l3lab/ntp-mathlib huggingface dataset:
@[simp, norm_cast]
theorem int_cast_abs (n : ℤ) : |↑n| = Complex.abs n := by
  

Proof state:
n : ℤ
⊢ |↑n| = abs ↑n

Next tactic:
rw [← ofReal_int_cast, abs_ofReal]
theorem normSq_eq_abs (x : ℂ) : normSq x = (Complex.abs x) ^ 2 := by
  

Proof state:
x : ℂ
⊢ normSq x = abs x ^ 2

Next tactic:
simp [abs, sq, abs_def, Real.mul_self_sqrt (normSq_nonneg _)]
@[simp]
theorem range_normSq : range normSq = Ici 0 :=
  Subset.antisymm (range_subset_iff.2 normSq_nonneg) fun x hx =>
    ⟨Real.sqrt x, by 

Proof state:
x : ℝ
hx : x ∈ Ici 0
⊢ normSq ↑(Real.sqrt x) = x

Next tactic:
rw [normSq_ofReal, Real.mul_self_sqrt hx]
theorem isCauSeq_re (f : CauSeq ℂ Complex.abs) : IsCauSeq abs' fun n => (f n).re := fun ε ε0 =>
  (f.cauchy ε0).imp fun i H j ij =>
    lt_of_le_of_lt (by 

Proof state:
f : CauSeq ℂ ⇑abs
ε : ℝ
ε0 : ε > 0
i : ℕ
H : ∀ j ≥ i, abs (↑f j - ↑f i) < ε
j : ℕ
ij : j ≥ i
⊢ abs' ((fun n => (↑f n).re) j - (fun n => (↑f n).re) i) ≤ abs (↑f j - ↑f i)

Next tactic:
simpa using abs_re_le_abs (f j - f i)
theorem isCauSeq_im (f : CauSeq ℂ Complex.abs) : IsCauSeq abs' fun n => (f n).im := fun ε ε0 =>
  (f.cauchy ε0).imp fun i H j ij =>
    lt_of_le_of_lt (by 

Proof state:
f : CauSeq ℂ ⇑abs
ε : ℝ
ε0 : ε > 0
i : ℕ
H : ∀ j ≥ i, abs (↑f j - ↑f i) < ε
j : ℕ
ij : j ≥ i
⊢ abs' ((fun n => (↑f n).im) j - (fun n => (↑f n).im) i) ≤ abs (↑f j - ↑f i)

Next tactic:
simpa using abs_im_le_abs (f j - f i)
theorem equiv_limAux (f : CauSeq ℂ Complex.abs) :
    f ≈ CauSeq.const Complex.abs (limAux f) := fun ε ε0 =>
  (exists_forall_ge_and
  (CauSeq.equiv_lim ⟨_, isCauSeq_re f⟩ _ (half_pos ε0))
        (CauSeq.equiv_lim ⟨_, isCauSeq_im f⟩ _ (half_pos ε0))).imp
    fun i H j ij => by
    

Proof state:
f : CauSeq ℂ ⇑abs
ε : ℝ
ε0 : ε > 0
i : ℕ
H :
  ∀ j ≥ i,
    abs'
          (↑({ val := fun n => (↑f n).re, property := (_ : IsCauSeq abs' fun n => (↑f n).re) } -
                CauSeq.const abs'
                  (CauSeq.lim { val := fun n => (↑f n).re, property := (_ : IsCauSeq abs' fun n => (↑f n).re) }))
            j) <
        ε / 2 ∧
      abs'
          (↑({ val := fun n => (↑f n).im, property := (_ : IsCauSeq abs' fun n => (↑f n).im) } -
                CauSeq.const abs'
                  (CauSeq.lim { val := fun n => (↑f n).im, property := (_ : IsCauSeq abs' fun n => (↑f n).im) }))
            j) <
        ε / 2
j : ℕ
ij : j ≥ i
⊢ abs (↑(f - CauSeq.const (⇑abs) (limAux f)) j) < ε

Next tactic:
cases' H _ ij with H₁ H₂
theorem equiv_limAux (f : CauSeq ℂ Complex.abs) :
    f ≈ CauSeq.const Complex.abs (limAux f) := fun ε ε0 =>
  (exists_forall_ge_and
  (CauSeq.equiv_lim ⟨_, isCauSeq_re f⟩ _ (half_pos ε0))
        (CauSeq.equiv_lim ⟨_, isCauSeq_im f⟩ _ (half_pos ε0))).imp
    fun i H j ij => by
    cases' H _ ij with H₁ H₂
    

Proof state:
case intro
f : CauSeq ℂ ⇑abs
ε : ℝ
ε0 : ε > 0
i : ℕ
H :
  ∀ j ≥ i,
    abs'
          (↑({ val := fun n => (↑f n).re, property := (_ : IsCauSeq abs' fun n => (↑f n).re) } -
                CauSeq.const abs'
                  (CauSeq.lim { val := fun n => (↑f n).re, property := (_ : IsCauSeq abs' fun n => (↑f n).re) }))
            j) <
        ε / 2 ∧
      abs'
          (↑({ val := fun n => (↑f n).im, property := (_ : IsCauSeq abs' fun n => (↑f n).im) } -
                CauSeq.const abs'
                  (CauSeq.lim { val := fun n => (↑f n).im, property := (_ : IsCauSeq abs' fun n => (↑f n).im) }))
            j) <
        ε / 2
j : ℕ
ij : j ≥ i
H₁ :
  abs'
      (↑({ val := fun n => (↑f n).re, property := (_ : IsCauSeq abs' fun n => (↑f n).re) } -
            CauSeq.const abs'
              (CauSeq.lim { val := fun n => (↑f n).re, property := (_ : IsCauSeq abs' fun n => (↑f n).re) }))
        j) <
    ε / 2
H₂ :
  abs'
      (↑({ val := fun n => (↑f n).im, property := (_ : IsCauSeq abs' fun n => (↑f n).im) } -
            CauSeq.const abs'
              (CauSeq.lim { val := fun n => (↑f n).im, property := (_ : IsCauSeq abs' fun n => (↑f n).im) }))
        j) <
    ε / 2
⊢ abs (↑(f - CauSeq.const (⇑abs) (limAux f)) j) < ε

Next tactic:
apply lt_of_le_of_lt (abs_le_abs_re_add_abs_im _)
theorem equiv_limAux (f : CauSeq ℂ Complex.abs) :
    f ≈ CauSeq.const Complex.abs (limAux f) := fun ε ε0 =>
  (exists_forall_ge_and
  (CauSeq.equiv_lim ⟨_, isCauSeq_re f⟩ _ (half_pos ε0))
        (CauSeq.equiv_lim ⟨_, isCauSeq_im f⟩ _ (half_pos ε0))).imp
    fun i H j ij => by
    cases' H _ ij with H₁ H₂
    apply lt_of_le_of_lt (abs_le_abs_re_add_abs_im _)
    

Proof state:
case intro
f : CauSeq ℂ ⇑abs
ε : ℝ
ε0 : ε > 0
i : ℕ
H :
  ∀ j ≥ i,
    abs'
          (↑({ val := fun n => (↑f n).re, property := (_ : IsCauSeq abs' fun n => (↑f n).re) } -
                CauSeq.const abs'
                  (CauSeq.lim { val := fun n => (↑f n).re, property := (_ : IsCauSeq abs' fun n => (↑f n).re) }))
            j) <
        ε / 2 ∧
      abs'
          (↑({ val := fun n => (↑f n).im, property := (_ : IsCauSeq abs' fun n => (↑f n).im) } -
                CauSeq.const abs'
                  (CauSeq.lim { val := fun n => (↑f n).im, property := (_ : IsCauSeq abs' fun n => (↑f n).im) }))
            j) <
        ε / 2
j : ℕ
ij : j ≥ i
H₁ :
  abs'
      (↑({ val := fun n => (↑f n).re, property := (_ : IsCauSeq abs' fun n => (↑f n).re) } -
            CauSeq.const abs'
              (CauSeq.lim { val := fun n => (↑f n).re, property := (_ : IsCauSeq abs' fun n => (↑f n).re) }))
        j) <
    ε / 2
H₂ :
  abs'
      (↑({ val := fun n => (↑f n).im, property := (_ : IsCauSeq abs' fun n => (↑f n).im) } -
            CauSeq.const abs'
              (CauSeq.lim { val := fun n => (↑f n).im, property := (_ : IsCauSeq abs' fun n => (↑f n).im) }))
        j) <
    ε / 2
⊢ abs' (↑(f - CauSeq.const (⇑abs) (limAux f)) j).re + abs' (↑(f - CauSeq.const (⇑abs) (limAux f)) j).im < ε

Next tactic:
dsimp [limAux] at *
theorem equiv_limAux (f : CauSeq ℂ Complex.abs) :
    f ≈ CauSeq.const Complex.abs (limAux f) := fun ε ε0 =>
  (exists_forall_ge_and
  (CauSeq.equiv_lim ⟨_, isCauSeq_re f⟩ _ (half_pos ε0))
        (CauSeq.equiv_lim ⟨_, isCauSeq_im f⟩ _ (half_pos ε0))).imp
    fun i H j ij => by
    cases' H _ ij with H₁ H₂
    apply lt_of_le_of_lt (abs_le_abs_re_add_abs_im _)
    dsimp [limAux] at *
    

Proof state:
case intro
f : CauSeq ℂ ⇑abs
ε : ℝ
ε0 : ε > 0
i : ℕ
H :
  ∀ j ≥ i,
    abs' ((↑f j).re - CauSeq.lim { val := fun n => (↑f n).re, property := (_ : IsCauSeq abs' fun n => (↑f n).re) }) <
        ε / 2 ∧
      abs' ((↑f j).im - CauSeq.lim { val := fun n => (↑f n).im, property := (_ : IsCauSeq abs' fun n => (↑f n).im) }) <
        ε / 2
j : ℕ
ij : j ≥ i
H₁ :
  abs' ((↑f j).re - CauSeq.lim { val := fun n => (↑f n).re, property := (_ : IsCauSeq abs' fun n => (↑f n).re) }) <
    ε / 2
H₂ :
  abs' ((↑f j).im - CauSeq.lim { val := fun n => (↑f n).im, property := (_ : IsCauSeq abs' fun n => (↑f n).im) }) <
    ε / 2
⊢ abs' ((↑f j).re - CauSeq.lim (cauSeqRe f)) + abs' ((↑f j).im - CauSeq.lim (cauSeqIm f)) < ε

Next tactic:
have := add_lt_add H₁ H₂
theorem equiv_limAux (f : CauSeq ℂ Complex.abs) :
    f ≈ CauSeq.const Complex.abs (limAux f) := fun ε ε0 =>
  (exists_forall_ge_and
  (CauSeq.equiv_lim ⟨_, isCauSeq_re f⟩ _ (half_pos ε0))
        (CauSeq.equiv_lim ⟨_, isCauSeq_im f⟩ _ (half_pos ε0))).imp
    fun i H j ij => by
    cases' H _ ij with H₁ H₂
    apply lt_of_le_of_lt (abs_le_abs_re_add_abs_im _)
    dsimp [limAux] at *
    have := add_lt_add H₁ H₂
    

Proof state:
case intro
f : CauSeq ℂ ⇑abs
ε : ℝ
ε0 : ε > 0
i : ℕ
H :
  ∀ j ≥ i,
    abs' ((↑f j).re - CauSeq.lim { val := fun n => (↑f n).re, property := (_ : IsCauSeq abs' fun n => (↑f n).re) }) <
        ε / 2 ∧
      abs' ((↑f j).im - CauSeq.lim { val := fun n => (↑f n).im, property := (_ : IsCauSeq abs' fun n => (↑f n).im) }) <
        ε / 2
j : ℕ
ij : j ≥ i
H₁ :
  abs' ((↑f j).re - CauSeq.lim { val := fun n => (↑f n).re, property := (_ : IsCauSeq abs' fun n => (↑f n).re) }) <
    ε / 2
H₂ :
  abs' ((↑f j).im - CauSeq.lim { val := fun n => (↑f n).im, property := (_ : IsCauSeq abs' fun n => (↑f n).im) }) <
    ε / 2
this :
  abs' ((↑f j).re - CauSeq.lim { val := fun n => (↑f n).re, property := (_ : IsCauSeq abs' fun n => (↑f n).re) }) +
      abs' ((↑f j).im - CauSeq.lim { val := fun n => (↑f n).im, property := (_ : IsCauSeq abs' fun n => (↑f n).im) }) <
    ε / 2 + ε / 2
⊢ abs' ((↑f j).re - CauSeq.lim (cauSeqRe f)) + abs' ((↑f j).im - CauSeq.lim (cauSeqIm f)) < ε

Next tactic:
rwa [add_halves] at this
theorem lim_eq_lim_im_add_lim_re (f : CauSeq ℂ Complex.abs) :
    lim f = ↑(lim (cauSeqRe f)) + ↑(lim (cauSeqIm f)) * I :=
  lim_eq_of_equiv_const <|
    calc
      f ≈ _ := equiv_limAux f
      _ = CauSeq.const Complex.abs (↑(lim (cauSeqRe f)) + ↑(lim (cauSeqIm f)) * I) :=
        CauSeq.ext fun _ =>
          Complex.ext (by 

Proof state:
f : CauSeq ℂ ⇑abs
x✝ : ℕ
⊢ (↑(const (⇑abs) (limAux f)) x✝).re =
    (↑(const (⇑abs) (↑(CauSeq.lim (cauSeqRe f)) + ↑(CauSeq.lim (cauSeqIm f)) * I)) x✝).re

Next tactic:
simp [limAux, cauSeqRe, ofReal']
theorem lim_eq_lim_im_add_lim_re (f : CauSeq ℂ Complex.abs) :
    lim f = ↑(lim (cauSeqRe f)) + ↑(lim (cauSeqIm f)) * I :=
  lim_eq_of_equiv_const <|
    calc
      f ≈ _ := equiv_limAux f
      _ = CauSeq.const Complex.abs (↑(lim (cauSeqRe f)) + ↑(lim (cauSeqIm f)) * I) :=
        CauSeq.ext fun _ =>
          Complex.ext (by simp [limAux, cauSeqRe, ofReal']) (by 

Proof state:
f : CauSeq ℂ ⇑abs
x✝ : ℕ
⊢ (↑(const (⇑abs) (limAux f)) x✝).im =
    (↑(const (⇑abs) (↑(CauSeq.lim (cauSeqRe f)) + ↑(CauSeq.lim (cauSeqIm f)) * I)) x✝).im

Next tactic:
simp [limAux, cauSeqIm, ofReal']
theorem lim_re (f : CauSeq ℂ Complex.abs) : lim (cauSeqRe f) = (lim f).re := by
  

Proof state:
f : CauSeq ℂ ⇑abs
⊢ CauSeq.lim (cauSeqRe f) = (CauSeq.lim f).re

Next tactic:
rw [lim_eq_lim_im_add_lim_re]
theorem lim_re (f : CauSeq ℂ Complex.abs) : lim (cauSeqRe f) = (lim f).re := by
  rw [lim_eq_lim_im_add_lim_re]; 

Proof state:
f : CauSeq ℂ ⇑abs
⊢ CauSeq.lim (cauSeqRe f) = (↑(CauSeq.lim (cauSeqRe f)) + ↑(CauSeq.lim (cauSeqIm f)) * I).re

Next tactic:
simp [ofReal']
theorem lim_im (f : CauSeq ℂ Complex.abs) : lim (cauSeqIm f) = (lim f).im := by
  

Proof state:
f : CauSeq ℂ ⇑abs
⊢ CauSeq.lim (cauSeqIm f) = (CauSeq.lim f).im

Next tactic:
rw [lim_eq_lim_im_add_lim_re]
theorem lim_im (f : CauSeq ℂ Complex.abs) : lim (cauSeqIm f) = (lim f).im := by
  rw [lim_eq_lim_im_add_lim_re]; 

Proof state:
f : CauSeq ℂ ⇑abs
⊢ CauSeq.lim (cauSeqIm f) = (↑(CauSeq.lim (cauSeqRe f)) + ↑(CauSeq.lim (cauSeqIm f)) * I).im

Next tactic:
simp [ofReal']
theorem isCauSeq_conj (f : CauSeq ℂ Complex.abs) :
    IsCauSeq Complex.abs fun n => conj (f n) := fun ε ε0 =>
  let ⟨i, hi⟩ := f.2 ε ε0
  ⟨i, fun j hj => by
    

Proof state:
f : CauSeq ℂ ⇑abs
ε : ℝ
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abs (↑f j - ↑f i) < ε
j : ℕ
hj : j ≥ i
⊢ abs ((fun n => (starRingEnd ℂ) (↑f n)) j - (fun n => (starRingEnd ℂ) (↑f n)) i) < ε

Next tactic:
rw [← RingHom.map_sub, abs_conj]
theorem isCauSeq_conj (f : CauSeq ℂ Complex.abs) :
    IsCauSeq Complex.abs fun n => conj (f n) := fun ε ε0 =>
  let ⟨i, hi⟩ := f.2 ε ε0
  ⟨i, fun j hj => by
    rw [← RingHom.map_sub, abs_conj]; 

Proof state:
f : CauSeq ℂ ⇑abs
ε : ℝ
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abs (↑f j - ↑f i) < ε
j : ℕ
hj : j ≥ i
⊢ abs (↑f j - ↑f i) < ε

Next tactic:
exact hi j hj
theorem lim_conj (f : CauSeq ℂ Complex.abs) : lim (cauSeqConj f) = conj (lim f) :=
  Complex.ext (by 

Proof state:
f : CauSeq ℂ ⇑abs
⊢ (CauSeq.lim (cauSeqConj f)).re = ((starRingEnd ℂ) (CauSeq.lim f)).re

Next tactic:
simp [cauSeqConj, (lim_re _).symm, cauSeqRe]
theorem lim_conj (f : CauSeq ℂ Complex.abs) : lim (cauSeqConj f) = conj (lim f) :=
  Complex.ext (by simp [cauSeqConj, (lim_re _).symm, cauSeqRe])
    (by 

Proof state:
f : CauSeq ℂ ⇑abs
⊢ (CauSeq.lim (cauSeqConj f)).im = ((starRingEnd ℂ) (CauSeq.lim f)).im

Next tactic:
simp [cauSeqConj, (lim_im _).symm, cauSeqIm, (lim_neg _).symm]
theorem lim_conj (f : CauSeq ℂ Complex.abs) : lim (cauSeqConj f) = conj (lim f) :=
  Complex.ext (by simp [cauSeqConj, (lim_re _).symm, cauSeqRe])
    (by simp [cauSeqConj, (lim_im _).symm, cauSeqIm, (lim_neg _).symm]; 

Proof state:
f : CauSeq ℂ ⇑abs
⊢ CauSeq.lim { val := fun n => -(↑f n).im, property := (_ : (fun f => IsCauSeq abs' f) fun n => -(↑f n).im) } =
    CauSeq.lim (-{ val := fun n => (↑f n).im, property := (_ : IsCauSeq abs' fun n => (↑f n).im) })

Next tactic:
rfl
lemma re_neg_ne_zero_of_one_lt_re {s : ℂ} (hs : 1 < s.re) : (-s).re ≠ 0 :=
  ne_iff_lt_or_gt.mpr <| Or.inl <| neg_re s ▸ by 

Proof state:
s : ℂ
hs : 1 < s.re
⊢ -s.re < 0

Next tactic:
linarith
/-- If `x` is in the convex hull of some finset `t` whose elements are not affine-independent,
then it is in the convex hull of a strict subset of `t`. -/
theorem mem_convexHull_erase [DecidableEq E] {t : Finset E} (h : ¬AffineIndependent 𝕜 ((↑) : t → E))
    {x : E} (m : x ∈ convexHull 𝕜 (↑t : Set E)) :
    ∃ y : (↑t : Set E), x ∈ convexHull 𝕜 (↑(t.erase y) : Set E) := by
  

Proof state:
𝕜 : Type u_1
E : Type u
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : DecidableEq E
t : Finset E
h : ¬AffineIndependent 𝕜 Subtype.val
x : E
m : x ∈ (convexHull 𝕜) ↑t
⊢ ∃ y, x ∈ (convexHull 𝕜) ↑(erase t ↑y)

Next tactic:
simp only [Finset.convexHull_eq, mem_setOf_eq] at m ⊢
/-- If `x` is in the convex hull of some finset `t` whose elements are not affine-independent,
then it is in the convex hull of a strict subset of `t`. -/
theorem mem_convexHull_erase [DecidableEq E] {t : Finset E} (h : ¬AffineIndependent 𝕜 ((↑) : t → E))
    {x : E} (m : x ∈ convexHull 𝕜 (↑t : Set E)) :
    ∃ y : (↑t : Set E), x ∈ convexHull 𝕜 (↑(t.erase y) : Set E) := by
  simp only [Finset.convexHull_eq, mem_setOf_eq] at m ⊢
  

Proof state:
𝕜 : Type u_1
E : Type u
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : DecidableEq E
t : Finset E
h : ¬AffineIndependent 𝕜 Subtype.val
x : E
m : ∃ w, (∀ y ∈ t, 0 ≤ w y) ∧ ∑ y in t, w y = 1 ∧ centerMass t w id = x
⊢ ∃ y w, (∀ y_1 ∈ erase t ↑y, 0 ≤ w y_1) ∧ ∑ y in erase t ↑y, w y = 1 ∧ centerMass (erase t ↑y) w id = x

Next tactic:
obtain ⟨f, fpos, fsum, rfl⟩ := m
/-- If `x` is in the convex hull of some finset `t` whose elements are not affine-independent,
then it is in the convex hull of a strict subset of `t`. -/
theorem mem_convexHull_erase [DecidableEq E] {t : Finset E} (h : ¬AffineIndependent 𝕜 ((↑) : t → E))
    {x : E} (m : x ∈ convexHull 𝕜 (↑t : Set E)) :
    ∃ y : (↑t : Set E), x ∈ convexHull 𝕜 (↑(t.erase y) : Set E) := by
  simp only [Finset.convexHull_eq, mem_setOf_eq] at m ⊢
  obtain ⟨f, fpos, fsum, rfl⟩ := m
  

Proof state:
case intro.intro.intro
𝕜 : Type u_1
E : Type u
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : DecidableEq E
t : Finset E
h : ¬AffineIndependent 𝕜 Subtype.val
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : ∑ y in t, f y = 1
⊢ ∃ y w, (∀ y_1 ∈ erase t ↑y, 0 ≤ w y_1) ∧ ∑ y in erase t ↑y, w y = 1 ∧ centerMass (erase t ↑y) w id = centerMass t f id

Next tactic:
obtain ⟨g, gcombo, gsum, gpos⟩ := exists_nontrivial_relation_sum_zero_of_not_affine_ind h
/-- If `x` is in the convex hull of some finset `t` whose elements are not affine-independent,
then it is in the convex hull of a strict subset of `t`. -/
theorem mem_convexHull_erase [DecidableEq E] {t : Finset E} (h : ¬AffineIndependent 𝕜 ((↑) : t → E))
    {x : E} (m : x ∈ convexHull 𝕜 (↑t : Set E)) :
    ∃ y : (↑t : Set E), x ∈ convexHull 𝕜 (↑(t.erase y) : Set E) := by
  simp only [Finset.convexHull_eq, mem_setOf_eq] at m ⊢
  obtain ⟨f, fpos, fsum, rfl⟩ := m
  obtain ⟨g, gcombo, gsum, gpos⟩ := exists_nontrivial_relation_sum_zero_of_not_affine_ind h
  

Proof state:
case intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : DecidableEq E
t : Finset E
h : ¬AffineIndependent 𝕜 Subtype.val
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : ∑ y in t, f y = 1
g : E → 𝕜
gcombo : ∑ e in t, g e • e = 0
gsum : ∑ e in t, g e = 0
gpos : ∃ x ∈ t, g x ≠ 0
⊢ ∃ y w, (∀ y_1 ∈ erase t ↑y, 0 ≤ w y_1) ∧ ∑ y in erase t ↑y, w y = 1 ∧ centerMass (erase t ↑y) w id = centerMass t f id

Next tactic:
replace gpos := exists_pos_of_sum_zero_of_exists_nonzero g gsum gpos
/-- If `x` is in the convex hull of some finset `t` whose elements are not affine-independent,
then it is in the convex hull of a strict subset of `t`. -/
theorem mem_convexHull_erase [DecidableEq E] {t : Finset E} (h : ¬AffineIndependent 𝕜 ((↑) : t → E))
    {x : E} (m : x ∈ convexHull 𝕜 (↑t : Set E)) :
    ∃ y : (↑t : Set E), x ∈ convexHull 𝕜 (↑(t.erase y) : Set E) := by
  simp only [Finset.convexHull_eq, mem_setOf_eq] at m ⊢
  obtain ⟨f, fpos, fsum, rfl⟩ := m
  obtain ⟨g, gcombo, gsum, gpos⟩ := exists_nontrivial_relation_sum_zero_of_not_affine_ind h
  replace gpos := exists_pos_of_sum_zero_of_exists_nonzero g gsum gpos
  

Proof state:
case intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : DecidableEq E
t : Finset E
h : ¬AffineIndependent 𝕜 Subtype.val
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : ∑ y in t, f y = 1
g : E → 𝕜
gcombo : ∑ e in t, g e • e = 0
gsum : ∑ e in t, g e = 0
gpos : ∃ i ∈ t, 0 < g i
⊢ ∃ y w, (∀ y_1 ∈ erase t ↑y, 0 ≤ w y_1) ∧ ∑ y in erase t ↑y, w y = 1 ∧ centerMass (erase t ↑y) w id = centerMass t f id

Next tactic:
clear h
/-- If `x` is in the convex hull of some finset `t` whose elements are not affine-independent,
then it is in the convex hull of a strict subset of `t`. -/
theorem mem_convexHull_erase [DecidableEq E] {t : Finset E} (h : ¬AffineIndependent 𝕜 ((↑) : t → E))
    {x : E} (m : x ∈ convexHull 𝕜 (↑t : Set E)) :
    ∃ y : (↑t : Set E), x ∈ convexHull 𝕜 (↑(t.erase y) : Set E) := by
  simp only [Finset.convexHull_eq, mem_setOf_eq] at m ⊢
  obtain ⟨f, fpos, fsum, rfl⟩ := m
  obtain ⟨g, gcombo, gsum, gpos⟩ := exists_nontrivial_relation_sum_zero_of_not_affine_ind h
  replace gpos := exists_pos_of_sum_zero_of_exists_nonzero g gsum gpos
  clear h
  

Proof state:
case intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : ∑ y in t, f y = 1
g : E → 𝕜
gcombo : ∑ e in t, g e • e = 0
gsum : ∑ e in t, g e = 0
gpos : ∃ i ∈ t, 0 < g i
⊢ ∃ y w, (∀ y_1 ∈ erase t ↑y, 0 ≤ w y_1) ∧ ∑ y in erase t ↑y, w y = 1 ∧ centerMass (erase t ↑y) w id = centerMass t f id

Next tactic:
let s := @Finset.filter _ (fun z => 0 < g z) (fun _ => LinearOrder.decidableLT _ _) t
/-- If `x` is in the convex hull of some finset `t` whose elements are not affine-independent,
then it is in the convex hull of a strict subset of `t`. -/
theorem mem_convexHull_erase [DecidableEq E] {t : Finset E} (h : ¬AffineIndependent 𝕜 ((↑) : t → E))
    {x : E} (m : x ∈ convexHull 𝕜 (↑t : Set E)) :
    ∃ y : (↑t : Set E), x ∈ convexHull 𝕜 (↑(t.erase y) : Set E) := by
  simp only [Finset.convexHull_eq, mem_setOf_eq] at m ⊢
  obtain ⟨f, fpos, fsum, rfl⟩ := m
  obtain ⟨g, gcombo, gsum, gpos⟩ := exists_nontrivial_relation_sum_zero_of_not_affine_ind h
  replace gpos := exists_pos_of_sum_zero_of_exists_nonzero g gsum gpos
  clear h
  let s := @Finset.filter _ (fun z => 0 < g z) (fun _ => LinearOrder.decidableLT _ _) t
  

Proof state:
case intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : ∑ y in t, f y = 1
g : E → 𝕜
gcombo : ∑ e in t, g e • e = 0
gsum : ∑ e in t, g e = 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := filter (fun z => 0 < g z) t
⊢ ∃ y w, (∀ y_1 ∈ erase t ↑y, 0 ≤ w y_1) ∧ ∑ y in erase t ↑y, w y = 1 ∧ centerMass (erase t ↑y) w id = centerMass t f id

Next tactic:
obtain ⟨i₀, mem, w⟩ : ∃ i₀ ∈ s, ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i := by
    apply s.exists_min_image fun z => f z / g z
    obtain ⟨x, hx, hgx⟩ : ∃ x ∈ t, 0 < g x := gpos
    exact ⟨x, mem_filter.mpr ⟨hx, hgx⟩⟩
/-- If `x` is in the convex hull of some finset `t` whose elements are not affine-independent,
then it is in the convex hull of a strict subset of `t`. -/
theorem mem_convexHull_erase [DecidableEq E] {t : Finset E} (h : ¬AffineIndependent 𝕜 ((↑) : t → E))
    {x : E} (m : x ∈ convexHull 𝕜 (↑t : Set E)) :
    ∃ y : (↑t : Set E), x ∈ convexHull 𝕜 (↑(t.erase y) : Set E) := by
  simp only [Finset.convexHull_eq, mem_setOf_eq] at m ⊢
  obtain ⟨f, fpos, fsum, rfl⟩ := m
  obtain ⟨g, gcombo, gsum, gpos⟩ := exists_nontrivial_relation_sum_zero_of_not_affine_ind h
  replace gpos := exists_pos_of_sum_zero_of_exists_nonzero g gsum gpos
  clear h
  let s := @Finset.filter _ (fun z => 0 < g z) (fun _ => LinearOrder.decidableLT _ _) t
  obtain ⟨i₀, mem, w⟩ : ∃ i₀ ∈ s, ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i := by
    

Proof state:
𝕜 : Type u_1
E : Type u
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : ∑ y in t, f y = 1
g : E → 𝕜
gcombo : ∑ e in t, g e • e = 0
gsum : ∑ e in t, g e = 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := filter (fun z => 0 < g z) t
⊢ ∃ i₀ ∈ s, ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i

Next tactic:
apply s.exists_min_image fun z => f z / g z
/-- If `x` is in the convex hull of some finset `t` whose elements are not affine-independent,
then it is in the convex hull of a strict subset of `t`. -/
theorem mem_convexHull_erase [DecidableEq E] {t : Finset E} (h : ¬AffineIndependent 𝕜 ((↑) : t → E))
    {x : E} (m : x ∈ convexHull 𝕜 (↑t : Set E)) :
    ∃ y : (↑t : Set E), x ∈ convexHull 𝕜 (↑(t.erase y) : Set E) := by
  simp only [Finset.convexHull_eq, mem_setOf_eq] at m ⊢
  obtain ⟨f, fpos, fsum, rfl⟩ := m
  obtain ⟨g, gcombo, gsum, gpos⟩ := exists_nontrivial_relation_sum_zero_of_not_affine_ind h
  replace gpos := exists_pos_of_sum_zero_of_exists_nonzero g gsum gpos
  clear h
  let s := @Finset.filter _ (fun z => 0 < g z) (fun _ => LinearOrder.decidableLT _ _) t
  obtain ⟨i₀, mem, w⟩ : ∃ i₀ ∈ s, ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i := by
    apply s.exists_min_image fun z => f z / g z
    

Proof state:
𝕜 : Type u_1
E : Type u
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : ∑ y in t, f y = 1
g : E → 𝕜
gcombo : ∑ e in t, g e • e = 0
gsum : ∑ e in t, g e = 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := filter (fun z => 0 < g z) t
⊢ Finset.Nonempty s

Next tactic:
obtain ⟨x, hx, hgx⟩ : ∃ x ∈ t, 0 < g x := gpos
/-- If `x` is in the convex hull of some finset `t` whose elements are not affine-independent,
then it is in the convex hull of a strict subset of `t`. -/
theorem mem_convexHull_erase [DecidableEq E] {t : Finset E} (h : ¬AffineIndependent 𝕜 ((↑) : t → E))
    {x : E} (m : x ∈ convexHull 𝕜 (↑t : Set E)) :
    ∃ y : (↑t : Set E), x ∈ convexHull 𝕜 (↑(t.erase y) : Set E) := by
  simp only [Finset.convexHull_eq, mem_setOf_eq] at m ⊢
  obtain ⟨f, fpos, fsum, rfl⟩ := m
  obtain ⟨g, gcombo, gsum, gpos⟩ := exists_nontrivial_relation_sum_zero_of_not_affine_ind h
  replace gpos := exists_pos_of_sum_zero_of_exists_nonzero g gsum gpos
  clear h
  let s := @Finset.filter _ (fun z => 0 < g z) (fun _ => LinearOrder.decidableLT _ _) t
  obtain ⟨i₀, mem, w⟩ : ∃ i₀ ∈ s, ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i := by
    apply s.exists_min_image fun z => f z / g z
    obtain ⟨x, hx, hgx⟩ : ∃ x ∈ t, 0 < g x := gpos
    

Proof state:
case intro.intro
𝕜 : Type u_1
E : Type u
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : ∑ y in t, f y = 1
g : E → 𝕜
gcombo : ∑ e in t, g e • e = 0
gsum : ∑ e in t, g e = 0
s : Finset E := filter (fun z => 0 < g z) t
x : E
hx : x ∈ t
hgx : 0 < g x
⊢ Finset.Nonempty s

Next tactic:
exact ⟨x, mem_filter.mpr ⟨hx, hgx⟩⟩