awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t
h0 :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range (s.faceOpposite j).points))
    ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)
pos : ∀ (ht1 : t = 1), False
neg : ∀ (ht1 : ¬t = 1), False
⊢ False

Tactic:
by_cases ht1 : t = 1
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t
h0 :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range (s.faceOpposite j).points))
    ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)
ht1 : t = 1
⊢ Membership.mem (γ := Set ℝ) (Set.Icc 0 1) 1

Tactic:
subst ht1
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t
h0 :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range (s.faceOpposite j).points))
    ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)
ht1 : t = 1
ht₁ : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) 1
⊢ Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range (s.faceOpposite j).points))
    ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) 1 : P)

Tactic:
subst ht1
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) 1
h0 :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range (s.faceOpposite j).points))
    ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) 1 : P)
⊢ Membership.mem (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range (s.faceOpposite j).points)) (s.touchpoint signs i)

Tactic:
rw [AffineMap.lineMap_apply_one] at h0

Premises used: AffineMap.lineMap_apply_one
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) 1
h0 :
  Membership.mem (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range (s.faceOpposite j).points)) (s.touchpoint signs i)
⊢ False

Tactic:
h.touchpoint_notMem_affineSpan_of_ne hne h0

Premises used: Affine.Simplex.ExcenterExists.touchpoint_notMem_affineSpan_of_ne
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) 1
h0 :
  Membership.mem (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range (s.faceOpposite j).points)) (s.touchpoint signs i)
⊢ False

Tactic:
exact h.touchpoint_notMem_affineSpan_of_ne hne h0

Premises used: Affine.Simplex.ExcenterExists.touchpoint_notMem_affineSpan_of_ne
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t
h0 :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range (s.faceOpposite j).points))
    ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)
ht1 : ¬t = 1
⊢ False

Tactic:
(h.isTangentAt_touchpoint j).isTangent.notMem_of_dist_lt ?_ h0

Premises used: Affine.Simplex.ExcenterExists.isTangentAt_touchpoint, EuclideanGeometry.Sphere.IsTangent.notMem_of_dist_lt, EuclideanGeometry.Sphere.IsTangentAt.isTangent
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t
h0 :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range (s.faceOpposite j).points))
    ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)
ht1 : ¬t = 1
neg :
  dist ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t) (s.exsphere signs).center <
    (s.exsphere signs).radius
⊢ False

Tactic:
refine (h.isTangentAt_touchpoint j).isTangent.notMem_of_dist_lt ?_ h0

Premises used: Affine.Simplex.ExcenterExists.isTangentAt_touchpoint, EuclideanGeometry.Sphere.IsTangent.notMem_of_dist_lt, EuclideanGeometry.Sphere.IsTangentAt.isTangent
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t
h0 :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range (s.faceOpposite j).points))
    ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)
ht1 : ¬t = 1
neg : |t| < 1
⊢ dist ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t) (s.exsphere signs).center <
    (s.exsphere signs).radius

Tactic:
simp only [exsphere_center, dist_lineMap_left, Real.norm_eq_abs, h.dist_excenter,
            exsphere_radius, h.exradius_pos, mul_lt_iff_lt_one_left]

Premises used: Affine.Simplex.ExcenterExists.dist_excenter, dist_lineMap_left, mul_lt_iff_lt_one_left, Affine.Simplex.exsphere_center, Affine.Simplex.exsphere_radius, Real.norm_eq_abs, Affine.Simplex.ExcenterExists.exradius_pos
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t
h0 :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range (s.faceOpposite j).points))
    ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)
ht1 : ¬t = 1
neg : -1 < t ∧ t < 1
⊢ |t| < 1

Tactic:
rw [abs_lt]

Premises used: abs_lt
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t
h0 :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range (s.faceOpposite j).points))
    ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)
ht1 : ¬t = 1
⊢ 0 ≤ t

Tactic:
rcases ht with ⟨ht0, ht1'⟩
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t
h0 :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range (s.faceOpposite j).points))
    ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)
ht1 : ¬t = 1
ht0 : 0 ≤ t
⊢ t ≤ 1

Tactic:
rcases ht with ⟨ht0, ht1'⟩
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
h0 :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range (s.faceOpposite j).points))
    ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)
ht1 : ¬t = 1
ht0 : 0 ≤ t
ht1' : t ≤ 1
linarith : t ≤ -1 → -1 + (0 - t) + (t - -1) = 0
⊢ -1 < t

Tactic:
linarith
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
h0 :
  Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range (s.faceOpposite j).points))
    ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)
ht1 : ¬t = 1
ht0 : 0 ≤ t
ht1' : t ≤ 1
⊢ -1 < t ∧ t < 1

Tactic:
exact ⟨by linarith, ht1'.lt_of_ne ht1⟩
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t
h0 :
  Eq (α := ℝ)
    (|(s.signedInfDist j : (a : P) → ℝ)
          ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)| :
      ℝ)
    (0 : ℝ)
⊢ Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points))
    ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)

Tactic:
AffineMap.lineMap_mem _ h.excenter_mem_affineSpan_range
          (s.touchpoint_mem_affineSpan_simplex _ _)

Premises used: AffineMap.lineMap_mem, Affine.Simplex.ExcenterExists.excenter_mem_affineSpan_range, Affine.Simplex.touchpoint_mem_affineSpan_simplex
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t
h0 :
  Eq (α := ℝ)
    (|(s.signedInfDist j : (a : P) → ℝ)
          ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)| :
      ℝ)
    (0 : ℝ)
⊢ Membership.mem (α := P) (γ := AffineSubspace ℝ P) (affineSpan ℝ (Set.range s.points))
    ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P)

Tactic:
exact AffineMap.lineMap_mem _ h.excenter_mem_affineSpan_range
          (s.touchpoint_mem_affineSpan_simplex _ _)

Premises used: AffineMap.lineMap_mem, Affine.Simplex.ExcenterExists.excenter_mem_affineSpan_range, Affine.Simplex.touchpoint_mem_affineSpan_simplex
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t
refine_2 :
  Continuous.{0, u_2} (X := ℝ) (Y := P)
    (⇑(↑(ContinuousAffineMap.lineMap (s.excenter signs) (s.touchpoint signs i)) : ℝ →ᵃ[ℝ] P) : (a : ℝ) → P)
⊢ Continuous.{0, u_2} (X := ℝ) (Y := P) (⇑(AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i)) : (a : ℝ) → P)

Tactic:
rw [← ContinuousAffineMap.lineMap_toAffineMap]

Premises used: ContinuousAffineMap.lineMap_toAffineMap
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t
⊢ Continuous.{0, u_2} (X := ℝ) (Y := P)
    (⇑(↑(ContinuousAffineMap.lineMap (s.excenter signs) (s.touchpoint signs i)) : ℝ →ᵃ[ℝ] P) : (a : ℝ) → P)

Tactic:
ContinuousAffineMap.cont _

Premises used: ContinuousAffineMap.cont
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) t
⊢ Continuous.{0, u_2} (X := ℝ) (Y := P)
    (⇑(↑(ContinuousAffineMap.lineMap (s.excenter signs) (s.touchpoint signs i)) : ℝ →ᵃ[ℝ] P) : (a : ℝ) → P)

Tactic:
exact ContinuousAffineMap.cont _

Premises used: ContinuousAffineMap.cont
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
⊢ ContinuousOn (Y := SignType)
    (fun (t : ℝ) ↦
      (SignType.sign : (a : ℝ) → SignType)
        ((s.signedInfDist j : (a : P) → ℝ)
            ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P) :
          ℝ))
    (Set.Icc 0 1)

Tactic:
have hc : ContinuousOn (fun (t : ℝ) ↦ SignType.sign
      (s.signedInfDist j (AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) t)))
      (Set.Icc 0 1) := by
    refine continuousOn_of_forall_continuousAt
      fun t ht ↦ ((continuousAt_sign_of_ne_zero ?_).comp
        (((s.signedInfDist j).cont.comp ?_).continuousAt))
    · intro h0
      rw [← abs_eq_zero, abs_signedInfDist_eq_dist_of_mem_affineSpan_range] at h0
      · rw [orthogonalProjectionSpan, dist_orthogonalProjection_eq_zero_iff] at h0
        by_cases ht1 : t = 1
        · subst ht1
          rw [AffineMap.lineMap_apply_one] at h0
          exact h.touchpoint_notMem_affineSpan_of_ne hne h0
        · refine (h.isTangentAt_touchpoint j).isTangent.notMem_of_dist_lt ?_ h0
          simp only [exsphere_center, dist_lineMap_left, Real.norm_eq_abs, h.dist_excenter,
            exsphere_radius, h.exradius_pos, mul_lt_iff_lt_one_left]
          rw [abs_lt]
          rcases ht with ⟨ht0, ht1'⟩
          exact ⟨by linarith, ht1'.lt_of_ne ht1⟩
      · exact AffineMap.lineMap_mem _ h.excenter_mem_affineSpan_range
          (s.touchpoint_mem_affineSpan_simplex _ _)
    · rw [← ContinuousAffineMap.lineMap_toAffineMap]
      exact ContinuousAffineMap.cont _
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
hc :
  ContinuousOn (Y := SignType)
    (fun (t : ℝ) ↦
      (SignType.sign : (a : ℝ) → SignType)
        ((s.signedInfDist j : (a : P) → ℝ)
            ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P) :
          ℝ))
    (Set.Icc 0 1)
⊢ Eq (α := SignType)
    ((SignType.sign : (a : ℝ) → SignType)
        ((s.signedInfDist j : (a : P) → ℝ)
            ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) r : P) :
          ℝ) :
      SignType)
    ((SignType.sign : (a : ℝ) → SignType) ((s.signedInfDist j : (a : P) → ℝ) (s.excenter signs) : ℝ) : SignType)

Tactic:
((isConnected_Icc zero_le_one).image _ hc).isPreconnected.subsingleton
    (Set.mem_image_of_mem _ hr) ?_

Premises used: IsConnected.image, isConnected_Icc, IsPreconnected.subsingleton, zero_le_one, IsConnected.isPreconnected, Set.mem_image_of_mem
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
hc :
  ContinuousOn (Y := SignType)
    (fun (t : ℝ) ↦
      (SignType.sign : (a : ℝ) → SignType)
        ((s.signedInfDist j : (a : P) → ℝ)
            ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P) :
          ℝ))
    (Set.Icc 0 1)
refine :
  Membership.mem (α := SignType) (γ := Set SignType)
    ((fun (t : ℝ) ↦
        (SignType.sign : (a : ℝ) → SignType)
          ((s.signedInfDist j : (a : P) → ℝ)
              ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P) :
            ℝ)) ''
      Set.Icc 0 1)
    ((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.signedInfDist j : (a : P) → ℝ)
            ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) r : P) :
          ℝ) :
      SignType)
    ((SignType.sign : (a : ℝ) → SignType) ((s.signedInfDist j : (a : P) → ℝ) (s.excenter signs) : ℝ) : SignType)

Tactic:
refine ((isConnected_Icc zero_le_one).image _ hc).isPreconnected.subsingleton
    (Set.mem_image_of_mem _ hr) ?_

Premises used: IsConnected.image, isConnected_Icc, IsPreconnected.subsingleton, zero_le_one, IsConnected.isPreconnected, Set.mem_image_of_mem
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
hc :
  ContinuousOn (Y := SignType)
    (fun (t : ℝ) ↦
      (SignType.sign : (a : ℝ) → SignType)
        ((s.signedInfDist j : (a : P) → ℝ)
            ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P) :
          ℝ))
    (Set.Icc 0 1)
⊢ Membership.mem (α := SignType) (γ := Set SignType)
    ((fun (t : ℝ) ↦
        (SignType.sign : (a : ℝ) → SignType)
          ((s.signedInfDist j : (a : P) → ℝ)
              ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P) :
            ℝ)) ''
      Set.Icc 0 1)
    ((SignType.sign : (a : ℝ) → SignType)
        ((s.signedInfDist j : (a : P) → ℝ)
            ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) 0 : P) :
          ℝ) :
      SignType)

Tactic:
Set.mem_image_of_mem _ (Set.left_mem_Icc.2 (zero_le_one' ℝ))

Premises used: Iff.mpr, Real, zero_le_one', Set.left_mem_Icc, Set.mem_image_of_mem
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
hc :
  ContinuousOn (Y := SignType)
    (fun (t : ℝ) ↦
      (SignType.sign : (a : ℝ) → SignType)
        ((s.signedInfDist j : (a : P) → ℝ)
            ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P) :
          ℝ))
    (Set.Icc 0 1)
h₁ : s.excenter signs = (AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) 0
⊢ Membership.mem (α := SignType) (γ := Set SignType)
    ((fun (t : ℝ) ↦
        (SignType.sign : (a : ℝ) → SignType)
          ((s.signedInfDist j : (a : P) → ℝ)
              ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P) :
            ℝ)) ''
      Set.Icc 0 1)
    ((SignType.sign : (a : ℝ) → SignType) ((s.signedInfDist j : (a : P) → ℝ) (s.excenter signs) : ℝ) : SignType)

Tactic:
convert! Set.mem_image_of_mem _ (Set.left_mem_Icc.2 (zero_le_one' ℝ))

Premises used: Iff.mpr, Real, zero_le_one', Set.left_mem_Icc, Set.mem_image_of_mem
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
hc :
  ContinuousOn (Y := SignType)
    (fun (t : ℝ) ↦
      (SignType.sign : (a : ℝ) → SignType)
        ((s.signedInfDist j : (a : P) → ℝ)
            ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) t : P) :
          ℝ))
    (Set.Icc 0 1)
⊢ s.excenter signs = (AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i) : (a : ℝ) → P) 0

Tactic:
simp
-- Module: Mathlib.Geometry.Euclidean.Incenter
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
r : ℝ
hr : Membership.mem (γ := Set ℝ) (Set.Icc 0 1) r
⊢ Eq (α := SignType)
    ((SignType.sign : (a : ℝ) → SignType)
        ((s.signedInfDist j : (a : P) → ℝ) ((AffineMap.lineMap s.incenter (s.touchpoint ∅ i) : (a : ℝ) → P) r : P) :
          ℝ) :
      SignType)
    ((SignType.sign : (a : ℝ) → SignType) ((s.signedInfDist j : (a : P) → ℝ) s.incenter : ℝ) : SignType)

Tactic:
s.excenterExists_empty.sign_signedInfDist_lineMap_excenter_touchpoint hne hr

Premises used: Affine.Simplex.incenter, Affine.Simplex.ExcenterExists.sign_signedInfDist_lineMap_excenter_touchpoint, Affine.Simplex.excenterExists_empty