chore(topology/algebra): remove dead code (#9467)

This code wasn't used and its historically intended use will soon be redone much better. The second file is only a dead import and a misleading comment (referring to the dead code of the *other* file).

src/topology/algebra/group.lean

@@ -465,78 +465,6 @@ class add_group_with_zero_nhd (G : Type u) extends add_comm_group G :=
 (zero_Z : pure 0 ≤ Z)
 (sub_Z : tendsto (λp:G×G, p.1 - p.2) (Z ×ᶠ Z) Z)
 
-namespace add_group_with_zero_nhd
-variables (G) [add_group_with_zero_nhd G]
-
-local notation `Z` := add_group_with_zero_nhd.Z
-
-@[priority 100] -- see Note [lower instance priority]
-instance : topological_space G :=
-topological_space.mk_of_nhds $ λa, map (λx, x + a) (Z G)
-
-variables {G}
-
-lemma neg_Z : tendsto (λa:G, - a) (Z G) (Z G) :=
-have tendsto (λa, (0:G)) (Z G) (Z G),
-  by refine le_trans (assume h, _) zero_Z; simp [univ_mem'] {contextual := tt},
-have tendsto (λa:G, 0 - a) (Z G) (Z G), from
-  sub_Z.comp (tendsto.prod_mk this tendsto_id),
-by simpa
-
-lemma add_Z : tendsto (λp:G×G, p.1 + p.2) (Z G ×ᶠ Z G) (Z G) :=
-suffices tendsto (λp:G×G, p.1 - -p.2) (Z G ×ᶠ Z G) (Z G),
-  by simpa [sub_eq_add_neg],
-sub_Z.comp (tendsto.prod_mk tendsto_fst (neg_Z.comp tendsto_snd))
-
-lemma exists_Z_half {s : set G} (hs : s ∈ Z G) : ∃ V ∈ Z G, ∀ (v ∈ V) (w ∈ V), v + w ∈ s :=
-begin
-  have : ((λa:G×G, a.1 + a.2) ⁻¹' s) ∈ Z G ×ᶠ Z G := add_Z (by simpa using hs),
-  rcases mem_prod_self_iff.1 this with ⟨V, H, H'⟩,
-  exact ⟨V, H, prod_subset_iff.1 H'⟩
-end
-
-lemma nhds_eq (a : G) : 𝓝 a = map (λx, x + a) (Z G) :=
-topological_space.nhds_mk_of_nhds _ _
-  (assume a, calc pure a = map (λx, x + a) (pure 0) : by simp
-    ... ≤ _ : map_mono zero_Z)
-  (assume b s hs,
-    let ⟨t, ht, eqt⟩ := exists_Z_half hs in
-    have t0 : (0:G) ∈ t, by simpa using zero_Z ht,
-    begin
-      refine ⟨(λx:G, x + b) '' t, image_mem_map ht, _, _⟩,
-      { refine set.image_subset_iff.2 (assume b hbt, _),
-        simpa using eqt 0 t0 b hbt },
-      { rintros _ ⟨c, hb, rfl⟩,
-        refine (Z G).sets_of_superset ht (assume x hxt, _),
-        simpa [add_assoc] using eqt _ hxt _ hb }
-    end)
-
-lemma nhds_zero_eq_Z : 𝓝 0 = Z G := by simp [nhds_eq]; exact filter.map_id
-
-@[priority 100] -- see Note [lower instance priority]
-instance : has_continuous_add G :=
-⟨ continuous_iff_continuous_at.2 $ assume ⟨a, b⟩,
-  begin
-    rw [continuous_at, nhds_prod_eq, nhds_eq, nhds_eq, nhds_eq, filter.prod_map_map_eq,
-      tendsto_map'_iff],
-    suffices :  tendsto ((λx:G, (a + b) + x) ∘ (λp:G×G,p.1 + p.2)) (Z G ×ᶠ Z G)
-      (map (λx:G, (a + b) + x) (Z G)),
-    { simpa [(∘), add_comm, add_left_comm] },
-    exact tendsto_map.comp add_Z
-  end ⟩
-
-@[priority 100] -- see Note [lower instance priority]
-instance : topological_add_group G :=
-⟨continuous_iff_continuous_at.2 $ assume a,
-  begin
-    rw [continuous_at, nhds_eq, nhds_eq, tendsto_map'_iff],
-    suffices : tendsto ((λx:G, x - a) ∘ (λx:G, -x)) (Z G) (map (λx:G, x - a) (Z G)),
-    { simpa [(∘), add_comm, sub_eq_add_neg] using this },
-    exact tendsto_map.comp neg_Z
-  end⟩
-
-end add_group_with_zero_nhd
-
 section filter_mul
 
 section

src/topology/algebra/uniform_group.lean

@@ -7,7 +7,6 @@ import topology.uniform_space.uniform_embedding
 import topology.uniform_space.complete_separated
 import topology.algebra.group
 import tactic.abel
-import deprecated.group
 
 /-!
 # Uniform structure on topological groups
@@ -16,9 +15,6 @@ import deprecated.group
   construct a canonical uniformity for a topological add group.
 
 * extension of ℤ-bilinear maps to complete groups (useful for ring completions)
-
-* `add_group_with_zero_nhd`: construct the topological structure from a group with a neighbourhood
-  around zero. Then with `topological_add_group.to_uniform_space` one can derive a `uniform_space`.
 -/
 
 noncomputable theory