awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
e : Fin (n + 1) ≃ Fin (n + 1)
⊢ HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) •
        ((affineCombination ℝ univ (s.points ∘ (⇑e.symm : (a : Fin (n + 1)) → Fin (n + 1))) : (a : Fin (n + 1) → ℝ) → P)
            (centroidWeights ℝ univ) -ᵥ
          s.circumcenter) +ᵥ
      s.circumcenter =
    HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) •
        ((affineCombination ℝ univ s.points : (a : Fin (n + 1) → ℝ) → P) (centroidWeights ℝ univ) -ᵥ s.circumcenter) +ᵥ
      s.circumcenter
have e_a : Eq.{u_2 + 1} (α := P)
  ((affineCombination ℝ univ (s.points ∘ (⇑e.symm : (a : Fin (n + 1)) → Fin (n + 1))) : (a : Fin (n + 1) → ℝ) → P)
      (centroidWeights ℝ univ) :
    P)
  ((affineCombination ℝ univ s.points : (a : Fin (n + 1) → ℝ) → P) (centroidWeights ℝ univ) : P)
Tactic: congr 3
-- 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
e : Fin (n + 1) ≃ Fin (n + 1)
⊢ Eq.{u_2 + 1} (α := P)
    ((affineCombination ℝ univ (s.points ∘ (⇑e.symm : (a : Fin (n + 1)) → Fin (n + 1))) : (a : Fin (n + 1) → ℝ) → P)
        (centroidWeights ℝ univ) :
      P)
    ((affineCombination ℝ univ s.points : (a : Fin (n + 1) → ℝ) → P) (centroidWeights ℝ univ) : P)
have h : univ = Finset.map e.toEmbedding univ
Tactic: convert! Finset.univ.affineCombination_map e.toEmbedding _ _
-- 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
e : Fin (n + 1) ≃ Fin (n + 1)
h : univ = Finset.map e.toEmbedding univ
⊢ Eq.{u_2 + 1} (α := P)
    ((affineCombination ℝ univ (s.points ∘ (⇑e.symm : (a : Fin (n + 1)) → Fin (n + 1))) : (a : Fin (n + 1) → ℝ) → P)
        (centroidWeights ℝ univ) :
      P)
    ((affineCombination ℝ univ s.points : (a : Fin (n + 1) → ℝ) → P) (centroidWeights ℝ univ) : P)
have h₁ : s.points =
  Function.comp (β := Fin (n + 1)) (s.points ∘ (⇑e.symm : (a : Fin (n + 1)) → Fin (n + 1)))
    (⇑e.toEmbedding : (a : Fin (n + 1)) → Fin (n + 1))
Tactic: convert! Finset.univ.affineCombination_map e.toEmbedding _ _
-- 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
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
s : Simplex ℝ P n
f : P →ᵃⁱ[ℝ] P₂
⊢ (s.map f.toAffineMap ⋯).mongePoint = (f : (a : P) → P₂) s.mongePoint
have simp_rw : HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) •
      (Finset.centroid ℝ univ (s.map f.toAffineMap sorry).points -ᵥ (s.map f.toAffineMap sorry).circumcenter) +ᵥ
    (s.map f.toAffineMap sorry).circumcenter =
  (f : (a : P) → P₂)
    (HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) • (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) +ᵥ
      s.circumcenter)
Tactic: simp_rw [mongePoint_eq_smul_vsub_vadd_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
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
s : Simplex ℝ P n
f : P →ᵃⁱ[ℝ] P₂
⊢ HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) •
        (Finset.centroid ℝ univ (s.map f.toAffineMap ⋯).points -ᵥ (s.map f.toAffineMap ⋯).circumcenter) +ᵥ
      (s.map f.toAffineMap ⋯).circumcenter =
    (f : (a : P) → P₂)
      (HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) • (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) +ᵥ
        s.circumcenter)
have rw : HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) •
      ((s.map f.toAffineMap sorry).centroid -ᵥ (s.map f.toAffineMap sorry).circumcenter) +ᵥ
    (s.map f.toAffineMap sorry).circumcenter =
  (f : (a : P) → P₂)
    (HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) • (s.centroid -ᵥ s.circumcenter) +ᵥ s.circumcenter)
Tactic: rw [← Simplex.centroid, ← Simplex.centroid]
-- 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)
⊢ (n + 1) • (s.mongePoint -ᵥ s.circumcenter) = ∑ i : Fin (n + 2 + 1), (s.points i -ᵥ s.circumcenter)
have rw : ((n + 1) • HDiv.hDiv (α := ℝ) (↑(n + 2 + 1) : ℝ) (↑(n + 2 - 1) : ℝ)) •
    (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) =
  ∑ i : Fin (n + 2 + 1), (s.points i -ᵥ s.circumcenter)
Tactic: rw [mongePoint_eq_smul_vsub_vadd_circumcenter, vadd_vsub, ← smul_assoc]
-- 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)
⊢ ((n + 1) • HDiv.hDiv (α := ℝ) (↑(n + 2 + 1) : ℝ) (↑(n + 2 - 1) : ℝ)) •
      (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) =
    ∑ i : Fin (n + 2 + 1), (s.points i -ᵥ s.circumcenter)
have simp : (HAdd.hAdd (α := ℝ) (↑n : ℝ) 1 * ((HAdd.hAdd (α := ℝ) (↑n : ℝ) 2 + 1) / HAdd.hAdd (α := ℝ) (↑n : ℝ) 1)) •
    (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) =
  ∑ i : Fin (n + 2 + 1), (s.points i -ᵥ s.circumcenter)
Tactic: simp only [Nat.cast_add, Nat.cast_ofNat, Nat.cast_one, Nat.add_one_sub_one, nsmul_eq_mul]
-- 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)
⊢ (HAdd.hAdd (α := ℝ) (↑n : ℝ) 1 * ((HAdd.hAdd (α := ℝ) (↑n : ℝ) 2 + 1) / HAdd.hAdd (α := ℝ) (↑n : ℝ) 1)) •
      (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) =
    ∑ i : Fin (n + 2 + 1), (s.points i -ᵥ s.circumcenter)
have field_simp : (HAdd.hAdd (α := ℝ) (↑n : ℝ) 2 + 1) • (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) =
  ∑ i : Fin (n + 2 + 1), (s.points i -ᵥ s.circumcenter)
Tactic: field_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)
⊢ Invertible (HAdd.hAdd (α := ℝ) (↑n : ℝ) 2 + 1)
have norm_cast : Invertible (α := ℝ) (↑(n + 2 + 1) : ℝ)
Tactic: norm_cast
-- 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)
⊢ (HAdd.hAdd (α := ℝ) (↑n : ℝ) 2 + 1) • (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) =
    ∑ i : Fin (n + 2 + 1), (s.points i -ᵥ s.circumcenter)
have have : ∀ (h : Invertible (HAdd.hAdd (α := ℝ) (↑n : ℝ) 2 + 1)),
  (HAdd.hAdd (α := ℝ) (↑n : ℝ) 2 + 1) • (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) =
    ∑ i : Fin (n + 2 + 1), (s.points i -ᵥ s.circumcenter)
Tactic: have h : Invertible (n + 2 + 1 : ℝ) := by norm_cast; apply invertibleOfPos
-- 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)
h : Invertible (HAdd.hAdd (α := ℝ) (↑n : ℝ) 2 + 1)
⊢ (HAdd.hAdd (α := ℝ) (↑n : ℝ) 2 + 1) • (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) =
    ∑ i : Fin (n + 2 + 1), (s.points i -ᵥ s.circumcenter)
have rw : Finset.centroid ℝ univ s.points -ᵥ s.circumcenter =
  ∑ x : Fin (n + 2 + 1),
    @Invertible.invOf _ MulOne.toMul MulOne.toOne (HAdd.hAdd (α := ℝ) (↑n : ℝ) 2 + 1) _ • (s.points x -ᵥ s.circumcenter)
Tactic: rw [smul_eq_iff_eq_invOf_smul, smul_sum]
-- 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)
h : Invertible (HAdd.hAdd (α := ℝ) (↑n : ℝ) 2 + 1)
⊢ Finset.centroid ℝ univ s.points -ᵥ s.circumcenter =
    ∑ x : Fin (n + 2 + 1),
      @Invertible.invOf _ MulOne.toMul MulOne.toOne (HAdd.hAdd (α := ℝ) (↑n : ℝ) 2 + 1) _ •
        (s.points x -ᵥ s.circumcenter)
have rw : (affineCombination ℝ univ s.points : (a : Fin (n + 2 + 1) → ℝ) → P) (centroidWeights ℝ univ) -ᵥ s.circumcenter =
  ∑ x : Fin (n + 2 + 1),
    @Invertible.invOf _ MulOne.toMul MulOne.toOne (HAdd.hAdd (α := ℝ) (↑n : ℝ) 2 + 1) _ • (s.points x -ᵥ s.circumcenter)
Tactic: rw [univ_centroid_eq, centroid_eq_affineCombination]
-- 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)
h : Invertible (HAdd.hAdd (α := ℝ) (↑n : ℝ) 2 + 1)
⊢ (affineCombination ℝ univ s.points : (a : Fin (n + 2 + 1) → ℝ) → P) (centroidWeights ℝ univ) -ᵥ s.circumcenter =
    ∑ x : Fin (n + 2 + 1),
      @Invertible.invOf _ MulOne.toMul MulOne.toOne (HAdd.hAdd (α := ℝ) (↑n : ℝ) 2 + 1) _ •
        (s.points x -ᵥ s.circumcenter)
have rw : Eq.{u_1 + 1} (α := V) (∑ i : Fin (n + 2 + 1), centroidWeights ℝ univ i • (s.points i -ᵥ s.circumcenter))
  (∑ x : Fin (n + 2 + 1),
    @Invertible.invOf _ MulOne.toMul MulOne.toOne (HAdd.hAdd (α := ℝ) (↑n : ℝ) 2 + 1) _ •
      (s.points x -ᵥ s.circumcenter))
Tactic: rw [← Finset.sum_smul_vsub_const_eq_affineCombination_vsub _ _ _ _ (by simp)]
-- 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
S : AffineSubspace ℝ P
hS : affineSpan ℝ (Set.range s.points) ≤ S
⊢ (↑((s.restrict S hS).mongePoint (n := n)) : P) = s.mongePoint
have this : Nonempty.{u_2 + 1} ↥S
Tactic: haveI := Nonempty.map (AffineSubspace.inclusion hS) inferInstance
-- 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
S : AffineSubspace ℝ P
hS : affineSpan ℝ (Set.range s.points) ≤ S
this : Nonempty.{u_2 + 1} ↥S
⊢ (↑((s.restrict S hS).mongePoint (n := n)) : P) = s.mongePoint
have simp_rw : (↑(HVAdd.hVAdd (β := ↥S)
        (HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) •
          (Finset.centroid ℝ univ (s.restrict S hS).points -ᵥ (s.restrict S hS).circumcenter (n := n)))
        ((s.restrict S hS).circumcenter (n := n))) :
    P) =
  HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) • (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) +ᵥ
    s.circumcenter
Tactic: simp_rw [mongePoint]
-- 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
S : AffineSubspace ℝ P
hS : affineSpan ℝ (Set.range s.points) ≤ S
this : Nonempty.{u_2 + 1} ↥S
⊢ (↑(HVAdd.hVAdd (β := ↥S)
          (HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) •
            (Finset.centroid ℝ univ (s.restrict S hS).points -ᵥ (s.restrict S hS).circumcenter (n := n)))
          ((s.restrict S hS).circumcenter (n := n))) :
      P) =
    HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) • (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) +ᵥ
      s.circumcenter
have rw : (↑(HVAdd.hVAdd (β := ↥S)
        (HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) •
          ((s.restrict S hS).centroid -ᵥ (s.restrict S hS).circumcenter (n := n)))
        ((s.restrict S hS).circumcenter (n := n))) :
    P) =
  HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) • (s.centroid -ᵥ s.circumcenter) +ᵥ s.circumcenter
Tactic: rw [← Simplex.centroid, ← Simplex.centroid]
-- 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
s₁ s₂ : Simplex ℝ P n
h : Set.range s₁.points = Set.range s₂.points
⊢ s₁.mongePoint = s₂.mongePoint
have simp_rw : HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) • (Finset.centroid ℝ univ s₁.points -ᵥ s₁.circumcenter) +ᵥ
    s₁.circumcenter =
  HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) • (Finset.centroid ℝ univ s₂.points -ᵥ s₂.circumcenter) +ᵥ
    s₂.circumcenter
Tactic: simp_rw [mongePoint_eq_smul_vsub_vadd_circumcenter, centroid_eq_of_range_eq h,
    circumcenter_eq_of_range_eq 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
s₁ s₂ : Simplex ℝ P n
h : Set.range s₁.points = Set.range s₂.points
simp_rw :
  HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) • (Finset.centroid ℝ univ s₁.points -ᵥ s₁.circumcenter) +ᵥ
      s₁.circumcenter =
    HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) • (Finset.centroid ℝ univ s₂.points -ᵥ s₂.circumcenter) +ᵥ
      s₂.circumcenter
⊢ s₁.mongePoint = s₂.mongePoint
have simp_rw₁ : HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) • (Finset.centroid ℝ univ s₂.points -ᵥ s₁.circumcenter) +ᵥ
    s₁.circumcenter =
  HDiv.hDiv (α := ℝ) (↑(n + 1) : ℝ) (↑(n - 1) : ℝ) • (Finset.centroid ℝ univ s₂.points -ᵥ s₂.circumcenter) +ᵥ
    s₂.circumcenter
Tactic: simp_rw [mongePoint_eq_smul_vsub_vadd_circumcenter, centroid_eq_of_range_eq h,
    circumcenter_eq_of_range_eq h]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (goal)
Goal:
n : ℕ
⊢ Eq (α := ℝ) (∑ i : PointsWithCircumcenterIndex (n + 2), mongePointWeightsWithCircumcenter n i) 1
have simp_rw : HMul.hMul (α := ℝ) (↑(n + 2 + 1) : ℝ) (↑(n + 1) : ℝ)⁻¹ + HDiv.hDiv (α := ℝ) (-2) (↑(n + 1) : ℝ) = 1
Tactic: simp_rw [sum_pointsWithCircumcenter, mongePointWeightsWithCircumcenter, sum_const, card_fin,
    nsmul_eq_mul]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (goal)
Goal:
n : ℕ
⊢ HMul.hMul (α := ℝ) (↑(n + 2 + 1) : ℝ) (↑(n + 1) : ℝ)⁻¹ + HDiv.hDiv (α := ℝ) (-2) (↑(n + 1) : ℝ) = 1
have simp : HAdd.hAdd (α := ℝ) (↑n : ℝ) 2 + 1 + -2 = HAdd.hAdd (α := ℝ) (↑n : ℝ) 1
Tactic: simp [field]
-- 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)
⊢ s.mongePoint =
    (affineCombination ℝ univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → P)
      (mongePointWeightsWithCircumcenter n)
have rw : Eq.{u_2 + 1} (α := P)
  ((affineCombination ℝ univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → P)
      (HDiv.hDiv (α := ℝ) (↑(n + 2 + 1) : ℝ) (↑(n + 2 - 1) : ℝ) •
          HSub.hSub (α := PointsWithCircumcenterIndex (n + 2) → ℝ) (centroidWeightsWithCircumcenter univ)
            (circumcenterWeightsWithCircumcenter (n + 2)) +
        circumcenterWeightsWithCircumcenter (n + 2)) :
    P)
  ((affineCombination ℝ univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → P)
      (mongePointWeightsWithCircumcenter n) :
    P)
Tactic: rw [mongePoint_eq_smul_vsub_vadd_circumcenter,
    centroid_eq_affineCombination_of_pointsWithCircumcenter,
    circumcenter_eq_affineCombination_of_pointsWithCircumcenter, affineCombination_vsub, ← map_smul,
    weightedVSub_vadd_affineCombination]
-- 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)
⊢ Eq.{u_2 + 1} (α := P)
    ((affineCombination ℝ univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → P)
        (HDiv.hDiv (α := ℝ) (↑(n + 2 + 1) : ℝ) (↑(n + 2 - 1) : ℝ) •
            HSub.hSub (α := PointsWithCircumcenterIndex (n + 2) → ℝ) (centroidWeightsWithCircumcenter univ)
              (circumcenterWeightsWithCircumcenter (n + 2)) +
          circumcenterWeightsWithCircumcenter (n + 2)) :
      P)
    ((affineCombination ℝ univ s.pointsWithCircumcenter : (a : PointsWithCircumcenterIndex (n + 2) → ℝ) → P)
        (mongePointWeightsWithCircumcenter n) :
      P)
have h : ∀ (i : PointsWithCircumcenterIndex (n + 2)),
  HAdd.hAdd (β := PointsWithCircumcenterIndex (n + 2) → ℝ)
      (HDiv.hDiv (α := ℝ) (↑(n + 2 + 1) : ℝ) (↑(n + 2 - 1) : ℝ) •
        HSub.hSub (α := PointsWithCircumcenterIndex (n + 2) → ℝ) (centroidWeightsWithCircumcenter univ)
          (circumcenterWeightsWithCircumcenter (n + 2)))
      (circumcenterWeightsWithCircumcenter (n + 2)) i =
    mongePointWeightsWithCircumcenter n i
Tactic: congr with i
-- 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)
i : PointsWithCircumcenterIndex (n + 2)
⊢ HDiv.hDiv (α := ℝ) (↑(n + 2 + 1) : ℝ) (↑(n + 2 - 1) : ℝ) *
        (centroidWeightsWithCircumcenter univ i - circumcenterWeightsWithCircumcenter (n + 2) i) +
      circumcenterWeightsWithCircumcenter (n + 2) i =
    mongePointWeightsWithCircumcenter n i
have h : ∀ (a : Fin (n + 2 + 1)),
  HDiv.hDiv (α := ℝ) (↑(n + 2 + 1) : ℝ) (↑(n + 2 - 1) : ℝ) *
        (centroidWeightsWithCircumcenter univ (pointIndex a) -
          circumcenterWeightsWithCircumcenter (n + 2) (pointIndex a)) +
      circumcenterWeightsWithCircumcenter (n + 2) (pointIndex a) =
    mongePointWeightsWithCircumcenter n (pointIndex a)
Tactic: cases i
Cleared hypotheses: i
-- 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)
i : PointsWithCircumcenterIndex (n + 2)
h :
  ∀ (a : Fin (n + 2 + 1)),
    HDiv.hDiv (α := ℝ) (↑(n + 2 + 1) : ℝ) (↑(n + 2 - 1) : ℝ) *
          (centroidWeightsWithCircumcenter univ (pointIndex a) -
            circumcenterWeightsWithCircumcenter (n + 2) (pointIndex a)) +
        circumcenterWeightsWithCircumcenter (n + 2) (pointIndex a) =
      mongePointWeightsWithCircumcenter n (pointIndex a)
⊢ HDiv.hDiv (α := ℝ) (↑(n + 2 + 1) : ℝ) (↑(n + 2 - 1) : ℝ) *
        (centroidWeightsWithCircumcenter univ i - circumcenterWeightsWithCircumcenter (n + 2) i) +
      circumcenterWeightsWithCircumcenter (n + 2) i =
    mongePointWeightsWithCircumcenter n i
have h₁ : HDiv.hDiv (α := ℝ) (↑(n + 2 + 1) : ℝ) (↑(n + 2 - 1) : ℝ) *
      (centroidWeightsWithCircumcenter (n := n + 2) univ circumcenterIndex -
        circumcenterWeightsWithCircumcenter (n + 2) circumcenterIndex) +
    circumcenterWeightsWithCircumcenter (n + 2) circumcenterIndex =
  mongePointWeightsWithCircumcenter n circumcenterIndex
Tactic: cases i
Cleared hypotheses: i
-- 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)
a : Fin (n + 2 + 1)
⊢ HDiv.hDiv (α := ℝ) (↑(n + 2 + 1) : ℝ) (↑(n + 1) : ℝ) *
        HSub.hSub (α := ℝ)
          (if Membership.mem (γ := Finset (Fin (n + 2 + 1))) univ a then (↑(card (α := Fin (n + 2 + 1)) univ) : ℝ)⁻¹
          else 0)
          0 +
      0 =
    (↑(n + 1) : ℝ)⁻¹
have h : HDiv.hDiv (α := ℝ) (↑(n + 2 + 1) : ℝ) (↑(n + 1) : ℝ) * HSub.hSub (α := ℝ) (↑(n + 2 + 1) : ℝ)⁻¹ 0 + 0 = (↑(n + 1) : ℝ)⁻¹
Tactic: rw [if_pos (mem_univ _), card_fin]
-- 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)
⊢ HDiv.hDiv (α := ℝ) (↑(n + 2 + 1) : ℝ) (↑(n + 1) : ℝ) * HSub.hSub (α := ℝ) 0 1 + 1 =
    HDiv.hDiv (α := ℝ) (-2) (↑(n + 1) : ℝ)
have h : -1 + HAdd.hAdd (α := ℝ) (-2) (-(↑n : ℝ)) + HAdd.hAdd (α := ℝ) (↑n : ℝ) 1 = -2
Tactic: simp [field]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (goal)
Goal:
n : ℕ
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
⊢ mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ =
    mongePointWeightsWithCircumcenter n - centroidWeightsWithCircumcenter {i₁, i₂}ᶜ
have h₁ : ∀ (i : PointsWithCircumcenterIndex (n + 2)),
  mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ i =
    HSub.hSub (β := PointsWithCircumcenterIndex (n + 2) → ℝ) (mongePointWeightsWithCircumcenter n)
      (centroidWeightsWithCircumcenter {i₁, i₂}ᶜ) i
Tactic: ext i
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (goal)
Goal:
n : ℕ
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
i : PointsWithCircumcenterIndex (n + 2)
⊢ mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ i =
    HSub.hSub (β := PointsWithCircumcenterIndex (n + 2) → ℝ) (mongePointWeightsWithCircumcenter n)
      (centroidWeightsWithCircumcenter {i₁, i₂}ᶜ) i
have h₁ : ∀ (i : Fin (n + 2 + 1)),
  mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ (pointIndex i) =
    HSub.hSub (β := PointsWithCircumcenterIndex (n + 2) → ℝ) (mongePointWeightsWithCircumcenter n)
      (centroidWeightsWithCircumcenter {i₁, i₂}ᶜ) (pointIndex i)
Tactic: obtain i | i := i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (goal)
Goal:
n : ℕ
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
i : PointsWithCircumcenterIndex (n + 2)
h₁ :
  ∀ (i : Fin (n + 2 + 1)),
    mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ (pointIndex i) =
      HSub.hSub (β := PointsWithCircumcenterIndex (n + 2) → ℝ) (mongePointWeightsWithCircumcenter n)
        (centroidWeightsWithCircumcenter {i₁, i₂}ᶜ) (pointIndex i)
⊢ mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ i =
    HSub.hSub (β := PointsWithCircumcenterIndex (n + 2) → ℝ) (mongePointWeightsWithCircumcenter n)
      (centroidWeightsWithCircumcenter {i₁, i₂}ᶜ) i
have h₂ : mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ circumcenterIndex =
  HSub.hSub (β := PointsWithCircumcenterIndex (n + 2) → ℝ) (mongePointWeightsWithCircumcenter n)
    (centroidWeightsWithCircumcenter {i₁, i₂}ᶜ) circumcenterIndex
Tactic: obtain i | i := i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.MongePoint (hypothesis)
Goal:
n : ℕ
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
i : Fin (n + 2 + 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
        (↑(card (α := Fin (n + 2 + 1)) {i₁, i₂}ᶜ) : ℝ)⁻¹
      else 0)
have hu : card (α := Fin (n + 3)) {i₁, i₂}ᶜ = n + 1
Tactic: have hu : #{i₁, i₂}ᶜ = n + 1 := by
      simp [card_compl, Fintype.card_fin, h]
-- 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
        (↑(card (α := Fin (n + 2 + 1)) {i₁, i₂}ᶜ) : ℝ)⁻¹
      else 0)
have h₁ : (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: rw [hu]