l3lab/ntp-mathlib huggingface dataset:
/-- Three points are affinely independent if and only if they are not collinear. -/
theorem affineIndependent_iff_not_collinear_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
    (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
    AffineIndependent k p ↔ ¬Collinear k ({p i₁, p i₂, p i₃} : Set P) := by
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    -- Porting note: Originally `by decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> 

Proof state:
case tail.head.head.tail.tail.head
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
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) }
⊢ Finset.univ =
    {{ val := 1, isLt := (_ : (fun a => a < 3) 1) }, { val := 0, isLt := (_ : 0 < 3) },
      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }}

Next tactic:
simp (config := {decide := true}) only at h₁₂ h₁₃ h₂₃ ⊢
/-- Three points are affinely independent if and only if they are not collinear. -/
theorem affineIndependent_iff_not_collinear_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
    (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
    AffineIndependent k p ↔ ¬Collinear k ({p i₁, p i₂, p i₃} : Set P) := by
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    -- Porting note: Originally `by decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> 

Proof state:
case tail.head.tail.head.head
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
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) }
⊢ Finset.univ =
    {{ val := 1, isLt := (_ : (fun a => a < 3) 1) }, { val := 1, isLt := (_ : (fun a => a < 3) 1) },
      { val := 0, isLt := (_ : 0 < 3) }}

Next tactic:
simp (config := {decide := true}) only at h₁₂ h₁₃ h₂₃ ⊢
/-- Three points are affinely independent if and only if they are not collinear. -/
theorem affineIndependent_iff_not_collinear_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
    (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
    AffineIndependent k p ↔ ¬Collinear k ({p i₁, p i₂, p i₃} : Set P) := by
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    -- Porting note: Originally `by decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> 

Proof state:
case tail.head.tail.head.tail.head
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
h₁₂ h₁₃ h₂₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
⊢ Finset.univ =
    {{ val := 1, isLt := (_ : (fun a => a < 3) 1) }, { val := 1, isLt := (_ : (fun a => a < 3) 1) },
      { val := 1, isLt := (_ : (fun a => a < 3) 1) }}

Next tactic:
simp (config := {decide := true}) only at h₁₂ h₁₃ h₂₃ ⊢
/-- Three points are affinely independent if and only if they are not collinear. -/
theorem affineIndependent_iff_not_collinear_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
    (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
    AffineIndependent k p ↔ ¬Collinear k ({p i₁, p i₂, p i₃} : Set P) := by
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    -- Porting note: Originally `by decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> 

Proof state:
case tail.head.tail.head.tail.tail.head
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
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) }
⊢ Finset.univ =
    {{ 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) }}

Next tactic:
simp (config := {decide := true}) only at h₁₂ h₁₃ h₂₃ ⊢
/-- Three points are affinely independent if and only if they are not collinear. -/
theorem affineIndependent_iff_not_collinear_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
    (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
    AffineIndependent k p ↔ ¬Collinear k ({p i₁, p i₂, p i₃} : Set P) := by
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    -- Porting note: Originally `by decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> 

Proof state:
case tail.head.tail.tail.head.head
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
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) }
⊢ Finset.univ =
    {{ 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 (config := {decide := true}) only at h₁₂ h₁₃ h₂₃ ⊢
/-- Three points are affinely independent if and only if they are not collinear. -/
theorem affineIndependent_iff_not_collinear_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
    (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
    AffineIndependent k p ↔ ¬Collinear k ({p i₁, p i₂, p i₃} : Set P) := by
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    -- Porting note: Originally `by decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> 

Proof state:
case tail.head.tail.tail.head.tail.head
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
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) }
⊢ Finset.univ =
    {{ 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 (config := {decide := true}) only at h₁₂ h₁₃ h₂₃ ⊢
/-- Three points are affinely independent if and only if they are not collinear. -/
theorem affineIndependent_iff_not_collinear_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
    (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
    AffineIndependent k p ↔ ¬Collinear k ({p i₁, p i₂, p i₃} : Set P) := by
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    -- Porting note: Originally `by decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> 

Proof state:
case tail.head.tail.tail.head.tail.tail.head
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
h₁₂ h₁₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { 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) }
⊢ Finset.univ =
    {{ 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 (config := {decide := true}) only at h₁₂ h₁₃ h₂₃ ⊢
/-- Three points are affinely independent if and only if they are not collinear. -/
theorem affineIndependent_iff_not_collinear_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
    (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
    AffineIndependent k p ↔ ¬Collinear k ({p i₁, p i₂, p i₃} : Set P) := by
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    -- Porting note: Originally `by decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> 

Proof state:
case tail.tail.head.head.head
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
h₁₂ h₁₃ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 0, isLt := (_ : 0 < 3) }
h₂₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 0, isLt := (_ : 0 < 3) }
⊢ Finset.univ =
    {{ val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }, { val := 0, isLt := (_ : 0 < 3) },
      { val := 0, isLt := (_ : 0 < 3) }}

Next tactic:
simp (config := {decide := true}) only at h₁₂ h₁₃ h₂₃ ⊢
/-- Three points are affinely independent if and only if they are not collinear. -/
theorem affineIndependent_iff_not_collinear_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
    (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
    AffineIndependent k p ↔ ¬Collinear k ({p i₁, p i₂, p i₃} : Set P) := by
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    -- Porting note: Originally `by decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> 

Proof state:
case tail.tail.head.head.tail.head
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
h₁₂ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 0, isLt := (_ : 0 < 3) }
h₁₃ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
h₂₃ : { val := 0, isLt := (_ : 0 < 3) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
⊢ Finset.univ =
    {{ val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }, { val := 0, isLt := (_ : 0 < 3) },
      { val := 1, isLt := (_ : (fun a => a < 3) 1) }}

Next tactic:
simp (config := {decide := true}) only at h₁₂ h₁₃ h₂₃ ⊢
/-- Three points are affinely independent if and only if they are not collinear. -/
theorem affineIndependent_iff_not_collinear_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
    (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
    AffineIndependent k p ↔ ¬Collinear k ({p i₁, p i₂, p i₃} : Set P) := by
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    -- Porting note: Originally `by decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> 

Proof state:
case tail.tail.head.head.tail.tail.head
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
h₁₂ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 0, isLt := (_ : 0 < 3) }
h₁₃ :
  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠
    { 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) }
⊢ Finset.univ =
    {{ 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) }}

Next tactic:
simp (config := {decide := true}) only at h₁₂ h₁₃ h₂₃ ⊢
/-- Three points are affinely independent if and only if they are not collinear. -/
theorem affineIndependent_iff_not_collinear_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
    (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
    AffineIndependent k p ↔ ¬Collinear k ({p i₁, p i₂, p i₃} : Set P) := by
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    -- Porting note: Originally `by decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> 

Proof state:
case tail.tail.head.tail.head.head
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
h₁₂ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
h₁₃ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 0, isLt := (_ : 0 < 3) }
h₂₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 0, isLt := (_ : 0 < 3) }
⊢ Finset.univ =
    {{ val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }, { val := 1, isLt := (_ : (fun a => a < 3) 1) },
      { val := 0, isLt := (_ : 0 < 3) }}

Next tactic:
simp (config := {decide := true}) only at h₁₂ h₁₃ h₂₃ ⊢
/-- Three points are affinely independent if and only if they are not collinear. -/
theorem affineIndependent_iff_not_collinear_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
    (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
    AffineIndependent k p ↔ ¬Collinear k ({p i₁, p i₂, p i₃} : Set P) := by
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    -- Porting note: Originally `by decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> 

Proof state:
case tail.tail.head.tail.head.tail.head
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
h₁₂ h₁₃ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
h₂₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
⊢ Finset.univ =
    {{ val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }, { val := 1, isLt := (_ : (fun a => a < 3) 1) },
      { val := 1, isLt := (_ : (fun a => a < 3) 1) }}

Next tactic:
simp (config := {decide := true}) only at h₁₂ h₁₃ h₂₃ ⊢
/-- Three points are affinely independent if and only if they are not collinear. -/
theorem affineIndependent_iff_not_collinear_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
    (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
    AffineIndependent k p ↔ ¬Collinear k ({p i₁, p i₂, p i₃} : Set P) := by
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    -- Porting note: Originally `by decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> 

Proof state:
case tail.tail.head.tail.head.tail.tail.head
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
h₁₂ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
h₁₃ :
  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₂₃ : { val := 1, isLt := (_ : (fun a => a < 3) 1) } ≠ { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
⊢ Finset.univ =
    {{ 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) }}

Next tactic:
simp (config := {decide := true}) only at h₁₂ h₁₃ h₂₃ ⊢
/-- Three points are affinely independent if and only if they are not collinear. -/
theorem affineIndependent_iff_not_collinear_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
    (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
    AffineIndependent k p ↔ ¬Collinear k ({p i₁, p i₂, p i₃} : Set P) := by
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    -- Porting note: Originally `by decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> 

Proof state:
case tail.tail.head.tail.tail.head.head
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
h₁₂ :
  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₁₃ h₂₃ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 0, isLt := (_ : 0 < 3) }
⊢ Finset.univ =
    {{ 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 (config := {decide := true}) only at h₁₂ h₁₃ h₂₃ ⊢
/-- Three points are affinely independent if and only if they are not collinear. -/
theorem affineIndependent_iff_not_collinear_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
    (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
    AffineIndependent k p ↔ ¬Collinear k ({p i₁, p i₂, p i₃} : Set P) := by
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    -- Porting note: Originally `by decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> 

Proof state:
case tail.tail.head.tail.tail.head.tail.head
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
h₁₂ :
  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
h₁₃ h₂₃ : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠ { val := 1, isLt := (_ : (fun a => a < 3) 1) }
⊢ Finset.univ =
    {{ 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 (config := {decide := true}) only at h₁₂ h₁₃ h₂₃ ⊢
/-- Three points are affinely independent if and only if they are not collinear. -/
theorem affineIndependent_iff_not_collinear_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
    (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
    AffineIndependent k p ↔ ¬Collinear k ({p i₁, p i₂, p i₃} : Set P) := by
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    -- Porting note: Originally `by decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> 

Proof state:
case tail.tail.head.tail.tail.head.tail.tail.head
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
h₁₂ h₁₃ h₂₃ :
  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ≠
    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }
⊢ Finset.univ =
    {{ 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 (config := {decide := true}) only at h₁₂ h₁₃ h₂₃ ⊢
/-- Three points are affinely independent if and only if they are not collinear. -/
theorem affineIndependent_iff_not_collinear_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
    (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
    AffineIndependent k p ↔ ¬Collinear k ({p i₁, p i₂, p i₃} : Set P) := by
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    -- Porting note: Originally `by decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> simp (config := {decide := true}) only at h₁₂ h₁₃ h₂₃ ⊢
  

Proof state:
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
hu : Finset.univ = {i₁, i₂, i₃}
⊢ AffineIndependent k p ↔ ¬Collinear k {p i₁, p i₂, p i₃}

Next tactic:
rw [affineIndependent_iff_not_collinear, ← Set.image_univ, ← Finset.coe_univ, hu,
    Finset.coe_insert, Finset.coe_insert, Finset.coe_singleton, Set.image_insert_eq, Set.image_pair]
/-- If three points are not collinear, the first and second are different. -/
theorem ne₁₂_of_not_collinear {p₁ p₂ p₃ : P} (h : ¬Collinear k ({p₁, p₂, p₃} : Set P)) :
    p₁ ≠ p₂ := by
  

Proof state:
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ p₃ : P
h : ¬Collinear k {p₁, p₂, p₃}
⊢ p₁ ≠ p₂

Next tactic:
rintro rfl
/-- If three points are not collinear, the first and second are different. -/
theorem ne₁₂_of_not_collinear {p₁ p₂ p₃ : P} (h : ¬Collinear k ({p₁, p₂, p₃} : Set P)) :
    p₁ ≠ p₂ := by
  rintro rfl
  

Proof state:
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₃ : P
h : ¬Collinear k {p₁, p₁, p₃}
⊢ False

Next tactic:
simp [collinear_pair] at h
/-- If three points are not collinear, the first and third are different. -/
theorem ne₁₃_of_not_collinear {p₁ p₂ p₃ : P} (h : ¬Collinear k ({p₁, p₂, p₃} : Set P)) :
    p₁ ≠ p₃ := by
  

Proof state:
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ p₃ : P
h : ¬Collinear k {p₁, p₂, p₃}
⊢ p₁ ≠ p₃

Next tactic:
rintro rfl
/-- If three points are not collinear, the first and third are different. -/
theorem ne₁₃_of_not_collinear {p₁ p₂ p₃ : P} (h : ¬Collinear k ({p₁, p₂, p₃} : Set P)) :
    p₁ ≠ p₃ := by
  rintro rfl
  

Proof state:
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ : P
h : ¬Collinear k {p₁, p₂, p₁}
⊢ False

Next tactic:
simp [collinear_pair] at h
/-- If three points are not collinear, the second and third are different. -/
theorem ne₂₃_of_not_collinear {p₁ p₂ p₃ : P} (h : ¬Collinear k ({p₁, p₂, p₃} : Set P)) :
    p₂ ≠ p₃ := by
  

Proof state:
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ p₃ : P
h : ¬Collinear k {p₁, p₂, p₃}
⊢ p₂ ≠ p₃

Next tactic:
rintro rfl
/-- If three points are not collinear, the second and third are different. -/
theorem ne₂₃_of_not_collinear {p₁ p₂ p₃ : P} (h : ¬Collinear k ({p₁, p₂, p₃} : Set P)) :
    p₂ ≠ p₃ := by
  rintro rfl
  

Proof state:
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ : P
h : ¬Collinear k {p₁, p₂, p₂}
⊢ False

Next tactic:
simp [collinear_pair] at h
/-- A point in a collinear set of points lies in the affine span of any two distinct points of
that set. -/
theorem Collinear.mem_affineSpan_of_mem_of_ne {s : Set P} (h : Collinear k s) {p₁ p₂ p₃ : P}
    (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) (hp₃ : p₃ ∈ s) (hp₁p₂ : p₁ ≠ p₂) : p₃ ∈ line[k, p₁, p₂] := by
  

Proof state:
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
h : Collinear k s
p₁ p₂ p₃ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
hp₃ : p₃ ∈ s
hp₁p₂ : p₁ ≠ p₂
⊢ p₃ ∈ affineSpan k {p₁, p₂}

Next tactic:
rw [collinear_iff_of_mem hp₁] at h
/-- A point in a collinear set of points lies in the affine span of any two distinct points of
that set. -/
theorem Collinear.mem_affineSpan_of_mem_of_ne {s : Set P} (h : Collinear k s) {p₁ p₂ p₃ : P}
    (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) (hp₃ : p₃ ∈ s) (hp₁p₂ : p₁ ≠ p₂) : p₃ ∈ line[k, p₁, p₂] := by
  rw [collinear_iff_of_mem hp₁] at h
  

Proof state:
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p₁ : P
h : ∃ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₁
p₂ p₃ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
hp₃ : p₃ ∈ s
hp₁p₂ : p₁ ≠ p₂
⊢ p₃ ∈ affineSpan k {p₁, p₂}

Next tactic:
rcases h with ⟨v, h⟩