chore(analysis/convex/basic): add finset.convex_hull_eq (#2956)

Author: urkud

src/analysis/convex/basic.lean

@@ -796,13 +796,13 @@ begin
   exact ⟨z i, hz i hit, Hi⟩
 end
 
-lemma set.finite.convex_hull_eq {s : set E} (hs : finite s) :
-  convex_hull s = {x : E | ∃ (w : E → ℝ) (hw₀ : ∀ y ∈ s, 0 ≤ w y) (hw₁ : hs.to_finset.sum w = 1),
-    hs.to_finset.center_mass w id = x} :=
+lemma finset.convex_hull_eq (s : finset E) :
+  convex_hull ↑s = {x : E | ∃ (w : E → ℝ) (hw₀ : ∀ y ∈ s, 0 ≤ w y) (hw₁ : s.sum w = 1),
+    s.center_mass w id = x} :=
 begin
   refine subset.antisymm (convex_hull_min _ _) _,
   { intros x hx,
-    replace hx : x ∈ hs.to_finset, from finite.mem_to_finset.2 hx,
+    rw [finset.mem_coe] at hx,
     refine ⟨_, _, _, finset.center_mass_ite_eq _ _ _ hx⟩,
     { intros, split_ifs, exacts [zero_le_one, le_refl 0] },
     { rw [finset.sum_ite_eq, if_pos hx] } },
@@ -814,10 +814,16 @@ begin
       apply_rules [add_nonneg, mul_nonneg, hwx₀, hwy₀], },
     { simp only [finset.sum_add_distrib, finset.mul_sum.symm, mul_one, *] } },
   { rintros _ ⟨w, hw₀, hw₁, rfl⟩,
-    exact hs.to_finset.center_mass_mem_convex_hull (λ x hx, hw₀ _ $ finite.mem_to_finset.1 hx)
-      (hw₁.symm ▸ zero_lt_one) (λ x hx, finite.mem_to_finset.1 hx) }
+    exact s.center_mass_mem_convex_hull (λ x hx, hw₀ _  hx)
+      (hw₁.symm ▸ zero_lt_one) (λ x hx, hx) }
 end
 
+lemma set.finite.convex_hull_eq {s : set E} (hs : finite s) :
+  convex_hull s = {x : E | ∃ (w : E → ℝ) (hw₀ : ∀ y ∈ s, 0 ≤ w y) (hw₁ : hs.to_finset.sum w = 1),
+    hs.to_finset.center_mass w id = x} :=
+by simpa only [set.finite.coe_to_finset, set.finite.mem_to_finset, exists_prop]
+  using hs.to_finset.convex_hull_eq
+
 lemma convex_hull_eq_union_convex_hull_finite_subsets (s : set E) :
   convex_hull s = ⋃ (t : finset E) (w : ↑t ⊆ s), convex_hull ↑t :=
 begin