feat(analysis/seminorm): Add has_add and has_scalar nnreal (#11414)

Add instances of `has_add` and `has_scalar nnreal` type classes for `seminorm`.

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>



Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

src/analysis/seminorm.lean

@@ -66,9 +66,9 @@ Absorbent and balanced sets in a vector space over a normed field.
 -/
 
 open normed_field set
-open_locale pointwise topological_space
+open_locale pointwise topological_space nnreal
 
-variables {𝕜 E ι : Type*}
+variables {R 𝕜 E ι : Type*}
 
 section semi_normed_ring
 variables [semi_normed_ring 𝕜]
@@ -278,6 +278,71 @@ variables (p : seminorm 𝕜 E) (c : 𝕜) (x y : E) (r : ℝ)
 protected lemma smul : p (c • x) = ∥c∥ * p x := p.smul' _ _
 protected lemma triangle : p (x + y) ≤ p x + p y := p.triangle' _ _
 
+/-- Any action on `ℝ` which factors through `ℝ≥0` applies to a seminorm. -/
+instance [has_scalar R ℝ] [has_scalar R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ] :
+  has_scalar R (seminorm 𝕜 E) :=
+{ smul := λ r p,
+  { to_fun := λ x, r • p x,
+    smul' := λ _ _, begin
+      simp only [←smul_one_smul ℝ≥0 r (_ : ℝ), nnreal.smul_def, smul_eq_mul],
+      rw [p.smul, mul_left_comm],
+    end,
+    triangle' := λ _ _, begin
+      simp only [←smul_one_smul ℝ≥0 r (_ : ℝ), nnreal.smul_def, smul_eq_mul],
+      exact (mul_le_mul_of_nonneg_left (p.triangle _ _) (nnreal.coe_nonneg _)).trans_eq
+        (mul_add _ _ _),
+    end } }
+
+lemma coe_smul [has_scalar R ℝ] [has_scalar R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ]
+  (r : R) (p : seminorm 𝕜 E) : ⇑(r • p) = r • p := rfl
+
+@[simp] lemma smul_apply [has_scalar R ℝ] [has_scalar R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ]
+  (r : R) (p : seminorm 𝕜 E) (x : E) : (r • p) x = r • p x := rfl
+
+instance : has_add (seminorm 𝕜 E) :=
+{ add := λ p q,
+  { to_fun := λ x, p x + q x,
+    smul' := λ a x, by rw [p.smul, q.smul, mul_add],
+    triangle' := λ _ _, has_le.le.trans_eq (add_le_add (p.triangle _ _) (q.triangle _ _))
+      (add_add_add_comm _ _ _ _) } }
+
+lemma coe_add (p q : seminorm 𝕜 E) : ⇑(p + q) = p + q := rfl
+
+@[simp] lemma add_apply (p q : seminorm 𝕜 E) (x : E) : (p + q) x = p x + q x := rfl
+
+instance : add_monoid (seminorm 𝕜 E) :=
+fun_like.coe_injective.add_monoid_smul _ rfl coe_add (λ p n, coe_smul n p)
+
+instance : ordered_cancel_add_comm_monoid (seminorm 𝕜 E) :=
+{ nsmul := (•),  -- to avoid introducing a diamond
+  ..seminorm.add_monoid,
+  ..(fun_like.coe_injective.ordered_cancel_add_comm_monoid _ rfl coe_add
+      : ordered_cancel_add_comm_monoid (seminorm 𝕜 E)) }
+
+instance [monoid R] [mul_action R ℝ] [has_scalar R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ] :
+  mul_action R (seminorm 𝕜 E) :=
+fun_like.coe_injective.mul_action _ coe_smul
+
+variables (𝕜 E)
+
+/-- `coe_fn` as an `add_monoid_hom`. Helper definition for showing that `seminorm 𝕜 E` is
+a module. -/
+@[simps]
+def coe_fn_add_monoid_hom : add_monoid_hom (seminorm 𝕜 E) (E → ℝ) := ⟨coe_fn, coe_zero, coe_add⟩
+
+lemma coe_fn_add_monoid_hom_injective : function.injective (coe_fn_add_monoid_hom 𝕜 E) :=
+show @function.injective (seminorm 𝕜 E) (E → ℝ) coe_fn, from fun_like.coe_injective
+
+variables {𝕜 E}
+
+instance [monoid R] [distrib_mul_action R ℝ] [has_scalar R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ] :
+  distrib_mul_action R (seminorm 𝕜 E) :=
+(coe_fn_add_monoid_hom_injective 𝕜 E).distrib_mul_action _ coe_smul
+
+instance [semiring R] [module R ℝ] [has_scalar R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ] :
+  module R (seminorm 𝕜 E) :=
+(coe_fn_add_monoid_hom_injective 𝕜 E).module R _ coe_smul
+
 -- TODO: define `has_Sup` too, from the skeleton at
 -- https://github.com/leanprover-community/mathlib/pull/11329#issuecomment-1008915345
 noncomputable instance : has_sup (seminorm 𝕜 E) :=