awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (goal)
Goal:
n : ℕ
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
i : Fin (n + 2 + 1)
hu : card (α := Fin (n + 3)) {i₁, i₂}ᶜ = n + 1
⊢ (if i = i₁ ∨ i = i₂ then (↑(n + 1) : ℝ)⁻¹ else 0) =
    HSub.hSub (α := ℝ) (↑(n + 1) : ℝ)⁻¹
      (if Membership.mem (γ := Finset (Fin (n + 2 + 1))) {i₁, i₂}ᶜ i then (↑(n + 1) : ℝ)⁻¹ else 0)
have pos : ∀ (hi : i = i₁ ∨ i = i₂),
  (if i = i₁ ∨ i = i₂ then (↑(n + 1) : ℝ)⁻¹ else 0) =
    HSub.hSub (α := ℝ) (↑(n + 1) : ℝ)⁻¹
      (if Membership.mem (γ := Finset (Fin (n + 2 + 1))) {i₁, i₂}ᶜ i then (↑(n + 1) : ℝ)⁻¹ else 0)
Tactic: by_cases hi : i = i₁ ∨ i = i₂
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (goal)
Goal:
n : ℕ
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
i : Fin (n + 2 + 1)
hu : card (α := Fin (n + 3)) {i₁, i₂}ᶜ = n + 1
pos :
  ∀ (hi : i = i₁ ∨ i = i₂),
    (if i = i₁ ∨ i = i₂ then (↑(n + 1) : ℝ)⁻¹ else 0) =
      HSub.hSub (α := ℝ) (↑(n + 1) : ℝ)⁻¹
        (if Membership.mem (γ := Finset (Fin (n + 2 + 1))) {i₁, i₂}ᶜ i then (↑(n + 1) : ℝ)⁻¹ else 0)
⊢ (if i = i₁ ∨ i = i₂ then (↑(n + 1) : ℝ)⁻¹ else 0) =
    HSub.hSub (α := ℝ) (↑(n + 1) : ℝ)⁻¹
      (if Membership.mem (γ := Finset (Fin (n + 2 + 1))) {i₁, i₂}ᶜ i then (↑(n + 1) : ℝ)⁻¹ else 0)
have neg : ∀ (hi : ¬(i = i₁ ∨ i = i₂)),
  (if i = i₁ ∨ i = i₂ then (↑(n + 1) : ℝ)⁻¹ else 0) =
    HSub.hSub (α := ℝ) (↑(n + 1) : ℝ)⁻¹
      (if Membership.mem (γ := Finset (Fin (n + 2 + 1))) {i₁, i₂}ᶜ i then (↑(n + 1) : ℝ)⁻¹ else 0)
Tactic: by_cases hi : i = i₁ ∨ i = i₂
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (goal)
Goal:
n : ℕ
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
⊢ Eq (α := ℝ) (∑ i : PointsWithCircumcenterIndex (n + 2), mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ i) 0
have rw : Eq (α := ℝ)
  (∑ i : PointsWithCircumcenterIndex (n + 2),
    HSub.hSub (β := PointsWithCircumcenterIndex (n + 2) → ℝ) (mongePointWeightsWithCircumcenter n)
      (centroidWeightsWithCircumcenter {i₁, i₂}ᶜ) i)
  0
Tactic: rw [mongePointVSubFaceCentroidWeightsWithCircumcenter_eq_sub h]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (goal)
Goal:
n : ℕ
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
⊢ Eq (α := ℝ)
    (∑ i : PointsWithCircumcenterIndex (n + 2),
      HSub.hSub (β := PointsWithCircumcenterIndex (n + 2) → ℝ) (mongePointWeightsWithCircumcenter n)
        (centroidWeightsWithCircumcenter {i₁, i₂}ᶜ) i)
    0
have simp_rw : HSub.hSub (α := ℝ) 1 (∑ x : PointsWithCircumcenterIndex (n + 2), centroidWeightsWithCircumcenter {i₁, i₂}ᶜ x) = 0
Tactic: simp_rw [Pi.sub_apply, sum_sub_distrib, sum_mongePointWeightsWithCircumcenter]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (goal)
Goal:
n : ℕ
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
⊢ HSub.hSub (α := ℝ) 1 (∑ x : PointsWithCircumcenterIndex (n + 2), centroidWeightsWithCircumcenter {i₁, i₂}ᶜ x) = 0
have rw : Finset.Nonempty (α := Fin (n + 2 + 1)) {i₁, i₂}ᶜ
Tactic: rw [sum_centroidWeightsWithCircumcenter, sub_self]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (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
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
⊢ s.mongePoint -ᵥ Finset.centroid ℝ {i₁, i₂}ᶜ s.points =
    (weightedVSub univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → V)
      (mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂)
have simp_rw : (affineCombination ℝ univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → P)
      (mongePointWeightsWithCircumcenter n) -ᵥ
    Finset.centroid ℝ {i₁, i₂}ᶜ s.points =
  (weightedVSub univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → V)
    (mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂)
Tactic: simp_rw [mongePoint_eq_affineCombination_of_pointsWithCircumcenter,
    centroid_eq_affineCombination_of_pointsWithCircumcenter, affineCombination_vsub,
    mongePointVSubFaceCentroidWeightsWithCircumcenter_eq_sub h]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (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
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
simp_rw :
  (affineCombination ℝ univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → P)
        (mongePointWeightsWithCircumcenter n) -ᵥ
      Finset.centroid ℝ {i₁, i₂}ᶜ s.points =
    (weightedVSub univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → V)
      (mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂)
⊢ s.mongePoint -ᵥ Finset.centroid ℝ {i₁, i₂}ᶜ s.points =
    (weightedVSub univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → V)
      (mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂)
have simp_rw₁ : VSub.vsub.{u_1, u_2} (P := P)
    ((affineCombination ℝ univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → P)
        (mongePointWeightsWithCircumcenter n) :
      P)
    ((affineCombination ℝ univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → P)
        (centroidWeightsWithCircumcenter {i₁, i₂}ᶜ) :
      P) =
  (weightedVSub univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → V)
    (mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂)
Tactic: simp_rw [mongePoint_eq_affineCombination_of_pointsWithCircumcenter,
    centroid_eq_affineCombination_of_pointsWithCircumcenter, affineCombination_vsub,
    mongePointVSubFaceCentroidWeightsWithCircumcenter_eq_sub h]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (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
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
simp_rw :
  (affineCombination ℝ univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → P)
        (mongePointWeightsWithCircumcenter n) -ᵥ
      Finset.centroid ℝ {i₁, i₂}ᶜ s.points =
    (weightedVSub univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → V)
      (mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂)
simp_rw₁ :
  VSub.vsub.{u_1, u_2} (P := P)
      ((affineCombination ℝ univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → P)
          (mongePointWeightsWithCircumcenter n) :
        P)
      ((affineCombination ℝ univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → P)
          (centroidWeightsWithCircumcenter {i₁, i₂}ᶜ) :
        P) =
    (weightedVSub univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → V)
      (mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂)
⊢ s.mongePoint -ᵥ Finset.centroid ℝ {i₁, i₂}ᶜ s.points =
    (weightedVSub univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → V)
      (mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂)
have simp_rw₂ : Eq (α := V)
  ((weightedVSub univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → V)
      (mongePointWeightsWithCircumcenter n - centroidWeightsWithCircumcenter {i₁, i₂}ᶜ) :
    V)
  ((weightedVSub univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → V)
      (mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂) :
    V)
Tactic: simp_rw [mongePoint_eq_affineCombination_of_pointsWithCircumcenter,
    centroid_eq_affineCombination_of_pointsWithCircumcenter, affineCombination_vsub,
    mongePointVSubFaceCentroidWeightsWithCircumcenter_eq_sub h]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (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
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
⊢ ⟪s.mongePoint -ᵥ Finset.centroid ℝ {i₁, i₂}ᶜ s.points, s.points i₁ -ᵥ s.points i₂⟫ = 0
have pos : ∀ (h : i₁ = i₂), ⟪s.mongePoint -ᵥ Finset.centroid ℝ {i₁, i₂}ᶜ s.points, s.points i₁ -ᵥ s.points i₂⟫ = 0
Tactic: by_cases h : i₁ = i₂
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (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
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
pos : ∀ (h : i₁ = i₂), ⟪s.mongePoint -ᵥ Finset.centroid ℝ {i₁, i₂}ᶜ s.points, s.points i₁ -ᵥ s.points i₂⟫ = 0
⊢ ⟪s.mongePoint -ᵥ Finset.centroid ℝ {i₁, i₂}ᶜ s.points, s.points i₁ -ᵥ s.points i₂⟫ = 0
have neg : ∀ (h : ¬i₁ = i₂), ⟪s.mongePoint -ᵥ Finset.centroid ℝ {i₁, i₂}ᶜ s.points, s.points i₁ -ᵥ s.points i₂⟫ = 0
Tactic: by_cases h : i₁ = i₂
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (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
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
⊢ ⟪s.mongePoint -ᵥ Finset.centroid ℝ {i₁, i₂}ᶜ s.points, s.points i₁ -ᵥ s.points i₂⟫ = 0
have neg : inner ℝ (E := V)
    ((weightedVSub univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → V)
        (mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂) :
      V)
    ((weightedVSub univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → V)
        (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) :
      V) =
  0
Tactic: simp_rw [mongePoint_vsub_face_centroid_eq_weightedVSub_of_pointsWithCircumcenter s h,
    point_eq_affineCombination_of_pointsWithCircumcenter, affineCombination_vsub]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (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
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
⊢ inner ℝ (E := V)
      ((weightedVSub univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → V)
          (mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂) :
        V)
      ((weightedVSub univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → V)
          (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) :
        V) =
    0
have hs : Eq (α := ℝ)
  (∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i) 0
Tactic: have hs : ∑ i, (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0 := by
    simp
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (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
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs :
  Eq (α := ℝ)
    (∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i) 0
⊢ inner ℝ (E := V)
      ((weightedVSub univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → V)
          (mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂) :
        V)
      ((weightedVSub univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → V)
          (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) :
        V) =
    0
have neg : -HAdd.hAdd (α := ℝ)
        (∑ i : Fin (n + 2 + 1),
          ∑ i₂_1 : PointsWithCircumcenterIndex (n + 2),
            mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ (pointIndex i) *
                (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i₂_1 *
              (dist (s.pointsWithCircumcenter (pointIndex i)) (s.pointsWithCircumcenter i₂_1) *
                dist (s.pointsWithCircumcenter (pointIndex i)) (s.pointsWithCircumcenter i₂_1)))
        (∑ i₂_1 : PointsWithCircumcenterIndex (n + 2),
          mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ circumcenterIndex *
              (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i₂_1 *
            (dist s.circumcenter (s.pointsWithCircumcenter i₂_1) *
              dist s.circumcenter (s.pointsWithCircumcenter i₂_1))) /
    2 =
  0
Tactic: rw [inner_weightedVSub _ (sum_mongePointVSubFaceCentroidWeightsWithCircumcenter h) _ hs,
    sum_pointsWithCircumcenter, pointsWithCircumcenter_eq_circumcenter]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (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
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs :
  Eq (α := ℝ)
    (∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i) 0
⊢ -HAdd.hAdd (α := ℝ)
          (∑ i : Fin (n + 2 + 1),
            ∑ i₂_1 : PointsWithCircumcenterIndex (n + 2),
              mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ (pointIndex i) *
                  (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i₂_1 *
                (dist (s.pointsWithCircumcenter (pointIndex i)) (s.pointsWithCircumcenter i₂_1) *
                  dist (s.pointsWithCircumcenter (pointIndex i)) (s.pointsWithCircumcenter i₂_1)))
          (∑ i₂_1 : PointsWithCircumcenterIndex (n + 2),
            mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ circumcenterIndex *
                (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i₂_1 *
              (dist s.circumcenter (s.pointsWithCircumcenter i₂_1) *
                dist s.circumcenter (s.pointsWithCircumcenter i₂_1))) /
      2 =
    0
have neg : -HAdd.hAdd (α := ℝ)
        (∑ x : Fin (n + 2 + 1),
          ∑ x_1 : PointsWithCircumcenterIndex (n + 2),
            (if x = i₁ ∨ x = i₂ then (↑(n + 1) : ℝ)⁻¹ else 0) *
                (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x_1 *
              (dist (s.points x) (s.pointsWithCircumcenter x_1) * dist (s.points x) (s.pointsWithCircumcenter x_1)))
        (∑ x : PointsWithCircumcenterIndex (n + 2),
          HDiv.hDiv (α := ℝ) (-2) (↑(n + 1) : ℝ) *
              (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x *
            (dist s.circumcenter (s.pointsWithCircumcenter x) * dist s.circumcenter (s.pointsWithCircumcenter x))) /
    2 =
  0
Tactic: simp only [mongePointVSubFaceCentroidWeightsWithCircumcenter, pointsWithCircumcenter_point]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (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
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs :
  Eq (α := ℝ)
    (∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i) 0
fs : Finset (Fin (n + 3)) := {i₁, i₂}
i : Fin (n + 3)
hi : i ∉ fs
⊢ i ≠ i₁ ∧ i ≠ i₂
have left : i ≠ i₁
Tactic: constructor
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (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
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs :
  Eq (α := ℝ)
    (∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i) 0
fs : Finset (Fin (n + 3)) := {i₁, i₂}
i : Fin (n + 3)
hi : i ∉ fs
left : i ≠ i₁
⊢ i ≠ i₁ ∧ i ≠ i₂
have right : i ≠ i₂
Tactic: constructor
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (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
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs :
  Eq (α := ℝ)
    (∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i) 0
fs : Finset (Fin (n + 3)) := {i₁, i₂}
⊢ -HAdd.hAdd (α := ℝ)
          (∑ x : Fin (n + 2 + 1),
            ∑ x_1 : PointsWithCircumcenterIndex (n + 2),
              (if x = i₁ ∨ x = i₂ then (↑(n + 1) : ℝ)⁻¹ else 0) *
                  (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x_1 *
                (dist (s.points x) (s.pointsWithCircumcenter x_1) * dist (s.points x) (s.pointsWithCircumcenter x_1)))
          (∑ x : PointsWithCircumcenterIndex (n + 2),
            HDiv.hDiv (α := ℝ) (-2) (↑(n + 1) : ℝ) *
                (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x *
              (dist s.circumcenter (s.pointsWithCircumcenter x) * dist s.circumcenter (s.pointsWithCircumcenter x))) /
      2 =
    0
have hfs : ∀ i ∉ fs, i ≠ i₁ ∧ i ≠ i₂
Tactic: have hfs : ∀ i : Fin (n + 3), i ∉ fs → i ≠ i₁ ∧ i ≠ i₂ := by
    intro i hi
    constructor <;> · intro hj; simp [fs, ← hj] at hi
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (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
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs :
  Eq (α := ℝ)
    (∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i) 0
fs : Finset (Fin (n + 3)) := {i₁, i₂}
hfs : ∀ i ∉ fs, i ≠ i₁ ∧ i ≠ i₂
⊢ -HAdd.hAdd (α := ℝ)
          (∑ x : Fin (n + 2 + 1),
            ∑ x_1 : PointsWithCircumcenterIndex (n + 2),
              (if x = i₁ ∨ x = i₂ then (↑(n + 1) : ℝ)⁻¹ else 0) *
                  (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x_1 *
                (dist (s.points x) (s.pointsWithCircumcenter x_1) * dist (s.points x) (s.pointsWithCircumcenter x_1)))
          (∑ x : PointsWithCircumcenterIndex (n + 2),
            HDiv.hDiv (α := ℝ) (-2) (↑(n + 1) : ℝ) *
                (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x *
              (dist s.circumcenter (s.pointsWithCircumcenter x) * dist s.circumcenter (s.pointsWithCircumcenter x))) /
      2 =
    0
have neg : -HAdd.hAdd (α := ℝ)
        (∑ x ∈ fs,
          ∑ x_1 : PointsWithCircumcenterIndex (n + 2),
            (if x = i₁ ∨ x = i₂ then (↑(n + 1) : ℝ)⁻¹ else 0) *
                (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x_1 *
              (dist (s.points x) (s.pointsWithCircumcenter x_1) * dist (s.points x) (s.pointsWithCircumcenter x_1)))
        (∑ x : PointsWithCircumcenterIndex (n + 2),
          HDiv.hDiv (α := ℝ) (-2) (↑(n + 1) : ℝ) *
              (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x *
            (dist s.circumcenter (s.pointsWithCircumcenter x) * dist s.circumcenter (s.pointsWithCircumcenter x))) /
    2 =
  0
Tactic: rw [← sum_subset fs.subset_univ _]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (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
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs :
  Eq (α := ℝ)
    (∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i) 0
fs : Finset (Fin (n + 3)) := {i₁, i₂}
hfs : ∀ i ∉ fs, i ≠ i₁ ∧ i ≠ i₂
neg :
  -HAdd.hAdd (α := ℝ)
          (∑ x ∈ fs,
            ∑ x_1 : PointsWithCircumcenterIndex (n + 2),
              (if x = i₁ ∨ x = i₂ then (↑(n + 1) : ℝ)⁻¹ else 0) *
                  (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x_1 *
                (dist (s.points x) (s.pointsWithCircumcenter x_1) * dist (s.points x) (s.pointsWithCircumcenter x_1)))
          (∑ x : PointsWithCircumcenterIndex (n + 2),
            HDiv.hDiv (α := ℝ) (-2) (↑(n + 1) : ℝ) *
                (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x *
              (dist s.circumcenter (s.pointsWithCircumcenter x) * dist s.circumcenter (s.pointsWithCircumcenter x))) /
      2 =
    0
⊢ -HAdd.hAdd (α := ℝ)
          (∑ x : Fin (n + 2 + 1),
            ∑ x_1 : PointsWithCircumcenterIndex (n + 2),
              (if x = i₁ ∨ x = i₂ then (↑(n + 1) : ℝ)⁻¹ else 0) *
                  (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x_1 *
                (dist (s.points x) (s.pointsWithCircumcenter x_1) * dist (s.points x) (s.pointsWithCircumcenter x_1)))
          (∑ x : PointsWithCircumcenterIndex (n + 2),
            HDiv.hDiv (α := ℝ) (-2) (↑(n + 1) : ℝ) *
                (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x *
              (dist s.circumcenter (s.pointsWithCircumcenter x) * dist s.circumcenter (s.pointsWithCircumcenter x))) /
      2 =
    0
have rw : ∀ (x : Fin (n + 3)),
  Membership.mem (γ := Finset (Fin (n + 3))) univ x →
    x ∉ fs →
      Eq (α := ℝ)
        (∑ x_1 : PointsWithCircumcenterIndex (n + 2),
          (if x = i₁ ∨ x = i₂ then (↑(n + 1) : ℝ)⁻¹ else 0) *
              (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x_1 *
            (dist (s.points x) (s.pointsWithCircumcenter x_1) * dist (s.points x) (s.pointsWithCircumcenter x_1)))
        0
Tactic: rw [← sum_subset fs.subset_univ _]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (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
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs :
  Eq (α := ℝ)
    (∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i) 0
fs : Finset (Fin (n + 3)) := {i₁, i₂}
hfs : ∀ i ∉ fs, i ≠ i₁ ∧ i ≠ i₂
⊢ -HAdd.hAdd (α := ℝ)
          (∑ x ∈ fs,
            ∑ x_1 : PointsWithCircumcenterIndex (n + 2),
              (if x = i₁ ∨ x = i₂ then (↑(n + 1) : ℝ)⁻¹ else 0) *
                  (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x_1 *
                (dist (s.points x) (s.pointsWithCircumcenter x_1) * dist (s.points x) (s.pointsWithCircumcenter x_1)))
          (∑ x : PointsWithCircumcenterIndex (n + 2),
            HDiv.hDiv (α := ℝ) (-2) (↑(n + 1) : ℝ) *
                (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x *
              (dist s.circumcenter (s.pointsWithCircumcenter x) * dist s.circumcenter (s.pointsWithCircumcenter x))) /
      2 =
    0
have neg : -(∑ x ∈ fs,
          (∑ x_1 : Fin (n + 2 + 1),
              (if x = i₁ ∨ x = i₂ then (↑(n + 1) : ℝ)⁻¹ else 0) *
                  HSub.hSub (α := ℝ) (if x_1 = i₁ then 1 else 0) (if x_1 = i₂ then 1 else 0) *
                (dist (s.points x) (s.points x_1) * dist (s.points x) (s.points x_1)) +
            (if x = i₁ ∨ x = i₂ then (↑(n + 1) : ℝ)⁻¹ else 0) * HSub.hSub (α := ℝ) 0 0 *
              (dist (s.points x) s.circumcenter * dist (s.points x) s.circumcenter)) +
        (∑ x : Fin (n + 2 + 1),
            HDiv.hDiv (α := ℝ) (-2) (↑(n + 1) : ℝ) *
                HSub.hSub (α := ℝ) (if x = i₁ then 1 else 0) (if x = i₂ then 1 else 0) *
              (dist s.circumcenter (s.points x) * dist s.circumcenter (s.points x)) +
          HDiv.hDiv (α := ℝ) (-2) (↑(n + 1) : ℝ) * HSub.hSub (α := ℝ) 0 0 *
            (dist s.circumcenter s.circumcenter * dist s.circumcenter s.circumcenter))) /
    2 =
  0
Tactic: simp_rw [sum_pointsWithCircumcenter, pointsWithCircumcenter_eq_circumcenter,
      pointsWithCircumcenter_point, Pi.sub_apply, pointWeightsWithCircumcenter]