awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Simplex (goal)
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
t : Triangle ℝ P
h :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
i₂ i₃ : Fin (HAdd.hAdd (α := ℕ) 2 1)
h₂₃ : i₂ ≠ i₃
h₁₂ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, ⋯⟩ ≠ i₂
h₁₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, ⋯⟩ ≠ i₃
1 :
  ∀ (h₂₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩ ≠ i₃)
    (h₁₂ :
      Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, ⋯⟩)
        ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩)),
    ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
        (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
        (t.points i₃) <
      π / 2
h_1 :
  ∀ (h₂₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, ⋯⟩ ≠ i₃)
    (h₁₂ :
      Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, ⋯⟩)
        ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, ⋯⟩)),
    ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
        (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
        (t.points i₃) <
      π / 2
⊢ ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) (t.points i₂)
      (t.points i₃) <
    π / 2
have 1₁ : ∀ (h₂₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, sorry⟩ ≠ i₃)
  (h₁₂ :
    Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, sorry⟩)
      ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, sorry⟩)),
  ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨1, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨2, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) (t.points i₃) <
    π / 2
Tactic: fin_cases i₂
Cleared hypotheses: i₂, h₂₃, h₁₂
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Simplex (goal)
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
t : Triangle ℝ P
h :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
i₂ i₃ : Fin (HAdd.hAdd (α := ℕ) 2 1)
h₂₃ : i₂ ≠ i₃
h₁₂ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, ⋯⟩ ≠ i₂
h₁₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, ⋯⟩ ≠ i₃
⊢ ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) (t.points i₂)
      (t.points i₃) <
    π / 2
have 2 : ∀ (h₂₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, sorry⟩ ≠ i₃)
  (h₁₂ :
    Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, sorry⟩)
      ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, sorry⟩)),
  ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨2, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) (t.points i₃) <
    π / 2
Tactic: fin_cases i₂
Cleared hypotheses: i₂, h₂₃, h₁₂
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Simplex (goal)
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
t : Triangle ℝ P
h :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
i₂ i₃ : Fin (HAdd.hAdd (α := ℕ) 2 1)
h₂₃ : i₂ ≠ i₃
h₁₂ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, ⋯⟩ ≠ i₂
h₁₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, ⋯⟩ ≠ i₃
2 :
  ∀ (h₂₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩ ≠ i₃)
    (h₁₂ :
      Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, ⋯⟩)
        ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩)),
    ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
        (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
        (t.points i₃) <
      π / 2
⊢ ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) (t.points i₂)
      (t.points i₃) <
    π / 2
have 2₁ : ∀ (h₂₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, sorry⟩ ≠ i₃)
  (h₁₂ :
    Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, sorry⟩)
      ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, sorry⟩)),
  ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨2, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨1, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) (t.points i₃) <
    π / 2
Tactic: fin_cases i₂
Cleared hypotheses: i₂, h₂₃, h₁₂
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Simplex (goal)
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
t : Triangle ℝ P
h :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
i₂ i₃ : Fin (HAdd.hAdd (α := ℕ) 2 1)
h₂₃ : i₂ ≠ i₃
h₁₂ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, ⋯⟩ ≠ i₂
h₁₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, ⋯⟩ ≠ i₃
2 :
  ∀ (h₂₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩ ≠ i₃)
    (h₁₂ :
      Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, ⋯⟩)
        ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩)),
    ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
        (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
        (t.points i₃) <
      π / 2
h_1 :
  ∀ (h₂₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, ⋯⟩ ≠ i₃)
    (h₁₂ :
      Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, ⋯⟩)
        ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, ⋯⟩)),
    ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
        (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
        (t.points i₃) <
      π / 2
⊢ ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) (t.points i₂)
      (t.points i₃) <
    π / 2
have 2₁ : ∀ (h₂₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, sorry⟩ ≠ i₃)
  (h₁₂ :
    Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, sorry⟩)
      ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, sorry⟩)),
  ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨2, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨2, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) (t.points i₃) <
    π / 2
Tactic: fin_cases i₂
Cleared hypotheses: i₂, h₂₃, h₁₂
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Simplex (goal)
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
t : Triangle ℝ P
h :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
i₃ : Fin (HAdd.hAdd (α := ℕ) 2 1)
h₁₃ h₂₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩ ≠ i₃
h₁₂ :
  Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩)
    ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩)
⊢ ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) (t.points i₃) <
    π / 2
have 0 : ∀
  (h₁₃ h₂₃ :
    Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, sorry⟩)
      ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, sorry⟩)),
  ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) <
    π / 2
Tactic: fin_cases i₃
Cleared hypotheses: i₃, h₁₃, h₂₃
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Simplex (goal)
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
t : Triangle ℝ P
h :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
i₃ : Fin (HAdd.hAdd (α := ℕ) 2 1)
h₁₃ h₂₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩ ≠ i₃
h₁₂ :
  Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩)
    ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩)
0 :
  ∀
    (h₁₃ h₂₃ :
      Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩)
        ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩)),
    ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
        (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
        (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) <
      π / 2
⊢ ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) (t.points i₃) <
    π / 2
have 0₁ : ∀
  (h₁₃ h₂₃ :
    Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, sorry⟩)
      ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, sorry⟩)),
  ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨1, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) <
    π / 2
Tactic: fin_cases i₃
Cleared hypotheses: i₃, h₁₃, h₂₃
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Simplex (goal)
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
t : Triangle ℝ P
h :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
i₃ : Fin (HAdd.hAdd (α := ℕ) 2 1)
h₁₃ h₂₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩ ≠ i₃
h₁₂ :
  Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩)
    ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩)
0 :
  ∀
    (h₁₃ h₂₃ :
      Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩)
        ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩)),
    ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
        (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
        (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) <
      π / 2
h_1 :
  ∀
    (h₁₃ h₂₃ :
      Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩)
        ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, ⋯⟩)),
    ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
        (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
        (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) <
      π / 2
⊢ ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) (t.points i₃) <
    π / 2
have 0₁ : ∀
  (h₁₃ h₂₃ :
    Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, sorry⟩)
      ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, sorry⟩)),
  ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨2, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) <
    π / 2
Tactic: fin_cases i₃
Cleared hypotheses: i₃, h₁₃, h₂₃
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Simplex (goal)
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
t : Triangle ℝ P
h :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
i₃ : Fin (HAdd.hAdd (α := ℕ) 2 1)
h₁₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩ ≠ i₃
h₂₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, ⋯⟩ ≠ i₃
h₁₂ :
  Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩)
    ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, ⋯⟩)
⊢ ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) (t.points i₃) <
    π / 2
have 0 : ∀
  (h₁₃ :
    Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, sorry⟩)
      ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, sorry⟩))
  (h₂₃ :
    Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, sorry⟩)
      ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, sorry⟩)),
  ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨1, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) <
    π / 2
Tactic: fin_cases i₃
Cleared hypotheses: i₃, h₁₃, h₂₃
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Simplex (goal)
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
t : Triangle ℝ P
h :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
i₃ : Fin (HAdd.hAdd (α := ℕ) 2 1)
h₁₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩ ≠ i₃
h₂₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, ⋯⟩ ≠ i₃
h₁₂ :
  Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩)
    ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, ⋯⟩)
0 :
  ∀
    (h₁₃ :
      Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩)
        ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩))
    (h₂₃ :
      Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, ⋯⟩)
        ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩)),
    ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
        (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
        (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) <
      π / 2
⊢ ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) (t.points i₃) <
    π / 2
have 0₁ : ∀
  (h₁₃ :
    Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, sorry⟩)
      ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, sorry⟩))
  (h₂₃ :
    Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, sorry⟩)
      ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨1, sorry⟩)),
  ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨1, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨1, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) <
    π / 2
Tactic: fin_cases i₃
Cleared hypotheses: i₃, h₁₃, h₂₃
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Simplex (goal)
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
t : Triangle ℝ P
h :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
i₃ : Fin (HAdd.hAdd (α := ℕ) 2 1)
h₁₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩ ≠ i₃
h₂₃ : (fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, ⋯⟩ ≠ i₃
h₁₂ :
  Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, ⋯⟩)
    ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, ⋯⟩)
⊢ ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) (t.points i₃) <
    π / 2
have 0 : ∀
  (h₁₃ :
    Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, sorry⟩)
      ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, sorry⟩))
  (h₂₃ :
    Ne (α := Fin (HAdd.hAdd (α := ℕ) 2 1)) ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨2, sorry⟩)
      ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) ⟨0, sorry⟩)),
  ∠ (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨2, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))))
      (t.points ((fun (i : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ i) (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))) <
    π / 2
Tactic: fin_cases i₃
Cleared hypotheses: i₃, h₁₃, h₂₃