chore(data/finsupp/basic): lemmas about sub and neg on filter and erase (#9228)

Author: eric-wieser

src/data/finsupp/basic.lean

@@ -832,6 +832,11 @@ begin
   rw [add_apply, erase_ne hs, erase_ne hs, erase_ne hs, add_apply],
 end
 
+/-- `finsupp.erase` as an `add_monoid_hom`. -/
+@[simps]
+def erase_add_hom (a : α) : (α →₀ M) →+ (α →₀ M) :=
+{ to_fun := erase a, map_zero' := erase_zero a, map_add' := erase_add a }
+
 @[elab_as_eliminator]
 protected theorem induction {p : (α →₀ M) → Prop} (f : α →₀ M)
   (h0 : p 0) (ha : ∀a b (f : α →₀ M), a ∉ f.support → b ≠ 0 → p f → p (single a b + f)) :
@@ -1890,6 +1895,19 @@ ext $ λ _, rfl
 @[simp] lemma single_sub {a : α} {b₁ b₂ : G} : single a (b₁ - b₂) = single a b₁ - single a b₂ :=
 (single_add_hom a : G →+ _).map_sub b₁ b₂
 
+@[simp] lemma erase_neg (a : α) (f : α →₀ G) : erase a (-f) = -erase a f :=
+(erase_add_hom a : (_ →₀ G) →+ _).map_neg f
+
+@[simp] lemma erase_sub (a : α) (f₁ f₂ : α →₀ G) : erase a (f₁ - f₂) = erase a f₁ - erase a f₂ :=
+(erase_add_hom a : (_ →₀ G) →+ _).map_sub f₁ f₂
+
+@[simp] lemma filter_neg (p : α → Prop) (f : α →₀ G) : filter p (-f) = -filter p f :=
+(filter_add_hom p : (_ →₀ G) →+ _).map_neg f
+
+@[simp] lemma filter_sub (p : α → Prop) (f₁ f₂ : α →₀ G) :
+  filter p (f₁ - f₂) = filter p f₁ - filter p f₂ :=
+(filter_add_hom p : (_ →₀ G) →+ _).map_sub f₁ f₂
+
 end group
 
 end subtype_domain