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
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
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
f : P →ᵃⁱ[ℝ] P₂
⊢ (s.map f.toAffineMap ⋯).touchpointWeights signs = s.touchpointWeights signs
have h₁ : ∀ (i : Fin (n + 1)), (s.map f.toAffineMap sorry).touchpointWeights signs i = s.touchpointWeights signs i
Tactic: ext i : 1
-- 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
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
f : P →ᵃⁱ[ℝ] P₂
i : Fin (n + 1)
⊢ (s.map f.toAffineMap ⋯).touchpointWeights signs i = s.touchpointWeights signs i
have h₁ : (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P)
    ((s.map f.toAffineMap sorry).touchpointWeights signs i) =
  s.touchpoint signs i
Tactic: rw [← affineCombination_eq_touchpoint_iff
    ((s.map f.toAffineMap f.injective).sum_touchpointWeights _ _)]
-- 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
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
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
f : P →ᵃⁱ[ℝ] P₂
i : Fin (n + 1)
⊢ (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P)
      ((s.map f.toAffineMap ⋯).touchpointWeights signs i) =
    s.touchpoint signs i
have hc : (Finset.affineCombination ℝ Finset.univ (s.map f.toAffineMap sorry).points : (a : Fin (n + 1) → ℝ) → P₂)
    ((s.map f.toAffineMap sorry).touchpointWeights signs i) =
  (s.map f.toAffineMap sorry).touchpoint signs i
Tactic: have hc := (s.map f.toAffineMap f.injective).affineCombination_touchpointWeights signs 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
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
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
f : P →ᵃⁱ[ℝ] P₂
i : Fin (n + 1)
hc :
  (Finset.affineCombination ℝ Finset.univ (s.map f.toAffineMap ⋯).points : (a : Fin (n + 1) → ℝ) → P₂)
      ((s.map f.toAffineMap ⋯).touchpointWeights signs i) =
    (s.map f.toAffineMap ⋯).touchpoint signs i
⊢ (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P)
      ((s.map f.toAffineMap ⋯).touchpointWeights signs i) =
    s.touchpoint signs i
have h₁ : ∀
  (hc :
    Eq.{u_4 + 1} (α := P₂)
      ((Finset.affineCombination ℝ Finset.univ (s.map f.toAffineMap sorry).points : (a : Fin (n + 1) → ℝ) → P₂)
          ((s.map f.toAffineMap sorry).touchpointWeights signs i) :
        P₂)
      ((f : (a : P) → P₂) (s.touchpoint signs i) : P₂)),
  (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P)
      ((s.map f.toAffineMap sorry).touchpointWeights signs i) =
    s.touchpoint signs i
Tactic: rwa [h.touchpoint_map, map_points, ← Finset.univ.map_affineCombination _ _
    ((s.map f.toAffineMap f.injective).sum_touchpointWeights _ _), AffineIsometry.coe_toAffineMap,
    AffineIsometry.map_eq_iff] at hc
Cleared hypotheses: hc
-- 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
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
f : P →ᵃⁱ[ℝ] P₂
i : Fin (n + 1)
hc :
  (Finset.affineCombination ℝ Finset.univ (s.map f.toAffineMap ⋯).points : (a : Fin (n + 1) → ℝ) → P₂)
      ((s.map f.toAffineMap ⋯).touchpointWeights signs i) =
    (s.map f.toAffineMap ⋯).touchpoint signs i
h₁ :
  ∀
    (hc :
      Eq.{u_4 + 1} (α := P₂)
        ((Finset.affineCombination ℝ Finset.univ (s.map f.toAffineMap ⋯).points : (a : Fin (n + 1) → ℝ) → P₂)
            ((s.map f.toAffineMap ⋯).touchpointWeights signs i) :
          P₂)
        ((f : (a : P) → P₂) (s.touchpoint signs i) : P₂)),
    (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P)
        ((s.map f.toAffineMap ⋯).touchpointWeights signs i) =
      s.touchpoint signs i
⊢ (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P)
      ((s.map f.toAffineMap ⋯).touchpointWeights signs i) =
    s.touchpoint signs i
have h₂ : ∀
  (hc :
    Eq.{u_4 + 1} (α := P₂)
      ((Finset.affineCombination ℝ Finset.univ ((⇑f.toAffineMap : (a : P) → P₂) ∘ s.points) :
            (a : Fin (n + 1) → ℝ) → P₂)
          ((s.map f.toAffineMap sorry).touchpointWeights signs i) :
        P₂)
      ((f : (a : P) → P₂) (s.touchpoint signs i) : P₂)),
  (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P)
      ((s.map f.toAffineMap sorry).touchpointWeights signs i) =
    s.touchpoint signs i
Tactic: rwa [h.touchpoint_map, map_points, ← Finset.univ.map_affineCombination _ _
    ((s.map f.toAffineMap f.injective).sum_touchpointWeights _ _), AffineIsometry.coe_toAffineMap,
    AffineIsometry.map_eq_iff] at hc
Cleared hypotheses: hc
-- 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
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
f : P →ᵃⁱ[ℝ] P₂
i : Fin (n + 1)
hc :
  (Finset.affineCombination ℝ Finset.univ (s.map f.toAffineMap ⋯).points : (a : Fin (n + 1) → ℝ) → P₂)
      ((s.map f.toAffineMap ⋯).touchpointWeights signs i) =
    (s.map f.toAffineMap ⋯).touchpoint signs i
h₁ :
  ∀
    (hc :
      Eq.{u_4 + 1} (α := P₂)
        ((Finset.affineCombination ℝ Finset.univ (s.map f.toAffineMap ⋯).points : (a : Fin (n + 1) → ℝ) → P₂)
            ((s.map f.toAffineMap ⋯).touchpointWeights signs i) :
          P₂)
        ((f : (a : P) → P₂) (s.touchpoint signs i) : P₂)),
    (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P)
        ((s.map f.toAffineMap ⋯).touchpointWeights signs i) =
      s.touchpoint signs i
h₂ :
  ∀
    (hc :
      Eq.{u_4 + 1} (α := P₂)
        ((Finset.affineCombination ℝ Finset.univ ((⇑f.toAffineMap : (a : P) → P₂) ∘ s.points) :
              (a : Fin (n + 1) → ℝ) → P₂)
            ((s.map f.toAffineMap ⋯).touchpointWeights signs i) :
          P₂)
        ((f : (a : P) → P₂) (s.touchpoint signs i) : P₂)),
    (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P)
        ((s.map f.toAffineMap ⋯).touchpointWeights signs i) =
      s.touchpoint signs i
⊢ (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P)
      ((s.map f.toAffineMap ⋯).touchpointWeights signs i) =
    s.touchpoint signs i
have h₃ : ∀
  (hc :
    Eq.{u_4 + 1} (α := P₂)
      ((f.toAffineMap : (a : P) → P₂)
          ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) fun (j : Fin (n + 1)) ↦
              (s.map f.toAffineMap sorry).touchpointWeights signs i j :
            P) :
        P₂)
      ((f : (a : P) → P₂) (s.touchpoint signs i) : P₂)),
  (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P)
      ((s.map f.toAffineMap sorry).touchpointWeights signs i) =
    s.touchpoint signs i
Tactic: rwa [h.touchpoint_map, map_points, ← Finset.univ.map_affineCombination _ _
    ((s.map f.toAffineMap f.injective).sum_touchpointWeights _ _), AffineIsometry.coe_toAffineMap,
    AffineIsometry.map_eq_iff] at hc
Cleared hypotheses: hc
-- 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
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
f : P →ᵃⁱ[ℝ] P₂
i : Fin (n + 1)
hc :
  (Finset.affineCombination ℝ Finset.univ (s.map f.toAffineMap ⋯).points : (a : Fin (n + 1) → ℝ) → P₂)
      ((s.map f.toAffineMap ⋯).touchpointWeights signs i) =
    (s.map f.toAffineMap ⋯).touchpoint signs i
h₁ :
  ∀
    (hc :
      Eq.{u_4 + 1} (α := P₂)
        ((Finset.affineCombination ℝ Finset.univ (s.map f.toAffineMap ⋯).points : (a : Fin (n + 1) → ℝ) → P₂)
            ((s.map f.toAffineMap ⋯).touchpointWeights signs i) :
          P₂)
        ((f : (a : P) → P₂) (s.touchpoint signs i) : P₂)),
    (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P)
        ((s.map f.toAffineMap ⋯).touchpointWeights signs i) =
      s.touchpoint signs i
h₂ :
  ∀
    (hc :
      Eq.{u_4 + 1} (α := P₂)
        ((Finset.affineCombination ℝ Finset.univ ((⇑f.toAffineMap : (a : P) → P₂) ∘ s.points) :
              (a : Fin (n + 1) → ℝ) → P₂)
            ((s.map f.toAffineMap ⋯).touchpointWeights signs i) :
          P₂)
        ((f : (a : P) → P₂) (s.touchpoint signs i) : P₂)),
    (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P)
        ((s.map f.toAffineMap ⋯).touchpointWeights signs i) =
      s.touchpoint signs i
h₃ :
  ∀
    (hc :
      Eq.{u_4 + 1} (α := P₂)
        ((f.toAffineMap : (a : P) → P₂)
            ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) fun (j : Fin (n + 1)) ↦
                (s.map f.toAffineMap ⋯).touchpointWeights signs i j :
              P) :
          P₂)
        ((f : (a : P) → P₂) (s.touchpoint signs i) : P₂)),
    (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P)
        ((s.map f.toAffineMap ⋯).touchpointWeights signs i) =
      s.touchpoint signs i
⊢ (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P)
      ((s.map f.toAffineMap ⋯).touchpointWeights signs i) =
    s.touchpoint signs i
have h₄ : ∀
  (hc :
    Eq.{u_4 + 1} (α := P₂)
      ((f : (a : P) → P₂)
          ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) fun (j : Fin (n + 1)) ↦
              (s.map f.toAffineMap sorry).touchpointWeights signs i j :
            P) :
        P₂)
      ((f : (a : P) → P₂) (s.touchpoint signs i) : P₂)),
  (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P)
      ((s.map f.toAffineMap sorry).touchpointWeights signs i) =
    s.touchpoint signs i
Tactic: rwa [h.touchpoint_map, map_points, ← Finset.univ.map_affineCombination _ _
    ((s.map f.toAffineMap f.injective).sum_touchpointWeights _ _), AffineIsometry.coe_toAffineMap,
    AffineIsometry.map_eq_iff] at hc
Cleared hypotheses: hc
-- 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))
h : s.ExcenterExists signs
S : AffineSubspace ℝ P
hS : affineSpan ℝ (Set.range s.points) ≤ S
⊢ touchpointWeights.{u_1, u_2} (P := ↥S) (s.restrict S hS) signs = s.touchpointWeights signs
have h₁ : ExcenterExists.{u_1, u_2} (P := ↥S) (s.restrict S hS) signs
Tactic: rw [← s.excenterExists_restrict S hS] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (hypothesis)
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
S : AffineSubspace ℝ P
hS : affineSpan ℝ (Set.range s.points) ≤ S
h : ExcenterExists.{u_1, u_2} (P := ↥S) (s.restrict S hS) signs
⊢ touchpointWeights.{u_1, u_2} (P := ↥S) (s.restrict S hS) signs = s.touchpointWeights signs
have this : Nonempty.{u_2 + 1} ↥S
Tactic: haveI := Nonempty.map (AffineSubspace.inclusion hS) inferInstance
-- 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))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hne : i ≠ j
⊢ Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (s.touchpointWeights signs i j) : SignType)
    ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeights signs j) : SignType)
have hs : Eq (α := SignType)
  ((SignType.sign : (a : ℝ) → SignType) ((s.signedInfDist j : (a : P) → ℝ) (s.touchpoint signs i) : ℝ) : SignType)
  ((SignType.sign : (a : ℝ) → SignType) ((s.signedInfDist j : (a : P) → ℝ) (s.excenter signs) : ℝ) : SignType)
Tactic: have hs := h.sign_signedInfDist_touchpoint hne
-- 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))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hne : i ≠ j
hs :
  Eq (α := SignType)
    ((SignType.sign : (a : ℝ) → SignType) ((s.signedInfDist j : (a : P) → ℝ) (s.touchpoint signs i) : ℝ) : SignType)
    ((SignType.sign : (a : ℝ) → SignType) ((s.signedInfDist j : (a : P) → ℝ) (s.excenter signs) : ℝ) : SignType)
⊢ Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (s.touchpointWeights signs i j) : SignType)
    ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeights signs j) : SignType)
have hs₁ : Eq (α := SignType)
  ((SignType.sign : (a : ℝ) → SignType)
      (s.touchpointWeights signs i j *
        ‖s.points j -ᵥ
            (↑(((s.faceOpposite j).orthogonalProjectionSpan :
                    (a : P) → ↥(affineSpan ℝ (Set.range (s.faceOpposite j).points)))
                  (s.points j)) :
              P)‖) :
    SignType)
  ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeights signs j) : SignType)
Tactic: rw [← s.affineCombination_touchpointWeights signs i, h.sign_signedInfDist_excenter,
    s.signedInfDist_affineCombination j (by simp)] at hs
Cleared hypotheses: 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
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hne : i ≠ j
hs :
  Eq (α := SignType)
    ((SignType.sign : (a : ℝ) → SignType)
        (s.touchpointWeights signs i j *
          ‖s.points j -ᵥ
              (↑(((s.faceOpposite j).orthogonalProjectionSpan :
                      (a : P) → ↥(affineSpan ℝ (Set.range (s.faceOpposite j).points)))
                    (s.points j)) :
                P)‖) :
      SignType)
    ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeights signs j) : SignType)
⊢ Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (s.touchpointWeights signs i j) : SignType)
    ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeights signs j) : SignType)
have rw : (SignType.sign : (a : ℝ) → SignType) (s.touchpointWeights signs i j) =
  HMul.hMul (α := SignType) (β := SignType)
    ((SignType.sign : (a : ℝ) → SignType) (s.touchpointWeights signs i j) : SignType)
    ((SignType.sign : (a : ℝ) → SignType)
        ‖s.points j -ᵥ
            (↑(((s.faceOpposite j).orthogonalProjectionSpan :
                    (a : P) → ↥(affineSpan ℝ (Set.range (s.faceOpposite j).points)))
                  (s.points j)) :
              P)‖ :
      SignType)
Tactic: rw [← hs, sign_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
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hne : i ≠ j
hs :
  Eq (α := SignType)
    ((SignType.sign : (a : ℝ) → SignType)
        (s.touchpointWeights signs i j *
          ‖s.points j -ᵥ
              (↑(((s.faceOpposite j).orthogonalProjectionSpan :
                      (a : P) → ↥(affineSpan ℝ (Set.range (s.faceOpposite j).points)))
                    (s.points j)) :
                P)‖) :
      SignType)
    ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeights signs j) : SignType)
⊢ (SignType.sign : (a : ℝ) → SignType) (s.touchpointWeights signs i j) =
    HMul.hMul (α := SignType) (β := SignType)
      ((SignType.sign : (a : ℝ) → SignType) (s.touchpointWeights signs i j) : SignType)
      ((SignType.sign : (a : ℝ) → SignType)
          ‖s.points j -ᵥ
              (↑(((s.faceOpposite j).orthogonalProjectionSpan :
                      (a : P) → ↥(affineSpan ℝ (Set.range (s.faceOpposite j).points)))
                    (s.points j)) :
                P)‖ :
        SignType)
have h₁ : Eq (α := SignType)
  ((SignType.sign : (a : ℝ) → SignType)
      ‖s.points j -ᵥ
          (↑(((s.faceOpposite j).orthogonalProjectionSpan :
                  (a : P) → ↥(affineSpan ℝ (Set.range (s.faceOpposite j).points)))
                (s.points j)) :
            P)‖ :
    SignType)
  (1 : SignType)
Tactic: convert! (mul_one _).symm
-- 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 j : Fin (n + 1)
hne : i ≠ j
hs :
  Eq (α := SignType)
    ((SignType.sign : (a : ℝ) → SignType)
        (s.touchpointWeights signs i j *
          ‖s.points j -ᵥ
              (↑(((s.faceOpposite j).orthogonalProjectionSpan :
                      (a : P) → ↥(affineSpan ℝ (Set.range (s.faceOpposite j).points)))
                    (s.points j)) :
                P)‖) :
      SignType)
    ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeights signs j) : SignType)
⊢ Eq (α := SignType)
    ((SignType.sign : (a : ℝ) → SignType)
        ‖s.points j -ᵥ
            (↑(((s.faceOpposite j).orthogonalProjectionSpan :
                    (a : P) → ↥(affineSpan ℝ (Set.range (s.faceOpposite j).points)))
                  (s.points j)) :
              P)‖ :
      SignType)
    (1 : SignType)
have h₁ : 0 <
  dist (s.points j)
    (↑(((s.faceOpposite j).orthogonalProjectionSpan : (a : P) → ↥(affineSpan ℝ (Set.range (s.faceOpposite j).points)))
          (s.points j)) :
      P)
Tactic: rw [sign_eq_one_iff, ← dist_eq_norm_vsub]
-- 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 j : Fin (n + 1)
hne : i ≠ j
⊢ Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (s.touchpointWeights ∅ i j) : SignType) (1 : SignType)
have rw : Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeights ∅ j) : SignType) (1 : SignType)
Tactic: rw [s.excenterExists_empty.sign_touchpointWeights hne]
-- 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.touchpointWeights signs i i = 0
have refine : Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (s.points '' {i}ᶜ))
  ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) (s.touchpointWeights signs i) : P)
Tactic: refine s.independent.eq_zero_of_affineCombination_mem_affineSpan
    (s.sum_touchpointWeights signs i) ?_ (Finset.mem_univ _)
    (Set.notMem_compl_iff.2 (Set.mem_singleton _))
-- 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)
⊢ Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (s.points '' {i}ᶜ))
    ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) (s.touchpointWeights signs i) : P)
have rw : Membership.mem (γ := AffineSubspace ℝ P) (affineSpan ℝ (s.points '' {i}ᶜ)) (s.touchpoint signs i)
Tactic: rw [s.affineCombination_touchpointWeights]
-- 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)
⊢ Membership.mem (γ := AffineSubspace ℝ P) (affineSpan ℝ (s.points '' {i}ᶜ)) (s.touchpoint signs i)
have h : s.points '' {i}ᶜ = Set.range (s.faceOpposite i).points
Tactic: convert! s.touchpoint_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
inst_5 : n.AtLeastTwo
i : Fin (n + 1)
⊢ s.touchpoint ∅ i ∈ (s.faceOpposite i).interior
have rw : (∀ (i_1 : Fin (n + 1)), Membership.mem (γ := Finset (Fin (n + 1))) {i}ᶜ i_1 → 0 < s.touchpointWeights ∅ i i_1) ∧
  ∀ i_1 ∉ {i}ᶜ, s.touchpointWeights ∅ i i_1 = 0
Tactic: rw [faceOpposite, ← affineCombination_touchpointWeights,
    s.affineCombination_mem_interior_face_iff_pos _ (s.sum_touchpointWeights _ _)]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (goal)
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
inst_5 : n.AtLeastTwo
i : Fin (n + 1)
⊢ (∀ (i_1 : Fin (n + 1)), Membership.mem (γ := Finset (Fin (n + 1))) {i}ᶜ i_1 → 0 < s.touchpointWeights ∅ i i_1) ∧
    ∀ i_1 ∉ {i}ᶜ, s.touchpointWeights ∅ i i_1 = 0
have simp : ∀ (i_1 : Fin (n + 1)), ¬i_1 = i → 0 < s.touchpointWeights ∅ i i_1
Tactic: simp only [Finset.mem_compl, Finset.mem_singleton, Decidable.not_not, forall_eq,
    touchpointWeights_eq_zero, and_true]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (goal)
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
inst_5 : n.AtLeastTwo
i j : Fin (n + 1)
hne : i ≠ j
⊢ Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (s.touchpointWeights {i} i j) : SignType) (1 : SignType)
have rw : Eq (α := SignType) ((SignType.sign : (a : ℝ) → SignType) (s.excenterWeights {i} j) : SignType) (1 : SignType)
Tactic: rw [(s.excenterExists_singleton i).sign_touchpointWeights hne,
    s.sign_excenterWeights_singleton_pos hne]