l3lab/ntp-mathlib huggingface dataset:
/-- In a tetrahedron with vertices `x`, `y`, `p`, `q`, any segment `[u, v]` joining the opposite
edges `[x, p]` and `[y, q]` passes through any triangle of vertices `p`, `q`, `z` where
`z ∈ [x, y]`. -/
theorem not_disjoint_segment_convexHull_triple {p q u v x y z : E} (hz : z ∈ segment 𝕜 x y)
    (hu : u ∈ segment 𝕜 x p) (hv : v ∈ segment 𝕜 y q) :
    ¬Disjoint (segment 𝕜 u v) (convexHull 𝕜 {p, q, z}) := by
  rw [not_disjoint_iff]
  obtain ⟨az, bz, haz, hbz, habz, rfl⟩ := hz
  obtain rfl | haz' := haz.eq_or_lt
  · rw [zero_add] at habz
    rw [zero_smul, zero_add, habz, one_smul]
    refine' ⟨v, by apply right_mem_segment, segment_subset_convexHull _ _ hv⟩ <;> simp
  obtain ⟨av, bv, hav, hbv, habv, rfl⟩ := hv
  obtain rfl | hav' := hav.eq_or_lt
  · rw [zero_add] at habv
    rw [zero_smul, zero_add, habv, one_smul]
    exact ⟨q, right_mem_segment _ _ _, subset_convexHull _ _ <| by simp⟩
  obtain ⟨au, bu, hau, hbu, habu, rfl⟩ := hu
  have hab : 0 < az * av + bz * au :=
    add_pos_of_pos_of_nonneg (mul_pos haz' hav') (mul_nonneg hbz hau)
  refine'
    ⟨(az * av / (az * av + bz * au)) • (au • x + bu • p) +
        (bz * au / (az * av + bz * au)) • (av • y + bv • q),
      ⟨_, _, _, _, _, rfl⟩, _⟩
  · exact div_nonneg (mul_nonneg haz hav) hab.le
  · exact div_nonneg (mul_nonneg hbz hau) hab.le
  · rw [← add_div, div_self hab.ne']
  rw [smul_add, smul_add, add_add_add_comm, add_comm, ← mul_smul, ← mul_smul]
  classical
    let w : Fin 3 → 𝕜 := ![az * av * bu, bz * au * bv, au * av]
    let z : Fin 3 → E := ![p, q, az • x + bz • y]
    have hw₀ : ∀ i, 0 ≤ w i := by
      rintro i
      fin_cases i
      · exact mul_nonneg (mul_nonneg haz hav) hbu
      · exact mul_nonneg (mul_nonneg hbz hau) hbv
      · exact mul_nonneg hau hav
    have hw : ∑ i, w i = az * av + bz * au := by
      trans az * av * bu + (bz * au * bv + au * av)
      · simp [Fin.sum_univ_succ, Fin.sum_univ_zero]
      rw [← one_mul (au * av), ← habz, add_mul, ← add_assoc, add_add_add_comm, mul_assoc, ← mul_add,
        mul_assoc, ← mul_add, mul_comm av, ← add_mul, ← mul_add, add_comm bu, add_comm bv, habu,
        habv, one_mul, mul_one]
    have hz : ∀ i, z i ∈ ({p, q, az • x + bz • y} : Set E) := fun i => by fin_cases i <;> 

Proof state:
case tail.head
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
p q x y : E
az bz : 𝕜
haz : 0 ≤ az
hbz : 0 ≤ bz
habz : az + bz = 1
haz' : 0 < az
av bv : 𝕜
hav : 0 ≤ av
hbv : 0 ≤ bv
habv : av + bv = 1
hav' : 0 < av
au bu : 𝕜
hau : 0 ≤ au
hbu : 0 ≤ bu
habu : au + bu = 1
hab : 0 < az * av + bz * au
w : Fin 3 → 𝕜 := ![az * av * bu, bz * au * bv, au * av]
z : Fin 3 → E := ![p, q, az • x + bz • y]
hw₀ : ∀ (i : Fin 3), 0 ≤ w i
hw : ∑ i : Fin 3, w i = az * av + bz * au
⊢ z { val := 1, isLt := (_ : (fun a => a < 3) 1) } ∈ {p, q, az • x + bz • y}

Next tactic:
simp
/-- In a tetrahedron with vertices `x`, `y`, `p`, `q`, any segment `[u, v]` joining the opposite
edges `[x, p]` and `[y, q]` passes through any triangle of vertices `p`, `q`, `z` where
`z ∈ [x, y]`. -/
theorem not_disjoint_segment_convexHull_triple {p q u v x y z : E} (hz : z ∈ segment 𝕜 x y)
    (hu : u ∈ segment 𝕜 x p) (hv : v ∈ segment 𝕜 y q) :
    ¬Disjoint (segment 𝕜 u v) (convexHull 𝕜 {p, q, z}) := by
  rw [not_disjoint_iff]
  obtain ⟨az, bz, haz, hbz, habz, rfl⟩ := hz
  obtain rfl | haz' := haz.eq_or_lt
  · rw [zero_add] at habz
    rw [zero_smul, zero_add, habz, one_smul]
    refine' ⟨v, by apply right_mem_segment, segment_subset_convexHull _ _ hv⟩ <;> simp
  obtain ⟨av, bv, hav, hbv, habv, rfl⟩ := hv
  obtain rfl | hav' := hav.eq_or_lt
  · rw [zero_add] at habv
    rw [zero_smul, zero_add, habv, one_smul]
    exact ⟨q, right_mem_segment _ _ _, subset_convexHull _ _ <| by simp⟩
  obtain ⟨au, bu, hau, hbu, habu, rfl⟩ := hu
  have hab : 0 < az * av + bz * au :=
    add_pos_of_pos_of_nonneg (mul_pos haz' hav') (mul_nonneg hbz hau)
  refine'
    ⟨(az * av / (az * av + bz * au)) • (au • x + bu • p) +
        (bz * au / (az * av + bz * au)) • (av • y + bv • q),
      ⟨_, _, _, _, _, rfl⟩, _⟩
  · exact div_nonneg (mul_nonneg haz hav) hab.le
  · exact div_nonneg (mul_nonneg hbz hau) hab.le
  · rw [← add_div, div_self hab.ne']
  rw [smul_add, smul_add, add_add_add_comm, add_comm, ← mul_smul, ← mul_smul]
  classical
    let w : Fin 3 → 𝕜 := ![az * av * bu, bz * au * bv, au * av]
    let z : Fin 3 → E := ![p, q, az • x + bz • y]
    have hw₀ : ∀ i, 0 ≤ w i := by
      rintro i
      fin_cases i
      · exact mul_nonneg (mul_nonneg haz hav) hbu
      · exact mul_nonneg (mul_nonneg hbz hau) hbv
      · exact mul_nonneg hau hav
    have hw : ∑ i, w i = az * av + bz * au := by
      trans az * av * bu + (bz * au * bv + au * av)
      · simp [Fin.sum_univ_succ, Fin.sum_univ_zero]
      rw [← one_mul (au * av), ← habz, add_mul, ← add_assoc, add_add_add_comm, mul_assoc, ← mul_add,
        mul_assoc, ← mul_add, mul_comm av, ← add_mul, ← mul_add, add_comm bu, add_comm bv, habu,
        habv, one_mul, mul_one]
    have hz : ∀ i, z i ∈ ({p, q, az • x + bz • y} : Set E) := fun i => by fin_cases i <;> 

Proof state:
case tail.tail.head
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
p q x y : E
az bz : 𝕜
haz : 0 ≤ az
hbz : 0 ≤ bz
habz : az + bz = 1
haz' : 0 < az
av bv : 𝕜
hav : 0 ≤ av
hbv : 0 ≤ bv
habv : av + bv = 1
hav' : 0 < av
au bu : 𝕜
hau : 0 ≤ au
hbu : 0 ≤ bu
habu : au + bu = 1
hab : 0 < az * av + bz * au
w : Fin 3 → 𝕜 := ![az * av * bu, bz * au * bv, au * av]
z : Fin 3 → E := ![p, q, az • x + bz • y]
hw₀ : ∀ (i : Fin 3), 0 ≤ w i
hw : ∑ i : Fin 3, w i = az * av + bz * au
⊢ z { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } ∈ {p, q, az • x + bz • y}

Next tactic:
simp
/-- In a tetrahedron with vertices `x`, `y`, `p`, `q`, any segment `[u, v]` joining the opposite
edges `[x, p]` and `[y, q]` passes through any triangle of vertices `p`, `q`, `z` where
`z ∈ [x, y]`. -/
theorem not_disjoint_segment_convexHull_triple {p q u v x y z : E} (hz : z ∈ segment 𝕜 x y)
    (hu : u ∈ segment 𝕜 x p) (hv : v ∈ segment 𝕜 y q) :
    ¬Disjoint (segment 𝕜 u v) (convexHull 𝕜 {p, q, z}) := by
  rw [not_disjoint_iff]
  obtain ⟨az, bz, haz, hbz, habz, rfl⟩ := hz
  obtain rfl | haz' := haz.eq_or_lt
  · rw [zero_add] at habz
    rw [zero_smul, zero_add, habz, one_smul]
    refine' ⟨v, by apply right_mem_segment, segment_subset_convexHull _ _ hv⟩ <;> simp
  obtain ⟨av, bv, hav, hbv, habv, rfl⟩ := hv
  obtain rfl | hav' := hav.eq_or_lt
  · rw [zero_add] at habv
    rw [zero_smul, zero_add, habv, one_smul]
    exact ⟨q, right_mem_segment _ _ _, subset_convexHull _ _ <| by simp⟩
  obtain ⟨au, bu, hau, hbu, habu, rfl⟩ := hu
  have hab : 0 < az * av + bz * au :=
    add_pos_of_pos_of_nonneg (mul_pos haz' hav') (mul_nonneg hbz hau)
  refine'
    ⟨(az * av / (az * av + bz * au)) • (au • x + bu • p) +
        (bz * au / (az * av + bz * au)) • (av • y + bv • q),
      ⟨_, _, _, _, _, rfl⟩, _⟩
  · exact div_nonneg (mul_nonneg haz hav) hab.le
  · exact div_nonneg (mul_nonneg hbz hau) hab.le
  · rw [← add_div, div_self hab.ne']
  rw [smul_add, smul_add, add_add_add_comm, add_comm, ← mul_smul, ← mul_smul]
  classical
    let w : Fin 3 → 𝕜 := ![az * av * bu, bz * au * bv, au * av]
    let z : Fin 3 → E := ![p, q, az • x + bz • y]
    have hw₀ : ∀ i, 0 ≤ w i := by
      rintro i
      fin_cases i
      · exact mul_nonneg (mul_nonneg haz hav) hbu
      · exact mul_nonneg (mul_nonneg hbz hau) hbv
      · exact mul_nonneg hau hav
    have hw : ∑ i, w i = az * av + bz * au := by
      trans az * av * bu + (bz * au * bv + au * av)
      · simp [Fin.sum_univ_succ, Fin.sum_univ_zero]
      rw [← one_mul (au * av), ← habz, add_mul, ← add_assoc, add_add_add_comm, mul_assoc, ← mul_add,
        mul_assoc, ← mul_add, mul_comm av, ← add_mul, ← mul_add, add_comm bu, add_comm bv, habu,
        habv, one_mul, mul_one]
    have hz : ∀ i, z i ∈ ({p, q, az • x + bz • y} : Set E) := fun i => by fin_cases i <;> simp
    convert Finset.centerMass_mem_convexHull (Finset.univ : Finset (Fin 3)) (fun i _ => hw₀ i)
        (by 

Proof state:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
p q x y : E
az bz : 𝕜
haz : 0 ≤ az
hbz : 0 ≤ bz
habz : az + bz = 1
haz' : 0 < az
av bv : 𝕜
hav : 0 ≤ av
hbv : 0 ≤ bv
habv : av + bv = 1
hav' : 0 < av
au bu : 𝕜
hau : 0 ≤ au
hbu : 0 ≤ bu
habu : au + bu = 1
hab : 0 < az * av + bz * au
w : Fin 3 → 𝕜 := ![az * av * bu, bz * au * bv, au * av]
z : Fin 3 → E := ![p, q, az • x + bz • y]
hw₀ : ∀ (i : Fin 3), 0 ≤ w i
hw : ∑ i : Fin 3, w i = az * av + bz * au
hz : ∀ (i : Fin 3), z i ∈ {p, q, az • x + bz • y}
⊢ 0 < ∑ i : Fin 3, w i

Next tactic:
rwa [hw]
/-- **Stone's Separation Theorem** -/
theorem exists_convex_convex_compl_subset (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hst : Disjoint s t) :
    ∃ C : Set E, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ := by
  

Proof state:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
hst : Disjoint s t
⊢ ∃ C, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ

Next tactic:
let S : Set (Set E) := { C | Convex 𝕜 C ∧ Disjoint C t }
/-- **Stone's Separation Theorem** -/
theorem exists_convex_convex_compl_subset (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hst : Disjoint s t) :
    ∃ C : Set E, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ := by
  let S : Set (Set E) := { C | Convex 𝕜 C ∧ Disjoint C t }
  

Proof state:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
hst : Disjoint s t
S : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}
⊢ ∃ C, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ

Next tactic:
obtain ⟨C, hC, hsC, hCmax⟩ :=
    zorn_subset_nonempty S
      (fun c hcS hc ⟨_, _⟩ =>
        ⟨⋃₀ c,
          ⟨hc.directedOn.convex_sUnion fun s hs => (hcS hs).1,
            disjoint_sUnion_left.2 fun c hc => (hcS hc).2⟩,
          fun s => subset_sUnion_of_mem⟩)
      s ⟨hs, hst⟩
/-- **Stone's Separation Theorem** -/
theorem exists_convex_convex_compl_subset (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hst : Disjoint s t) :
    ∃ C : Set E, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ := by
  let S : Set (Set E) := { C | Convex 𝕜 C ∧ Disjoint C t }
  obtain ⟨C, hC, hsC, hCmax⟩ :=
    zorn_subset_nonempty S
      (fun c hcS hc ⟨_, _⟩ =>
        ⟨⋃₀ c,
          ⟨hc.directedOn.convex_sUnion fun s hs => (hcS hs).1,
            disjoint_sUnion_left.2 fun c hc => (hcS hc).2⟩,
          fun s => subset_sUnion_of_mem⟩)
      s ⟨hs, hst⟩
  

Proof state:
case intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
hst : Disjoint s t
S : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}
C : Set E
hC : C ∈ S
hsC : s ⊆ C
hCmax : ∀ a ∈ S, C ⊆ a → a = C
⊢ ∃ C, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ

Next tactic:
refine'
    ⟨C, hC.1, convex_iff_segment_subset.2 fun x hx y hy z hz hzC => _, hsC, hC.2.subset_compl_left⟩
/-- **Stone's Separation Theorem** -/
theorem exists_convex_convex_compl_subset (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hst : Disjoint s t) :
    ∃ C : Set E, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ := by
  let S : Set (Set E) := { C | Convex 𝕜 C ∧ Disjoint C t }
  obtain ⟨C, hC, hsC, hCmax⟩ :=
    zorn_subset_nonempty S
      (fun c hcS hc ⟨_, _⟩ =>
        ⟨⋃₀ c,
          ⟨hc.directedOn.convex_sUnion fun s hs => (hcS hs).1,
            disjoint_sUnion_left.2 fun c hc => (hcS hc).2⟩,
          fun s => subset_sUnion_of_mem⟩)
      s ⟨hs, hst⟩
  refine'
    ⟨C, hC.1, convex_iff_segment_subset.2 fun x hx y hy z hz hzC => _, hsC, hC.2.subset_compl_left⟩
  

Proof state:
case intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
hst : Disjoint s t
S : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}
C : Set E
hC : C ∈ S
hsC : s ⊆ C
hCmax : ∀ a ∈ S, C ⊆ a → a = C
x : E
hx : x ∈ Cᶜ
y : E
hy : y ∈ Cᶜ
z : E
hz : z ∈ segment 𝕜 x y
hzC : z ∈ C
⊢ False

Next tactic:
suffices h : ∀ c ∈ Cᶜ, ∃ a ∈ C, (segment 𝕜 c a ∩ t).Nonempty
/-- **Stone's Separation Theorem** -/
theorem exists_convex_convex_compl_subset (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hst : Disjoint s t) :
    ∃ C : Set E, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ := by
  let S : Set (Set E) := { C | Convex 𝕜 C ∧ Disjoint C t }
  obtain ⟨C, hC, hsC, hCmax⟩ :=
    zorn_subset_nonempty S
      (fun c hcS hc ⟨_, _⟩ =>
        ⟨⋃₀ c,
          ⟨hc.directedOn.convex_sUnion fun s hs => (hcS hs).1,
            disjoint_sUnion_left.2 fun c hc => (hcS hc).2⟩,
          fun s => subset_sUnion_of_mem⟩)
      s ⟨hs, hst⟩
  refine'
    ⟨C, hC.1, convex_iff_segment_subset.2 fun x hx y hy z hz hzC => _, hsC, hC.2.subset_compl_left⟩
  suffices h : ∀ c ∈ Cᶜ, ∃ a ∈ C, (segment 𝕜 c a ∩ t).Nonempty
  · 

Proof state:
case intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
hst : Disjoint s t
S : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}
C : Set E
hC : C ∈ S
hsC : s ⊆ C
hCmax : ∀ a ∈ S, C ⊆ a → a = C
x : E
hx : x ∈ Cᶜ
y : E
hy : y ∈ Cᶜ
z : E
hz : z ∈ segment 𝕜 x y
hzC : z ∈ C
h : ∀ c ∈ Cᶜ, ∃ a ∈ C, Set.Nonempty (segment 𝕜 c a ∩ t)
⊢ False

Next tactic:
obtain ⟨p, hp, u, hu, hut⟩ := h x hx
/-- **Stone's Separation Theorem** -/
theorem exists_convex_convex_compl_subset (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hst : Disjoint s t) :
    ∃ C : Set E, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ := by
  let S : Set (Set E) := { C | Convex 𝕜 C ∧ Disjoint C t }
  obtain ⟨C, hC, hsC, hCmax⟩ :=
    zorn_subset_nonempty S
      (fun c hcS hc ⟨_, _⟩ =>
        ⟨⋃₀ c,
          ⟨hc.directedOn.convex_sUnion fun s hs => (hcS hs).1,
            disjoint_sUnion_left.2 fun c hc => (hcS hc).2⟩,
          fun s => subset_sUnion_of_mem⟩)
      s ⟨hs, hst⟩
  refine'
    ⟨C, hC.1, convex_iff_segment_subset.2 fun x hx y hy z hz hzC => _, hsC, hC.2.subset_compl_left⟩
  suffices h : ∀ c ∈ Cᶜ, ∃ a ∈ C, (segment 𝕜 c a ∩ t).Nonempty
  · obtain ⟨p, hp, u, hu, hut⟩ := h x hx
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
hst : Disjoint s t
S : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}
C : Set E
hC : C ∈ S
hsC : s ⊆ C
hCmax : ∀ a ∈ S, C ⊆ a → a = C
x : E
hx : x ∈ Cᶜ
y : E
hy : y ∈ Cᶜ
z : E
hz : z ∈ segment 𝕜 x y
hzC : z ∈ C
h : ∀ c ∈ Cᶜ, ∃ a ∈ C, Set.Nonempty (segment 𝕜 c a ∩ t)
p : E
hp : p ∈ C
u : E
hu : u ∈ segment 𝕜 x p
hut : u ∈ t
⊢ False

Next tactic:
obtain ⟨q, hq, v, hv, hvt⟩ := h y hy
/-- **Stone's Separation Theorem** -/
theorem exists_convex_convex_compl_subset (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hst : Disjoint s t) :
    ∃ C : Set E, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ := by
  let S : Set (Set E) := { C | Convex 𝕜 C ∧ Disjoint C t }
  obtain ⟨C, hC, hsC, hCmax⟩ :=
    zorn_subset_nonempty S
      (fun c hcS hc ⟨_, _⟩ =>
        ⟨⋃₀ c,
          ⟨hc.directedOn.convex_sUnion fun s hs => (hcS hs).1,
            disjoint_sUnion_left.2 fun c hc => (hcS hc).2⟩,
          fun s => subset_sUnion_of_mem⟩)
      s ⟨hs, hst⟩
  refine'
    ⟨C, hC.1, convex_iff_segment_subset.2 fun x hx y hy z hz hzC => _, hsC, hC.2.subset_compl_left⟩
  suffices h : ∀ c ∈ Cᶜ, ∃ a ∈ C, (segment 𝕜 c a ∩ t).Nonempty
  · obtain ⟨p, hp, u, hu, hut⟩ := h x hx
    obtain ⟨q, hq, v, hv, hvt⟩ := h y hy
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
hst : Disjoint s t
S : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}
C : Set E
hC : C ∈ S
hsC : s ⊆ C
hCmax : ∀ a ∈ S, C ⊆ a → a = C
x : E
hx : x ∈ Cᶜ
y : E
hy : y ∈ Cᶜ
z : E
hz : z ∈ segment 𝕜 x y
hzC : z ∈ C
h : ∀ c ∈ Cᶜ, ∃ a ∈ C, Set.Nonempty (segment 𝕜 c a ∩ t)
p : E
hp : p ∈ C
u : E
hu : u ∈ segment 𝕜 x p
hut : u ∈ t
q : E
hq : q ∈ C
v : E
hv : v ∈ segment 𝕜 y q
hvt : v ∈ t
⊢ False

Next tactic:
refine'
      not_disjoint_segment_convexHull_triple hz hu hv
        (hC.2.symm.mono (ht.segment_subset hut hvt) <| convexHull_min _ hC.1)
/-- **Stone's Separation Theorem** -/
theorem exists_convex_convex_compl_subset (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hst : Disjoint s t) :
    ∃ C : Set E, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ := by
  let S : Set (Set E) := { C | Convex 𝕜 C ∧ Disjoint C t }
  obtain ⟨C, hC, hsC, hCmax⟩ :=
    zorn_subset_nonempty S
      (fun c hcS hc ⟨_, _⟩ =>
        ⟨⋃₀ c,
          ⟨hc.directedOn.convex_sUnion fun s hs => (hcS hs).1,
            disjoint_sUnion_left.2 fun c hc => (hcS hc).2⟩,
          fun s => subset_sUnion_of_mem⟩)
      s ⟨hs, hst⟩
  refine'
    ⟨C, hC.1, convex_iff_segment_subset.2 fun x hx y hy z hz hzC => _, hsC, hC.2.subset_compl_left⟩
  suffices h : ∀ c ∈ Cᶜ, ∃ a ∈ C, (segment 𝕜 c a ∩ t).Nonempty
  · obtain ⟨p, hp, u, hu, hut⟩ := h x hx
    obtain ⟨q, hq, v, hv, hvt⟩ := h y hy
    refine'
      not_disjoint_segment_convexHull_triple hz hu hv
        (hC.2.symm.mono (ht.segment_subset hut hvt) <| convexHull_min _ hC.1)
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
hst : Disjoint s t
S : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}
C : Set E
hC : C ∈ S
hsC : s ⊆ C
hCmax : ∀ a ∈ S, C ⊆ a → a = C
x : E
hx : x ∈ Cᶜ
y : E
hy : y ∈ Cᶜ
z : E
hz : z ∈ segment 𝕜 x y
hzC : z ∈ C
h : ∀ c ∈ Cᶜ, ∃ a ∈ C, Set.Nonempty (segment 𝕜 c a ∩ t)
p : E
hp : p ∈ C
u : E
hu : u ∈ segment 𝕜 x p
hut : u ∈ t
q : E
hq : q ∈ C
v : E
hv : v ∈ segment 𝕜 y q
hvt : v ∈ t
⊢ {p, q, z} ⊆ C

Next tactic:
simpa [insert_subset_iff, hp, hq, singleton_subset_iff.2 hzC]
/-- **Stone's Separation Theorem** -/
theorem exists_convex_convex_compl_subset (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hst : Disjoint s t) :
    ∃ C : Set E, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ := by
  let S : Set (Set E) := { C | Convex 𝕜 C ∧ Disjoint C t }
  obtain ⟨C, hC, hsC, hCmax⟩ :=
    zorn_subset_nonempty S
      (fun c hcS hc ⟨_, _⟩ =>
        ⟨⋃₀ c,
          ⟨hc.directedOn.convex_sUnion fun s hs => (hcS hs).1,
            disjoint_sUnion_left.2 fun c hc => (hcS hc).2⟩,
          fun s => subset_sUnion_of_mem⟩)
      s ⟨hs, hst⟩
  refine'
    ⟨C, hC.1, convex_iff_segment_subset.2 fun x hx y hy z hz hzC => _, hsC, hC.2.subset_compl_left⟩
  suffices h : ∀ c ∈ Cᶜ, ∃ a ∈ C, (segment 𝕜 c a ∩ t).Nonempty
  · obtain ⟨p, hp, u, hu, hut⟩ := h x hx
    obtain ⟨q, hq, v, hv, hvt⟩ := h y hy
    refine'
      not_disjoint_segment_convexHull_triple hz hu hv
        (hC.2.symm.mono (ht.segment_subset hut hvt) <| convexHull_min _ hC.1)
    simpa [insert_subset_iff, hp, hq, singleton_subset_iff.2 hzC]
  

Proof state:
case h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
hst : Disjoint s t
S : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}
C : Set E
hC : C ∈ S
hsC : s ⊆ C
hCmax : ∀ a ∈ S, C ⊆ a → a = C
x : E
hx : x ∈ Cᶜ
y : E
hy : y ∈ Cᶜ
z : E
hz : z ∈ segment 𝕜 x y
hzC : z ∈ C
⊢ ∀ c ∈ Cᶜ, ∃ a ∈ C, Set.Nonempty (segment 𝕜 c a ∩ t)

Next tactic:
rintro c hc
/-- **Stone's Separation Theorem** -/
theorem exists_convex_convex_compl_subset (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hst : Disjoint s t) :
    ∃ C : Set E, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ := by
  let S : Set (Set E) := { C | Convex 𝕜 C ∧ Disjoint C t }
  obtain ⟨C, hC, hsC, hCmax⟩ :=
    zorn_subset_nonempty S
      (fun c hcS hc ⟨_, _⟩ =>
        ⟨⋃₀ c,
          ⟨hc.directedOn.convex_sUnion fun s hs => (hcS hs).1,
            disjoint_sUnion_left.2 fun c hc => (hcS hc).2⟩,
          fun s => subset_sUnion_of_mem⟩)
      s ⟨hs, hst⟩
  refine'
    ⟨C, hC.1, convex_iff_segment_subset.2 fun x hx y hy z hz hzC => _, hsC, hC.2.subset_compl_left⟩
  suffices h : ∀ c ∈ Cᶜ, ∃ a ∈ C, (segment 𝕜 c a ∩ t).Nonempty
  · obtain ⟨p, hp, u, hu, hut⟩ := h x hx
    obtain ⟨q, hq, v, hv, hvt⟩ := h y hy
    refine'
      not_disjoint_segment_convexHull_triple hz hu hv
        (hC.2.symm.mono (ht.segment_subset hut hvt) <| convexHull_min _ hC.1)
    simpa [insert_subset_iff, hp, hq, singleton_subset_iff.2 hzC]
  rintro c hc
  

Proof state:
case h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
hst : Disjoint s t
S : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}
C : Set E
hC : C ∈ S
hsC : s ⊆ C
hCmax : ∀ a ∈ S, C ⊆ a → a = C
x : E
hx : x ∈ Cᶜ
y : E
hy : y ∈ Cᶜ
z : E
hz : z ∈ segment 𝕜 x y
hzC : z ∈ C
c : E
hc : c ∈ Cᶜ
⊢ ∃ a ∈ C, Set.Nonempty (segment 𝕜 c a ∩ t)

Next tactic:
by_contra! h
/-- **Stone's Separation Theorem** -/
theorem exists_convex_convex_compl_subset (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hst : Disjoint s t) :
    ∃ C : Set E, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ := by
  let S : Set (Set E) := { C | Convex 𝕜 C ∧ Disjoint C t }
  obtain ⟨C, hC, hsC, hCmax⟩ :=
    zorn_subset_nonempty S
      (fun c hcS hc ⟨_, _⟩ =>
        ⟨⋃₀ c,
          ⟨hc.directedOn.convex_sUnion fun s hs => (hcS hs).1,
            disjoint_sUnion_left.2 fun c hc => (hcS hc).2⟩,
          fun s => subset_sUnion_of_mem⟩)
      s ⟨hs, hst⟩
  refine'
    ⟨C, hC.1, convex_iff_segment_subset.2 fun x hx y hy z hz hzC => _, hsC, hC.2.subset_compl_left⟩
  suffices h : ∀ c ∈ Cᶜ, ∃ a ∈ C, (segment 𝕜 c a ∩ t).Nonempty
  · obtain ⟨p, hp, u, hu, hut⟩ := h x hx
    obtain ⟨q, hq, v, hv, hvt⟩ := h y hy
    refine'
      not_disjoint_segment_convexHull_triple hz hu hv
        (hC.2.symm.mono (ht.segment_subset hut hvt) <| convexHull_min _ hC.1)
    simpa [insert_subset_iff, hp, hq, singleton_subset_iff.2 hzC]
  rintro c hc
  by_contra! h
  

Proof state:
case h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
hst : Disjoint s t
S : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}
C : Set E
hC : C ∈ S
hsC : s ⊆ C
hCmax : ∀ a ∈ S, C ⊆ a → a = C
x : E
hx : x ∈ Cᶜ
y : E
hy : y ∈ Cᶜ
z : E
hz : z ∈ segment 𝕜 x y
hzC : z ∈ C
c : E
hc : c ∈ Cᶜ
h : ∀ a ∈ C, segment 𝕜 c a ∩ t = ∅
⊢ False

Next tactic:
suffices h : Disjoint (convexHull 𝕜 (insert c C)) t
/-- **Stone's Separation Theorem** -/
theorem exists_convex_convex_compl_subset (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hst : Disjoint s t) :
    ∃ C : Set E, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ := by
  let S : Set (Set E) := { C | Convex 𝕜 C ∧ Disjoint C t }
  obtain ⟨C, hC, hsC, hCmax⟩ :=
    zorn_subset_nonempty S
      (fun c hcS hc ⟨_, _⟩ =>
        ⟨⋃₀ c,
          ⟨hc.directedOn.convex_sUnion fun s hs => (hcS hs).1,
            disjoint_sUnion_left.2 fun c hc => (hcS hc).2⟩,
          fun s => subset_sUnion_of_mem⟩)
      s ⟨hs, hst⟩
  refine'
    ⟨C, hC.1, convex_iff_segment_subset.2 fun x hx y hy z hz hzC => _, hsC, hC.2.subset_compl_left⟩
  suffices h : ∀ c ∈ Cᶜ, ∃ a ∈ C, (segment 𝕜 c a ∩ t).Nonempty
  · obtain ⟨p, hp, u, hu, hut⟩ := h x hx
    obtain ⟨q, hq, v, hv, hvt⟩ := h y hy
    refine'
      not_disjoint_segment_convexHull_triple hz hu hv
        (hC.2.symm.mono (ht.segment_subset hut hvt) <| convexHull_min _ hC.1)
    simpa [insert_subset_iff, hp, hq, singleton_subset_iff.2 hzC]
  rintro c hc
  by_contra! h
  suffices h : Disjoint (convexHull 𝕜 (insert c C)) t
  · 

Proof state:
case h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
hst : Disjoint s t
S : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}
C : Set E
hC : C ∈ S
hsC : s ⊆ C
hCmax : ∀ a ∈ S, C ⊆ a → a = C
x : E
hx : x ∈ Cᶜ
y : E
hy : y ∈ Cᶜ
z : E
hz : z ∈ segment 𝕜 x y
hzC : z ∈ C
c : E
hc : c ∈ Cᶜ
h✝ : ∀ a ∈ C, segment 𝕜 c a ∩ t = ∅
h : Disjoint ((convexHull 𝕜) (insert c C)) t
⊢ False

Next tactic:
rw [←
      hCmax _ ⟨convex_convexHull _ _, h⟩ ((subset_insert _ _).trans <| subset_convexHull _ _)] at hc