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
m n : ℕ
inst_4 : NeZero m
inst_5 : NeZero n
s : Simplex ℝ P n
e : Fin (n + 1) ≃ Fin (m + 1)
signs : Finset (Fin (m + 1))
simp_rw :
  HSMul.hSMul (α := ℝ) (∑ i : Fin (m + 1), (s.reindex e).excenterWeightsUnnorm signs i)⁻¹
      ((s.reindex e).excenterWeightsUnnorm signs) =
    HSMul.hSMul (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs) i)⁻¹
        (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs)) ∘
      (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1))
⊢ (s.reindex e).excenterWeights signs =
    s.excenterWeights (Finset.map e.symm.toEmbedding signs) ∘ (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1))
have simp_rw₁ : HSMul.hSMul (α := ℝ) (β := (i : Fin (m + 1)) → ℝ)
    (∑ x : Fin (m + 1),
        (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs) ∘ (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1)))
          x)⁻¹
    (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs) ∘ (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1))) =
  HSMul.hSMul (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs) i)⁻¹
      (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs)) ∘
    (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1))
Tactic: simp_rw [excenterWeights, excenterWeightsUnnorm_reindex, Finset.sum_comp_equiv,
    Finset.map_univ_equiv, Pi.smul_comp]
-- 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
m n : ℕ
inst_4 : NeZero m
inst_5 : NeZero n
s : Simplex ℝ P n
e : Fin (n + 1) ≃ Fin (m + 1)
signs : Finset (Fin (m + 1))
simp_rw :
  HSMul.hSMul (α := ℝ) (∑ i : Fin (m + 1), (s.reindex e).excenterWeightsUnnorm signs i)⁻¹
      ((s.reindex e).excenterWeightsUnnorm signs) =
    HSMul.hSMul (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs) i)⁻¹
        (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs)) ∘
      (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1))
simp_rw₁ :
  HSMul.hSMul (α := ℝ) (β := (i : Fin (m + 1)) → ℝ)
      (∑ x : Fin (m + 1),
          (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs) ∘ (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1)))
            x)⁻¹
      (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs) ∘ (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1))) =
    HSMul.hSMul (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs) i)⁻¹
        (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs)) ∘
      (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1))
⊢ (s.reindex e).excenterWeights signs =
    s.excenterWeights (Finset.map e.symm.toEmbedding signs) ∘ (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1))
have simp_rw₂ : HSMul.hSMul (β := (i : Fin (m + 1)) → ℝ)
    ((Finset.map e.symm.toEmbedding Finset.univ).sum (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs)))⁻¹
    (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs) ∘ (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1))) =
  HSMul.hSMul (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs) i)⁻¹
      (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs)) ∘
    (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1))
Tactic: simp_rw [excenterWeights, excenterWeightsUnnorm_reindex, Finset.sum_comp_equiv,
    Finset.map_univ_equiv, Pi.smul_comp]
-- 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
m n : ℕ
inst_4 : NeZero m
inst_5 : NeZero n
s : Simplex ℝ P n
e : Fin (n + 1) ≃ Fin (m + 1)
signs : Finset (Fin (m + 1))
simp_rw :
  HSMul.hSMul (α := ℝ) (∑ i : Fin (m + 1), (s.reindex e).excenterWeightsUnnorm signs i)⁻¹
      ((s.reindex e).excenterWeightsUnnorm signs) =
    HSMul.hSMul (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs) i)⁻¹
        (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs)) ∘
      (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1))
simp_rw₁ :
  HSMul.hSMul (α := ℝ) (β := (i : Fin (m + 1)) → ℝ)
      (∑ x : Fin (m + 1),
          (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs) ∘ (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1)))
            x)⁻¹
      (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs) ∘ (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1))) =
    HSMul.hSMul (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs) i)⁻¹
        (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs)) ∘
      (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1))
simp_rw₂ :
  HSMul.hSMul (β := (i : Fin (m + 1)) → ℝ)
      ((Finset.map e.symm.toEmbedding Finset.univ).sum
          (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs)))⁻¹
      (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs) ∘ (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1))) =
    HSMul.hSMul (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs) i)⁻¹
        (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs)) ∘
      (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1))
⊢ (s.reindex e).excenterWeights signs =
    s.excenterWeights (Finset.map e.symm.toEmbedding signs) ∘ (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1))
have simp_rw₃ : HSMul.hSMul (α := ℝ) (β := (i : Fin (m + 1)) → ℝ)
    (∑ x : Fin (n + 1), s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs) x)⁻¹
    (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs) ∘ (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1))) =
  HSMul.hSMul (α := ℝ) (∑ x : Fin (n + 1), s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs) x)⁻¹
      (s.excenterWeightsUnnorm (Finset.map e.symm.toEmbedding signs)) ∘
    (⇑e.symm : (a : Fin (m + 1)) → Fin (n + 1))
Tactic: simp_rw [excenterWeights, excenterWeightsUnnorm_reindex, Finset.sum_comp_equiv,
    Finset.map_univ_equiv, Pi.smul_comp]
-- 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
V₂ : Type u_3
P₂ : Type u_4
inst_4 : NormedAddCommGroup V₂
inst_5 : InnerProductSpace ℝ V₂
inst_6 : MetricSpace P₂
inst_7 : NormedAddTorsor V₂ P₂
n : ℕ
inst_8 : NeZero n
s : Simplex ℝ P n
f : P →ᵃⁱ[ℝ] P₂
⊢ (s.map f.toAffineMap ⋯).excenterWeights = s.excenterWeights
have h : ∀ (x : Finset (Fin (n + 1))) (x_1 : Fin (n + 1)),
  (s.map f.toAffineMap sorry).excenterWeights x x_1 = s.excenterWeights x x_1
Tactic: ext
-- 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
S : AffineSubspace ℝ P
hS : affineSpan ℝ (Set.range s.points) ≤ S
⊢ excenterWeights.{u_1, u_2} (P := ↥S) (s.restrict S hS) = s.excenterWeights
have h : ∀ (x : Finset (Fin (n + 1))) (x_1 : Fin (n + 1)),
  excenterWeights.{u_1, u_2} (P := ↥S) (s.restrict S hS) x x_1 = s.excenterWeights x x_1
Tactic: ext
-- 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)
⊢ HSMul.hSMul (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i)⁻¹ (s.excenterWeightsUnnorm signs) i ≠ 0
have refine : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i)⁻¹ 0
Tactic: refine mul_ne_zero ?_ (s.excenterWeightsUnnorm_ne_zero _ _)
-- 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)
⊢ Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i)⁻¹ 0
have rw : ∀ (h : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i) 0) (i : Fin (n + 1)),
  Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i)⁻¹ 0
Tactic: rw [ExcenterExists] at h
Cleared hypotheses: h, i
-- 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))
⊢ s.excenterWeightsUnnorm signsᶜ = -s.excenterWeightsUnnorm signs
have h : ∀ (i : Fin (n + 1)), s.excenterWeightsUnnorm signsᶜ i = (-s.excenterWeightsUnnorm signs) i
Tactic: ext i
-- 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))
i : Fin (n + 1)
⊢ s.excenterWeightsUnnorm signsᶜ i = (-s.excenterWeightsUnnorm signs) i
have pos : ∀ (h : i ∈ signs), s.excenterWeightsUnnorm signsᶜ i = (-s.excenterWeightsUnnorm signs) i
Tactic: by_cases h : i ∈ signs
-- 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))
i : Fin (n + 1)
pos : ∀ (h : i ∈ signs), s.excenterWeightsUnnorm signsᶜ i = (-s.excenterWeightsUnnorm signs) i
⊢ s.excenterWeightsUnnorm signsᶜ i = (-s.excenterWeightsUnnorm signs) i
have neg : ∀ (h : i ∉ signs), s.excenterWeightsUnnorm signsᶜ i = (-s.excenterWeightsUnnorm signs) i
Tactic: by_cases h : i ∈ signs
-- 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))
inst_5 : Decidable (s.ExcenterExists signs)
⊢ Eq (α := ℝ)
    (∑ x : Fin (n + 1),
      HSMul.hSMul (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i)⁻¹ (s.excenterWeightsUnnorm signs) x)
    (if Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i) 0 then 1 else 0)
have pos : ∀ (h : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i) 0),
  Eq (α := ℝ)
    (∑ x : Fin (n + 1),
      HSMul.hSMul (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i)⁻¹ (s.excenterWeightsUnnorm signs) x)
    1
Tactic: split_ifs with 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
signs : Finset (Fin (n + 1))
inst_5 : Decidable (s.ExcenterExists signs)
pos :
  ∀ (h : Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i) 0),
    Eq (α := ℝ)
      (∑ x : Fin (n + 1),
        HSMul.hSMul (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i)⁻¹ (s.excenterWeightsUnnorm signs) x)
      1
⊢ Eq (α := ℝ)
    (∑ x : Fin (n + 1),
      HSMul.hSMul (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i)⁻¹ (s.excenterWeightsUnnorm signs) x)
    (if Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i) 0 then 1 else 0)
have neg : ∀ (h : ¬Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i) 0),
  Eq (α := ℝ)
    (∑ x : Fin (n + 1),
      HSMul.hSMul (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i)⁻¹ (s.excenterWeightsUnnorm signs) x)
    0
Tactic: split_ifs with 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
signs : Finset (Fin (n + 1))
inst_5 : Decidable (s.ExcenterExists signs)
h : ¬Ne (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i) 0
⊢ Eq (α := ℝ)
    (∑ x : Fin (n + 1),
      HSMul.hSMul (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i)⁻¹ (s.excenterWeightsUnnorm signs) x)
    0
have h₁ : Eq (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i) 0
Tactic: simp only [ne_eq, not_not] at h
Cleared hypotheses: 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
⊢ LT.lt (α := ℝ) 0 (∑ i : Fin (n + 1), s.excenterWeightsUnnorm ∅ i)
have simp_rw : LT.lt (α := ℝ) 0 (∑ x : Fin (n + 1), (s.height x)⁻¹)
Tactic: simp_rw [excenterWeightsUnnorm_empty_apply]
-- 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
i : Fin (n + 1)
⊢ 0 < s.excenterWeights ∅ i
have simp : 0 < (∑ x : Fin (n + 1), (s.height x)⁻¹)⁻¹ * (s.height i)⁻¹
Tactic: simp only [excenterWeights, excenterWeightsUnnorm_empty_apply, Pi.smul_apply, smul_eq_mul]
-- 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
i : Fin (n + 1)
⊢ Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeights ∅ i) : SignType) (1 : SignType)
have rw : 0 < s.excenterWeights ∅ i
Tactic: rw [sign_eq_one_iff]
-- 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
this :
  ∀ (i : Fin (n + 1)),
    i ≠ 0 →
      Eq (α := ℝ)
        (∑ j : Fin (n + 1),
          ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height j)⁻¹ 2 • (s.points j -ᵥ s.altitudeFoot j)⟫)
        0
⊢ Eq.{u_1 + 1} (α := V) (∑ i : Fin (n + 1), HPow.hPow (β := ℕ) (s.height i)⁻¹ 2 • (s.points i -ᵥ s.altitudeFoot i)) 0
have rw : Membership.mem (γ := Submodule ℝ V) (vectorSpan ℝ (Set.range s.points) ⊓ (vectorSpan ℝ (Set.range s.points))ᗮ)
  (∑ i : Fin (n + 1), HPow.hPow (β := ℕ) (s.height i)⁻¹ 2 • (s.points i -ᵥ s.altitudeFoot 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 (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
this :
  ∀ (i : Fin (n + 1)),
    i ≠ 0 →
      Eq (α := ℝ)
        (∑ j : Fin (n + 1),
          ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height j)⁻¹ 2 • (s.points j -ᵥ s.altitudeFoot j)⟫)
        0
⊢ Membership.mem (γ := Submodule ℝ V) (vectorSpan ℝ (Set.range s.points) ⊓ (vectorSpan ℝ (Set.range s.points))ᗮ)
    (∑ i : Fin (n + 1), HPow.hPow (β := ℕ) (s.height i)⁻¹ 2 • (s.points i -ᵥ s.altitudeFoot i))
have refine : ∑ i : Fin (n + 1), HPow.hPow (β := ℕ) (s.height i)⁻¹ 2 • (s.points i -ᵥ s.altitudeFoot i) ∈
  (↑(vectorSpan ℝ (Set.range s.points))ᗮ : Set V)
Tactic: refine ⟨Submodule.sum_smul_mem _ _ fun i hi ↦
              vsub_mem_vectorSpan_of_mem_affineSpan_of_mem_affineSpan
                (mem_affineSpan _ (Set.mem_range_self _))
                (altitudeFoot_mem_affineSpan  _ _),
            ?_⟩
-- 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
this :
  ∀ (i : Fin (n + 1)),
    i ≠ 0 →
      Eq (α := ℝ)
        (∑ j : Fin (n + 1),
          ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height j)⁻¹ 2 • (s.points j -ᵥ s.altitudeFoot j)⟫)
        0
⊢ ∑ i : Fin (n + 1), HPow.hPow (β := ℕ) (s.height i)⁻¹ 2 • (s.points i -ᵥ s.altitudeFoot i) ∈
    (↑(vectorSpan ℝ (Set.range s.points))ᗮ : Set V)
have rw : ∀ (i : { x : Fin (n + 1) // x ≠ 0 }),
  ∑ i : Fin (n + 1), HPow.hPow (β := ℕ) (s.height i)⁻¹ 2 • (s.points i -ᵥ s.altitudeFoot i) ∈
    (↑(ℝ ∙ (s.points (↑i : Fin (n + 1)) -ᵥ s.points 0))ᗮ : Set V)
Tactic: rw [vectorSpan_range_eq_span_range_vsub_right_ne _ _ 0, Submodule.span_range_eq_iSup,
      ← Submodule.iInf_orthogonal, Submodule.coe_iInf, Set.mem_iInter]
-- 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
this :
  ∀ (i : Fin (n + 1)),
    i ≠ 0 →
      Eq (α := ℝ)
        (∑ j : Fin (n + 1),
          ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height j)⁻¹ 2 • (s.points j -ᵥ s.altitudeFoot j)⟫)
        0
i : { x : Fin (n + 1) // x ≠ 0 }
⊢ ∑ i : Fin (n + 1), HPow.hPow (β := ℕ) (s.height i)⁻¹ 2 • (s.points i -ᵥ s.altitudeFoot i) ∈
    (↑(ℝ ∙ (s.points (↑i : Fin (n + 1)) -ᵥ s.points 0))ᗮ : Set V)
have rcases : ∀ (i : Fin (n + 1)) (hi : i ≠ 0),
  ∑ i : Fin (n + 1), HPow.hPow (β := ℕ) (s.height i)⁻¹ 2 • (s.points i -ᵥ s.altitudeFoot i) ∈
    (↑(ℝ ∙ (s.points (↑(Subtype.mk (p := fun (x : Fin (n + 1)) ↦ x ≠ 0) i hi) : Fin (n + 1)) -ᵥ s.points 0))ᗮ : Set V)
Tactic: rcases i with ⟨i, hi⟩
Cleared hypotheses: i
-- 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
⊢ Eq.{u_1 + 1} (α := V) (∑ i : Fin (n + 1), HPow.hPow (β := ℕ) (s.height i)⁻¹ 2 • (s.points i -ᵥ s.altitudeFoot i)) 0
have suffices : ∀ (i : Fin (n + 1)),
  i ≠ 0 →
    Eq (α := ℝ)
      (∑ j : Fin (n + 1),
        ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height j)⁻¹ 2 • (s.points j -ᵥ s.altitudeFoot j)⟫)
      0
Tactic: suffices ∀ i, i ≠ 0 →
      ∑ j, ⟪s.points i -ᵥ s.points 0, (s.height j)⁻¹ ^ 2 • (s.points j -ᵥ s.altitudeFoot j)⟫ = 0 by
    rw [← Submodule.mem_bot ℝ,
      ← Submodule.inf_orthogonal_eq_bot (vectorSpan ℝ (Set.range s.points))]
    refine ⟨Submodule.sum_smul_mem _ _ fun i hi ↦
              vsub_mem_vectorSpan_of_mem_affineSpan_of_mem_affineSpan
                (mem_affineSpan _ (Set.mem_range_self _))
                (altitudeFoot_mem_affineSpan  _ _),
            ?_⟩
    rw [vectorSpan_range_eq_span_range_vsub_right_ne _ _ 0, Submodule.span_range_eq_iSup,
      ← Submodule.iInf_orthogonal, Submodule.coe_iInf, Set.mem_iInter]
    intro i
    rcases i with ⟨i, hi⟩
    simpa only [SetLike.mem_coe, Submodule.mem_orthogonal_singleton_iff_inner_right, inner_sum]
      using this i 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
i : Fin (n + 1)
hi : i ≠ 0
⊢ Eq (α := ℝ)
    (∑ j : Fin (n + 1),
      ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height j)⁻¹ 2 • (s.points j -ᵥ s.altitudeFoot j)⟫)
    0
have rw : ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height 0)⁻¹ 2 • (s.points 0 -ᵥ s.altitudeFoot 0)⟫ +
      ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height i)⁻¹ 2 • (s.points i -ᵥ s.altitudeFoot i)⟫ +
    ∑ x ∈ Finset.erase (Finset.erase Finset.univ 0) i,
      ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height x)⁻¹ 2 • (s.points x -ᵥ s.altitudeFoot x)⟫ =
  0
Tactic: rw [← Finset.add_sum_erase _ _ (Finset.mem_univ 0),
    ← Finset.add_sum_erase _ _ (Finset.mem_erase.2 ⟨hi, Finset.mem_univ _⟩), ← add_assoc]
-- 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
i : Fin (n + 1)
hi : i ≠ 0
⊢ ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height 0)⁻¹ 2 • (s.points 0 -ᵥ s.altitudeFoot 0)⟫ +
        ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height i)⁻¹ 2 • (s.points i -ᵥ s.altitudeFoot i)⟫ +
      ∑ x ∈ Finset.erase (Finset.erase Finset.univ 0) i,
        ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height x)⁻¹ 2 • (s.points x -ᵥ s.altitudeFoot x)⟫ =
    0
have h : Eq (α := ℝ)
  (∑ x ∈ Finset.erase (Finset.erase Finset.univ 0) i,
    ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height x)⁻¹ 2 • (s.points x -ᵥ s.altitudeFoot x)⟫)
  0
Tactic: convert! add_zero _
-- 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
i : Fin (n + 1)
hi : i ≠ 0
h :
  Eq (α := ℝ)
    (∑ x ∈ Finset.erase (Finset.erase Finset.univ 0) i,
      ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height x)⁻¹ 2 • (s.points x -ᵥ s.altitudeFoot x)⟫)
    0
⊢ ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height 0)⁻¹ 2 • (s.points 0 -ᵥ s.altitudeFoot 0)⟫ +
        ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height i)⁻¹ 2 • (s.points i -ᵥ s.altitudeFoot i)⟫ +
      ∑ x ∈ Finset.erase (Finset.erase Finset.univ 0) i,
        ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height x)⁻¹ 2 • (s.points x -ᵥ s.altitudeFoot x)⟫ =
    0
have h₁ : 0 =
  ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height 0)⁻¹ 2 • (s.points 0 -ᵥ s.altitudeFoot 0)⟫ +
    ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height i)⁻¹ 2 • (s.points i -ᵥ s.altitudeFoot i)⟫
Tactic: convert! add_zero _
-- 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
i : Fin (n + 1)
hi : i ≠ 0
⊢ Eq (α := ℝ)
    (∑ x ∈ Finset.erase (Finset.erase Finset.univ 0) i,
      ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height x)⁻¹ 2 • (s.points x -ᵥ s.altitudeFoot x)⟫)
    0
have h : ∀ (j : Fin (n + 1)) (hj : j ∈ Finset.erase (Finset.erase Finset.univ 0) i),
  ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height j)⁻¹ 2 • (s.points j -ᵥ s.altitudeFoot j)⟫ = 0
Tactic: convert! Finset.sum_const_zero with j hj
-- 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
i : Fin (n + 1)
hi : i ≠ 0
j : Fin (n + 1)
hj : j ∈ Finset.erase (Finset.erase Finset.univ 0) i
⊢ ⟪s.points i -ᵥ s.points 0, HPow.hPow (β := ℕ) (s.height j)⁻¹ 2 • (s.points j -ᵥ s.altitudeFoot j)⟫ = 0
have h : HPow.hPow (β := ℕ) (s.height j)⁻¹ 2 * ⟪s.points i -ᵥ s.points 0, s.points j -ᵥ s.altitudeFoot j⟫ = 0
Tactic: rw [real_inner_smul_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
i : Fin (n + 1)
hi : i ≠ 0
j : Fin (n + 1)
hj : j ∈ Finset.erase (Finset.erase Finset.univ 0) i
⊢ HPow.hPow (β := ℕ) (s.height j)⁻¹ 2 * ⟪s.points i -ᵥ s.points 0, s.points j -ᵥ s.altitudeFoot j⟫ = 0
have h : ⟪s.points i -ᵥ s.points 0, s.points j -ᵥ s.altitudeFoot j⟫ = 0
Tactic: convert! mul_zero _