠ i → ∃ j, Simplex.points t j = p i
j₂ : Fin 3
h₂ : Simplex.points t j₂ = p i₂
j₃ : Fin 3
h₃ : Simplex.points t j₃ = p i₃
⊢ j₂ ≠ j₃

Next tactic:
intro he
/-- This is an auxiliary lemma giving information about the relation
of two triangles in an orthocentric system; it abstracts some
reasoning, with no geometric content, that is common to some other
lemmas.  Suppose the orthocentric system is generated by triangle `t`,
and we are given three points `p` in the orthocentric system.  Then
either we can find indices `i₁`, `i₂` and `i₃` for `p` such that `p
i₁` is the orthocenter of `t` and `p i₂` and `p i₃` are points `j₂`
and `j₃` of `t`, or `p` has the same points as `t`. -/
theorem exists_of_range_subset_orthocentricSystem {t : Triangle ℝ P}
    (ho : t.orthocenter ∉ Set.range t.points) {p : Fin 3 → P}
    (hps : Set.range p ⊆ insert t.orthocenter (Set.range t.points)) (hpi : Function.Injective p) :
    (∃ i₁ i₂ i₃ j₂ j₃ : Fin 3,
      i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ (∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃) ∧
        p i₁ = t.orthocenter ∧ j₂ ≠ j₃ ∧ t.points j₂ = p i₂ ∧ t.points j₃ = p i₃) ∨
      Set.range p = Set.range t.points := by
  by_cases h : t.orthocenter ∈ Set.range p
  · left
    rcases h with ⟨i₁, h₁⟩
    obtain ⟨i₂, i₃, h₁₂, h₁₃, h₂₃, h₁₂₃⟩ :
        ∃ i₂ i₃ : Fin 3, i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
      clear h₁
      fin_cases i₁ <;> decide
    have h : ∀ i, i₁ ≠ i → ∃ j : Fin 3, t.points j = p i := by
      intro i hi
      replace hps := Set.mem_of_mem_insert_of_ne
        (Set.mem_of_mem_of_subset (Set.mem_range_self i) hps) (h₁ ▸ hpi.ne hi.symm)
      exact hps
    rcases h i₂ h₁₂ with ⟨j₂, h₂⟩
    rcases h i₃ h₁₃ with ⟨j₃, h₃⟩
    have hj₂₃ : j₂ ≠ j₃ := by
      intro he
      

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
t : Triangle ℝ P
ho : Triangle.orthocenter t ∉ Set.range t.points
p : Fin 3 → P
hps : Set.range p ⊆ insert (Triangle.orthocenter t) (Set.range t.points)
hpi : Function.Injective p
i₁ : Fin 3
h₁ : p i₁ = Triangle.orthocenter t
i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h : ∀ (i : Fin 3), i₁ ≠ i → ∃ j, Simplex.points t j = p i
j₂ : Fin 3
h₂ : Simplex.points t j₂ = p i₂
j₃ : Fin 3
h₃ : Simplex.points t j₃ = p i₃
he : j₂ = j₃
⊢ False

Next tactic:
rw [he, h₃] at h₂
/-- This is an auxiliary lemma giving information about the relation
of two triangles in an orthocentric system; it abstracts some
reasoning, with no geometric content, that is common to some other
lemmas.  Suppose the orthocentric system is generated by triangle `t`,
and we are given three points `p` in the orthocentric system.  Then
either we can find indices `i₁`, `i₂` and `i₃` for `p` such that `p
i₁` is the orthocenter of `t` and `p i₂` and `p i₃` are points `j₂`
and `j₃` of `t`, or `p` has the same points as `t`. -/
theorem exists_of_range_subset_orthocentricSystem {t : Triangle ℝ P}
    (ho : t.orthocenter ∉ Set.range t.points) {p : Fin 3 → P}
    (hps : Set.range p ⊆ insert t.orthocenter (Set.range t.points)) (hpi : Function.Injective p) :
    (∃ i₁ i₂ i₃ j₂ j₃ : Fin 3,
      i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ (∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃) ∧
        p i₁ = t.orthocenter ∧ j₂ ≠ j₃ ∧ t.points j₂ = p i₂ ∧ t.points j₃ = p i₃) ∨
      Set.range p = Set.range t.points := by
  by_cases h : t.orthocenter ∈ Set.range p
  · left
    rcases h with ⟨i₁, h₁⟩
    obtain ⟨i₂, i₃, h₁₂, h₁₃, h₂₃, h₁₂₃⟩ :
        ∃ i₂ i₃ : Fin 3, i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
      clear h₁
      fin_cases i₁ <;> decide
    have h : ∀ i, i₁ ≠ i → ∃ j : Fin 3, t.points j = p i := by
      intro i hi
      replace hps := Set.mem_of_mem_insert_of_ne
        (Set.mem_of_mem_of_subset (Set.mem_range_self i) hps) (h₁ ▸ hpi.ne hi.symm)
      exact hps
    rcases h i₂ h₁₂ with ⟨j₂, h₂⟩
    rcases h i₃ h₁₃ with ⟨j₃, h₃⟩
    have hj₂₃ : j₂ ≠ j₃ := by
      intro he
      rw [he, h₃] at h₂
      

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
t : Triangle ℝ P
ho : Triangle.orthocenter t ∉ Set.range t.points
p : Fin 3 → P
hps : Set.range p ⊆ insert (Triangle.orthocenter t) (Set.range t.points)
hpi : Function.Injective p
i₁ : Fin 3
h₁ : p i₁ = Triangle.orthocenter t
i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h : ∀ (i : Fin 3), i₁ ≠ i → ∃ j, Simplex.points t j = p i
j₂ j₃ : Fin 3
h₂ : p i₃ = p i₂
h₃ : Simplex.points t j₃ = p i₃
he : j₂ = j₃
⊢ False

Next tactic:
exact h₂₃.symm (hpi h₂)
/-- This is an auxiliary lemma giving information about the relation
of two triangles in an orthocentric system; it abstracts some
reasoning, with no geometric content, that is common to some other
lemmas.  Suppose the orthocentric system is generated by triangle `t`,
and we are given three points `p` in the orthocentric system.  Then
either we can find indices `i₁`, `i₂` and `i₃` for `p` such that `p
i₁` is the orthocenter of `t` and `p i₂` and `p i₃` are points `j₂`
and `j₃` of `t`, or `p` has the same points as `t`. -/
theorem exists_of_range_subset_orthocentricSystem {t : Triangle ℝ P}
    (ho : t.orthocenter ∉ Set.range t.points) {p : Fin 3 → P}
    (hps : Set.range p ⊆ insert t.orthocenter (Set.range t.points)) (hpi : Function.Injective p) :
    (∃ i₁ i₂ i₃ j₂ j₃ : Fin 3,
      i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ (∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃) ∧
        p i₁ = t.orthocenter ∧ j₂ ≠ j₃ ∧ t.points j₂ = p i₂ ∧ t.points j₃ = p i₃) ∨
      Set.range p = Set.range t.points := by
  by_cases h : t.orthocenter ∈ Set.range p
  · left
    rcases h with ⟨i₁, h₁⟩
    obtain ⟨i₂, i₃, h₁₂, h₁₃, h₂₃, h₁₂₃⟩ :
        ∃ i₂ i₃ : Fin 3, i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
      clear h₁
      fin_cases i₁ <;> decide
    have h : ∀ i, i₁ ≠ i → ∃ j : Fin 3, t.points j = p i := by
      intro i hi
      replace hps := Set.mem_of_mem_insert_of_ne
        (Set.mem_of_mem_of_subset (Set.mem_range_self i) hps) (h₁ ▸ hpi.ne hi.symm)
      exact hps
    rcases h i₂ h₁₂ with ⟨j₂, h₂⟩
    rcases h i₃ h₁₃ with ⟨j₃, h₃⟩
    have hj₂₃ : j₂ ≠ j₃ := by
      intro he
      rw [he, h₃] at h₂
      exact h₂₃.symm (hpi h₂)
    

Proof state:
case pos.h.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
t : Triangle ℝ P
ho : Triangle.orthocenter t ∉ Set.range t.points
p : Fin 3 → P
hps : Set.range p ⊆ insert (Triangle.orthocenter t) (Set.range t.points)
hpi : Function.Injective p
i₁ : Fin 3
h₁ : p i₁ = Triangle.orthocenter t
i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h : ∀ (i : Fin 3), i₁ ≠ i → ∃ j, Simplex.points t j = p i
j₂ : Fin 3
h₂ : Simplex.points t j₂ = p i₂
j₃ : Fin 3
h₃ : Simplex.points t j₃ = p i₃
hj₂₃ : j₂ ≠ j₃
⊢ ∃ i₁ i₂ i₃ j₂ j₃,
    i₁ ≠ i₂ ∧
      i₁ ≠ i₃ ∧
        i₂ ≠ i₃ ∧
          (∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃) ∧
            p i₁ = Triangle.orthocenter t ∧ j₂ ≠ j₃ ∧ Simplex.points t j₂ = p i₂ ∧ Simplex.points t j₃ = p i₃

Next tactic:
exact ⟨i₁, i₂, i₃, j₂, j₃, h₁₂, h₁₃, h₂₃, h₁₂₃, h₁, hj₂₃, h₂, h₃⟩
/-- This is an auxiliary lemma giving information about the relation
of two triangles in an orthocentric system; it abstracts some
reasoning, with no geometric content, that is common to some other
lemmas.  Suppose the orthocentric system is generated by triangle `t`,
and we are given three points `p` in the orthocentric system.  Then
either we can find indices `i₁`, `i₂` and `i₃` for `p` such that `p
i₁` is the orthocenter of `t` and `p i₂` and `p i₃` are points `j₂`
and `j₃` of `t`, or `p` has the same points as `t`. -/
theorem exists_of_range_subset_orthocentricSystem {t : Triangle ℝ P}
    (ho : t.orthocenter ∉ Set.range t.points) {p : Fin 3 → P}
    (hps : Set.range p ⊆ insert t.orthocenter (Set.range t.points)) (hpi : Function.Injective p) :
    (∃ i₁ i₂ i₃ j₂ j₃ : Fin 3,
      i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ (∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃) ∧
        p i₁ = t.orthocenter ∧ j₂ ≠ j₃ ∧ t.points j₂ = p i₂ ∧ t.points j₃ = p i₃) ∨
      Set.range p = Set.range t.points := by
  by_cases h : t.orthocenter ∈ Set.range p
  · left
    rcases h with ⟨i₁, h₁⟩
    obtain ⟨i₂, i₃, h₁₂, h₁₃, h₂₃, h₁₂₃⟩ :
        ∃ i₂ i₃ : Fin 3, i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
      clear h₁
      fin_cases i₁ <;> decide
    have h : ∀ i, i₁ ≠ i → ∃ j : Fin 3, t.points j = p i := by
      intro i hi
      replace hps := Set.mem_of_mem_insert_of_ne
        (Set.mem_of_mem_of_subset (Set.mem_range_self i) hps) (h₁ ▸ hpi.ne hi.symm)
      exact hps
    rcases h i₂ h₁₂ with ⟨j₂, h₂⟩
    rcases h i₃ h₁₃ with ⟨j₃, h₃⟩
    have hj₂₃ : j₂ ≠ j₃ := by
      intro he
      rw [he, h₃] at h₂
      exact h₂₃.symm (hpi h₂)
    exact ⟨i₁, i₂, i₃, j₂, j₃, h₁₂, h₁₃, h₂₃, h₁₂₃, h₁, hj₂₃, h₂, h₃⟩
  · 

Proof state:
case neg
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
ho : Triangle.orthocenter t ∉ Set.range t.points
p : Fin 3 → P
hps : Set.range p ⊆ insert (Triangle.orthocenter t) (Set.range t.points)
hpi : Function.Injective p
h : Triangle.orthocenter t ∉ Set.range p
⊢ (∃ i₁ i₂ i₃ j₂ j₃,
      i₁ ≠ i₂ ∧
        i₁ ≠ i₃ ∧
          i₂ ≠ i₃ ∧
            (∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃) ∧
              p i₁ = Triangle.orthocenter t ∧ j₂ ≠ j₃ ∧ Simplex.points t j₂ = p i₂ ∧ Simplex.points t j₃ = p i₃) ∨
    Set.range p = Set.range t.points

Next tactic:
right
/-- This is an auxiliary lemma giving information about the relation
of two triangles in an orthocentric system; it abstracts some
reasoning, with no geometric content, that is common to some other
lemmas.  Suppose the orthocentric system is generated by triangle `t`,
and we are given three points `p` in the orthocentric system.  Then
either we can find indices `i₁`, `i₂` and `i₃` for `p` such that `p
i₁` is the orthocenter of `t` and `p i₂` and `p i₃` are points `j₂`
and `j₃` of `t`, or `p` has the same points as `t`. -/
theorem exists_of_range_subset_orthocentricSystem {t : Triangle ℝ P}
    (ho : t.orthocenter ∉ Set.range t.points) {p : Fin 3 → P}
    (hps : Set.range p ⊆ insert t.orthocenter (Set.range t.points)) (hpi : Function.Injective p) :
    (∃ i₁ i₂ i₃ j₂ j₃ : Fin 3,
      i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ (∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃) ∧
        p i₁ = t.orthocenter ∧ j₂ ≠ j₃ ∧ t.points j₂ = p i₂ ∧ t.points j₃ = p i₃) ∨
      Set.range p = Set.range t.points := by
  by_cases h : t.orthocenter ∈ Set.range p
  · left
    rcases h with ⟨i₁, h₁⟩
    obtain ⟨i₂, i₃, h₁₂, h₁₃, h₂₃, h₁₂₃⟩ :
        ∃ i₂ i₃ : Fin 3, i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
      clear h₁
      fin_cases i₁ <;> decide
    have h : ∀ i, i₁ ≠ i → ∃ j : Fin 3, t.points j = p i := by
      intro i hi
      replace hps := Set.mem_of_mem_insert_of_ne
        (Set.mem_of_mem_of_subset (Set.mem_range_self i) hps) (h₁ ▸ hpi.ne hi.symm)
      exact hps
    rcases h i₂ h₁₂ with ⟨j₂, h₂⟩
    rcases h i₃ h₁₃ with ⟨j₃, h₃⟩
    have hj₂₃ : j₂ ≠ j₃ := by
      intro he
      rw [he, h₃] at h₂
      exact h₂₃.symm (hpi h₂)
    exact ⟨i₁, i₂, i₃, j₂, j₃, h₁₂, h₁₃, h₂₃, h₁₂₃, h₁, hj₂₃, h₂, h₃⟩
  · right
    

Proof state:
case neg.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
t : Triangle ℝ P
ho : Triangle.orthocenter t ∉ Set.range t.points
p : Fin 3 → P
hps : Set.range p ⊆ insert (Triangle.orthocenter t) (Set.range t.points)
hpi : Function.Injective p
h : Triangle.orthocenter t ∉ Set.range p
⊢ Set.range p = Set.range t.points

Next tactic:
have hs := Set.subset_diff_singleton hps h