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 <;> 

Proof state:
case tail.tail.head.head.tail.tail.head.head
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
p₁ p₂ p : P
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₂ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₁₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 0, isLt := (_ : 0 < 3) }
h₂₃ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 0, isLt := (_ : 0 < 3) }
⊢ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =
        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∨
      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ ⊢
/-- 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 <;> 

Proof state:
case tail.tail.head.head.tail.tail.head.tail.head
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
p₁ p₂ p : P
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₂ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₁₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
h₂₃ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
⊢ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =
        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∨
      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ ⊢
/-- 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 <;> 

Proof state:
case tail.tail.head.head.tail.tail.head.tail.tail.head
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
p₁ p₂ p : P
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₂ h₁₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₂₃ :
  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
⊢ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =
        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∨
      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =
        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ ⊢
/-- 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 <;> 

Proof state:
case tail.tail.head.tail.head.head.head
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
p₁ p₂ p : P
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₂ h₁₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 0, isLt := (_ : 0 < 3) }
h₂₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 0, isLt := (_ : 0 < 3) }
⊢ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } ∨
      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ ⊢
/-- 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 <;> 

Proof state:
case tail.tail.head.tail.head.head.tail.head
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
p₁ p₂ p : P
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₂ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 0, isLt := (_ : 0 < 3) }
h₁₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
h₂₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
⊢ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } ∨
      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ ⊢
/-- 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 <;> 

Proof state:
case tail.tail.head.tail.head.head.tail.tail.head
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
p₁ p₂ p : P
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₂ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 0, isLt := (_ : 0 < 3) }
h₁₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₂₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
⊢ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } ∨
      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =
        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ ⊢
/-- 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 <;> 

Proof state:
case tail.tail.head.tail.head.tail.head.head
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
p₁ p₂ p : P
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₂ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
h₁₃ h₂₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 0, isLt := (_ : 0 < 3) }
⊢ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∨
      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ ⊢
/-- 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 <;> 

Proof state:
case tail.tail.head.tail.head.tail.head.tail.head
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
p₁ p₂ p : P
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₂ h₁₃ h₂₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
⊢ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∨
      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ ⊢
/-- 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 <;> 

Proof state:
case tail.tail.head.tail.head.tail.head.tail.tail.head
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
p₁ p₂ p : P
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₂ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
h₁₃ h₂₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
⊢ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∨
      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =
        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ ⊢
/-- 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 <;> 

Proof state:
case tail.tail.head.tail.head.tail.tail.head.head
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
p₁ p₂ p : P
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₂ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₁₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 0, isLt := (_ : 0 < 3) }
h₂₃ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 0, isLt := (_ : 0 < 3) }
⊢ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =
        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∨
      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ ⊢
/-- 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 <;> 

Proof state:
case tail.tail.head.tail.head.tail.tail.head.tail.head
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
p₁ p₂ p : P
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₂ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₁₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
h₂₃ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
⊢ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∨
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =
        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∨
      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }

Next tactic:
simp at h₁₂ h₁₃ h₂₃ ⊢