l3lab/ntp-mathlib huggingface dataset:
/-- Suppose lines from two vertices of a triangle to interior points of the opposite side meet at
`p`. Then `p` lies in the interior of the first (and by symmetry the other) segment from a
vertex to the point on the opposite side. -/
theorem sbtw_of_sbtw_of_sbtw_of_mem_affineSpan_pair [NoZeroSMulDivisors R V]
    {t : Affine.Triangle R P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) {p₁ p₂ p : P}
    (h₁ : Sbtw R (t.points i₂) p₁ (t.points i₃)) (h₂ : Sbtw R (t.points i₁) p₂ (t.points i₃))
    (h₁' : p ∈ line[R, t.points i₁, p₁]) (h₂' : p ∈ line[R, t.points i₂, p₂]) :
    Sbtw R (t.points i₁) p p₁ := by
  -- Should not be needed; see comments on local instances in `Data.Sign`.
  letI : DecidableRel ((· < ·) : R → R → Prop) := LinearOrderedRing.decidableLT
  have h₁₃ : i₁ ≠ i₃ := by
    rintro rfl
    simp at h₂
  have h₂₃ : i₂ ≠ i₃ := by
    rintro rfl
    simp at h₁
  have h3 : ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    intro i
    fin_cases i <;> fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> simp at h₁₂ h₁₃ h₂₃ ⊢
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> simp (config := {decide := true}) at h₁₂ h₁₃ h₂₃ ⊢
  have hp : p ∈ affineSpan R (Set.range t.points) := by
    

Proof state:
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
p₁ p₂ p : P
h₁ : Sbtw R (Affine.Simplex.points t i₂) p₁ (Affine.Simplex.points t i₃)
h₂ : Sbtw R (Affine.Simplex.points t i₁) p₂ (Affine.Simplex.points t i₃)
h₁' : p ∈ affineSpan R {Affine.Simplex.points t i₁, p₁}
h₂' : p ∈ affineSpan R {Affine.Simplex.points t i₂, p₂}
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
⊢ p ∈ affineSpan R (Set.range t.points)

Next tactic:
have hle : line[R, t.points i₁, p₁] ≤ affineSpan R (Set.range t.points) := by
      refine' affineSpan_pair_le_of_mem_of_mem (mem_affineSpan R (Set.mem_range_self _)) _
      have hle : line[R, t.points i₂, t.points i₃] ≤ affineSpan R (Set.range t.points) := by
        refine' affineSpan_mono R _
        simp [Set.insert_subset_iff]
      rw [AffineSubspace.le_def'] at hle
      exact hle _ h₁.wbtw.mem_affineSpan
/-- Suppose lines from two vertices of a triangle to interior points of the opposite side meet at
`p`. Then `p` lies in the interior of the first (and by symmetry the other) segment from a
vertex to the point on the opposite side. -/
theorem sbtw_of_sbtw_of_sbtw_of_mem_affineSpan_pair [NoZeroSMulDivisors R V]
    {t : Affine.Triangle R P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) {p₁ p₂ p : P}
    (h₁ : Sbtw R (t.points i₂) p₁ (t.points i₃)) (h₂ : Sbtw R (t.points i₁) p₂ (t.points i₃))
    (h₁' : p ∈ line[R, t.points i₁, p₁]) (h₂' : p ∈ line[R, t.points i₂, p₂]) :
    Sbtw R (t.points i₁) p p₁ := by
  -- Should not be needed; see comments on local instances in `Data.Sign`.
  letI : DecidableRel ((· < ·) : R → R → Prop) := LinearOrderedRing.decidableLT
  have h₁₃ : i₁ ≠ i₃ := by
    rintro rfl
    simp at h₂
  have h₂₃ : i₂ ≠ i₃ := by
    rintro rfl
    simp at h₁
  have h3 : ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    intro i
    fin_cases i <;> fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> simp at h₁₂ h₁₃ h₂₃ ⊢
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> simp (config := {decide := true}) at h₁₂ h₁₃ h₂₃ ⊢
  have hp : p ∈ affineSpan R (Set.range t.points) := by
    have hle : line[R, t.points i₁, p₁] ≤ affineSpan R (Set.range t.points) := by
      

Proof state:
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
p₁ p₂ p : P
h₁ : Sbtw R (Affine.Simplex.points t i₂) p₁ (Affine.Simplex.points t i₃)
h₂ : Sbtw R (Affine.Simplex.points t i₁) p₂ (Affine.Simplex.points t i₃)
h₁' : p ∈ affineSpan R {Affine.Simplex.points t i₁, p₁}
h₂' : p ∈ affineSpan R {Affine.Simplex.points t i₂, p₂}
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
⊢ affineSpan R {Affine.Simplex.points t i₁, p₁} ≤ affineSpan R (Set.range t.points)

Next tactic:
refine' affineSpan_pair_le_of_mem_of_mem (mem_affineSpan R (Set.mem_range_self _)) _
/-- Suppose lines from two vertices of a triangle to interior points of the opposite side meet at
`p`. Then `p` lies in the interior of the first (and by symmetry the other) segment from a
vertex to the point on the opposite side. -/
theorem sbtw_of_sbtw_of_sbtw_of_mem_affineSpan_pair [NoZeroSMulDivisors R V]
    {t : Affine.Triangle R P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) {p₁ p₂ p : P}
    (h₁ : Sbtw R (t.points i₂) p₁ (t.points i₃)) (h₂ : Sbtw R (t.points i₁) p₂ (t.points i₃))
    (h₁' : p ∈ line[R, t.points i₁, p₁]) (h₂' : p ∈ line[R, t.points i₂, p₂]) :
    Sbtw R (t.points i₁) p p₁ := by
  -- Should not be needed; see comments on local instances in `Data.Sign`.
  letI : DecidableRel ((· < ·) : R → R → Prop) := LinearOrderedRing.decidableLT
  have h₁₃ : i₁ ≠ i₃ := by
    rintro rfl
    simp at h₂
  have h₂₃ : i₂ ≠ i₃ := by
    rintro rfl
    simp at h₁
  have h3 : ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    intro i
    fin_cases i <;> fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> simp at h₁₂ h₁₃ h₂₃ ⊢
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> simp (config := {decide := true}) at h₁₂ h₁₃ h₂₃ ⊢
  have hp : p ∈ affineSpan R (Set.range t.points) := by
    have hle : line[R, t.points i₁, p₁] ≤ affineSpan R (Set.range t.points) := by
      refine' affineSpan_pair_le_of_mem_of_mem (mem_affineSpan R (Set.mem_range_self _)) _
      

Proof state:
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
p₁ p₂ p : P
h₁ : Sbtw R (Affine.Simplex.points t i₂) p₁ (Affine.Simplex.points t i₃)
h₂ : Sbtw R (Affine.Simplex.points t i₁) p₂ (Affine.Simplex.points t i₃)
h₁' : p ∈ affineSpan R {Affine.Simplex.points t i₁, p₁}
h₂' : p ∈ affineSpan R {Affine.Simplex.points t i₂, p₂}
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
⊢ p₁ ∈ affineSpan R (Set.range t.points)

Next tactic:
have hle : line[R, t.points i₂, t.points i₃] ≤ affineSpan R (Set.range t.points) := by
        refine' affineSpan_mono R _
        simp [Set.insert_subset_iff]
/-- Suppose lines from two vertices of a triangle to interior points of the opposite side meet at
`p`. Then `p` lies in the interior of the first (and by symmetry the other) segment from a
vertex to the point on the opposite side. -/
theorem sbtw_of_sbtw_of_sbtw_of_mem_affineSpan_pair [NoZeroSMulDivisors R V]
    {t : Affine.Triangle R P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) {p₁ p₂ p : P}
    (h₁ : Sbtw R (t.points i₂) p₁ (t.points i₃)) (h₂ : Sbtw R (t.points i₁) p₂ (t.points i₃))
    (h₁' : p ∈ line[R, t.points i₁, p₁]) (h₂' : p ∈ line[R, t.points i₂, p₂]) :
    Sbtw R (t.points i₁) p p₁ := by
  -- Should not be needed; see comments on local instances in `Data.Sign`.
  letI : DecidableRel ((· < ·) : R → R → Prop) := LinearOrderedRing.decidableLT
  have h₁₃ : i₁ ≠ i₃ := by
    rintro rfl
    simp at h₂
  have h₂₃ : i₂ ≠ i₃ := by
    rintro rfl
    simp at h₁
  have h3 : ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    intro i
    fin_cases i <;> fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> simp at h₁₂ h₁₃ h₂₃ ⊢
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> simp (config := {decide := true}) at h₁₂ h₁₃ h₂₃ ⊢
  have hp : p ∈ affineSpan R (Set.range t.points) := by
    have hle : line[R, t.points i₁, p₁] ≤ affineSpan R (Set.range t.points) := by
      refine' affineSpan_pair_le_of_mem_of_mem (mem_affineSpan R (Set.mem_range_self _)) _
      have hle : line[R, t.points i₂, t.points i₃] ≤ affineSpan R (Set.range t.points) := by
        

Proof state:
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
p₁ p₂ p : P
h₁ : Sbtw R (Affine.Simplex.points t i₂) p₁ (Affine.Simplex.points t i₃)
h₂ : Sbtw R (Affine.Simplex.points t i₁) p₂ (Affine.Simplex.points t i₃)
h₁' : p ∈ affineSpan R {Affine.Simplex.points t i₁, p₁}
h₂' : p ∈ affineSpan R {Affine.Simplex.points t i₂, p₂}
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
⊢ affineSpan R {Affine.Simplex.points t i₂, Affine.Simplex.points t i₃} ≤ affineSpan R (Set.range t.points)

Next tactic:
refine' affineSpan_mono R _
/-- Suppose lines from two vertices of a triangle to interior points of the opposite side meet at
`p`. Then `p` lies in the interior of the first (and by symmetry the other) segment from a
vertex to the point on the opposite side. -/
theorem sbtw_of_sbtw_of_sbtw_of_mem_affineSpan_pair [NoZeroSMulDivisors R V]
    {t : Affine.Triangle R P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) {p₁ p₂ p : P}
    (h₁ : Sbtw R (t.points i₂) p₁ (t.points i₃)) (h₂ : Sbtw R (t.points i₁) p₂ (t.points i₃))
    (h₁' : p ∈ line[R, t.points i₁, p₁]) (h₂' : p ∈ line[R, t.points i₂, p₂]) :
    Sbtw R (t.points i₁) p p₁ := by
  -- Should not be needed; see comments on local instances in `Data.Sign`.
  letI : DecidableRel ((· < ·) : R → R → Prop) := LinearOrderedRing.decidableLT
  have h₁₃ : i₁ ≠ i₃ := by
    rintro rfl
    simp at h₂
  have h₂₃ : i₂ ≠ i₃ := by
    rintro rfl
    simp at h₁
  have h3 : ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    intro i
    fin_cases i <;> fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> simp at h₁₂ h₁₃ h₂₃ ⊢
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> simp (config := {decide := true}) at h₁₂ h₁₃ h₂₃ ⊢
  have hp : p ∈ affineSpan R (Set.range t.points) := by
    have hle : line[R, t.points i₁, p₁] ≤ affineSpan R (Set.range t.points) := by
      refine' affineSpan_pair_le_of_mem_of_mem (mem_affineSpan R (Set.mem_range_self _)) _
      have hle : line[R, t.points i₂, t.points i₃] ≤ affineSpan R (Set.range t.points) := by
        refine' affineSpan_mono R _
        

Proof state:
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
p₁ p₂ p : P
h₁ : Sbtw R (Affine.Simplex.points t i₂) p₁ (Affine.Simplex.points t i₃)
h₂ : Sbtw R (Affine.Simplex.points t i₁) p₂ (Affine.Simplex.points t i₃)
h₁' : p ∈ affineSpan R {Affine.Simplex.points t i₁, p₁}
h₂' : p ∈ affineSpan R {Affine.Simplex.points t i₂, p₂}
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
⊢ {Affine.Simplex.points t i₂, Affine.Simplex.points t i₃} ⊆ Set.range t.points

Next tactic:
simp [Set.insert_subset_iff]
/-- Suppose lines from two vertices of a triangle to interior points of the opposite side meet at
`p`. Then `p` lies in the interior of the first (and by symmetry the other) segment from a
vertex to the point on the opposite side. -/
theorem sbtw_of_sbtw_of_sbtw_of_mem_affineSpan_pair [NoZeroSMulDivisors R V]
    {t : Affine.Triangle R P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) {p₁ p₂ p : P}
    (h₁ : Sbtw R (t.points i₂) p₁ (t.points i₃)) (h₂ : Sbtw R (t.points i₁) p₂ (t.points i₃))
    (h₁' : p ∈ line[R, t.points i₁, p₁]) (h₂' : p ∈ line[R, t.points i₂, p₂]) :
    Sbtw R (t.points i₁) p p₁ := by
  -- Should not be needed; see comments on local instances in `Data.Sign`.
  letI : DecidableRel ((· < ·) : R → R → Prop) := LinearOrderedRing.decidableLT
  have h₁₃ : i₁ ≠ i₃ := by
    rintro rfl
    simp at h₂
  have h₂₃ : i₂ ≠ i₃ := by
    rintro rfl
    simp at h₁
  have h3 : ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    intro i
    fin_cases i <;> fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> simp at h₁₂ h₁₃ h₂₃ ⊢
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> simp (config := {decide := true}) at h₁₂ h₁₃ h₂₃ ⊢
  have hp : p ∈ affineSpan R (Set.range t.points) := by
    have hle : line[R, t.points i₁, p₁] ≤ affineSpan R (Set.range t.points) := by
      refine' affineSpan_pair_le_of_mem_of_mem (mem_affineSpan R (Set.mem_range_self _)) _
      have hle : line[R, t.points i₂, t.points i₃] ≤ affineSpan R (Set.range t.points) := by
        refine' affineSpan_mono R _
        simp [Set.insert_subset_iff]
      

Proof state:
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
p₁ p₂ p : P
h₁ : Sbtw R (Affine.Simplex.points t i₂) p₁ (Affine.Simplex.points t i₃)
h₂ : Sbtw R (Affine.Simplex.points t i₁) p₂ (Affine.Simplex.points t i₃)
h₁' : p ∈ affineSpan R {Affine.Simplex.points t i₁, p₁}
h₂' : p ∈ affineSpan R {Affine.Simplex.points t i₂, p₂}
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
hle : affineSpan R {Affine.Simplex.points t i₂, Affine.Simplex.points t i₃} ≤ affineSpan R (Set.range t.points)
⊢ p₁ ∈ affineSpan R (Set.range t.points)

Next tactic:
rw [AffineSubspace.le_def'] at hle
/-- Suppose lines from two vertices of a triangle to interior points of the opposite side meet at
`p`. Then `p` lies in the interior of the first (and by symmetry the other) segment from a
vertex to the point on the opposite side. -/
theorem sbtw_of_sbtw_of_sbtw_of_mem_affineSpan_pair [NoZeroSMulDivisors R V]
    {t : Affine.Triangle R P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) {p₁ p₂ p : P}
    (h₁ : Sbtw R (t.points i₂) p₁ (t.points i₃)) (h₂ : Sbtw R (t.points i₁) p₂ (t.points i₃))
    (h₁' : p ∈ line[R, t.points i₁, p₁]) (h₂' : p ∈ line[R, t.points i₂, p₂]) :
    Sbtw R (t.points i₁) p p₁ := by
  -- Should not be needed; see comments on local instances in `Data.Sign`.
  letI : DecidableRel ((· < ·) : R → R → Prop) := LinearOrderedRing.decidableLT
  have h₁₃ : i₁ ≠ i₃ := by
    rintro rfl
    simp at h₂
  have h₂₃ : i₂ ≠ i₃ := by
    rintro rfl
    simp at h₁
  have h3 : ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    intro i
    fin_cases i <;> fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> simp at h₁₂ h₁₃ h₂₃ ⊢
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> simp (config := {decide := true}) at h₁₂ h₁₃ h₂₃ ⊢
  have hp : p ∈ affineSpan R (Set.range t.points) := by
    have hle : line[R, t.points i₁, p₁] ≤ affineSpan R (Set.range t.points) := by
      refine' affineSpan_pair_le_of_mem_of_mem (mem_affineSpan R (Set.mem_range_self _)) _
      have hle : line[R, t.points i₂, t.points i₃] ≤ affineSpan R (Set.range t.points) := by
        refine' affineSpan_mono R _
        simp [Set.insert_subset_iff]
      rw [AffineSubspace.le_def'] at hle
      

Proof state:
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
p₁ p₂ p : P
h₁ : Sbtw R (Affine.Simplex.points t i₂) p₁ (Affine.Simplex.points t i₃)
h₂ : Sbtw R (Affine.Simplex.points t i₁) p₂ (Affine.Simplex.points t i₃)
h₁' : p ∈ affineSpan R {Affine.Simplex.points t i₁, p₁}
h₂' : p ∈ affineSpan R {Affine.Simplex.points t i₂, p₂}
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
hle : ∀ p ∈ affineSpan R {Affine.Simplex.points t i₂, Affine.Simplex.points t i₃}, p ∈ affineSpan R (Set.range t.points)
⊢ p₁ ∈ affineSpan R (Set.range t.points)

Next tactic:
exact hle _ h₁.wbtw.mem_affineSpan
/-- Suppose lines from two vertices of a triangle to interior points of the opposite side meet at
`p`. Then `p` lies in the interior of the first (and by symmetry the other) segment from a
vertex to the point on the opposite side. -/
theorem sbtw_of_sbtw_of_sbtw_of_mem_affineSpan_pair [NoZeroSMulDivisors R V]
    {t : Affine.Triangle R P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) {p₁ p₂ p : P}
    (h₁ : Sbtw R (t.points i₂) p₁ (t.points i₃)) (h₂ : Sbtw R (t.points i₁) p₂ (t.points i₃))
    (h₁' : p ∈ line[R, t.points i₁, p₁]) (h₂' : p ∈ line[R, t.points i₂, p₂]) :
    Sbtw R (t.points i₁) p p₁ := by
  -- Should not be needed; see comments on local instances in `Data.Sign`.
  letI : DecidableRel ((· < ·) : R → R → Prop) := LinearOrderedRing.decidableLT
  have h₁₃ : i₁ ≠ i₃ := by
    rintro rfl
    simp at h₂
  have h₂₃ : i₂ ≠ i₃ := by
    rintro rfl
    simp at h₁
  have h3 : ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    intro i
    fin_cases i <;> fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> simp at h₁₂ h₁₃ h₂₃ ⊢
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> simp (config := {decide := true}) at h₁₂ h₁₃ h₂₃ ⊢
  have hp : p ∈ affineSpan R (Set.range t.points) := by
    have hle : line[R, t.points i₁, p₁] ≤ affineSpan R (Set.range t.points) := by
      refine' affineSpan_pair_le_of_mem_of_mem (mem_affineSpan R (Set.mem_range_self _)) _
      have hle : line[R, t.points i₂, t.points i₃] ≤ affineSpan R (Set.range t.points) := by
        refine' affineSpan_mono R _
        simp [Set.insert_subset_iff]
      rw [AffineSubspace.le_def'] at hle
      exact hle _ h₁.wbtw.mem_affineSpan
    

Proof state:
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
p₁ p₂ p : P
h₁ : Sbtw R (Affine.Simplex.points t i₂) p₁ (Affine.Simplex.points t i₃)
h₂ : Sbtw R (Affine.Simplex.points t i₁) p₂ (Affine.Simplex.points t i₃)
h₁' : p ∈ affineSpan R {Affine.Simplex.points t i₁, p₁}
h₂' : p ∈ affineSpan R {Affine.Simplex.points t i₂, p₂}
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
hle : affineSpan R {Affine.Simplex.points t i₁, p₁} ≤ affineSpan R (Set.range t.points)
⊢ p ∈ affineSpan R (Set.range t.points)

Next tactic:
rw [AffineSubspace.le_def'] at hle
/-- Suppose lines from two vertices of a triangle to interior points of the opposite side meet at
`p`. Then `p` lies in the interior of the first (and by symmetry the other) segment from a
vertex to the point on the opposite side. -/
theorem sbtw_of_sbtw_of_sbtw_of_mem_affineSpan_pair [NoZeroSMulDivisors R V]
    {t : Affine.Triangle R P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) {p₁ p₂ p : P}
    (h₁ : Sbtw R (t.points i₂) p₁ (t.points i₃)) (h₂ : Sbtw R (t.points i₁) p₂ (t.points i₃))
    (h₁' : p ∈ line[R, t.points i₁, p₁]) (h₂' : p ∈ line[R, t.points i₂, p₂]) :
    Sbtw R (t.points i₁) p p₁ := by
  -- Should not be needed; see comments on local instances in `Data.Sign`.
  letI : DecidableRel ((· < ·) : R → R → Prop) := LinearOrderedRing.decidableLT
  have h₁₃ : i₁ ≠ i₃ := by
    rintro rfl
    simp at h₂
  have h₂₃ : i₂ ≠ i₃ := by
    rintro rfl
    simp at h₁
  have h3 : ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    intro i
    fin_cases i <;> fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> simp at h₁₂ h₁₃ h₂₃ ⊢
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> simp (config := {decide := true}) at h₁₂ h₁₃ h₂₃ ⊢
  have hp : p ∈ affineSpan R (Set.range t.points) := by
    have hle : line[R, t.points i₁, p₁] ≤ affineSpan R (Set.range t.points) := by
      refine' affineSpan_pair_le_of_mem_of_mem (mem_affineSpan R (Set.mem_range_self _)) _
      have hle : line[R, t.points i₂, t.points i₃] ≤ affineSpan R (Set.range t.points) := by
        refine' affineSpan_mono R _
        simp [Set.insert_subset_iff]
      rw [AffineSubspace.le_def'] at hle
      exact hle _ h₁.wbtw.mem_affineSpan
    rw [AffineSubspace.le_def'] at hle
    

Proof state:
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
p₁ p₂ p : P
h₁ : Sbtw R (Affine.Simplex.points t i₂) p₁ (Affine.Simplex.points t i₃)
h₂ : Sbtw R (Affine.Simplex.points t i₁) p₂ (Affine.Simplex.points t i₃)
h₁' : p ∈ affineSpan R {Affine.Simplex.points t i₁, p₁}
h₂' : p ∈ affineSpan R {Affine.Simplex.points t i₂, p₂}
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
hle : ∀ p ∈ affineSpan R {Affine.Simplex.points t i₁, p₁}, p ∈ affineSpan R (Set.range t.points)
⊢ p ∈ affineSpan R (Set.range t.points)

Next tactic:
exact hle _ h₁'