-- Mathlib/Analysis/Convex/Visible.lean
/-- If a point `x` sees a convex combination of points of a set `s` through `convexHull ℝ s ∌ x`,
then it sees all terms of that combination.

Note that the converse does not hold. -/
lemma IsVisible.of_convexHull_of_pos {ι : Type*} {t : Finset ι} {a : ι → V} {w : ι → 𝕜}
    (hw₀ : ∀ i ∈ t, 0 ≤ w i) (hw₁ : ∑ i ∈ t, w i = 1) (ha : ∀ i ∈ t, a i ∈ s)
    (hx : x ∉ convexHull 𝕜 s) (hw : IsVisible 𝕜 (convexHull 𝕜 s) x (∑ i ∈ t, w i • a i)) {i : ι}
    (hi : i ∈ t) (hwi : 0 < w i) : IsVisible 𝕜 (convexHull 𝕜 s) x (a i)

Proof state:
case inr.intro.intro.intro.intro.refine_1.refine_1
𝕜 : Type u_1
V : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝¹ : 0 < w i
hwi✝ : w i < 1
ε : 𝕜
hε : (lineMap x (a i)) ε ∈ (convexHull 𝕜) s
h : (lineMap x (a i)) ε ≠ x ∧ (lineMap x (a i)) ε ≠ a i
hε₀ : 0 < ε
hε₁ : ε < 1
this✝ : 0 < 1 - ε
hwi : 0 < 1 - w i
this :
  Wbtw 𝕜 ((lineMap x (a i)) ε) ((lineMap x (∑ j ∈ t, w j • a j)) ((w i)⁻¹ / ((1 - ε) / ε + (w i)⁻¹)))
    (∑ j ∈ t.erase i, (w j / (1 - w i)) • a j)
⊢ ∀ i_1 ∈ t.erase i, 0 ≤ w i_1 / (1 - w i)

Tactic:
intros j hj

Result:
case inr.intro.intro.intro.intro.refine_1.refine_1
𝕜 : Type u_1
V : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝¹ : 0 < w i
hwi✝ : w i < 1
ε : 𝕜
hε : (lineMap x (a i)) ε ∈ (convexHull 𝕜) s
h : (lineMap x (a i)) ε ≠ x ∧ (lineMap x (a i)) ε ≠ a i
hε₀ : 0 < ε
hε₁ : ε < 1
this✝ : 0 < 1 - ε
hwi : 0 < 1 - w i
this :
  Wbtw 𝕜 ((lineMap x (a i)) ε) ((lineMap x (∑ j ∈ t, w j • a j)) ((w i)⁻¹ / ((1 - ε) / ε + (w i)⁻¹)))
    (∑ j ∈ t.erase i, (w j / (1 - w i)) • a j)
j : ι
hj : j ∈ t.erase i
⊢ 0 ≤ w j / (1 - w i)

Proof state:
case inr.intro.intro.intro.intro.refine_1.refine_1
𝕜 : Type u_1
V : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝¹ : 0 < w i
hwi✝ : w i < 1
ε : 𝕜
hε : (lineMap x (a i)) ε ∈ (convexHull 𝕜) s
h : (lineMap x (a i)) ε ≠ x ∧ (lineMap x (a i)) ε ≠ a i
hε₀ : 0 < ε
hε₁ : ε < 1
this✝ : 0 < 1 - ε
hwi : 0 < 1 - w i
this :
  Wbtw 𝕜 ((lineMap x (a i)) ε) ((lineMap x (∑ j ∈ t, w j • a j)) ((w i)⁻¹ / ((1 - ε) / ε + (w i)⁻¹)))
    (∑ j ∈ t.erase i, (w j / (1 - w i)) • a j)
j : ι
hj : j ∈ t.erase i
⊢ 0 ≤ w j / (1 - w i)

Tactic:
have := hw₀ j <| erase_subset _ _ hj

Result:
case inr.intro.intro.intro.intro.refine_1.refine_1
𝕜 : Type u_1
V : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝¹ : 0 < w i
hwi✝ : w i < 1
ε : 𝕜
hε : (lineMap x (a i)) ε ∈ (convexHull 𝕜) s
h : (lineMap x (a i)) ε ≠ x ∧ (lineMap x (a i)) ε ≠ a i
hε₀ : 0 < ε
hε₁ : ε < 1
this✝¹ : 0 < 1 - ε
hwi : 0 < 1 - w i
this✝ :
  Wbtw 𝕜 ((lineMap x (a i)) ε) ((lineMap x (∑ j ∈ t, w j • a j)) ((w i)⁻¹ / ((1 - ε) / ε + (w i)⁻¹)))
    (∑ j ∈ t.erase i, (w j / (1 - w i)) • a j)
j : ι
hj : j ∈ t.erase i
this : 0 ≤ w j
⊢ 0 ≤ w j / (1 - w i)

Proof state:
𝕜 : Type u_1
V : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝¹ : 0 < w i
hwi✝ : w i < 1
ε : 𝕜
hε : (lineMap x (a i)) ε ∈ (convexHull 𝕜) s
h : (lineMap x (a i)) ε ≠ x ∧ (lineMap x (a i)) ε ≠ a i
hε₀ : 0 < ε
hε₁ : ε < 1
this : 0 < 1 - ε
hwi : 0 < 1 - w i
⊢ Wbtw 𝕜 ((lineMap x (a i)) ε) ((lineMap x (∑ j ∈ t, w j • a j)) ((w i)⁻¹ / ((1 - ε) / ε + (w i)⁻¹)))
    (∑ j ∈ t.erase i, (w j / (1 - w i)) • a j)

Tactic:
refine ⟨((1 - w i) / w i) / ((1 - ε) / ε + (1 - w i) / w i + 1), ⟨by positivity, ?_⟩, ?_⟩

Result:
case refine_1
𝕜 : Type u_1
V : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝¹ : 0 < w i
hwi✝ : w i < 1
ε : 𝕜
hε : (lineMap x (a i)) ε ∈ (convexHull 𝕜) s
h : (lineMap x (a i)) ε ≠ x ∧ (lineMap x (a i)) ε ≠ a i
hε₀ : 0 < ε
hε₁ : ε < 1
this : 0 < 1 - ε
hwi : 0 < 1 - w i
⊢ (1 - w i) / w i / ((1 - ε) / ε + (1 - w i) / w i + 1) ≤ 1

case refine_2
𝕜 : Type u_1
V : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝¹ : 0 < w i
hwi✝ : w i < 1
ε : 𝕜
hε : (lineMap x (a i)) ε ∈ (convexHull 𝕜) s
h : (lineMap x (a i)) ε ≠ x ∧ (lineMap x (a i)) ε ≠ a i
hε₀ : 0 < ε
hε₁ : ε < 1
this : 0 < 1 - ε
hwi : 0 < 1 - w i
⊢ (lineMap ((lineMap x (a i)) ε) (∑ j ∈ t.erase i, (w j / (1 - w i)) • a j))
      ((1 - w i) / w i / ((1 - ε) / ε + (1 - w i) / w i + 1)) =
    (lineMap x (∑ j ∈ t, w j • a j)) ((w i)⁻¹ / ((1 - ε) / ε + (w i)⁻¹))

Proof state:
case refine_2
𝕜 : Type u_1
V : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝¹ : 0 < w i
hwi✝ : w i < 1
ε : 𝕜
hε : (lineMap x (a i)) ε ∈ (convexHull 𝕜) s
h : (lineMap x (a i)) ε ≠ x ∧ (lineMap x (a i)) ε ≠ a i
hε₀ : 0 < ε
hε₁ : ε < 1
this : 0 < 1 - ε
hwi : 0 < 1 - w i
⊢ (lineMap ((lineMap x (a i)) ε) (∑ j ∈ t.erase i, (w j / (1 - w i)) • a j))
      ((1 - w i) / w i / ((1 - ε) / ε + (1 - w i) / w i + 1)) =
    (lineMap x (∑ j ∈ t, w j • a j)) ((w i)⁻¹ / ((1 - ε) / ε + (w i)⁻¹))

Tactic:
have :
  w i • a i + (1 - w i) • ∑ j ∈ t.erase i, (w j / (1 - w i)) • a j = ∑ j ∈ t, w j • a j := by
  rw [smul_sum]
  simp_rw [smul_smul, mul_div_cancel₀ _ hwi.ne']
  exact add_sum_erase _ (fun i ↦ w i • a i) hi

Result:
case refine_2
𝕜 : Type u_1
V : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝¹ : 0 < w i
hwi✝ : w i < 1
ε : 𝕜
hε : (lineMap x (a i)) ε ∈ (convexHull 𝕜) s
h : (lineMap x (a i)) ε ≠ x ∧ (lineMap x (a i)) ε ≠ a i
hε₀ : 0 < ε
hε₁ : ε < 1
this✝ : 0 < 1 - ε
hwi : 0 < 1 - w i
this : w i • a i + (1 - w i) • ∑ j ∈ t.erase i, (w j / (1 - w i)) • a j = ∑ j ∈ t, w j • a j
⊢ (lineMap ((lineMap x (a i)) ε) (∑ j ∈ t.erase i, (w j / (1 - w i)) • a j))
      ((1 - w i) / w i / ((1 - ε) / ε + (1 - w i) / w i + 1)) =
    (lineMap x (∑ j ∈ t, w j • a j)) ((w i)⁻¹ / ((1 - ε) / ε + (w i)⁻¹))