-- Mathlib/Analysis/Normed/Affine/Convex.lean
/-- We can intercalate a simplex between a point and one of its neighborhoods. -/
lemma exists_mem_interior_convexHull_affineBasis (hs : s ∈ 𝓝 x) :
    ∃ b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E,
      x ∈ interior (convexHull ℝ (range b)) ∧ convexHull ℝ (range b) ⊆ s

Proof state:
E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
inst✝³ : NormedSpace ℝ E✝
s✝ : Set E✝
x : E✝
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hs : s ∈ 𝓝 0
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
c : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E := -Finset.centroid ℝ Finset.univ ⇑b +ᵥ b
hc₀ : 0 ∈ interior ((convexHull ℝ) (range ⇑c))
cnorm : ℝ := Finset.univ.sup' ⋯ fun i => ‖c i‖
hcnorm : range ⇑c ⊆ closedBall 0 (cnorm + 1)
ε : ℝ
hε : ε > 0
hεs : ball 0 ε ⊆ s
ε' : ℝ := ε / 2 / (cnorm + 1)
⊢ 0 < cnorm + 1

Tactic:
have : 0 ≤ cnorm := Finset.le_sup'_of_le _ (Finset.mem_univ 0) (norm_nonneg _)

Result:
E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
inst✝³ : NormedSpace ℝ E✝
s✝ : Set E✝
x : E✝
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hs : s ∈ 𝓝 0
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
c : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E := -Finset.centroid ℝ Finset.univ ⇑b +ᵥ b
hc₀ : 0 ∈ interior ((convexHull ℝ) (range ⇑c))
cnorm : ℝ := Finset.univ.sup' ⋯ fun i => ‖c i‖
hcnorm : range ⇑c ⊆ closedBall 0 (cnorm + 1)
ε : ℝ
hε : ε > 0
hεs : ball 0 ε ⊆ s
ε' : ℝ := ε / 2 / (cnorm + 1)
this : 0 ≤ cnorm
⊢ 0 < cnorm + 1

Proof state:
case intro.intro.intro
E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
inst✝³ : NormedSpace ℝ E✝
s✝ : Set E✝
x : E✝
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hs : s ∈ 𝓝 0
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
c : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E := -Finset.centroid ℝ Finset.univ ⇑b +ᵥ b
hc₀ : 0 ∈ interior ((convexHull ℝ) (range ⇑c))
cnorm : ℝ := Finset.univ.sup' ⋯ fun i => ‖c i‖
hcnorm : range ⇑c ⊆ closedBall 0 (cnorm + 1)
ε : ℝ
hε : ε > 0
hεs : ball 0 ε ⊆ s
⊢ ∃ b, 0 ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s

Tactic:
set ε' : ℝ := ε / 2 / (cnorm + 1)

Result:
case intro.intro.intro
E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
inst✝³ : NormedSpace ℝ E✝
s✝ : Set E✝
x : E✝
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hs : s ∈ 𝓝 0
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
c : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E := -Finset.centroid ℝ Finset.univ ⇑b +ᵥ b
hc₀ : 0 ∈ interior ((convexHull ℝ) (range ⇑c))
cnorm : ℝ := Finset.univ.sup' ⋯ fun i => ‖c i‖
hcnorm : range ⇑c ⊆ closedBall 0 (cnorm + 1)
ε : ℝ
hε : ε > 0
hεs : ball 0 ε ⊆ s
ε' : ℝ := ε / 2 / (cnorm + 1)
⊢ ∃ b, 0 ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s

Proof state:
case inr.intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
hs : s ∈ 𝓝 x
this :
  ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {s : Set E}
    {x : E}, s ∈ 𝓝 x → x = 0 → ∃ b, x ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s
hx : ¬x = 0
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
hb : 0 ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ -x +ᵥ s
⊢ ∃ b, x ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s

Tactic:
use x +ᵥ b

Result:
case h
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
hs : s ∈ 𝓝 x
this :
  ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {s : Set E}
    {x : E}, s ∈ 𝓝 x → x = 0 → ∃ b, x ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s
hx : ¬x = 0
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
hb : 0 ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ -x +ᵥ s
⊢ x ∈ interior ((convexHull ℝ) (range ⇑(x +ᵥ b))) ∧ (convexHull ℝ) (range ⇑(x +ᵥ b)) ⊆ s

Proof state:
case intro.intro.intro.refine_2
E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
inst✝³ : NormedSpace ℝ E✝
s✝ : Set E✝
x : E✝
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hs : s ∈ 𝓝 0
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
c : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E := -Finset.centroid ℝ Finset.univ ⇑b +ᵥ b
hc₀ : 0 ∈ interior ((convexHull ℝ) (range ⇑c))
cnorm : ℝ := Finset.univ.sup' ⋯ fun i => ‖c i‖
hcnorm : range ⇑c ⊆ closedBall 0 (cnorm + 1)
ε : ℝ
hε : ε > 0
hεs : ball 0 ε ⊆ s
ε' : ℝ := ε / 2 / (cnorm + 1)
hc' : 0 < cnorm + 1
hε' : 0 < ε'
d : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E := Units.mk0 ε' ⋯ • c
hε₀ : 0 < ε / 2
hdnorm : range ⇑d ⊆ closedBall 0 (ε / 2)
⊢ (convexHull ℝ) (range ⇑d) ⊆ s

Tactic:
calc
  convexHull ℝ (range d) ⊆ closedBall 0 (ε / 2) := convexHull_min hdnorm (convex_closedBall ..)
  _ ⊆ ball 0 ε := closedBall_subset_ball (by linarith)
  _ ⊆ s := hεs

Result:
no goals

Proof state:
case inr
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
hs : s ∈ 𝓝 x
this :
  ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {s : Set E}
    {x : E}, s ∈ 𝓝 x → x = 0 → ∃ b, x ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s
hx : ¬x = 0
⊢ ∃ b, x ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s

Tactic:
obtain ⟨b, hb⟩ := this (s := -x +ᵥ s) (by simpa using vadd_mem_nhds_vadd (-x) hs) rfl

Result:
case inr.intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
hs : s ∈ 𝓝 x
this :
  ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {s : Set E}
    {x : E}, s ∈ 𝓝 x → x = 0 → ∃ b, x ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s
hx : ¬x = 0
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
hb : 0 ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ -x +ᵥ s
⊢ ∃ b, x ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s

Proof state:
case h
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
hs : s ∈ 𝓝 x
this :
  ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {s : Set E}
    {x : E}, s ∈ 𝓝 x → x = 0 → ∃ b, x ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s
hx : ¬x = 0
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
hb : 0 ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ -x +ᵥ s
⊢ x ∈ interior ((convexHull ℝ) (range ⇑(x +ᵥ b))) ∧ (convexHull ℝ) (range ⇑(x +ᵥ b)) ⊆ s

Tactic:
simpa [subset_vadd_set_iff, mem_vadd_set_iff_neg_vadd_mem, convexHull_vadd, interior_vadd,
  Pi.vadd_def, -vadd_eq_add, vadd_eq_add (a := -x), ← Set.vadd_set_range] using hb

Result:
no goals