feat(equiv/transfer_instances): other algebraic structures (#3870)

Some updates to `data.equiv.transfer_instances`.

1. Use `@[to_additive]`
2. Add algebraic equivalences between the original and transferred instances.
3. Transfer modules and algebras.



Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

Author: kim-em

src/data/equiv/transfer_instance.lean

@@ -5,6 +5,8 @@ Authors: Johannes Hölzl
 -/
 import data.equiv.basic
 import algebra.field
+import algebra.module
+import ring_theory.algebra
 import algebra.group.type_tags
 
 /-!
@@ -15,9 +17,11 @@ group structure and `α ≃ β` then `α` has a group structure, and
 similarly for monoids, semigroups, rings, integral domains, fields and
 so on.
 
+Note that most of these constructions can also be obtained using the `transport` tactic.
+
 ## Tags
 
-equiv, group, ring, field
+equiv, group, ring, field, module, algebra
 -/
 
 universes u v
@@ -30,89 +34,121 @@ section instances
 
 variables (e : α ≃ β)
 
-/-- Transfer `has_zero` across an `equiv` -/
-protected def has_zero [has_zero β] : has_zero α := ⟨e.symm 0⟩
-lemma zero_def [has_zero β] : @has_zero.zero _ (equiv.has_zero e) = e.symm 0 := rfl
-
 /-- Transfer `has_one` across an `equiv` -/
+@[to_additive "Transfer `has_zero` across an `equiv`"]
 protected def has_one [has_one β] : has_one α := ⟨e.symm 1⟩
+@[to_additive]
 lemma one_def [has_one β] : @has_one.one _ (equiv.has_one e) = e.symm 1 := rfl
 
 /-- Transfer `has_mul` across an `equiv` -/
+@[to_additive "Transfer `has_add` across an `equiv`"]
 protected def has_mul [has_mul β] : has_mul α := ⟨λ x y, e.symm (e x * e y)⟩
+@[to_additive]
 lemma mul_def [has_mul β] (x y : α) :
   @has_mul.mul _ (equiv.has_mul e) x y = e.symm (e x * e y) := rfl
 
-/-- Transfer `has_add` across an `equiv` -/
-protected def has_add [has_add β] : has_add α := ⟨λ x y, e.symm (e x + e y)⟩
-lemma add_def [has_add β] (x y : α) :
-  @has_add.add _ (equiv.has_add e) x y = e.symm (e x + e y) := rfl
-
 /-- Transfer `has_inv` across an `equiv` -/
+@[to_additive "Transfer `has_neg` across an `equiv`"]
 protected def has_inv [has_inv β] : has_inv α := ⟨λ x, e.symm (e x)⁻¹⟩
+@[to_additive]
 lemma inv_def [has_inv β] (x : α) : @has_inv.inv _ (equiv.has_inv e) x = e.symm (e x)⁻¹ := rfl
 
-/-- Transfer `has_neg` across an `equiv` -/
-protected def has_neg [has_neg β] : has_neg α := ⟨λ x, e.symm (-e x)⟩
-lemma neg_def [has_neg β] (x : α) : @has_neg.neg _ (equiv.has_neg e) x = e.symm (-e x) := rfl
+/-- Transfer `has_scalar` across an `equiv` -/
+protected def has_scalar {R : Type*} [has_scalar R β] : has_scalar R α  :=
+⟨λ r x, e.symm (r • (e x))⟩
+lemma smul_def {R : Type*} [has_scalar R β] (r : R) (x : α) :
+  @has_scalar.smul _ _ (equiv.has_scalar e) r x = e.symm (r • (e x)) := rfl
+
+/--
+An equivalence `e : α ≃ β` gives a multiplicative equivalence `α ≃* β`
+where the multiplicative structure on `α` is
+the one obtained by transporting a multiplicative structure on `β` back along `e`.
+-/
+@[to_additive
+"An equivalence `e : α ≃ β` gives a additive equivalence `α ≃+ β`
+where the additive structure on `α` is
+the one obtained by transporting an additive structure on `β` back along `e`."]
+def mul_equiv (e : α ≃ β) [has_mul β] :
+  by { letI := equiv.has_mul e, exact α ≃* β } :=
+begin
+  introsI,
+  exact
+  { map_mul' := λ x y, by { apply e.symm.injective, simp, refl, },
+    ..e }
+end
+
+@[simp, to_additive] lemma mul_equiv_apply (e : α ≃ β) [has_mul β] (a : α) :
+  (mul_equiv e) a = e a := rfl
+
+@[to_additive] lemma mul_equiv_symm_apply (e : α ≃ β) [has_mul β] (b : β) :
+  by { letI := equiv.has_mul e, exact (mul_equiv e).symm b = e.symm b } :=
+begin
+  intros, refl,
+end
+
+/--
+An equivalence `e : α ≃ β` gives a ring equivalence `α ≃+* β`
+where the ring structure on `α` is
+the one obtained by transporting a ring structure on `β` back along `e`.
+-/
+def ring_equiv (e : α ≃ β) [has_add β] [has_mul β] :
+  by { letI := equiv.has_add e, letI := equiv.has_mul e, exact α ≃+* β } :=
+begin
+  introsI,
+  exact
+  { map_add' := λ x y, by { apply e.symm.injective, simp, refl, },
+    map_mul' := λ x y, by { apply e.symm.injective, simp, refl, },
+    ..e }
+end
+
+@[simp] lemma ring_equiv_apply (e : α ≃ β) [has_add β] [has_mul β] (a : α) :
+  (ring_equiv e) a = e a := rfl
+
+lemma ring_equiv_symm_apply (e : α ≃ β) [has_add β] [has_mul β] (b : β) :
+  by { letI := equiv.has_add e, letI := equiv.has_mul e, exact (ring_equiv e).symm b = e.symm b } :=
+begin
+  intros, refl,
+end
 
 /-- Transfer `semigroup` across an `equiv` -/
+@[to_additive "Transfer `add_semigroup` across an `equiv`"]
 protected def semigroup [semigroup β] : semigroup α :=
 { mul_assoc := by simp [mul_def, mul_assoc],
   ..equiv.has_mul e }
 
 /-- Transfer `comm_semigroup` across an `equiv` -/
+@[to_additive "Transfer `add_comm_semigroup` across an `equiv`"]
 protected def comm_semigroup [comm_semigroup β] : comm_semigroup α :=
 { mul_comm := by simp [mul_def, mul_comm],
   ..equiv.semigroup e }
 
 /-- Transfer `monoid` across an `equiv` -/
+@[to_additive "Transfer `add_monoid` across an `equiv`"]
 protected def monoid [monoid β] : monoid α :=
 { one_mul := by simp [mul_def, one_def],
   mul_one := by simp [mul_def, one_def],
   ..equiv.semigroup e,
   ..equiv.has_one e }
 
 /-- Transfer `comm_monoid` across an `equiv` -/
+@[to_additive "Transfer `add_comm_monoid` across an `equiv`"]
 protected def comm_monoid [comm_monoid β] : comm_monoid α :=
 { ..equiv.comm_semigroup e,
   ..equiv.monoid e }
 
 /-- Transfer `group` across an `equiv` -/
+@[to_additive "Transfer `add_group` across an `equiv`"]
 protected def group [group β] : group α :=
 { mul_left_inv := by simp [mul_def, inv_def, one_def],
   ..equiv.monoid e,
   ..equiv.has_inv e }
 
 /-- Transfer `comm_group` across an `equiv` -/
+@[to_additive "Transfer `add_comm_group` across an `equiv`"]
 protected def comm_group [comm_group β] : comm_group α :=
 { ..equiv.group e,
   ..equiv.comm_semigroup e }
 
-/-- Transfer `add_semigroup` across an `equiv` -/
-protected def add_semigroup [add_semigroup β] : add_semigroup α :=
-@additive.add_semigroup _ (@equiv.semigroup _ _ e multiplicative.semigroup)
-
-/-- Transfer `add_comm_semigroup` across an `equiv` -/
-protected def add_comm_semigroup [add_comm_semigroup β] : add_comm_semigroup α :=
-@additive.add_comm_semigroup _ (@equiv.comm_semigroup _ _ e multiplicative.comm_semigroup)
-
-/-- Transfer `add_monoid` across an `equiv` -/
-protected def add_monoid [add_monoid β] : add_monoid α :=
-@additive.add_monoid _ (@equiv.monoid _ _ e multiplicative.monoid)
-
-/-- Transfer `add_comm_monoid` across an `equiv` -/
-protected def add_comm_monoid [add_comm_monoid β] : add_comm_monoid α :=
-@additive.add_comm_monoid _ (@equiv.comm_monoid _ _ e multiplicative.comm_monoid)
-
-/-- Transfer `add_group` across an `equiv` -/
-protected def add_group [add_group β] : add_group α :=
-@additive.add_group _ (@equiv.group _ _ e multiplicative.group)
-
-/-- Transfer `add_comm_group` across an `equiv` -/
-protected def add_comm_group [add_comm_group β] : add_comm_group α :=
-@additive.add_comm_group _ (@equiv.comm_group _ _ e multiplicative.comm_group)
-
 /-- Transfer `semiring` across an `equiv` -/
 protected def semiring [semiring β] : semiring α :=
 { right_distrib := by simp [mul_def, add_def, add_mul],
@@ -174,5 +210,115 @@ protected def field [field β] : field α :=
 { ..equiv.integral_domain e,
   ..equiv.division_ring e }
 
+variables (R : Type*)
+include R
+
+section
+variables [monoid R]
+
+/-- Transfer `mul_action` across an `equiv` -/
+protected def mul_action (e : α ≃ β) [mul_action R β] : mul_action R α :=
+{ one_smul := by simp [smul_def],
+  mul_smul := by simp [smul_def, mul_smul],
+  ..equiv.has_scalar e }
+
+/-- Transfer `distrib_mul_action` across an `equiv` -/
+protected def distrib_mul_action (e : α ≃ β) [add_comm_monoid β] :
+  begin
+    letI := equiv.add_comm_monoid e,
+    exact Π [distrib_mul_action R β], distrib_mul_action R α
+  end :=
+begin
+  intros,
+  letI := equiv.add_comm_monoid e,
+  exact (
+  { smul_zero := by simp [zero_def, smul_def],
+    smul_add := by simp [add_def, smul_def, smul_add],
+    ..equiv.mul_action R e } : distrib_mul_action R α)
+end
+
+end
+
+section
+variables [semiring R]
+
+/-- Transfer `semimodule` across an `equiv` -/
+protected def semimodule (e : α ≃ β) [add_comm_monoid β] :
+  begin
+    letI := equiv.add_comm_monoid e,
+    exact Π [semimodule R β], semimodule R α
+  end :=
+begin
+  introsI,
+  exact (
+  { zero_smul := by simp [zero_def, smul_def],
+    add_smul := by simp [add_def, smul_def, add_smul],
+    ..equiv.distrib_mul_action R e } : semimodule R α)
+end
+
+/--
+An equivalence `e : α ≃ β` gives a linear equivalence `α ≃ₗ[R] β`
+where the `R`-module structure on `α` is
+the one obtained by transporting an `R`-module structure on `β` back along `e`.
+-/
+def linear_equiv (e : α ≃ β) [add_comm_monoid β] [semimodule R β] :
+  begin
+    letI := equiv.add_comm_monoid e,
+    letI := equiv.semimodule R e,
+    exact α ≃ₗ[R] β
+  end :=
+begin
+  introsI,
+  exact
+  { map_smul' := λ r x, by { apply e.symm.injective, simp, refl, },
+    ..equiv.add_equiv e }
+end
+
+end
+
+section
+variables [comm_semiring R]
+
+/-- Transfer `algebra` across an `equiv` -/
+protected def algebra (e : α ≃ β) [semiring β] :
+  begin
+    letI := equiv.semiring e,
+    exact Π [algebra R β], algebra R α
+  end :=
+begin
+  introsI,
+  fapply ring_hom.to_algebra',
+  { exact ((ring_equiv e).symm : β →+* α).comp (algebra_map R β), },
+  { intros r x,
+    simp only [function.comp_app, ring_hom.coe_comp],
+    have p := ring_equiv_symm_apply e,
+    dsimp at p,
+    erw p, clear p,
+    apply (ring_equiv e).injective,
+    simp only [(ring_equiv e).map_mul],
+    simp [algebra.commutes], }
+end
+
+/--
+An equivalence `e : α ≃ β` gives an algebra equivalence `α ≃ₐ[R] β`
+where the `R`-algebra structure on `α` is
+the one obtained by transporting an `R`-algebra structure on `β` back along `e`.
+-/
+def alg_equiv (e : α ≃ β) [semiring β] [algebra R β] :
+  begin
+    letI := equiv.semiring e,
+    letI := equiv.algebra R e,
+    exact α ≃ₐ[R] β
+  end :=
+begin
+  introsI,
+  exact
+  { commutes' := λ r, by { apply e.symm.injective, simp, refl, },
+    ..equiv.ring_equiv e }
+end
+
+end
+
 end instances
 end equiv
+#lint