awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
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) ∨
      Eq (α := SignType)
        ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i)⁻¹ : SignType)
        (0 : SignType)
⊢ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ
have hi₁ : ∀ (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 [sign_eq_zero_iff, inv_eq_zero, hn₂, or_false] at hi
Cleared hypotheses: hi
-- 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
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)
⊢ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ
have hi₁ : ∀ (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
Cleared hypotheses: hi
-- 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
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)
⊢ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ
have refine_1 : signs₁ = signs₂ᶜ
Tactic: right
-- 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
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)
⊢ signs₁ = signs₂ᶜ
have refine_1 : ∀ (i : Fin (n + 1)), i ∈ signs₁ ↔ i ∈ signs₂ᶜ
Tactic: ext i
-- 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
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)
⊢ i ∈ signs₁ ↔ i ∈ signs₂ᶜ
have hi₁ : 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
Cleared hypotheses: hi
-- 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
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)
⊢ i ∈ signs₁ ↔ i ∈ signs₂ᶜ
have 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 ∈ signs₂ᶜ
Tactic: split_ifs at hi
Cleared hypotheses: hi
-- 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
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 ∈ signs₂ᶜ
⊢ i ∈ signs₁ ↔ i ∈ signs₂ᶜ
have 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 ∈ signs₂ᶜ
Tactic: split_ifs at hi
Cleared hypotheses: hi
-- 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
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 ∈ signs₂ᶜ
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 ∈ signs₂ᶜ
⊢ i ∈ signs₁ ↔ i ∈ signs₂ᶜ
have 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 ∈ signs₂ᶜ
Tactic: split_ifs at hi
Cleared hypotheses: hi
-- 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
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 ∈ signs₂ᶜ
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 ∈ signs₂ᶜ
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 ∈ signs₂ᶜ
⊢ i ∈ signs₁ ↔ i ∈ signs₂ᶜ
have 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 ∈ signs₂ᶜ
Tactic: split_ifs at hi
Cleared hypotheses: hi
-- 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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ
⊢ s.excenterWeights signs₁ = s.excenterWeights signs₂
have refine_2 : ∀ (h₂ : s.ExcenterExists signs₁), s.excenterWeights signs₁ = s.excenterWeights signs₁
Tactic: rcases h with rfl | rfl
Cleared hypotheses: signs₂, h₂, h
-- 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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ
refine_2 : ∀ (h₂ : s.ExcenterExists signs₁), s.excenterWeights signs₁ = s.excenterWeights signs₁
⊢ s.excenterWeights signs₁ = s.excenterWeights signs₂
have refine_2₁ : ∀ (h₁ : s.ExcenterExists signs₂ᶜ), s.excenterWeights signs₂ᶜ = s.excenterWeights signs₂
Tactic: rcases h with rfl | rfl
Cleared hypotheses: signs₁, h₁, h
-- 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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
⊢ s.excenter signs₁ = s.excenter signs₂ ↔ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ
have rw : s.excenterWeights signs₁ = s.excenterWeights signs₂ ↔ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ
Tactic: rw [excenter_eq_affineCombination,
    h₂.affineCombination_eq_excenter_iff (s.sum_excenterWeights_eq_one_iff.2 h₁)]
-- 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
⊢ s.excenter signs = s.incenter ↔ signs = ∅ ∨ signs = Finset.univ
have rw : signs = ∅ ∨ signs = ∅ᶜ ↔ signs = ∅ ∨ signs = Finset.univ
Tactic: rw [incenter, ← excenter, h.excenter_eq_excenter_iff s.excenterExists_empty]
-- 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
inst_5 : n.AtLeastTwo
i : Fin (n + 1)
h : s.excenter {i} = s.incenter
⊢ False
have h₁ : Eq (α := Finset (Fin (n + 1))) {i} ∅ ∨ Eq (α := Finset (Fin (n + 1))) {i} Finset.univ
Tactic: rw [(s.excenterExists_singleton i).excenter_eq_incenter_iff] at h
Cleared hypotheses: h
-- 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
inst_5 : n.AtLeastTwo
i j : Fin (n + 1)
hij : s.excenter {i} = s.excenter {j}
⊢ i = j
have hij₁ : Eq (α := Finset (Fin (n + 1))) {i} {j} ∨ Eq (α := Finset (Fin (n + 1))) {i} {j}ᶜ
Tactic: rw [(s.excenterExists_singleton i).excenter_eq_excenter_iff (s.excenterExists_singleton j)] at hij
Cleared hypotheses: hij
-- 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
inst_5 : n.AtLeastTwo
i j : Fin (n + 1)
hij : Eq (α := Finset (Fin (n + 1))) {i} {j} ∨ Eq (α := Finset (Fin (n + 1))) {i} {j}ᶜ
⊢ i = j
have inl : ∀ (hij : Eq (α := Finset (Fin (n + 1))) {i} {j}), i = j
Tactic: rcases hij with hij | hij
Cleared hypotheses: hij
-- 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
inst_5 : n.AtLeastTwo
i j : Fin (n + 1)
hij : Eq (α := Finset (Fin (n + 1))) {i} {j} ∨ Eq (α := Finset (Fin (n + 1))) {i} {j}ᶜ
inl : ∀ (hij : Eq (α := Finset (Fin (n + 1))) {i} {j}), i = j
⊢ i = j
have inr : ∀ (hij : Eq (α := Finset (Fin (n + 1))) {i} {j}ᶜ), i = j
Tactic: rcases hij with hij | hij
Cleared hypotheses: hij
-- 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
inst_5 : n.AtLeastTwo
i j : Fin (n + 1)
hij : Eq (α := Finset (Fin (n + 1))) {i} {j}ᶜ
⊢ i = j
have this : 2 ≤ n
Tactic: have : 2 ≤ n := Nat.AtLeastTwo.prop
-- 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
inst_5 : n.AtLeastTwo
i j : Fin (n + 1)
hij : Eq (α := Finset (Fin (n + 1))) {i} {j}ᶜ
this : 2 ≤ n
⊢ i = j
have inr : ∀ (k : Fin (n + 1)) (hki : k ≠ i) (hkj : k ≠ j), i = j
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
inst_5 : n.AtLeastTwo
i j : Fin (n + 1)
hij : Eq (α := Finset (Fin (n + 1))) {i} {j}ᶜ
this : 2 ≤ n
k : Fin (n + 1)
hki : k ≠ i
hkj : k ≠ j
⊢ i = j
have inr : ∀
  (hij :
    ∀ (a : Fin (n + 1)),
      Membership.mem (γ := Finset (Fin (n + 1))) {i} a ↔ Membership.mem (γ := Finset (Fin (n + 1))) {j}ᶜ a)
  (this : 2 ≤ n) (k : Fin (n + 1)) (hki : k ≠ i) (hkj : k ≠ j), i = j
Tactic: rw [Finset.ext_iff] at hij
Cleared hypotheses: hij, this, k, hki, hkj
-- 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
inst_5 : n.AtLeastTwo
i j : Fin (n + 1)
hij :
  ∀ (a : Fin (n + 1)),
    Membership.mem (γ := Finset (Fin (n + 1))) {i} a ↔ Membership.mem (γ := Finset (Fin (n + 1))) {j}ᶜ a
this : 2 ≤ n
k : Fin (n + 1)
hki : k ≠ i
hkj : k ≠ j
⊢ i = j
have hij₁ : Membership.mem (γ := Finset (Fin (n + 1))) {i} k ↔ Membership.mem (γ := Finset (Fin (n + 1))) {j}ᶜ k
Tactic: replace hij := hij k
Cleared hypotheses: hij
-- 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 : Fin (n + 1)
⊢ AffineSubspace.SSameSide (affineSpan ℝ (Set.range (s.faceOpposite i).points)) (s.excenter signs) (s.points i) ↔
    0 < s.excenterWeights signs i
have rw : AffineSubspace.SSameSide (affineSpan ℝ (Set.range (s.faceOpposite i).points))
    ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) (s.excenterWeights signs))
    (s.points i) ↔
  0 < s.excenterWeights signs i
Tactic: rw [excenter_eq_affineCombination,
    s.sSameSide_affineSpan_faceOpposite_point_right_iff h.sum_excenterWeights_eq_one]
-- 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 : Fin (n + 1)
rw :
  AffineSubspace.SSameSide (affineSpan ℝ (Set.range (s.faceOpposite i).points))
      ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) (s.excenterWeights signs))
      (s.points i) ↔
    0 < s.excenterWeights signs i
⊢ AffineSubspace.SSameSide (affineSpan ℝ (Set.range (s.faceOpposite i).points)) (s.excenter signs) (s.points i) ↔
    0 < s.excenterWeights signs i
have rw₁ : 0 < s.excenterWeights signs i ↔ 0 < s.excenterWeights signs i
Tactic: rw [excenter_eq_affineCombination,
    s.sSameSide_affineSpan_faceOpposite_point_right_iff h.sum_excenterWeights_eq_one]
-- 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 : Fin (n + 1)
⊢ AffineSubspace.SSameSide (affineSpan ℝ (Set.range (s.faceOpposite i).points)) (s.points i) (s.excenter signs) ↔
    0 < s.excenterWeights signs i
have rw : AffineSubspace.SSameSide (affineSpan ℝ (Set.range (s.faceOpposite i).points)) (s.points i)
    ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) (s.excenterWeights signs)) ↔
  0 < s.excenterWeights signs i
Tactic: rw [excenter_eq_affineCombination,
    s.sSameSide_affineSpan_faceOpposite_point_left_iff h.sum_excenterWeights_eq_one]
-- 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 : Fin (n + 1)
rw :
  AffineSubspace.SSameSide (affineSpan ℝ (Set.range (s.faceOpposite i).points)) (s.points i)
      ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) (s.excenterWeights signs)) ↔
    0 < s.excenterWeights signs i
⊢ AffineSubspace.SSameSide (affineSpan ℝ (Set.range (s.faceOpposite i).points)) (s.points i) (s.excenter signs) ↔
    0 < s.excenterWeights signs i
have rw₁ : 0 < s.excenterWeights signs i ↔ 0 < s.excenterWeights signs i
Tactic: rw [excenter_eq_affineCombination,
    s.sSameSide_affineSpan_faceOpposite_point_left_iff h.sum_excenterWeights_eq_one]