awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Geometry.Euclidean.Incenter
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 : Fin (n + 1)
hf : s.excenter signs ∉ affineSpan ℝ (Set.range (ι := Fin (HAdd.hAdd (α := ℕ) 0 1)) (points (s.face ⋯)))
⊢ s.excenter signs ≠ s.points i

Tactic:
simpa using hf
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
i : Fin (n + 1)
⊢ s.incenter ≠ s.points i

Tactic:
s.excenterExists_empty.excenter_ne_point i

Premises used: Affine.Simplex.incenter, Affine.Simplex.excenterExists_empty, Affine.Simplex.ExcenterExists.excenter_ne_point
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
inst_5 : n.AtLeastTwo
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
pos : ∀ (hij : i = j), s.excenter signs ∉ affineSpan ℝ {s.points i, s.points j}
neg : ∀ (hij : ¬i = j), s.excenter signs ∉ affineSpan ℝ {s.points i, s.points j}
⊢ s.excenter signs ∉ affineSpan ℝ {s.points i, s.points j}

Tactic:
by_cases hij : i = j
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
inst_5 : n.AtLeastTwo
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : i = j
pos : ¬s.excenter signs = s.points j
⊢ s.excenter signs ∉ affineSpan ℝ {s.points i, s.points j}

Tactic:
simp only [hij, Set.mem_singleton_iff, Set.insert_eq_of_mem,
      AffineSubspace.mem_affineSpan_singleton]

Premises used: Set.insert_eq_of_mem, Set.mem_singleton_iff, AffineSubspace.mem_affineSpan_singleton
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
inst_5 : n.AtLeastTwo
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : i = j
⊢ ¬s.excenter signs = s.points j

Tactic:
h.excenter_ne_point j

Premises used: Affine.Simplex.ExcenterExists.excenter_ne_point
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
inst_5 : n.AtLeastTwo
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : i = j
⊢ ¬s.excenter signs = s.points j

Tactic:
exact h.excenter_ne_point j

Premises used: Affine.Simplex.ExcenterExists.excenter_ne_point
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
inst_5 : n.AtLeastTwo
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : ¬i = j
⊢ Finset.card (α := Fin (n + 1)) {i, j} = HAdd.hAdd (α := ℕ) 1 1

Tactic:
simp_all
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
inst_5 : n.AtLeastTwo
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : ¬i = j
h₁ : {s.points i, s.points j} = Set.range (s.face (fs := {i, j}) ⋯).points
⊢ s.excenter signs ∉ affineSpan ℝ {s.points i, s.points j}

Tactic:
convert!
    h.excenter_notMem_affineSpan_face (fs := { i, j }) (m := 1) (by simp_all)
      Nat.AtLeastTwo.ne_one.symm

Premises used: Affine.Simplex.ExcenterExists.excenter_notMem_affineSpan_face, Ne.symm, Nat.AtLeastTwo.ne_one
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
inst_5 : n.AtLeastTwo
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : ¬i = j
⊢ {s.points i, s.points j} = Set.range (s.face (fs := {i, j}) ⋯).points

Tactic:
simp [Set.image_insert_eq]

Premises used: Set.image_insert_eq
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
inst_5 : n.AtLeastTwo
i j : Fin (n + 1)
⊢ s.incenter ∉ affineSpan ℝ {s.points i, s.points j}

Tactic:
s.excenterExists_empty.excenter_notMem_affineSpan_pair i j

Premises used: Set.Mem, Affine.Simplex.incenter, Affine.Simplex.ExcenterExists.excenter_notMem_affineSpan_pair, Affine.Simplex.excenterExists_empty
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
⊢ s.excenterWeights signs₁ = s.excenterWeights signs₂ ↔ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ

Tactic:
⟨fun h ↦ ?_, fun h ↦ ?_⟩

Premises used: Iff.intro
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
refine_1 : ∀ (h : s.excenterWeights signs₁ = s.excenterWeights signs₂), signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ
refine_2 : ∀ (h : signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ), s.excenterWeights signs₁ = s.excenterWeights signs₂
⊢ s.excenterWeights signs₁ = s.excenterWeights signs₂ ↔ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ

Tactic:
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : s.excenterWeights signs₁ = s.excenterWeights signs₂
⊢ ∀ (i : Fin (n + 1)),
    Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeights signs₁ i) : SignType)
      ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeights signs₂ i) : SignType)

Tactic:
simp [h]
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : s.excenterWeights signs₁ = s.excenterWeights signs₂
⊢ ∀ (i : Fin (n + 1)),
    Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeights signs₁ i) : SignType)
      ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeights signs₂ i) : SignType)

Tactic:
have hi : ∀ i, SignType.sign (s.excenterWeights signs₁ i) =
        SignType.sign (s.excenterWeights signs₂ i) := by
      simp [h]
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : s.excenterWeights signs₁ = s.excenterWeights signs₂
hi :
  ∀ (i : Fin (n + 1)),
    Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeights signs₁ i) : SignType)
      ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeights signs₂ i) : SignType)
⊢ ∀ (i : Fin (n + 1)),
    HMul.hMul (α := SignType) (β := SignType)
        ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i)⁻¹ : SignType)
        ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeightsUnnorm signs₁ i) : SignType) =
      HMul.hMul (α := SignType) (β := SignType)
        ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i)⁻¹ : SignType)
        ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeightsUnnorm signs₂ i) : SignType)

Tactic:
simp_rw [excenterWeights, Pi.smul_apply, smul_eq_mul, sign_mul] at hi

Premises used: sign_mul, Affine.Simplex.excenterWeights, smul_eq_mul, Pi.smul_apply
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : s.excenterWeights signs₁ = s.excenterWeights signs₂
hi :
  ∀ (i : Fin (n + 1)),
    HMul.hMul (α := SignType) (β := SignType)
        ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i)⁻¹ : SignType)
        ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeightsUnnorm signs₁ i) : SignType) =
      HMul.hMul (α := SignType) (β := SignType)
        ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i)⁻¹ : SignType)
        ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeightsUnnorm signs₂ i) : SignType)
⊢ Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i) 0

Tactic:
have hn₁ : ∑ i, s.excenterWeightsUnnorm signs₁ i ≠ 0 := h₁
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : s.excenterWeights signs₁ = s.excenterWeights signs₂
hi :
  ∀ (i : Fin (n + 1)),
    HMul.hMul (α := SignType) (β := SignType)
        ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i)⁻¹ : SignType)
        ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeightsUnnorm signs₁ i) : SignType) =
      HMul.hMul (α := SignType) (β := SignType)
        ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i)⁻¹ : SignType)
        ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeightsUnnorm signs₂ i) : SignType)
hn₁ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i) 0
⊢ Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i) 0

Tactic:
have hn₂ : ∑ i, s.excenterWeightsUnnorm signs₂ i ≠ 0 := h₂
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : s.excenterWeights signs₁ = s.excenterWeights signs₂
hi :
  ∀ (i : Fin (n + 1)),
    HMul.hMul (α := SignType) (β := SignType)
        ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i)⁻¹ : SignType)
        ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeightsUnnorm signs₁ i) : SignType) =
      HMul.hMul (α := SignType) (β := SignType)
        ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i)⁻¹ : SignType)
        ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeightsUnnorm signs₂ i) : SignType)
hn₁ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i) 0
hn₂ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i) 0
refine_1 :
  ∀
    (hs :
      Eq (α := SignType)
        ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i)⁻¹ : SignType)
        ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i)⁻¹ : SignType)),
    signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ
refine_1₁ :
  ∀
    (hs :
      Eq (α := SignType)
        ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i)⁻¹ : SignType)
        (-(SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i)⁻¹ : SignType)),
    signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ
⊢ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ

Tactic:
rcases sign_eq_sign_or_eq_neg (inv_ne_zero hn₁) (inv_ne_zero hn₂) with hs | hs

Premises used: inv_ne_zero, sign_eq_sign_or_eq_neg
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : s.excenterWeights signs₁ = s.excenterWeights signs₂
hi :
  ∀ (i : Fin (n + 1)),
    HMul.hMul (α := SignType) (β := SignType)
        ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i)⁻¹ : SignType)
        ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeightsUnnorm signs₁ i) : SignType) =
      HMul.hMul (α := SignType) (β := SignType)
        ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i)⁻¹ : SignType)
        ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeightsUnnorm signs₂ i) : SignType)
hn₁ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i) 0
hn₂ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i) 0
hs :
  Eq (α := SignType)
    ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i)⁻¹ : SignType)
    ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i)⁻¹ : SignType)
⊢ ∀ (i : Fin (n + 1)),
    Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeightsUnnorm signs₁ i) : SignType)
      ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeightsUnnorm signs₂ i) : SignType)

Tactic:
simp only [hs, mul_eq_mul_left_iff, sign_eq_zero_iff, inv_eq_zero, hn₂, or_false] at hi

Premises used: inv_eq_zero, mul_eq_mul_left_iff, sign_eq_zero_iff, or_false
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : s.excenterWeights signs₁ = s.excenterWeights signs₂
hn₁ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i) 0
hn₂ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i) 0
hs :
  Eq (α := SignType)
    ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i)⁻¹ : SignType)
    ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i)⁻¹ : SignType)
hi :
  ∀ (i : Fin (n + 1)),
    Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeightsUnnorm signs₁ i) : SignType)
      ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeightsUnnorm signs₂ i) : SignType)
⊢ ∀ (i : Fin (n + 1)),
    Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (if i ∈ signs₁ then -1 else 1) : SignType)
      ((SignType.sign : (a : ℝ) → SignType) (if i ∈ signs₂ then -1 else 1) : SignType)

Tactic:
simp only [excenterWeightsUnnorm, sign_mul, inv_pos, height_pos, sign_pos, mul_one] at hi

Premises used: sign_pos, inv_pos, sign_mul, Affine.Simplex.height_pos, Affine.Simplex.excenterWeightsUnnorm, mul_one
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : s.excenterWeights signs₁ = s.excenterWeights signs₂
hn₁ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i) 0
hn₂ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i) 0
hs :
  Eq (α := SignType)
    ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i)⁻¹ : SignType)
    ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i)⁻¹ : SignType)
hi :
  ∀ (i : Fin (n + 1)),
    Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (if i ∈ signs₁ then -1 else 1) : SignType)
      ((SignType.sign : (a : ℝ) → SignType) (if i ∈ signs₂ then -1 else 1) : SignType)
refine_1 : signs₁ = signs₂
⊢ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ

Tactic:
left
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : s.excenterWeights signs₁ = s.excenterWeights signs₂
hn₁ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i) 0
hn₂ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i) 0
hs :
  Eq (α := SignType)
    ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i)⁻¹ : SignType)
    ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i)⁻¹ : SignType)
hi :
  ∀ (i : Fin (n + 1)),
    Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (if i ∈ signs₁ then -1 else 1) : SignType)
      ((SignType.sign : (a : ℝ) → SignType) (if i ∈ signs₂ then -1 else 1) : SignType)
refine_1 : ∀ (i : Fin (n + 1)), i ∈ signs₁ ↔ i ∈ signs₂
⊢ signs₁ = signs₂

Tactic:
ext i
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : s.excenterWeights signs₁ = s.excenterWeights signs₂
hn₁ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i) 0
hn₂ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i) 0
hs :
  Eq (α := SignType)
    ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i)⁻¹ : SignType)
    ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i)⁻¹ : SignType)
hi :
  ∀ (i : Fin (n + 1)),
    Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (if i ∈ signs₁ then -1 else 1) : SignType)
      ((SignType.sign : (a : ℝ) → SignType) (if i ∈ signs₂ then -1 else 1) : SignType)
i : Fin (n + 1)
⊢ Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (if i ∈ signs₁ then -1 else 1) : SignType)
    ((SignType.sign : (a : ℝ) → SignType) (if i ∈ signs₂ then -1 else 1) : SignType)

Tactic:
replace hi := hi i
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : s.excenterWeights signs₁ = s.excenterWeights signs₂
hn₁ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i) 0
hn₂ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i) 0
hs :
  Eq (α := SignType)
    ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i)⁻¹ : SignType)
    ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i)⁻¹ : SignType)
i : Fin (n + 1)
hi :
  Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (if i ∈ signs₁ then -1 else 1) : SignType)
    ((SignType.sign : (a : ℝ) → SignType) (if i ∈ signs₂ then -1 else 1) : SignType)
pos :
  i ∈ signs₁ →
    i ∈ signs₂ →
      ∀
        (hi :
          Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (-1) : SignType)
            ((SignType.sign : (a : ℝ) → SignType) (-1) : SignType)),
        i ∈ signs₁ ↔ i ∈ signs₂
neg :
  i ∈ signs₁ →
    i ∉ signs₂ →
      ∀
        (hi :
          Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (-1) : SignType)
            ((SignType.sign : (a : ℝ) → SignType) 1 : SignType)),
        i ∈ signs₁ ↔ i ∈ signs₂
pos₁ :
  i ∉ signs₁ →
    i ∈ signs₂ →
      ∀
        (hi :
          Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) 1 : SignType)
            ((SignType.sign : (a : ℝ) → SignType) (-1) : SignType)),
        i ∈ signs₁ ↔ i ∈ signs₂
neg₁ :
  i ∉ signs₁ →
    i ∉ signs₂ →
      ∀
        (hi :
          Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) 1 : SignType)
            ((SignType.sign : (a : ℝ) → SignType) 1 : SignType)),
        i ∈ signs₁ ↔ i ∈ signs₂
⊢ i ∈ signs₁ ↔ i ∈ signs₂

Tactic:
split_ifs at hi
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : s.excenterWeights signs₁ = s.excenterWeights signs₂
hn₁ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i) 0
hn₂ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i) 0
hs :
  Eq (α := SignType)
    ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i)⁻¹ : SignType)
    ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i)⁻¹ : SignType)
i : Fin (n + 1)
h_1 : i ∈ signs₁
h_2 : i ∈ signs₂
hi :
  Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (-1) : SignType)
    ((SignType.sign : (a : ℝ) → SignType) (-1) : SignType)
⊢ i ∈ signs₁ ↔ i ∈ signs₂

Tactic:
simp_all
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : s.excenterWeights signs₁ = s.excenterWeights signs₂
hn₁ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i) 0
hn₂ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i) 0
hs :
  Eq (α := SignType)
    ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i)⁻¹ : SignType)
    ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i)⁻¹ : SignType)
i : Fin (n + 1)
h_1 : i ∈ signs₁
h_2 : i ∉ signs₂
hi :
  Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (-1) : SignType)
    ((SignType.sign : (a : ℝ) → SignType) 1 : SignType)
⊢ i ∈ signs₁ ↔ i ∈ signs₂

Tactic:
simp_all