l3lab/ntp-mathlib huggingface dataset:
/-- Any three points in a cospherical set are affinely independent. -/
theorem Cospherical.affineIndependent {s : Set P} (hs : Cospherical s) {p : Fin 3 → P}
    (hps : Set.range p ⊆ s) (hpi : Function.Injective p) : AffineIndependent ℝ p := by
  rw [affineIndependent_iff_not_collinear]
  intro hc
  rw [collinear_iff_of_mem (Set.mem_range_self (0 : Fin 3))] at hc
  rcases hc with ⟨v, hv⟩
  rw [Set.forall_range_iff] at hv
  have hv0 : v ≠ 0 := by
    intro h
    have he : p 1 = p 0 := by simpa [h] using hv 1
    exact (by decide : (1 : Fin 3) ≠ 0) (hpi he)
  rcases hs with ⟨c, r, hs⟩
  have hs' := fun i => hs (p i) (Set.mem_of_mem_of_subset (Set.mem_range_self _) hps)
  choose f hf using hv
  have hsd : ∀ i, dist (f i • v +ᵥ p 0) c = r := by
    intro i
    rw [← hf]
    exact hs' i
  have hf0 : f 0 = 0 := by
    have hf0' := hf 0
    rw [eq_comm, ← @vsub_eq_zero_iff_eq V, vadd_vsub, smul_eq_zero] at hf0'
    simpa [hv0] using hf0'
  have hfi : Function.Injective f := by
    intro i j h
    have hi := hf i
    

Proof state:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
v : V
hv0 : v ≠ 0
c : P
r : ℝ
hs : ∀ p ∈ s, dist p c = r
hs' : ∀ (i : Fin 3), dist (p i) c = r
f : Fin 3 → ℝ
hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0
hsd : ∀ (i : Fin 3), dist (f i • v +ᵥ p 0) c = r
hf0 : f 0 = 0
i j : Fin 3
h : f i = f j
hi : p i = f i • v +ᵥ p 0
⊢ i = j

Next tactic:
rw [h, ← hf j] at hi
/-- Any three points in a cospherical set are affinely independent. -/
theorem Cospherical.affineIndependent {s : Set P} (hs : Cospherical s) {p : Fin 3 → P}
    (hps : Set.range p ⊆ s) (hpi : Function.Injective p) : AffineIndependent ℝ p := by
  rw [affineIndependent_iff_not_collinear]
  intro hc
  rw [collinear_iff_of_mem (Set.mem_range_self (0 : Fin 3))] at hc
  rcases hc with ⟨v, hv⟩
  rw [Set.forall_range_iff] at hv
  have hv0 : v ≠ 0 := by
    intro h
    have he : p 1 = p 0 := by simpa [h] using hv 1
    exact (by decide : (1 : Fin 3) ≠ 0) (hpi he)
  rcases hs with ⟨c, r, hs⟩
  have hs' := fun i => hs (p i) (Set.mem_of_mem_of_subset (Set.mem_range_self _) hps)
  choose f hf using hv
  have hsd : ∀ i, dist (f i • v +ᵥ p 0) c = r := by
    intro i
    rw [← hf]
    exact hs' i
  have hf0 : f 0 = 0 := by
    have hf0' := hf 0
    rw [eq_comm, ← @vsub_eq_zero_iff_eq V, vadd_vsub, smul_eq_zero] at hf0'
    simpa [hv0] using hf0'
  have hfi : Function.Injective f := by
    intro i j h
    have hi := hf i
    rw [h, ← hf j] at hi
    

Proof state:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
v : V
hv0 : v ≠ 0
c : P
r : ℝ
hs : ∀ p ∈ s, dist p c = r
hs' : ∀ (i : Fin 3), dist (p i) c = r
f : Fin 3 → ℝ
hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0
hsd : ∀ (i : Fin 3), dist (f i • v +ᵥ p 0) c = r
hf0 : f 0 = 0
i j : Fin 3
h : f i = f j
hi : p i = p j
⊢ i = j

Next tactic:
exact hpi hi
/-- Any three points in a cospherical set are affinely independent. -/
theorem Cospherical.affineIndependent {s : Set P} (hs : Cospherical s) {p : Fin 3 → P}
    (hps : Set.range p ⊆ s) (hpi : Function.Injective p) : AffineIndependent ℝ p := by
  rw [affineIndependent_iff_not_collinear]
  intro hc
  rw [collinear_iff_of_mem (Set.mem_range_self (0 : Fin 3))] at hc
  rcases hc with ⟨v, hv⟩
  rw [Set.forall_range_iff] at hv
  have hv0 : v ≠ 0 := by
    intro h
    have he : p 1 = p 0 := by simpa [h] using hv 1
    exact (by decide : (1 : Fin 3) ≠ 0) (hpi he)
  rcases hs with ⟨c, r, hs⟩
  have hs' := fun i => hs (p i) (Set.mem_of_mem_of_subset (Set.mem_range_self _) hps)
  choose f hf using hv
  have hsd : ∀ i, dist (f i • v +ᵥ p 0) c = r := by
    intro i
    rw [← hf]
    exact hs' i
  have hf0 : f 0 = 0 := by
    have hf0' := hf 0
    rw [eq_comm, ← @vsub_eq_zero_iff_eq V, vadd_vsub, smul_eq_zero] at hf0'
    simpa [hv0] using hf0'
  have hfi : Function.Injective f := by
    intro i j h
    have hi := hf i
    rw [h, ← hf j] at hi
    exact hpi hi
  

Proof state:
case intro.intro.intro
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
v : V
hv0 : v ≠ 0
c : P
r : ℝ
hs : ∀ p ∈ s, dist p c = r
hs' : ∀ (i : Fin 3), dist (p i) c = r
f : Fin 3 → ℝ
hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0
hsd : ∀ (i : Fin 3), dist (f i • v +ᵥ p 0) c = r
hf0 : f 0 = 0
hfi : Function.Injective f
⊢ False

Next tactic:
simp_rw [← hsd 0, hf0, zero_smul, zero_vadd, dist_smul_vadd_eq_dist (p 0) c hv0] at hsd
/-- Any three points in a cospherical set are affinely independent. -/
theorem Cospherical.affineIndependent {s : Set P} (hs : Cospherical s) {p : Fin 3 → P}
    (hps : Set.range p ⊆ s) (hpi : Function.Injective p) : AffineIndependent ℝ p := by
  rw [affineIndependent_iff_not_collinear]
  intro hc
  rw [collinear_iff_of_mem (Set.mem_range_self (0 : Fin 3))] at hc
  rcases hc with ⟨v, hv⟩
  rw [Set.forall_range_iff] at hv
  have hv0 : v ≠ 0 := by
    intro h
    have he : p 1 = p 0 := by simpa [h] using hv 1
    exact (by decide : (1 : Fin 3) ≠ 0) (hpi he)
  rcases hs with ⟨c, r, hs⟩
  have hs' := fun i => hs (p i) (Set.mem_of_mem_of_subset (Set.mem_range_self _) hps)
  choose f hf using hv
  have hsd : ∀ i, dist (f i • v +ᵥ p 0) c = r := by
    intro i
    rw [← hf]
    exact hs' i
  have hf0 : f 0 = 0 := by
    have hf0' := hf 0
    rw [eq_comm, ← @vsub_eq_zero_iff_eq V, vadd_vsub, smul_eq_zero] at hf0'
    simpa [hv0] using hf0'
  have hfi : Function.Injective f := by
    intro i j h
    have hi := hf i
    rw [h, ← hf j] at hi
    exact hpi hi
  simp_rw [← hsd 0, hf0, zero_smul, zero_vadd, dist_smul_vadd_eq_dist (p 0) c hv0] at hsd
  

Proof state:
case intro.intro.intro
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
v : V
hv0 : v ≠ 0
c : P
r : ℝ
hs : ∀ p ∈ s, dist p c = r
hs' : ∀ (i : Fin 3), dist (p i) c = r
f : Fin 3 → ℝ
hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0
hf0 : f 0 = 0
hfi : Function.Injective f
hsd : ∀ (i : Fin 3), f i = 0 ∨ f i = -2 * inner v (p 0 -ᵥ c) / inner v v
⊢ False

Next tactic:
have hfn0 : ∀ i, i ≠ 0 → f i ≠ 0 := fun i => (hfi.ne_iff' hf0).2
/-- Any three points in a cospherical set are affinely independent. -/
theorem Cospherical.affineIndependent {s : Set P} (hs : Cospherical s) {p : Fin 3 → P}
    (hps : Set.range p ⊆ s) (hpi : Function.Injective p) : AffineIndependent ℝ p := by
  rw [affineIndependent_iff_not_collinear]
  intro hc
  rw [collinear_iff_of_mem (Set.mem_range_self (0 : Fin 3))] at hc
  rcases hc with ⟨v, hv⟩
  rw [Set.forall_range_iff] at hv
  have hv0 : v ≠ 0 := by
    intro h
    have he : p 1 = p 0 := by simpa [h] using hv 1
    exact (by decide : (1 : Fin 3) ≠ 0) (hpi he)
  rcases hs with ⟨c, r, hs⟩
  have hs' := fun i => hs (p i) (Set.mem_of_mem_of_subset (Set.mem_range_self _) hps)
  choose f hf using hv
  have hsd : ∀ i, dist (f i • v +ᵥ p 0) c = r := by
    intro i
    rw [← hf]
    exact hs' i
  have hf0 : f 0 = 0 := by
    have hf0' := hf 0
    rw [eq_comm, ← @vsub_eq_zero_iff_eq V, vadd_vsub, smul_eq_zero] at hf0'
    simpa [hv0] using hf0'
  have hfi : Function.Injective f := by
    intro i j h
    have hi := hf i
    rw [h, ← hf j] at hi
    exact hpi hi
  simp_rw [← hsd 0, hf0, zero_smul, zero_vadd, dist_smul_vadd_eq_dist (p 0) c hv0] at hsd
  have hfn0 : ∀ i, i ≠ 0 → f i ≠ 0 := fun i => (hfi.ne_iff' hf0).2
  

Proof state:
case intro.intro.intro
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
v : V
hv0 : v ≠ 0
c : P
r : ℝ
hs : ∀ p ∈ s, dist p c = r
hs' : ∀ (i : Fin 3), dist (p i) c = r
f : Fin 3 → ℝ
hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0
hf0 : f 0 = 0
hfi : Function.Injective f
hsd : ∀ (i : Fin 3), f i = 0 ∨ f i = -2 * inner v (p 0 -ᵥ c) / inner v v
hfn0 : ∀ (i : Fin 3), i ≠ 0 → f i ≠ 0
⊢ False

Next tactic:
have hfn0' : ∀ i, i ≠ 0 → f i = -2 * ⟪v, p 0 -ᵥ c⟫ / ⟪v, v⟫ := by
    intro i hi
    have hsdi := hsd i
    simpa [hfn0, hi] using hsdi
/-- Any three points in a cospherical set are affinely independent. -/
theorem Cospherical.affineIndependent {s : Set P} (hs : Cospherical s) {p : Fin 3 → P}
    (hps : Set.range p ⊆ s) (hpi : Function.Injective p) : AffineIndependent ℝ p := by
  rw [affineIndependent_iff_not_collinear]
  intro hc
  rw [collinear_iff_of_mem (Set.mem_range_self (0 : Fin 3))] at hc
  rcases hc with ⟨v, hv⟩
  rw [Set.forall_range_iff] at hv
  have hv0 : v ≠ 0 := by
    intro h
    have he : p 1 = p 0 := by simpa [h] using hv 1
    exact (by decide : (1 : Fin 3) ≠ 0) (hpi he)
  rcases hs with ⟨c, r, hs⟩
  have hs' := fun i => hs (p i) (Set.mem_of_mem_of_subset (Set.mem_range_self _) hps)
  choose f hf using hv
  have hsd : ∀ i, dist (f i • v +ᵥ p 0) c = r := by
    intro i
    rw [← hf]
    exact hs' i
  have hf0 : f 0 = 0 := by
    have hf0' := hf 0
    rw [eq_comm, ← @vsub_eq_zero_iff_eq V, vadd_vsub, smul_eq_zero] at hf0'
    simpa [hv0] using hf0'
  have hfi : Function.Injective f := by
    intro i j h
    have hi := hf i
    rw [h, ← hf j] at hi
    exact hpi hi
  simp_rw [← hsd 0, hf0, zero_smul, zero_vadd, dist_smul_vadd_eq_dist (p 0) c hv0] at hsd
  have hfn0 : ∀ i, i ≠ 0 → f i ≠ 0 := fun i => (hfi.ne_iff' hf0).2
  have hfn0' : ∀ i, i ≠ 0 → f i = -2 * ⟪v, p 0 -ᵥ c⟫ / ⟪v, v⟫ := by
    

Proof state:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
v : V
hv0 : v ≠ 0
c : P
r : ℝ
hs : ∀ p ∈ s, dist p c = r
hs' : ∀ (i : Fin 3), dist (p i) c = r
f : Fin 3 → ℝ
hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0
hf0 : f 0 = 0
hfi : Function.Injective f
hsd : ∀ (i : Fin 3), f i = 0 ∨ f i = -2 * inner v (p 0 -ᵥ c) / inner v v
hfn0 : ∀ (i : Fin 3), i ≠ 0 → f i ≠ 0
⊢ ∀ (i : Fin 3), i ≠ 0 → f i = -2 * inner v (p 0 -ᵥ c) / inner v v

Next tactic:
intro i hi
/-- Any three points in a cospherical set are affinely independent. -/
theorem Cospherical.affineIndependent {s : Set P} (hs : Cospherical s) {p : Fin 3 → P}
    (hps : Set.range p ⊆ s) (hpi : Function.Injective p) : AffineIndependent ℝ p := by
  rw [affineIndependent_iff_not_collinear]
  intro hc
  rw [collinear_iff_of_mem (Set.mem_range_self (0 : Fin 3))] at hc
  rcases hc with ⟨v, hv⟩
  rw [Set.forall_range_iff] at hv
  have hv0 : v ≠ 0 := by
    intro h
    have he : p 1 = p 0 := by simpa [h] using hv 1
    exact (by decide : (1 : Fin 3) ≠ 0) (hpi he)
  rcases hs with ⟨c, r, hs⟩
  have hs' := fun i => hs (p i) (Set.mem_of_mem_of_subset (Set.mem_range_self _) hps)
  choose f hf using hv
  have hsd : ∀ i, dist (f i • v +ᵥ p 0) c = r := by
    intro i
    rw [← hf]
    exact hs' i
  have hf0 : f 0 = 0 := by
    have hf0' := hf 0
    rw [eq_comm, ← @vsub_eq_zero_iff_eq V, vadd_vsub, smul_eq_zero] at hf0'
    simpa [hv0] using hf0'
  have hfi : Function.Injective f := by
    intro i j h
    have hi := hf i
    rw [h, ← hf j] at hi
    exact hpi hi
  simp_rw [← hsd 0, hf0, zero_smul, zero_vadd, dist_smul_vadd_eq_dist (p 0) c hv0] at hsd
  have hfn0 : ∀ i, i ≠ 0 → f i ≠ 0 := fun i => (hfi.ne_iff' hf0).2
  have hfn0' : ∀ i, i ≠ 0 → f i = -2 * ⟪v, p 0 -ᵥ c⟫ / ⟪v, v⟫ := by
    intro i hi
    

Proof state:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
v : V
hv0 : v ≠ 0
c : P
r : ℝ
hs : ∀ p ∈ s, dist p c = r
hs' : ∀ (i : Fin 3), dist (p i) c = r
f : Fin 3 → ℝ
hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0
hf0 : f 0 = 0
hfi : Function.Injective f
hsd : ∀ (i : Fin 3), f i = 0 ∨ f i = -2 * inner v (p 0 -ᵥ c) / inner v v
hfn0 : ∀ (i : Fin 3), i ≠ 0 → f i ≠ 0
i : Fin 3
hi : i ≠ 0
⊢ f i = -2 * inner v (p 0 -ᵥ c) / inner v v

Next tactic:
have hsdi := hsd i
/-- Any three points in a cospherical set are affinely independent. -/
theorem Cospherical.affineIndependent {s : Set P} (hs : Cospherical s) {p : Fin 3 → P}
    (hps : Set.range p ⊆ s) (hpi : Function.Injective p) : AffineIndependent ℝ p := by
  rw [affineIndependent_iff_not_collinear]
  intro hc
  rw [collinear_iff_of_mem (Set.mem_range_self (0 : Fin 3))] at hc
  rcases hc with ⟨v, hv⟩
  rw [Set.forall_range_iff] at hv
  have hv0 : v ≠ 0 := by
    intro h
    have he : p 1 = p 0 := by simpa [h] using hv 1
    exact (by decide : (1 : Fin 3) ≠ 0) (hpi he)
  rcases hs with ⟨c, r, hs⟩
  have hs' := fun i => hs (p i) (Set.mem_of_mem_of_subset (Set.mem_range_self _) hps)
  choose f hf using hv
  have hsd : ∀ i, dist (f i • v +ᵥ p 0) c = r := by
    intro i
    rw [← hf]
    exact hs' i
  have hf0 : f 0 = 0 := by
    have hf0' := hf 0
    rw [eq_comm, ← @vsub_eq_zero_iff_eq V, vadd_vsub, smul_eq_zero] at hf0'
    simpa [hv0] using hf0'
  have hfi : Function.Injective f := by
    intro i j h
    have hi := hf i
    rw [h, ← hf j] at hi
    exact hpi hi
  simp_rw [← hsd 0, hf0, zero_smul, zero_vadd, dist_smul_vadd_eq_dist (p 0) c hv0] at hsd
  have hfn0 : ∀ i, i ≠ 0 → f i ≠ 0 := fun i => (hfi.ne_iff' hf0).2
  have hfn0' : ∀ i, i ≠ 0 → f i = -2 * ⟪v, p 0 -ᵥ c⟫ / ⟪v, v⟫ := by
    intro i hi
    have hsdi := hsd i
    

Proof state:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
v : V
hv0 : v ≠ 0
c : P
r : ℝ
hs : ∀ p ∈ s, dist p c = r
hs' : ∀ (i : Fin 3), dist (p i) c = r
f : Fin 3 → ℝ
hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0
hf0 : f 0 = 0
hfi : Function.Injective f
hsd : ∀ (i : Fin 3), f i = 0 ∨ f i = -2 * inner v (p 0 -ᵥ c) / inner v v
hfn0 : ∀ (i : Fin 3), i ≠ 0 → f i ≠ 0
i : Fin 3
hi : i ≠ 0
hsdi : f i = 0 ∨ f i = -2 * inner v (p 0 -ᵥ c) / inner v v
⊢ f i = -2 * inner v (p 0 -ᵥ c) / inner v v

Next tactic:
simpa [hfn0, hi] using hsdi
/-- Any three points in a cospherical set are affinely independent. -/
theorem Cospherical.affineIndependent {s : Set P} (hs : Cospherical s) {p : Fin 3 → P}
    (hps : Set.range p ⊆ s) (hpi : Function.Injective p) : AffineIndependent ℝ p := by
  rw [affineIndependent_iff_not_collinear]
  intro hc
  rw [collinear_iff_of_mem (Set.mem_range_self (0 : Fin 3))] at hc
  rcases hc with ⟨v, hv⟩
  rw [Set.forall_range_iff] at hv
  have hv0 : v ≠ 0 := by
    intro h
    have he : p 1 = p 0 := by simpa [h] using hv 1
    exact (by decide : (1 : Fin 3) ≠ 0) (hpi he)
  rcases hs with ⟨c, r, hs⟩
  have hs' := fun i => hs (p i) (Set.mem_of_mem_of_subset (Set.mem_range_self _) hps)
  choose f hf using hv
  have hsd : ∀ i, dist (f i • v +ᵥ p 0) c = r := by
    intro i
    rw [← hf]
    exact hs' i
  have hf0 : f 0 = 0 := by
    have hf0' := hf 0
    rw [eq_comm, ← @vsub_eq_zero_iff_eq V, vadd_vsub, smul_eq_zero] at hf0'
    simpa [hv0] using hf0'
  have hfi : Function.Injective f := by
    intro i j h
    have hi := hf i
    rw [h, ← hf j] at hi
    exact hpi hi
  simp_rw [← hsd 0, hf0, zero_smul, zero_vadd, dist_smul_vadd_eq_dist (p 0) c hv0] at hsd
  have hfn0 : ∀ i, i ≠ 0 → f i ≠ 0 := fun i => (hfi.ne_iff' hf0).2
  have hfn0' : ∀ i, i ≠ 0 → f i = -2 * ⟪v, p 0 -ᵥ c⟫ / ⟪v, v⟫ := by
    intro i hi
    have hsdi := hsd i
    simpa [hfn0, hi] using hsdi
  

Proof state:
case intro.intro.intro
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
v : V
hv0 : v ≠ 0
c : P
r : ℝ
hs : ∀ p ∈ s, dist p c = r
hs' : ∀ (i : Fin 3), dist (p i) c = r
f : Fin 3 → ℝ
hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0
hf0 : f 0 = 0
hfi : Function.Injective f
hsd : ∀ (i : Fin 3), f i = 0 ∨ f i = -2 * inner v (p 0 -ᵥ c) / inner v v
hfn0 : ∀ (i : Fin 3), i ≠ 0 → f i ≠ 0
hfn0' : ∀ (i : Fin 3), i ≠ 0 → f i = -2 * inner v (p 0 -ᵥ c) / inner v v
⊢ False

Next tactic:
have hf12 : f 1 = f 2 := by rw [hfn0' 1 (by decide), hfn0' 2 (by decide)]
/-- Any three points in a cospherical set are affinely independent. -/
theorem Cospherical.affineIndependent {s : Set P} (hs : Cospherical s) {p : Fin 3 → P}
    (hps : Set.range p ⊆ s) (hpi : Function.Injective p) : AffineIndependent ℝ p := by
  rw [affineIndependent_iff_not_collinear]
  intro hc
  rw [collinear_iff_of_mem (Set.mem_range_self (0 : Fin 3))] at hc
  rcases hc with ⟨v, hv⟩
  rw [Set.forall_range_iff] at hv
  have hv0 : v ≠ 0 := by
    intro h
    have he : p 1 = p 0 := by simpa [h] using hv 1
    exact (by decide : (1 : Fin 3) ≠ 0) (hpi he)
  rcases hs with ⟨c, r, hs⟩
  have hs' := fun i => hs (p i) (Set.mem_of_mem_of_subset (Set.mem_range_self _) hps)
  choose f hf using hv
  have hsd : ∀ i, dist (f i • v +ᵥ p 0) c = r := by
    intro i
    rw [← hf]
    exact hs' i
  have hf0 : f 0 = 0 := by
    have hf0' := hf 0
    rw [eq_comm, ← @vsub_eq_zero_iff_eq V, vadd_vsub, smul_eq_zero] at hf0'
    simpa [hv0] using hf0'
  have hfi : Function.Injective f := by
    intro i j h
    have hi := hf i
    rw [h, ← hf j] at hi
    exact hpi hi
  simp_rw [← hsd 0, hf0, zero_smul, zero_vadd, dist_smul_vadd_eq_dist (p 0) c hv0] at hsd
  have hfn0 : ∀ i, i ≠ 0 → f i ≠ 0 := fun i => (hfi.ne_iff' hf0).2
  have hfn0' : ∀ i, i ≠ 0 → f i = -2 * ⟪v, p 0 -ᵥ c⟫ / ⟪v, v⟫ := by
    intro i hi
    have hsdi := hsd i
    simpa [hfn0, hi] using hsdi
  have hf12 : f 1 = f 2 := by 

Proof state:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
v : V
hv0 : v ≠ 0
c : P
r : ℝ
hs : ∀ p ∈ s, dist p c = r
hs' : ∀ (i : Fin 3), dist (p i) c = r
f : Fin 3 → ℝ
hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0
hf0 : f 0 = 0
hfi : Function.Injective f
hsd : ∀ (i : Fin 3), f i = 0 ∨ f i = -2 * inner v (p 0 -ᵥ c) / inner v v
hfn0 : ∀ (i : Fin 3), i ≠ 0 → f i ≠ 0
hfn0' : ∀ (i : Fin 3), i ≠ 0 → f i = -2 * inner v (p 0 -ᵥ c) / inner v v
⊢ f 1 = f 2

Next tactic:
rw [hfn0' 1 (by decide), hfn0' 2 (by decide)]
/-- Any three points in a cospherical set are affinely independent. -/
theorem Cospherical.affineIndependent {s : Set P} (hs : Cospherical s) {p : Fin 3 → P}
    (hps : Set.range p ⊆ s) (hpi : Function.Injective p) : AffineIndependent ℝ p := by
  rw [affineIndependent_iff_not_collinear]
  intro hc
  rw [collinear_iff_of_mem (Set.mem_range_self (0 : Fin 3))] at hc
  rcases hc with ⟨v, hv⟩
  rw [Set.forall_range_iff] at hv
  have hv0 : v ≠ 0 := by
    intro h
    have he : p 1 = p 0 := by simpa [h] using hv 1
    exact (by decide : (1 : Fin 3) ≠ 0) (hpi he)
  rcases hs with ⟨c, r, hs⟩
  have hs' := fun i => hs (p i) (Set.mem_of_mem_of_subset (Set.mem_range_self _) hps)
  choose f hf using hv
  have hsd : ∀ i, dist (f i • v +ᵥ p 0) c = r := by
    intro i
    rw [← hf]
    exact hs' i
  have hf0 : f 0 = 0 := by
    have hf0' := hf 0
    rw [eq_comm, ← @vsub_eq_zero_iff_eq V, vadd_vsub, smul_eq_zero] at hf0'
    simpa [hv0] using hf0'
  have hfi : Function.Injective f := by
    intro i j h
    have hi := hf i
    rw [h, ← hf j] at hi
    exact hpi hi
  simp_rw [← hsd 0, hf0, zero_smul, zero_vadd, dist_smul_vadd_eq_dist (p 0) c hv0] at hsd
  have hfn0 : ∀ i, i ≠ 0 → f i ≠ 0 := fun i => (hfi.ne_iff' hf0).2
  have hfn0' : ∀ i, i ≠ 0 → f i = -2 * ⟪v, p 0 -ᵥ c⟫ / ⟪v, v⟫ := by
    intro i hi
    have hsdi := hsd i
    simpa [hfn0, hi] using hsdi
  have hf12 : f 1 = f 2 := by rw [hfn0' 1 (by 

Proof state:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
v : V
hv0 : v ≠ 0
c : P
r : ℝ
hs : ∀ p ∈ s, dist p c = r
hs' : ∀ (i : Fin 3), dist (p i) c = r
f : Fin 3 → ℝ
hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0
hf0 : f 0 = 0
hfi : Function.Injective f
hsd : ∀ (i : Fin 3), f i = 0 ∨ f i = -2 * inner v (p 0 -ᵥ c) / inner v v
hfn0 : ∀ (i : Fin 3), i ≠ 0 → f i ≠ 0
hfn0' : ∀ (i : Fin 3), i ≠ 0 → f i = -2 * inner v (p 0 -ᵥ c) / inner v v
⊢ 1 ≠ 0

Next tactic:
decide
/-- Any three points in a cospherical set are affinely independent. -/
theorem Cospherical.affineIndependent {s : Set P} (hs : Cospherical s) {p : Fin 3 → P}
    (hps : Set.range p ⊆ s) (hpi : Function.Injective p) : AffineIndependent ℝ p := by
  rw [affineIndependent_iff_not_collinear]
  intro hc
  rw [collinear_iff_of_mem (Set.mem_range_self (0 : Fin 3))] at hc
  rcases hc with ⟨v, hv⟩
  rw [Set.forall_range_iff] at hv
  have hv0 : v ≠ 0 := by
    intro h
    have he : p 1 = p 0 := by simpa [h] using hv 1
    exact (by decide : (1 : Fin 3) ≠ 0) (hpi he)
  rcases hs with ⟨c, r, hs⟩
  have hs' := fun i => hs (p i) (Set.mem_of_mem_of_subset (Set.mem_range_self _) hps)
  choose f hf using hv
  have hsd : ∀ i, dist (f i • v +ᵥ p 0) c = r := by
    intro i
    rw [← hf]
    exact hs' i
  have hf0 : f 0 = 0 := by
    have hf0' := hf 0
    rw [eq_comm, ← @vsub_eq_zero_iff_eq V, vadd_vsub, smul_eq_zero] at hf0'
    simpa [hv0] using hf0'
  have hfi : Function.Injective f := by
    intro i j h
    have hi := hf i
    rw [h, ← hf j] at hi
    exact hpi hi
  simp_rw [← hsd 0, hf0, zero_smul, zero_vadd, dist_smul_vadd_eq_dist (p 0) c hv0] at hsd
  have hfn0 : ∀ i, i ≠ 0 → f i ≠ 0 := fun i => (hfi.ne_iff' hf0).2
  have hfn0' : ∀ i, i ≠ 0 → f i = -2 * ⟪v, p 0 -ᵥ c⟫ / ⟪v, v⟫ := by
    intro i hi
    have hsdi := hsd i
    simpa [hfn0, hi] using hsdi
  have hf12 : f 1 = f 2 := by rw [hfn0' 1 (by decide), hfn0' 2 (by 

Proof state:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
v : V
hv0 : v ≠ 0
c : P
r : ℝ
hs : ∀ p ∈ s, dist p c = r
hs' : ∀ (i : Fin 3), dist (p i) c = r
f : Fin 3 → ℝ
hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0
hf0 : f 0 = 0
hfi : Function.Injective f
hsd : ∀ (i : Fin 3), f i = 0 ∨ f i = -2 * inner v (p 0 -ᵥ c) / inner v v
hfn0 : ∀ (i : Fin 3), i ≠ 0 → f i ≠ 0
hfn0' : ∀ (i : Fin 3), i ≠ 0 → f i = -2 * inner v (p 0 -ᵥ c) / inner v v
⊢ 2 ≠ 0

Next tactic:
decide