awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : s.touchpoint signs i = s.touchpoint signs j
hne : ¬i = j
hn1 : n = 1
⊢ False
have pos : ∀ [inst_5 : NeZero (R := ℕ) 1] {s : Simplex ℝ P 1} {signs : Finset (Fin (HAdd.hAdd (α := ℕ) 1 1))}
  (h : s.ExcenterExists signs) ⦃i j : Fin (HAdd.hAdd (α := ℕ) 1 1)⦄ (hij : s.touchpoint signs i = s.touchpoint signs j)
  (hne : ¬i = j), False
Tactic: subst hn1
Cleared hypotheses: n, inst_4, s, signs, h, i, j, hij, hne, hn1
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (hypothesis)
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
inst_4 : NeZero (R := ℕ) 1
s : Simplex ℝ P 1
signs : Finset (Fin (HAdd.hAdd (α := ℕ) 1 1))
h : s.ExcenterExists signs
i j : Fin (HAdd.hAdd (α := ℕ) 1 1)
hij : s.touchpoint signs i = s.touchpoint signs j
hne : ¬i = j
⊢ False
have hij₁ : s.points i.rev = s.points j.rev
Tactic: rw [s.touchpoint_eq_point_rev signs i, s.touchpoint_eq_point_rev signs j] at hij
Cleared hypotheses: hij, hne
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
inst_4 : NeZero (R := ℕ) 1
s : Simplex ℝ P 1
signs : Finset (Fin (HAdd.hAdd (α := ℕ) 1 1))
h : s.ExcenterExists signs
i j : Fin (HAdd.hAdd (α := ℕ) 1 1)
hij : s.points i.rev = s.points j.rev
hne : ¬i = j
⊢ False
have pos : s.points i = s.points j
Tactic: apply s.independent.injective.ne hne
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
inst_4 : NeZero (R := ℕ) 1
s : Simplex ℝ P 1
signs : Finset (Fin (HAdd.hAdd (α := ℕ) 1 1))
h : s.ExcenterExists signs
i j : Fin (HAdd.hAdd (α := ℕ) 1 1)
hij : s.points i.rev = s.points j.rev
hne : ¬i = j
⊢ s.points i = s.points j
have h₁ : i = j.rev
Tactic: convert! hij.symm
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
inst_4 : NeZero (R := ℕ) 1
s : Simplex ℝ P 1
signs : Finset (Fin (HAdd.hAdd (α := ℕ) 1 1))
h : s.ExcenterExists signs
i j : Fin (HAdd.hAdd (α := ℕ) 1 1)
hij : s.points i.rev = s.points j.rev
hne : ¬i = j
h₁ : i = j.rev
⊢ s.points i = s.points j
have h₂ : j = i.rev
Tactic: convert! hij.symm
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : s.touchpoint signs i = s.touchpoint signs j
hne : ¬i = j
hn1 : ¬n = 1
this :
  Membership.mem (γ := Submodule ℝ V) (vectorSpan ℝ (Set.range s.points))ᗮ (s.excenter signs -ᵥ s.touchpoint signs i)
⊢ Membership.mem (γ := Submodule ℝ V) (vectorSpan ℝ (Set.range s.points)) (s.excenter signs -ᵥ s.touchpoint signs i)
have rw : Membership.mem (γ := Submodule ℝ V) (AffineSubspace.direction (affineSpan ℝ (Set.range s.points)))
  (s.excenter signs -ᵥ s.touchpoint signs i)
Tactic: rw [← direction_affineSpan]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (hypothesis)
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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : s.touchpoint signs i = s.touchpoint signs j
hne : ¬i = j
hn1 : ¬n = 1
this :
  Membership.mem (γ := Submodule ℝ V) (vectorSpan ℝ (Set.range s.points))ᗮ (s.excenter signs -ᵥ s.touchpoint signs i)
⊢ False
have h' : Membership.mem (γ := Submodule ℝ V) (vectorSpan ℝ (Set.range s.points)) (s.excenter signs -ᵥ s.touchpoint signs i)
Tactic: have h' : s.excenter signs -ᵥ s.touchpoint signs i ∈ (vectorSpan ℝ (Set.range s.points)) := by
        rw [← direction_affineSpan]
        exact AffineSubspace.vsub_mem_direction h.excenter_mem_affineSpan_range
          (s.touchpoint_mem_affineSpan_simplex _ _)
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : s.touchpoint signs i = s.touchpoint signs j
hne : ¬i = j
hn1 : ¬n = 1
this :
  Membership.mem (γ := Submodule ℝ V) (vectorSpan ℝ (Set.range s.points))ᗮ (s.excenter signs -ᵥ s.touchpoint signs i)
h' : Membership.mem (γ := Submodule ℝ V) (vectorSpan ℝ (Set.range s.points)) (s.excenter signs -ᵥ s.touchpoint signs i)
⊢ s.excenter signs -ᵥ s.touchpoint signs i = 0
have rw : Membership.mem (γ := Submodule ℝ V) (vectorSpan ℝ (Set.range s.points) ⊓ (vectorSpan ℝ (Set.range s.points))ᗮ)
  (s.excenter signs -ᵥ s.touchpoint signs i)
Tactic: rw [← Submodule.mem_bot ℝ,
          ← Submodule.inf_orthogonal_eq_bot (vectorSpan ℝ (Set.range s.points))]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (hypothesis)
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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : s.touchpoint signs i = s.touchpoint signs j
hne : ¬i = j
hn1 : ¬n = 1
this :
  Membership.mem (γ := Submodule ℝ V) (vectorSpan ℝ (Set.range s.points))ᗮ (s.excenter signs -ᵥ s.touchpoint signs i)
h' : Membership.mem (γ := Submodule ℝ V) (vectorSpan ℝ (Set.range s.points)) (s.excenter signs -ᵥ s.touchpoint signs i)
⊢ False
have h0 : s.excenter signs -ᵥ s.touchpoint signs i = 0
Tactic: have h0 : s.excenter signs -ᵥ s.touchpoint signs i = 0 := by
        rw [← Submodule.mem_bot ℝ,
          ← Submodule.inf_orthogonal_eq_bot (vectorSpan ℝ (Set.range s.points))]
        exact ⟨h', this⟩
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (hypothesis)
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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : s.touchpoint signs i = s.touchpoint signs j
hne : ¬i = j
hn1 : ¬n = 1
this :
  Membership.mem (γ := Submodule ℝ V) (vectorSpan ℝ (Set.range s.points))ᗮ (s.excenter signs -ᵥ s.touchpoint signs i)
h' : Membership.mem (γ := Submodule ℝ V) (vectorSpan ℝ (Set.range s.points)) (s.excenter signs -ᵥ s.touchpoint signs i)
h0 : s.excenter signs -ᵥ s.touchpoint signs i = 0
⊢ False
have h0₁ : s.exradius signs = 0
Tactic: rw [← norm_eq_zero, ← dist_eq_norm_vsub, h.dist_excenter] at h0
Cleared hypotheses: h0
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : s.touchpoint signs i = s.touchpoint signs j
hne : ¬i = j
hn1 : ¬n = 1
⊢ False
have neg : Membership.mem (γ := Submodule ℝ V) (vectorSpan ℝ (Set.range s.points))ᗮ (s.excenter signs -ᵥ s.touchpoint signs i)
Tactic: suffices s.excenter signs -ᵥ s.touchpoint signs i ∈ (vectorSpan ℝ (Set.range s.points))ᗮ by
      have h' : s.excenter signs -ᵥ s.touchpoint signs i ∈ (vectorSpan ℝ (Set.range s.points)) := by
        rw [← direction_affineSpan]
        exact AffineSubspace.vsub_mem_direction h.excenter_mem_affineSpan_range
          (s.touchpoint_mem_affineSpan_simplex _ _)
      have h0 : s.excenter signs -ᵥ s.touchpoint signs i = 0 := by
        rw [← Submodule.mem_bot ℝ,
          ← Submodule.inf_orthogonal_eq_bot (vectorSpan ℝ (Set.range s.points))]
        exact ⟨h', this⟩
      rw [← norm_eq_zero, ← dist_eq_norm_vsub, h.dist_excenter] at h0
      exact h.exradius_pos.ne' h0
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : s.touchpoint signs i = s.touchpoint signs j
hne : ¬i = j
hn1 : ¬n = 1
⊢ Membership.mem (γ := Submodule ℝ V) (vectorSpan ℝ (Set.range s.points))ᗮ (s.excenter signs -ᵥ s.touchpoint signs i)
have neg : ∀ (k : Fin (n + 1)) (hki : k ≠ i) (hkj : k ≠ j),
  Membership.mem (γ := Submodule ℝ V) (vectorSpan ℝ (Set.range s.points))ᗮ (s.excenter signs -ᵥ s.touchpoint signs i)
Tactic: obtain ⟨k, hki, hkj⟩ : ∃ k, k ≠ i ∧ k ≠ j := Fin.exists_ne_and_ne_of_two_lt i j (by lia)
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : s.touchpoint signs i = s.touchpoint signs j
hne : ¬i = j
hn1 : ¬n = 1
k : Fin (n + 1)
hki : k ≠ i
hkj : k ≠ j
⊢ Set.range s.points = Set.range (s.faceOpposite i).points ∪ Set.range (s.faceOpposite j).points
have simp : Set.range s.points = s.points '' ({i} ∩ {j})ᶜ
Tactic: simp only [range_faceOpposite_points, ← Set.image_union, ← Set.compl_inter]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : s.touchpoint signs i = s.touchpoint signs j
hne : ¬i = j
hn1 : ¬n = 1
k : Fin (n + 1)
hki : k ≠ i
hkj : k ≠ j
⊢ Set.range s.points = s.points '' ({i} ∩ {j})ᶜ
have h₁ : Eq (α := Set (Fin (n + 1))) ({i} ∩ {j})ᶜ Set.univ
Tactic: convert! Set.image_univ.symm
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (hypothesis)
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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : s.touchpoint signs i = s.touchpoint signs j
hne : ¬i = j
hn1 : ¬n = 1
k : Fin (n + 1)
hki : k ≠ i
hkj : k ≠ j
⊢ Membership.mem (γ := Submodule ℝ V) (vectorSpan ℝ (Set.range s.points))ᗮ (s.excenter signs -ᵥ s.touchpoint signs i)
have hu : Set.range s.points = Set.range (s.faceOpposite i).points ∪ Set.range (s.faceOpposite j).points
Tactic: have hu : Set.range s.points =
        Set.range (s.faceOpposite i).points ∪ Set.range (s.faceOpposite j).points := by
      simp only [range_faceOpposite_points, ← Set.image_union, ← Set.compl_inter]
      convert! Set.image_univ.symm
      simp [Ne.symm hne]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : s.touchpoint signs i = s.touchpoint signs j
hne : ¬i = j
hn1 : ¬n = 1
k : Fin (n + 1)
hki : k ≠ i
hkj : k ≠ j
hu : Set.range s.points = Set.range (s.faceOpposite i).points ∪ Set.range (s.faceOpposite j).points
⊢ Membership.mem (γ := Submodule ℝ V) (vectorSpan ℝ (Set.range s.points))ᗮ (s.excenter signs -ᵥ s.touchpoint signs i)
have neg : Membership.mem (γ := Submodule ℝ V) ((vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ ⊓ (vectorSpan ℝ (s.points '' {j}ᶜ))ᗮ)
  (s.excenter signs -ᵥ s.touchpoint signs i)
Tactic: rw [hu, range_faceOpposite_points, range_faceOpposite_points,
      AffineSubspace.vectorSpan_union_of_mem_of_mem ℝ (p := s.points k)
        (Set.mem_image_of_mem _ (by simp [hki])) (Set.mem_image_of_mem _ (by simp [hkj])),
      ← Submodule.inf_orthogonal]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : s.touchpoint signs i = s.touchpoint signs j
hne : ¬i = j
hn1 : ¬n = 1
k : Fin (n + 1)
hki : k ≠ i
hkj : k ≠ j
hu : Set.range s.points = Set.range (s.faceOpposite i).points ∪ Set.range (s.faceOpposite j).points
⊢ Membership.mem (γ := Submodule ℝ V) ((vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ ⊓ (vectorSpan ℝ (s.points '' {j}ᶜ))ᗮ)
    (s.excenter signs -ᵥ s.touchpoint signs i)
have neg : s.excenter signs -ᵥ s.touchpoint signs i ∈ (↑(vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ : Set V)
Tactic: refine ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : s.touchpoint signs i = s.touchpoint signs j
hne : ¬i = j
hn1 : ¬n = 1
k : Fin (n + 1)
hki : k ≠ i
hkj : k ≠ j
hu : Set.range s.points = Set.range (s.faceOpposite i).points ∪ Set.range (s.faceOpposite j).points
neg : s.excenter signs -ᵥ s.touchpoint signs i ∈ (↑(vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ : Set V)
⊢ Membership.mem (γ := Submodule ℝ V) ((vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ ⊓ (vectorSpan ℝ (s.points '' {j}ᶜ))ᗮ)
    (s.excenter signs -ᵥ s.touchpoint signs i)
have neg₁ : s.excenter signs -ᵥ s.touchpoint signs i ∈ (↑(vectorSpan ℝ (s.points '' {j}ᶜ))ᗮ : Set V)
Tactic: refine ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : s.touchpoint signs i = s.touchpoint signs j
hne : ¬i = j
hn1 : ¬n = 1
k : Fin (n + 1)
hki : k ≠ i
hkj : k ≠ j
hu : Set.range s.points = Set.range (s.faceOpposite i).points ∪ Set.range (s.faceOpposite j).points
⊢ s.excenter signs -ᵥ s.touchpoint signs i ∈ (↑(vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ : Set V)
have neg : s.excenter signs -ᵥ s.touchpoint signs i ∈
  (↑(AffineSubspace.direction (affineSpan ℝ (Set.range (s.faceOpposite i).points)))ᗮ : Set V)
Tactic: rw [← direction_affineSpan, ← range_faceOpposite_points]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : s.touchpoint signs i = s.touchpoint signs j
hne : ¬i = j
hn1 : ¬n = 1
k : Fin (n + 1)
hki : k ≠ i
hkj : k ≠ j
hu : Set.range s.points = Set.range (s.faceOpposite i).points ∪ Set.range (s.faceOpposite j).points
⊢ s.excenter signs -ᵥ s.touchpoint signs i ∈ (↑(vectorSpan ℝ (s.points '' {j}ᶜ))ᗮ : Set V)
have neg : s.excenter signs -ᵥ s.touchpoint signs j ∈
  (↑(AffineSubspace.direction (affineSpan ℝ (Set.range (s.faceOpposite j).points)))ᗮ : Set V)
Tactic: rw [hij, ← direction_affineSpan, ← range_faceOpposite_points]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hne : i ≠ j
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
⊢ ContinuousOn (Y := SignType)
    (fun (t : ℝ) ↦
      (SignType.sign : (a : ℝ) → SignType)
        ((s.signedInfDist j : (a : P) → ℝ)
            ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P) :
          ℝ))
    (Set.Icc 0 1)
have refine_1 : ∀ (t : ℝ) (ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t),
  Ne (α := ℝ)
    ((s.signedInfDist j : (a : P) → ℝ)
        ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P) :
      ℝ)
    (0 : ℝ)
Tactic: refine continuousOn_of_forall_continuousAt
      fun t ht ↦ ((continuousAt_sign_of_ne_zero ?_).comp
        (((s.signedInfDist j).cont.comp ?_).continuousAt))
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hne : i ≠ j
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
refine_1 :
  ∀ (t : ℝ) (ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t),
    Ne (α := ℝ)
      ((s.signedInfDist j : (a : P) → ℝ)
          ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P) :
        ℝ)
      (0 : ℝ)
⊢ ContinuousOn (Y := SignType)
    (fun (t : ℝ) ↦
      (SignType.sign : (a : ℝ) → SignType)
        ((s.signedInfDist j : (a : P) → ℝ)
            ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P) :
          ℝ))
    (Set.Icc 0 1)
have refine_2 : ∀ (t : ℝ) (ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t),
  Continuous.{0, u_2} (X := ℝ) (Y := P) (⇑(AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i)) : (a : ℝ) → P)
Tactic: refine continuousOn_of_forall_continuousAt
      fun t ht ↦ ((continuousAt_sign_of_ne_zero ?_).comp
        (((s.signedInfDist j).cont.comp ?_).continuousAt))
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hne : i ≠ j
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t
h0 :
  Eq (α := ℝ)
    ((s.signedInfDist j : (a : P) → ℝ)
        ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P) :
      ℝ)
    (0 : ℝ)
⊢ False
have refine_1 : ∀
  (h0 :
    dist.{u_2} (α := P) ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)
        ((↑(((s.faceOpposite j).orthogonalProjectionSpan :
                  (a : P) → ↥(affineSpan ℝ (Set.range (s.faceOpposite j).points)))
                ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t)) :
            P) :
          P) =
      0),
  False
Tactic: rw [← abs_eq_zero, abs_signedInfDist_eq_dist_of_mem_affineSpan_range] at h0
Cleared hypotheses: h0
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hne : i ≠ j
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t
h0 :
  Eq (α := ℝ)
    ((s.signedInfDist j : (a : P) → ℝ)
        ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P) :
      ℝ)
    (0 : ℝ)
refine_1 :
  ∀
    (h0 :
      dist.{u_2} (α := P) ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)
          ((↑(((s.faceOpposite j).orthogonalProjectionSpan :
                    (a : P) → ↥(affineSpan ℝ (Set.range (s.faceOpposite j).points)))
                  ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t)) :
              P) :
            P) =
        0),
    False
⊢ False
have refine_1₁ : ∀
  (h0 :
    Eq (α := ℝ)
      (|(s.signedInfDist j : (a : P) → ℝ)
            ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)| :
        ℝ)
      (0 : ℝ)),
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points))
    ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)
Tactic: rw [← abs_eq_zero, abs_signedInfDist_eq_dist_of_mem_affineSpan_range] at h0
Cleared hypotheses: h0
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (hypothesis)
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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hne : i ≠ j
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t
h0 :
  dist.{u_2} (α := P) ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)
      ((↑(((s.faceOpposite j).orthogonalProjectionSpan :
                (a : P) → ↥(affineSpan ℝ (Set.range (s.faceOpposite j).points)))
              ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t)) :
          P) :
        P) =
    0
⊢ False
have h0₁ : Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range (s.faceOpposite j).points))
  ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)
Tactic: rw [orthogonalProjectionSpan, dist_orthogonalProjection_eq_zero_iff] at h0
Cleared hypotheses: h0
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hne : i ≠ j
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t
h0 :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range (s.faceOpposite j).points))
    ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)
⊢ False
have pos : ∀ (ht1 : t = 1), False
Tactic: by_cases ht1 : t = 1
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hne : i ≠ j
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t
h0 :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range (s.faceOpposite j).points))
    ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)
pos : ∀ (ht1 : t = 1), False
⊢ False
have neg : ∀ (ht1 : ¬t = 1), False
Tactic: by_cases ht1 : t = 1