chore(algebra/group): make coe_norm_subgroup and submodule.norm_coe consistent (#11427)

The `simp` lemmas for norms in a subgroup and in a submodule disagreed: the first inserted a coercion to the larger group, the second deleted the coercion. Currently this is not a big deal, but it will become a real issue when defining `add_subgroup_class`. I want to make them consistent by pointing them in the same direction. The consensus in the [Zulip thread](https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/Simp.20normal.20form.3A.20coe_norm_subgroup.2C.20submodule.2Enorm_coe) suggests `simp` should insert a coercion here, so I went with that.

After making the changes, a few places need extra `simp [submodule.coe_norm]` on the local hypotheses, but nothing major.

Author: Vierkantor

src/analysis/normed/group/basic.lean

@@ -656,8 +656,20 @@ semi_normed_group.induced s.subtype
 
 /-- If `x` is an element of a subgroup `s` of a seminormed group `E`, its norm in `s` is equal to
 its norm in `E`. -/
-@[simp] lemma coe_norm_subgroup {E : Type*} [semi_normed_group E] {s : add_subgroup E} (x : s) :
-  ∥x∥ = ∥(x:E)∥ :=
+@[simp] lemma add_subgroup.coe_norm {E : Type*} [semi_normed_group E]
+  {s : add_subgroup E} (x : s) :
+  ∥(x : s)∥ = ∥(x:E)∥ :=
+rfl
+
+/-- If `x` is an element of a subgroup `s` of a seminormed group `E`, its norm in `s` is equal to
+its norm in `E`.
+
+This is a reversed version of the `simp` lemma `add_subgroup.coe_norm` for use by `norm_cast`.
+-/
+
+@[norm_cast] lemma add_subgroup.norm_coe {E : Type*} [semi_normed_group E] {s : add_subgroup E}
+  (x : s) :
+  ∥(x : E)∥ = ∥(x : s)∥ :=
 rfl
 
 /-- A submodule of a seminormed group is also a seminormed group, with the restriction of the norm.
@@ -668,18 +680,24 @@ instance submodule.semi_normed_group {𝕜 : Type*} {_ : ring 𝕜}
 { norm := λx, norm (x : E),
   dist_eq := λx y, dist_eq_norm (x : E) (y : E) }
 
-/-- If `x` is an element of a submodule `s` of a normed group `E`, its norm in `E` is equal to its
-norm in `s`.
+/-- If `x` is an element of a submodule `s` of a normed group `E`, its norm in `s` is equal to its
+norm in `E`.
 
 See note [implicit instance arguments]. -/
-@[simp, norm_cast] lemma submodule.norm_coe {𝕜 : Type*} {_ : ring 𝕜}
+@[simp] lemma submodule.coe_norm {𝕜 : Type*} {_ : ring 𝕜}
   {E : Type*} [semi_normed_group E] {_ : module 𝕜 E} {s : submodule 𝕜 E} (x : s) :
-  ∥(x : E)∥ = ∥x∥ :=
+  ∥(x : s)∥ = ∥(x : E)∥ :=
 rfl
 
-@[simp] lemma submodule.norm_mk {𝕜 : Type*} {_ : ring 𝕜}
-  {E : Type*} [semi_normed_group E] {_ : module 𝕜 E} {s : submodule 𝕜 E} (x : E) (hx : x ∈ s) :
-  ∥(⟨x, hx⟩ : s)∥ = ∥x∥ :=
+/-- If `x` is an element of a submodule `s` of a normed group `E`, its norm in `E` is equal to its
+norm in `s`.
+
+This is a reversed version of the `simp` lemma `submodule.coe_norm` for use by `norm_cast`.
+
+See note [implicit instance arguments]. -/
+@[norm_cast] lemma submodule.norm_coe {𝕜 : Type*} {_ : ring 𝕜}
+  {E : Type*} [semi_normed_group E] {_ : module 𝕜 E} {s : submodule 𝕜 E} (x : s) :
+  ∥(x : E)∥ = ∥(x : s)∥ :=
 rfl
 
 /-- seminormed group instance on the product of two seminormed groups, using the sup norm. -/

src/geometry/manifold/instances/sphere.lean

@@ -200,11 +200,13 @@ begin
   -- the core of the problem is these two algebraic identities:
   have h₁ : (2 ^ 2 / (1 - a) ^ 2 * ∥y∥ ^ 2 + 4)⁻¹ * 4 * (2 / (1 - a)) = 1,
   { field_simp,
+    simp only [submodule.coe_norm] at *,
     nlinarith },
   have h₂ : (2 ^ 2 / (1 - a) ^ 2 * ∥y∥ ^ 2 + 4)⁻¹ * (2 ^ 2 / (1 - a) ^ 2 * ∥y∥ ^ 2 - 4) = a,
   { field_simp,
     transitivity (1 - a) ^ 2 * (a * (2 ^ 2 * ∥y∥ ^ 2 + 4 * (1 - a) ^ 2)),
     { congr,
+      simp only [submodule.coe_norm] at *,
       nlinarith },
     ring },
   -- deduce the result
@@ -337,7 +339,7 @@ begin
   have h_set : ∀ p : sphere (0:E) 1, p = v' ↔ ⟪(p:E), v'⟫_ℝ = 1,
   { simp [subtype.ext_iff, inner_eq_norm_mul_iff_of_norm_one] },
   ext,
-  simp [h_set, hUv, hU'v', stereographic, real_inner_comm]
+  simp [h_set, hUv, hU'v', stereographic, real_inner_comm, ← submodule.coe_norm]
 end
 
 /-- The inclusion map (i.e., `coe`) from the sphere in `E` to `E` is smooth.  -/

src/measure_theory/function/simple_func_dense.lean

@@ -569,7 +569,7 @@ local attribute [instance] simple_func.module
 /-- If `E` is a normed space, `Lp.simple_func E p μ` is a normed space. Not declared as an
 instance as it is (as of writing) used only in the construction of the Bochner integral. -/
 protected def normed_space [fact (1 ≤ p)] : normed_space 𝕜 (Lp.simple_func E p μ) :=
-⟨ λc f, by { rw [coe_norm_subgroup, coe_norm_subgroup, coe_smul, norm_smul] } ⟩
+⟨ λc f, by { rw [add_subgroup.coe_norm, add_subgroup.coe_norm, coe_smul, norm_smul] } ⟩
 
 end instances