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
⊢ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ
have 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)
Tactic: have hi : ∀ i, SignType.sign (s.excenterWeights signs₁ i) =
        SignType.sign (s.excenterWeights signs₂ i) := by
      simp [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
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)
⊢ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ
have hi₁ : ∀ (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
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
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)
⊢ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ
have hn₁ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₁ i) 0
Tactic: have hn₁ : ∑ i, s.excenterWeightsUnnorm signs₁ i ≠ 0 := h₁
-- 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
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
⊢ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ
have hn₂ : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i) 0
Tactic: have hn₂ : ∑ i, s.excenterWeightsUnnorm signs₂ i ≠ 0 := h₂
-- 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
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
⊢ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ
have 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
Tactic: rcases sign_eq_sign_or_eq_neg (inv_ne_zero hn₁) (inv_ne_zero hn₂) with hs | hs
-- 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
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
⊢ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ
have 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
Tactic: rcases sign_eq_sign_or_eq_neg (inv_ne_zero hn₁) (inv_ne_zero hn₂) with hs | hs
-- 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
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)
⊢ 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 [hs, mul_eq_mul_left_iff, 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 = signs₂
Tactic: left
-- 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
have refine_1 : ∀ (i : Fin (n + 1)), i ∈ signs₁ ↔ i ∈ signs₂
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 ∈ signs₂
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 ∈ signs₂
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 ∈ signs₂
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 ∈ signs₂
⊢ i ∈ signs₁ ↔ i ∈ signs₂
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 ∈ signs₂
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 ∈ 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
⊢ i ∈ signs₁ ↔ i ∈ signs₂
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 ∈ signs₂
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 ∈ 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
⊢ i ∈ signs₁ ↔ i ∈ signs₂
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 ∈ signs₂
Tactic: split_ifs 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
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)
⊢ 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) ∨
    Eq (α := SignType)
      ((SignType.sign : (a : ℝ) → SignType) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs₂ i)⁻¹ : SignType)
      (0 : SignType)
Tactic: simp_rw [hs, neg_mul, ← mul_neg, mul_eq_mul_left_iff] at hi
Cleared hypotheses: hi