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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
r : ℝ
w : Fin (n + 1) → ℝ
h1 : Eq (α := ℝ) (∑ i : Fin (n + 1), w i) 1
hp :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points))
    ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) w : P)
h :
  ∀ (i : Fin (n + 1)),
    (s.signedInfDist i : (a : P) → ℝ)
        ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) w : P) =
      (if i ∈ signs then -1 else 1) * r
i : Fin (n + 1)
h' : w i = (if i ∈ signs then -1 else 1) * r / s.height i
rw : (if i ∈ signs then -1 else 1) * r / s.height i = r * ((if i ∈ signs then -1 else 1) * (s.height i)⁻¹)
rw₁ : (r * if i ∈ signs then -1 else 1) / s.height i = r * ((if i ∈ signs then -1 else 1) * (s.height i)⁻¹)
rw₂ : (r * if i ∈ signs then -1 else 1) * (s.height i)⁻¹ = r * ((if i ∈ signs then -1 else 1) * (s.height i)⁻¹)
rw₃ : r * ((if i ∈ signs then -1 else 1) * (s.height i)⁻¹) = r * ((if i ∈ signs then -1 else 1) * (s.height i)⁻¹)
rw₄ :
  r * ((if i ∈ signs then -1 else 1) * (dist (s.points i) (s.altitudeFoot i))⁻¹) =
    r * ((if i ∈ signs then -1 else 1) * (dist (s.points i) (s.altitudeFoot i))⁻¹)
rw₅ :
  r *
      ((if i ∈ signs then -1 else 1) *
        (dist (s.points i)
            (↑(((s.faceOpposite i).orthogonalProjectionSpan :
                    (a : P) → ↥(affineSpan ℝ (Set.range (s.faceOpposite i).points)))
                  (s.points i)) :
              P))⁻¹) =
    r *
      ((if i ∈ signs then -1 else 1) *
        (dist (s.points i)
            (↑(((s.faceOpposite i).orthogonalProjectionSpan :
                    (a : P) → ↥(affineSpan ℝ (Set.range (s.faceOpposite i).points)))
                  (s.points i)) :
              P))⁻¹)
⊢ w i = r * ((if i ∈ signs then -1 else 1) * (s.height i)⁻¹)
have rw₆ : r *
    ((if i ∈ signs then -1 else 1) *
      (dist (s.points i)
          (↑((orthogonalProjection (affineSpan ℝ (Set.range (s.faceOpposite i).points)) :
                  (a : P) → ↥(affineSpan ℝ (Set.range (s.faceOpposite i).points)))
                (s.points i)) :
            P))⁻¹) =
  r *
    ((if i ∈ signs then -1 else 1) *
      (dist (s.points i)
          (↑((orthogonalProjection (affineSpan ℝ (Set.range (s.faceOpposite i).points)) :
                  (a : P) → ↥(affineSpan ℝ (Set.range (s.faceOpposite i).points)))
                (s.points i)) :
            P))⁻¹)
Tactic: rw [h', mul_comm, div_eq_mul_inv, mul_assoc, height, altitudeFoot, orthogonalProjectionSpan]
-- 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))
r : ℝ
w : Fin (n + 1) → ℝ
h1 : Eq (α := ℝ) (∑ i : Fin (n + 1), w i) 1
hp :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points))
    ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) w : P)
h :
  ∀ (i : Fin (n + 1)),
    (s.signedInfDist i : (a : P) → ℝ)
        ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) w : P) =
      (if i ∈ signs then -1 else 1) * r
h' : ∀ (i : Fin (n + 1)), w i * dist (s.points i) (s.altitudeFoot i) = (if i ∈ signs then -1 else 1) * r
⊢ s.ExcenterExists signs ∧
    (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) w = s.excenter signs
have h'' : ∀ (i : Fin (n + 1)), w i = r * s.excenterWeightsUnnorm signs i
Tactic: have h'' : ∀ i, w i = r * s.excenterWeightsUnnorm signs i := by
      simp_rw [excenterWeightsUnnorm]
      intro i
      replace h' := h' i
      rw [← height, ← eq_div_iff (s.height_pos i).ne'] at h'
      rw [h', mul_comm, div_eq_mul_inv, mul_assoc, height, altitudeFoot, orthogonalProjectionSpan]
-- 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))
r : ℝ
w : Fin (n + 1) → ℝ
h1 : Eq (α := ℝ) (∑ i : Fin (n + 1), w i) 1
hp :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points))
    ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) w : P)
h :
  ∀ (i : Fin (n + 1)),
    (s.signedInfDist i : (a : P) → ℝ)
        ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) w : P) =
      (if i ∈ signs then -1 else 1) * r
h' : ∀ (i : Fin (n + 1)), w i * dist (s.points i) (s.altitudeFoot i) = (if i ∈ signs then -1 else 1) * r
h'' : ∀ (i : Fin (n + 1)), w i = r * s.excenterWeightsUnnorm signs i
⊢ w = s.excenterWeights signs
have h1₁ : r * ∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i = 1
Tactic: simp_rw [h'', ← Finset.mul_sum] at h1
Cleared hypotheses: h1
-- 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))
r : ℝ
w : Fin (n + 1) → ℝ
hp :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points))
    ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) w : P)
h :
  ∀ (i : Fin (n + 1)),
    (s.signedInfDist i : (a : P) → ℝ)
        ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) w : P) =
      (if i ∈ signs then -1 else 1) * r
h' : ∀ (i : Fin (n + 1)), w i * dist (s.points i) (s.altitudeFoot i) = (if i ∈ signs then -1 else 1) * r
h'' : ∀ (i : Fin (n + 1)), w i = r * s.excenterWeightsUnnorm signs i
h1 : r * ∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i = 1
⊢ w = s.excenterWeights signs
have h₁ : ∀ (j : Fin (n + 1)), w j = s.excenterWeights signs j
Tactic: ext j
-- 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))
r : ℝ
w : Fin (n + 1) → ℝ
hp :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points))
    ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) w : P)
h :
  ∀ (i : Fin (n + 1)),
    (s.signedInfDist i : (a : P) → ℝ)
        ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) w : P) =
      (if i ∈ signs then -1 else 1) * r
h' : ∀ (i : Fin (n + 1)), w i * dist (s.points i) (s.altitudeFoot i) = (if i ∈ signs then -1 else 1) * r
h'' : ∀ (i : Fin (n + 1)), w i = r * s.excenterWeightsUnnorm signs i
h1 : r * ∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i = 1
j : Fin (n + 1)
⊢ w j = s.excenterWeights signs j
have h₁ : (∑ i : Fin (n + 1), s.excenterWeightsUnnorm signs i)⁻¹ * s.excenterWeightsUnnorm signs j = s.excenterWeights signs j
Tactic: rw [h'', eq_inv_of_mul_eq_one_left h1]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Incenter (hypothesis)
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
r : ℝ
w : Fin (n + 1) → ℝ
h1 : Eq (α := ℝ) (∑ i : Fin (n + 1), w i) 1
hp :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points))
    ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) w : P)
h :
  ∀ (i : Fin (n + 1)),
    (s.signedInfDist i : (a : P) → ℝ)
        ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) w : P) =
      (if i ∈ signs then -1 else 1) * r
h' : ∀ (i : Fin (n + 1)), w i * dist (s.points i) (s.altitudeFoot i) = (if i ∈ signs then -1 else 1) * r
h'' : ∀ (i : Fin (n + 1)), w i = r * s.excenterWeightsUnnorm signs i
⊢ s.ExcenterExists signs ∧
    (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) w = s.excenter signs
have hw : w = s.excenterWeights signs
Tactic: have hw : w = s.excenterWeights signs := by
      simp_rw [h'', ← Finset.mul_sum] at h1
      ext j
      rw [h'', eq_inv_of_mul_eq_one_left h1]
      simp [excenterWeights]
-- 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))
r : ℝ
w : Fin (n + 1) → ℝ
h1 : Eq (α := ℝ) (∑ i : Fin (n + 1), w i) 1
hp :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points))
    ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) w : P)
h :
  ∀ (i : Fin (n + 1)),
    (s.signedInfDist i : (a : P) → ℝ)
        ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) w : P) =
      (if i ∈ signs then -1 else 1) * r
h' : ∀ (i : Fin (n + 1)), w i * dist (s.points i) (s.altitudeFoot i) = (if i ∈ signs then -1 else 1) * r
h'' : ∀ (i : Fin (n + 1)), w i = r * s.excenterWeightsUnnorm signs i
hw : w = s.excenterWeights signs
⊢ s.ExcenterExists signs ∧
    (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) w = s.excenter signs
have refine_1 : ∀ (h1 : Eq (α := ℝ) (∑ i : Fin (n + 1), s.excenterWeights signs i) 1)
  (hp :
    Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points))
      ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) (s.excenterWeights signs) : P))
  (h :
    ∀ (i : Fin (n + 1)),
      (s.signedInfDist i : (a : P) → ℝ)
          ((Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) (s.excenterWeights signs) :
            P) =
        (if i ∈ signs then -1 else 1) * r)
  (h' :
    ∀ (i : Fin (n + 1)),
      s.excenterWeights signs i * dist (s.points i) (s.altitudeFoot i) = (if i ∈ signs then -1 else 1) * r)
  (h'' : ∀ (i : Fin (n + 1)), s.excenterWeights signs i = r * s.excenterWeightsUnnorm signs i),
  s.ExcenterExists signs ∧
    (Finset.affineCombination ℝ Finset.univ s.points : (a : Fin (n + 1) → ℝ) → P) (s.excenterWeights signs) =
      s.excenter signs
Tactic: subst hw
Cleared hypotheses: w, h1, hp, h, h', h'', hw
-- 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
hp : Membership.mem (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points)) (s.excenter signs)
⊢ ∃ (r : ℝ),
    ∀ (i : Fin (n + 1)), (s.signedInfDist i : (a : P) → ℝ) (s.excenter signs) = (if i ∈ signs then -1 else 1) * r
have refine_2 : ∀ (i : Fin (n + 1)),
  (s.signedInfDist i : (a : P) → ℝ) (s.excenter signs) =
    (if i ∈ signs then -1 else 1) *
      ((↑((SignType.sign : (a : ℝ) → SignType) (∑ j : Fin (n + 1), s.excenterWeightsUnnorm signs j)) : ℝ) *
        s.exradius signs)
Tactic: refine ⟨SignType.sign (∑ j, s.excenterWeightsUnnorm signs j) * (s.exradius signs), fun 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))
h : s.ExcenterExists signs
hp : Membership.mem (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points)) (s.excenter signs)
i : Fin (n + 1)
⊢ (s.signedInfDist i : (a : P) → ℝ) (s.excenter signs) =
    (if i ∈ signs then -1 else 1) *
      ((↑((SignType.sign : (a : ℝ) → SignType) (∑ j : Fin (n + 1), s.excenterWeightsUnnorm signs j)) : ℝ) *
        s.exradius signs)
have refine_2 : HMul.hMul (α := ℝ) (if i ∈ signs then -1 else 1)
      (↑((SignType.sign : (a : ℝ) → SignType) (∑ j : Fin (n + 1), s.excenterWeightsUnnorm signs j)) : ℝ) *
    s.exradius signs =
  (if i ∈ signs then -1 else 1) *
    ((↑((SignType.sign : (a : ℝ) → SignType) (∑ j : Fin (n + 1), s.excenterWeightsUnnorm signs j)) : ℝ) *
      s.exradius signs)
Tactic: rw [h.signedInfDist_excenter]
-- 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
p : P
hp : Membership.mem (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points)) p
⊢ (∃ (r : ℝ), ∀ (i : Fin (n + 1)), (s.signedInfDist i : (a : P) → ℝ) p = r) ↔ p = s.incenter
have h : ∀ (x : ℝ) (a : Fin (n + 1)), x = (if Membership.mem (γ := Finset (Fin (n + 1))) ∅ a then -1 else 1) * x
Tactic: convert! s.exists_forall_signedInfDist_eq_iff_excenterExists_and_eq_excenter hp (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
p : P
hp : Membership.mem (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points)) p
h : ∀ (x : ℝ) (a : Fin (n + 1)), x = (if Membership.mem (γ := Finset (Fin (n + 1))) ∅ a then -1 else 1) * x
⊢ (∃ (r : ℝ), ∀ (i : Fin (n + 1)), (s.signedInfDist i : (a : P) → ℝ) p = r) ↔ p = s.incenter
have h₁ : p = s.incenter ↔ s.ExcenterExists ∅ ∧ p = s.excenter ∅
Tactic: convert! s.exists_forall_signedInfDist_eq_iff_excenterExists_and_eq_excenter hp (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
p : P
hp : Membership.mem (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points)) p
⊢ (∃ (r : ℝ),
      ∀ (i : Fin (n + 1)),
        dist p
            (↑(((s.faceOpposite i).orthogonalProjectionSpan :
                    (a : P) → ↥(affineSpan ℝ (Set.range (s.faceOpposite i).points)))
                  p) :
              P) =
          r) ↔
    ∃ (signs : Finset (Fin (n + 1))), s.ExcenterExists signs ∧ p = s.excenter signs
have simp_rw : (∃ (r : ℝ), ∀ (i : Fin (n + 1)), |(s.signedInfDist i : (a : P) → ℝ) p| = r) ↔
  ∃ (signs : Finset (Fin (n + 1))), s.ExcenterExists signs ∧ p = s.excenter signs
Tactic: simp_rw [← abs_signedInfDist_eq_dist_of_mem_affineSpan_range _ hp]
-- 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
p : P
hp : Membership.mem (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points)) p
⊢ (∃ (r : ℝ), ∀ (i : Fin (n + 1)), |(s.signedInfDist i : (a : P) → ℝ) p| = r) ↔
    ∃ (signs : Finset (Fin (n + 1))), s.ExcenterExists signs ∧ p = s.excenter signs
have refine_1 : (∃ (r : ℝ), ∀ (i : Fin (n + 1)), |(s.signedInfDist i : (a : P) → ℝ) p| = r) →
  ∃ (signs : Finset (Fin (n + 1))), s.ExcenterExists signs ∧ p = s.excenter signs
Tactic: refine ⟨?_, ?_⟩
-- 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
p : P
hp : Membership.mem (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points)) p
refine_1 :
  (∃ (r : ℝ), ∀ (i : Fin (n + 1)), |(s.signedInfDist i : (a : P) → ℝ) p| = r) →
    ∃ (signs : Finset (Fin (n + 1))), s.ExcenterExists signs ∧ p = s.excenter signs
⊢ (∃ (r : ℝ), ∀ (i : Fin (n + 1)), |(s.signedInfDist i : (a : P) → ℝ) p| = r) ↔
    ∃ (signs : Finset (Fin (n + 1))), s.ExcenterExists signs ∧ p = s.excenter signs
have refine_2 : (∃ (signs : Finset (Fin (n + 1))), s.ExcenterExists signs ∧ p = s.excenter signs) →
  ∃ (r : ℝ), ∀ (i : Fin (n + 1)), |(s.signedInfDist i : (a : P) → ℝ) p| = r
Tactic: refine ⟨?_, ?_⟩
-- 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
p : P
hp : Membership.mem (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points)) p
r : ℝ
h : ∀ (i : Fin (n + 1)), |(s.signedInfDist i : (a : P) → ℝ) p| = r
⊢ ∃ (signs : Finset (Fin (n + 1))), s.ExcenterExists signs ∧ p = s.excenter signs
have h' : ∀ (i : Fin (n + 1)), (s.signedInfDist i : (a : P) → ℝ) p = r ∨ (s.signedInfDist i : (a : P) → ℝ) p = -r
Tactic: have h' : ∀ i, s.signedInfDist i p = r ∨ s.signedInfDist i p = -r :=
      fun i ↦ eq_or_eq_neg_of_abs_eq (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
p : P
hp : Membership.mem (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points)) p
r : ℝ
h : ∀ (i : Fin (n + 1)), |(s.signedInfDist i : (a : P) → ℝ) p| = r
h' : ∀ (i : Fin (n + 1)), (s.signedInfDist i : (a : P) → ℝ) p = r ∨ (s.signedInfDist i : (a : P) → ℝ) p = -r
⊢ ∃ (signs : Finset (Fin (n + 1))), s.ExcenterExists signs ∧ p = s.excenter signs
have refine_1 : s.ExcenterExists {i : Fin (n + 1) | (s.signedInfDist i : (a : P) → ℝ) p = -r} ∧
  p = s.excenter {i : Fin (n + 1) | (s.signedInfDist i : (a : P) → ℝ) p = -r}
Tactic: refine ⟨{i ∈ (Finset.univ : Finset (Fin (n + 1))) | s.signedInfDist i p = -r}, ?_⟩
-- 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
p : P
hp : Membership.mem (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points)) p
r : ℝ
h : ∀ (i : Fin (n + 1)), |(s.signedInfDist i : (a : P) → ℝ) p| = r
h' : ∀ (i : Fin (n + 1)), (s.signedInfDist i : (a : P) → ℝ) p = r ∨ (s.signedInfDist i : (a : P) → ℝ) p = -r
⊢ s.ExcenterExists {i : Fin (n + 1) | (s.signedInfDist i : (a : P) → ℝ) p = -r} ∧
    p = s.excenter {i : Fin (n + 1) | (s.signedInfDist i : (a : P) → ℝ) p = -r}
have refine_1 : ∃ (r_1 : ℝ),
  ∀ (i : Fin (n + 1)),
    (s.signedInfDist i : (a : P) → ℝ) p =
      (if Membership.mem (γ := Finset (Fin (n + 1))) {i : Fin (n + 1) | (s.signedInfDist i : (a : P) → ℝ) p = -r} i then
          -1
        else 1) *
        r_1
Tactic: apply (s.exists_forall_signedInfDist_eq_iff_excenterExists_and_eq_excenter hp).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
n : ℕ
inst_4 : NeZero n
s : Simplex ℝ P n
p : P
hp : Membership.mem (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points)) p
r : ℝ
h : ∀ (i : Fin (n + 1)), |(s.signedInfDist i : (a : P) → ℝ) p| = r
h' : ∀ (i : Fin (n + 1)), (s.signedInfDist i : (a : P) → ℝ) p = r ∨ (s.signedInfDist i : (a : P) → ℝ) p = -r
⊢ ∃ (r_1 : ℝ),
    ∀ (i : Fin (n + 1)),
      (s.signedInfDist i : (a : P) → ℝ) p =
        (if
              Membership.mem (γ := Finset (Fin (n + 1))) {i : Fin (n + 1) | (s.signedInfDist i : (a : P) → ℝ) p = -r}
                i then
            -1
          else 1) *
          r_1
have refine_1 : ∀ (i : Fin (n + 1)),
  (s.signedInfDist i : (a : P) → ℝ) p =
    (if Membership.mem (γ := Finset (Fin (n + 1))) {i : Fin (n + 1) | (s.signedInfDist i : (a : P) → ℝ) p = -r} i then
        -1
      else 1) *
      r
Tactic: refine ⟨r, ?_⟩
-- 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
p : P
hp : Membership.mem (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points)) p
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs ∧ p = s.excenter signs
⊢ ∃ (r : ℝ), ∀ (i : Fin (n + 1)), |(s.signedInfDist i : (a : P) → ℝ) p| = r
have h₁ : ∃ (r : ℝ), ∀ (i : Fin (n + 1)), (s.signedInfDist i : (a : P) → ℝ) p = (if i ∈ signs then -1 else 1) * r
Tactic: replace h := (s.exists_forall_signedInfDist_eq_iff_excenterExists_and_eq_excenter hp).2 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
p : P
hp : Membership.mem (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points)) p
signs : Finset (Fin (n + 1))
h : ∃ (r : ℝ), ∀ (i : Fin (n + 1)), (s.signedInfDist i : (a : P) → ℝ) p = (if i ∈ signs then -1 else 1) * r
⊢ ∃ (r : ℝ), ∀ (i : Fin (n + 1)), |(s.signedInfDist i : (a : P) → ℝ) p| = r
have refine_2 : ∀ (r : ℝ) (h : ∀ (i : Fin (n + 1)), (s.signedInfDist i : (a : P) → ℝ) p = (if i ∈ signs then -1 else 1) * r),
  ∃ (r : ℝ), ∀ (i : Fin (n + 1)), |(s.signedInfDist i : (a : P) → ℝ) p| = r
Tactic: rcases h with ⟨r, 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
p : P
hp : Membership.mem (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points)) p
signs : Finset (Fin (n + 1))
r : ℝ
h : ∀ (i : Fin (n + 1)), (s.signedInfDist i : (a : P) → ℝ) p = (if i ∈ signs then -1 else 1) * r
⊢ ∃ (r : ℝ), ∀ (i : Fin (n + 1)), |(s.signedInfDist i : (a : P) → ℝ) p| = r
have refine_2 : ∀ (i : Fin (n + 1)), |(s.signedInfDist i : (a : P) → ℝ) p| = |r|
Tactic: refine ⟨|r|, ?_⟩
-- 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)
hij : s.touchpoint signs i = s.touchpoint signs j
⊢ i = j
have by_contra : ∀ (hne : ¬i = j), False
Tactic: by_contra 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))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : s.touchpoint signs i = s.touchpoint signs j
hne : ¬i = j
⊢ False
have pos : ∀ (hn1 : n = 1), False
Tactic: by_cases hn1 : n = 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
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)
hij : s.touchpoint signs i = s.touchpoint signs j
hne : ¬i = j
pos : ∀ (hn1 : n = 1), False
⊢ False
have neg : ∀ (hn1 : ¬n = 1), False
Tactic: by_cases hn1 : n = 1