pi_instances.lean

/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot

Pi instances for algebraic structures.
-/
import order.basic
import algebra.module
import data.finset
import ring_theory.subring
import tactic.pi_instances

namespace pi
universes u v w
variable {I : Type u}     -- The indexing type
variable {f : I → Type v} -- The family of types already equipped with instances
variables (x y : Π i, f i) (i : I)

instance has_zero [∀ i, has_zero $ f i] : has_zero (Π i : I, f i) := ⟨λ i, 0⟩
@[simp] lemma zero_apply [∀ i, has_zero $ f i] : (0 : Π i, f i) i = 0 := rfl

instance has_one [∀ i, has_one $ f i] : has_one (Π i : I, f i) := ⟨λ i, 1⟩
@[simp] lemma one_apply [∀ i, has_one $ f i] : (1 : Π i, f i) i = 1 := rfl

attribute [to_additive] pi.has_one
attribute [to_additive] pi.one_apply

instance has_add [∀ i, has_add $ f i] : has_add (Π i : I, f i) := ⟨λ x y, λ i, x i + y i⟩
@[simp] lemma add_apply [∀ i, has_add $ f i] : (x + y) i = x i + y i := rfl

instance has_mul [∀ i, has_mul $ f i] : has_mul (Π i : I, f i) := ⟨λ x y, λ i, x i * y i⟩
@[simp] lemma mul_apply [∀ i, has_mul $ f i] : (x * y) i = x i * y i := rfl

attribute [to_additive] pi.has_mul
attribute [to_additive] pi.mul_apply

instance has_inv [∀ i, has_inv $ f i] : has_inv (Π i : I, f i) := ⟨λ x, λ i, (x i)⁻¹⟩
@[simp] lemma inv_apply [∀ i, has_inv $ f i] : x⁻¹ i = (x i)⁻¹ := rfl

instance has_neg [∀ i, has_neg $ f i] : has_neg (Π i : I, f i) := ⟨λ x, λ i, -(x i)⟩
@[simp] lemma neg_apply [∀ i, has_neg $ f i] : (-x) i = -x i := rfl

attribute [to_additive] pi.has_inv
attribute [to_additive] pi.inv_apply

instance has_scalar {α : Type*} [∀ i, has_scalar α $ f i] : has_scalar α (Π i : I, f i) := ⟨λ s x, λ i, s • (x i)⟩
@[simp] lemma smul_apply {α : Type*} [∀ i, has_scalar α $ f i] (s : α) : (s • x) i = s • x i := rfl

instance semigroup          [∀ i, semigroup          $ f i] : semigroup          (Π i : I, f i) := by pi_instance
instance comm_semigroup     [∀ i, comm_semigroup     $ f i] : comm_semigroup     (Π i : I, f i) := by pi_instance
instance monoid             [∀ i, monoid             $ f i] : monoid             (Π i : I, f i) := by pi_instance
instance comm_monoid        [∀ i, comm_monoid        $ f i] : comm_monoid        (Π i : I, f i) := by pi_instance
instance group              [∀ i, group              $ f i] : group              (Π i : I, f i) := by pi_instance
instance comm_group         [∀ i, comm_group         $ f i] : comm_group         (Π i : I, f i) := by pi_instance
instance add_semigroup      [∀ i, add_semigroup      $ f i] : add_semigroup      (Π i : I, f i) := by pi_instance
instance add_comm_semigroup [∀ i, add_comm_semigroup $ f i] : add_comm_semigroup (Π i : I, f i) := by pi_instance
instance add_monoid         [∀ i, add_monoid         $ f i] : add_monoid         (Π i : I, f i) := by pi_instance
instance add_comm_monoid    [∀ i, add_comm_monoid    $ f i] : add_comm_monoid    (Π i : I, f i) := by pi_instance
instance add_group          [∀ i, add_group          $ f i] : add_group          (Π i : I, f i) := by pi_instance
instance add_comm_group     [∀ i, add_comm_group     $ f i] : add_comm_group     (Π i : I, f i) := by pi_instance
instance semiring           [∀ i, semiring           $ f i] : semiring           (Π i : I, f i) := by pi_instance
instance ring               [∀ i, ring               $ f i] : ring               (Π i : I, f i) := by pi_instance
instance comm_ring          [∀ i, comm_ring          $ f i] : comm_ring          (Π i : I, f i) := by pi_instance

instance mul_action     (α) {m : monoid α}                                      [∀ i, mul_action α $ f i]     : mul_action α (Π i : I, f i) :=
{ smul := λ c f i, c • f i,
  mul_smul := λ r s f, funext $ λ i, mul_smul _ _ _,
  one_smul := λ f, funext $ λ i, one_smul α _ }

instance distrib_mul_action (α) {m : monoid α}         [∀ i, add_monoid $ f i]      [∀ i, distrib_mul_action α $ f i] : distrib_mul_action α (Π i : I, f i) :=
{ smul_zero := λ c, funext $ λ i, smul_zero _,
  smul_add := λ c f g, funext $ λ i, smul_add _ _ _,
  ..pi.mul_action _ }

variables (I f)

instance semimodule     (α) {r : semiring α}       [∀ i, add_comm_monoid $ f i] [∀ i, semimodule α $ f i]     : semimodule α (Π i : I, f i) :=
{ add_smul := λ c f g, funext $ λ i, add_smul _ _ _,
  zero_smul := λ f, funext $ λ i, zero_smul α _,
  ..pi.distrib_mul_action _ }

variables {I f}

instance module         (α) {r : ring α}           [∀ i, add_comm_group $ f i]  [∀ i, module α $ f i]         : module α (Π i : I, f i)       := {..pi.semimodule I f α}

instance left_cancel_semigroup [∀ i, left_cancel_semigroup $ f i] : left_cancel_semigroup (Π i : I, f i) :=
by pi_instance

instance add_left_cancel_semigroup [∀ i, add_left_cancel_semigroup $ f i] : add_left_cancel_semigroup (Π i : I, f i) :=
by pi_instance

instance right_cancel_semigroup [∀ i, right_cancel_semigroup $ f i] : right_cancel_semigroup (Π i : I, f i) :=
by pi_instance

instance add_right_cancel_semigroup [∀ i, add_right_cancel_semigroup $ f i] : add_right_cancel_semigroup (Π i : I, f i) :=
by pi_instance

instance ordered_cancel_comm_monoid [∀ i, ordered_cancel_comm_monoid $ f i] : ordered_cancel_comm_monoid (Π i : I, f i) :=
by pi_instance

instance ordered_comm_group [∀ i, ordered_comm_group $ f i] : ordered_comm_group (Π i : I, f i) :=
{ add_lt_add_left := λ a b hab c, ⟨λ i, add_le_add_left (hab.1 i) (c i),
    λ h, hab.2 $ λ i, le_of_add_le_add_left (h i)⟩,
  add_le_add_left := λ x y hxy c i, add_le_add_left (hxy i) _,
  ..pi.add_comm_group,
  ..pi.partial_order }

attribute [to_additive add_semigroup]              pi.semigroup
attribute [to_additive add_comm_semigroup]         pi.comm_semigroup
attribute [to_additive add_monoid]                 pi.monoid
attribute [to_additive add_comm_monoid]            pi.comm_monoid
attribute [to_additive add_group]                  pi.group
attribute [to_additive add_comm_group]             pi.comm_group
attribute [to_additive add_left_cancel_semigroup]  pi.left_cancel_semigroup
attribute [to_additive add_right_cancel_semigroup] pi.right_cancel_semigroup

@[simp] lemma sub_apply [∀ i, add_group $ f i] : (x - y) i = x i - y i := rfl

@[to_additive]
lemma list_prod_apply {α : Type*} {β : α → Type*} [∀a, monoid (β a)] (a : α) :
  ∀ (l : list (Πa, β a)), l.prod a = (l.map (λf:Πa, β a, f a)).prod
| []       := rfl
| (f :: l) := by simp [mul_apply f l.prod a, list_prod_apply l]

@[to_additive]
lemma multiset_prod_apply {α : Type*} {β : α → Type*} [∀a, comm_monoid (β a)] (a : α)
  (s : multiset (Πa, β a)) : s.prod a = (s.map (λf:Πa, β a, f a)).prod :=
quotient.induction_on s $ assume l, begin simp [list_prod_apply a l] end

@[to_additive]
lemma finset_prod_apply {α : Type*} {β : α → Type*} {γ} [∀a, comm_monoid (β a)] (a : α)
  (s : finset γ) (g : γ → Πa, β a) : s.prod g a = s.prod (λc, g c a) :=
show (s.val.map g).prod a = (s.val.map (λc, g c a)).prod,
  by rw [multiset_prod_apply, multiset.map_map]

/-- A family of ring homomorphisms `f a : γ →+* β a` defines a ring homomorphism
`pi.ring_hom f : γ →+* Π a, β a` given by `pi.ring_hom f x b = f b x`. -/
protected def ring_hom
  {α : Type u} {β : α → Type v} [R : Π a : α, semiring (β a)]
  {γ : Type w} [semiring γ] (f : Π a : α, γ →+* β a) :
  γ →+* Π a, β a :=
{ to_fun := λ x b, f b x,
  map_add' := λ x y, funext $ λ z, (f z).map_add x y,
  map_mul' := λ x y, funext $ λ z, (f z).map_mul x y,
  map_one' := funext $ λ z, (f z).map_one,
  map_zero' := funext $ λ z, (f z).map_zero }

instance is_ring_hom_pi
  {α : Type u} {β : α → Type v} [R : Π a : α, ring (β a)]
  {γ : Type w} [ring γ]
  (f : Π a : α, γ → β a) [Rh : Π a : α, is_ring_hom (f a)] :
  is_ring_hom (λ x b, f b x) :=
(show γ →+* Π a, β a, from pi.ring_hom (λ a, ring_hom.of (f a))).is_ring_hom

end pi

namespace prod

variables {α : Type*} {β : Type*} {γ : Type*} {δ : Type*} {p q : α × β}

instance [has_add α] [has_add β] : has_add (α × β) :=
⟨λp q, (p.1 + q.1, p.2 + q.2)⟩
@[to_additive]
instance [has_mul α] [has_mul β] : has_mul (α × β) :=
⟨λp q, (p.1 * q.1, p.2 * q.2)⟩

@[simp, to_additive]
lemma fst_mul [has_mul α] [has_mul β] : (p * q).1 = p.1 * q.1 := rfl
@[simp, to_additive]
lemma snd_mul [has_mul α] [has_mul β] : (p * q).2 = p.2 * q.2 := rfl
@[simp, to_additive]
lemma mk_mul_mk [has_mul α] [has_mul β] (a₁ a₂ : α) (b₁ b₂ : β) :
  (a₁, b₁) * (a₂, b₂) = (a₁ * a₂, b₁ * b₂) := rfl

instance [has_zero α] [has_zero β] : has_zero (α × β) := ⟨(0, 0)⟩
@[to_additive]
instance [has_one α] [has_one β] : has_one (α × β) := ⟨(1, 1)⟩

@[simp, to_additive]
lemma fst_one [has_one α] [has_one β] : (1 : α × β).1 = 1 := rfl
@[simp, to_additive]
lemma snd_one [has_one α] [has_one β] : (1 : α × β).2 = 1 := rfl
@[to_additive]
lemma one_eq_mk [has_one α] [has_one β] : (1 : α × β) = (1, 1) := rfl

instance [has_neg α] [has_neg β] : has_neg (α × β) := ⟨λp, (- p.1, - p.2)⟩
@[to_additive]
instance [has_inv α] [has_inv β] : has_inv (α × β) := ⟨λp, (p.1⁻¹, p.2⁻¹)⟩

@[simp, to_additive]
lemma fst_inv [has_inv α] [has_inv β] : (p⁻¹).1 = (p.1)⁻¹ := rfl
@[simp, to_additive]
lemma snd_inv [has_inv α] [has_inv β] : (p⁻¹).2 = (p.2)⁻¹ := rfl
@[to_additive]
lemma inv_mk [has_inv α] [has_inv β] (a : α) (b : β) : (a, b)⁻¹ = (a⁻¹, b⁻¹) := rfl

instance [add_semigroup α] [add_semigroup β] : add_semigroup (α × β) :=
{ add_assoc := assume a b c, mk.inj_iff.mpr ⟨add_assoc _ _ _, add_assoc _ _ _⟩,
  .. prod.has_add }
@[to_additive add_semigroup]
instance [semigroup α] [semigroup β] : semigroup (α × β) :=
{ mul_assoc := assume a b c, mk.inj_iff.mpr ⟨mul_assoc _ _ _, mul_assoc _ _ _⟩,
  .. prod.has_mul }

instance [add_monoid α] [add_monoid β] : add_monoid (α × β) :=
{ zero_add := assume a, prod.rec_on a $ λa b, mk.inj_iff.mpr ⟨zero_add _, zero_add _⟩,
  add_zero := assume a, prod.rec_on a $ λa b, mk.inj_iff.mpr ⟨add_zero _, add_zero _⟩,
  .. prod.add_semigroup, .. prod.has_zero }
@[to_additive add_monoid]
instance [monoid α] [monoid β] : monoid (α × β) :=
{ one_mul := assume a, prod.rec_on a $ λa b, mk.inj_iff.mpr ⟨one_mul _, one_mul _⟩,
  mul_one := assume a, prod.rec_on a $ λa b, mk.inj_iff.mpr ⟨mul_one _, mul_one _⟩,
  .. prod.semigroup, .. prod.has_one }

instance [add_group α] [add_group β] : add_group (α × β) :=
{ add_left_neg := assume a, mk.inj_iff.mpr ⟨add_left_neg _, add_left_neg _⟩,
  .. prod.add_monoid, .. prod.has_neg }
@[to_additive add_group]
instance [group α] [group β] : group (α × β) :=
{ mul_left_inv := assume a, mk.inj_iff.mpr ⟨mul_left_inv _, mul_left_inv _⟩,
  .. prod.monoid, .. prod.has_inv }

instance [add_comm_semigroup α] [add_comm_semigroup β] : add_comm_semigroup (α × β) :=
{ add_comm := assume a b, mk.inj_iff.mpr ⟨add_comm _ _, add_comm _ _⟩,
  .. prod.add_semigroup }
@[to_additive add_comm_semigroup]
instance [comm_semigroup α] [comm_semigroup β] : comm_semigroup (α × β) :=
{ mul_comm := assume a b, mk.inj_iff.mpr ⟨mul_comm _ _, mul_comm _ _⟩,
  .. prod.semigroup }

instance [add_comm_monoid α] [add_comm_monoid β] : add_comm_monoid (α × β) :=
{ .. prod.add_comm_semigroup, .. prod.add_monoid }
@[to_additive add_comm_monoid]
instance [comm_monoid α] [comm_monoid β] : comm_monoid (α × β) :=
{ .. prod.comm_semigroup, .. prod.monoid }

instance [add_comm_group α] [add_comm_group β] : add_comm_group (α × β) :=
{ .. prod.add_comm_semigroup, .. prod.add_group }
@[to_additive add_comm_group]
instance [comm_group α] [comm_group β] : comm_group (α × β) :=
{ .. prod.comm_semigroup, .. prod.group }

@[to_additive is_add_monoid_hom]
lemma fst.is_monoid_hom [monoid α] [monoid β] : is_monoid_hom (prod.fst : α × β → α) :=
{ map_mul := λ _ _, rfl, map_one := rfl }
@[to_additive is_add_monoid_hom]
lemma snd.is_monoid_hom [monoid α] [monoid β] : is_monoid_hom (prod.snd : α × β → β) :=
{ map_mul := λ _ _, rfl, map_one := rfl }

@[simp] lemma fst_sub [add_group α] [add_group β] : (p - q).1 = p.1 - q.1 := rfl
@[simp] lemma snd_sub [add_group α] [add_group β] : (p - q).2 = p.2 - q.2 := rfl
@[simp] lemma mk_sub_mk [add_group α] [add_group β] {a₁ a₂ : α} {b₁ b₂ : β} :
  (a₁, b₁) - (a₂, b₂) = (a₁ - a₂, b₁ - b₂) := rfl

/-- Given monoids `α, β`, the natural projection homomorphism from `α × β` to `α`. -/
@[to_additive prod.add_monoid_hom.fst "Given add_monoids `α, β`, the natural projection homomorphism from `α × β` to `α`."]
def monoid_hom.fst [monoid α] [monoid β] : α × β →* α :=
⟨λ x, x.1, rfl, λ _ _, prod.fst_mul⟩

/-- Given monoids `α, β`, the natural projection homomorphism from `α × β` to `β`.-/
@[to_additive prod.add_monoid_hom.snd "Given add_monoids `α, β`, the natural projection homomorphism from `α × β` to `β`."]
def monoid_hom.snd [monoid α] [monoid β] : α × β →* β :=
⟨λ x, x.2, rfl, λ _ _, prod.snd_mul⟩

/-- Given monoids `α, β`, the natural inclusion homomorphism from `α` to `α × β`. -/
@[to_additive prod.add_monoid_hom.inl "Given add_monoids `α, β`, the natural inclusion homomorphism from `α` to `α × β`. There is an unbundled version, `prod.inl`, for arbitrary `α, β` such that `β` has a zero."]
def monoid_hom.inl [monoid α] [monoid β] : α →* α × β :=
⟨λ x, (x, 1), rfl, λ x y, show _ = (_, _), by rw mul_one⟩

/-- Given monoids `α, β`, the natural inclusion homomorphism from `β` to `α × β`. -/
@[to_additive prod.add_monoid_hom.inr "Given add_monoids `α, β`, the natural inclusion homomorphism from `β` to `α × β`. There is an unbundled version, `prod.inr`, for arbitrary `α, β` such that `α` has a zero."]
def monoid_hom.inr [monoid α] [monoid β] : β →* α × β :=
⟨λ x, (1, x), rfl, λ x y, show _ = (_, _), by rw mul_one⟩

@[to_additive is_add_group_hom]
lemma fst.is_group_hom [group α] [group β] : is_group_hom (prod.fst : α × β → α) :=
{ map_mul := λ _ _, rfl }
@[to_additive is_add_group_hom]
lemma snd.is_group_hom [group α] [group β] : is_group_hom (prod.snd : α × β → β) :=
{ map_mul := λ _ _, rfl }

attribute [instance] fst.is_monoid_hom fst.is_add_monoid_hom snd.is_monoid_hom snd.is_add_monoid_hom
fst.is_group_hom fst.is_add_group_hom snd.is_group_hom snd.is_add_group_hom

@[to_additive]
lemma fst_prod [comm_monoid α] [comm_monoid β] {t : finset γ} {f : γ → α × β} :
  (t.prod f).1 = t.prod (λc, (f c).1) :=
(t.prod_hom prod.fst).symm

@[to_additive]
lemma snd_prod [comm_monoid α] [comm_monoid β] {t : finset γ} {f : γ → α × β} :
  (t.prod f).2 = t.prod (λc, (f c).2) :=
(t.prod_hom prod.snd).symm

instance [semiring α] [semiring β] : semiring (α × β) :=
{ zero_mul := λ a, mk.inj_iff.mpr ⟨zero_mul _, zero_mul _⟩,
  mul_zero := λ a, mk.inj_iff.mpr ⟨mul_zero _, mul_zero _⟩,
  left_distrib := λ a b c, mk.inj_iff.mpr ⟨left_distrib _ _ _, left_distrib _ _ _⟩,
  right_distrib := λ a b c, mk.inj_iff.mpr ⟨right_distrib _ _ _, right_distrib _ _ _⟩,
  ..prod.add_comm_monoid, ..prod.monoid }

instance [ring α] [ring β] : ring (α × β) :=
{ ..prod.add_comm_group, ..prod.semiring }

instance [comm_ring α] [comm_ring β] : comm_ring (α × β) :=
{ ..prod.ring, ..prod.comm_monoid }

instance [nonzero_comm_ring α] [comm_ring β] : nonzero_comm_ring (α × β) :=
{ zero_ne_one := mt (congr_arg prod.fst) zero_ne_one,
  ..prod.comm_ring }

instance fst.is_semiring_hom [semiring α] [semiring β] : is_semiring_hom (prod.fst : α × β → α) :=
by refine_struct {..}; simp
instance snd.is_semiring_hom [semiring α] [semiring β] : is_semiring_hom (prod.snd : α × β → β) :=
by refine_struct {..}; simp

instance fst.is_ring_hom [ring α] [ring β] : is_ring_hom (prod.fst : α × β → α) :=
by refine_struct {..}; simp
instance snd.is_ring_hom [ring α] [ring β] : is_ring_hom (prod.snd : α × β → β) :=
by refine_struct {..}; simp

/-- Left injection function for the inner product
From a vector space (and also group and module) perspective the product is the same as the sum of
two vector spaces. `inl` and `inr` provide the corresponding injection functions.
-/
def inl [has_zero β] (a : α) : α × β := (a, 0)

/-- Right injection function for the inner product -/
def inr [has_zero α] (b : β) : α × β := (0, b)

lemma injective_inl [has_zero β] : function.injective (inl : α → α × β) :=
assume x y h, (prod.mk.inj_iff.mp h).1

lemma injective_inr [has_zero α] : function.injective (inr : β → α × β) :=
assume x y h, (prod.mk.inj_iff.mp h).2

@[simp] lemma inl_eq_inl [has_zero β] {a₁ a₂ : α} : (inl a₁ : α × β) = inl a₂ ↔ a₁ = a₂ :=
iff.intro (assume h, injective_inl h) (assume h, h ▸ rfl)

@[simp] lemma inr_eq_inr [has_zero α] {b₁ b₂ : β} : (inr b₁ : α × β) = inr b₂ ↔ b₁ = b₂ :=
iff.intro (assume h, injective_inr h) (assume h, h ▸ rfl)

@[simp] lemma inl_eq_inr [has_zero α] [has_zero β] {a : α} {b : β} :
  inl a = inr b ↔ a = 0 ∧ b = 0 :=
by constructor; simp [inl, inr] {contextual := tt}

@[simp] lemma inr_eq_inl [has_zero α] [has_zero β] {a : α} {b : β} :
  inr b = inl a ↔ a = 0 ∧ b = 0 :=
by constructor; simp [inl, inr] {contextual := tt}

@[simp] lemma fst_inl [has_zero β] (a : α) : (inl a : α × β).1 = a := rfl
@[simp] lemma snd_inl [has_zero β] (a : α) : (inl a : α × β).2 = 0 := rfl
@[simp] lemma fst_inr [has_zero α] (b : β) : (inr b : α × β).1 = 0 := rfl
@[simp] lemma snd_inr [has_zero α] (b : β) : (inr b : α × β).2 = b := rfl

instance [has_scalar α β] [has_scalar α γ] : has_scalar α (β × γ) := ⟨λa p, (a • p.1, a • p.2)⟩

@[simp] theorem smul_fst [has_scalar α β] [has_scalar α γ]
  (a : α) (x : β × γ) : (a • x).1 = a • x.1 := rfl
@[simp] theorem smul_snd [has_scalar α β] [has_scalar α γ]
  (a : α) (x : β × γ) : (a • x).2 = a • x.2 := rfl
@[simp] theorem smul_mk [has_scalar α β] [has_scalar α γ]
  (a : α) (b : β) (c : γ) : a • (b, c) = (a • b, a • c) := rfl

instance {r : semiring α} [add_comm_monoid β] [add_comm_monoid γ]
  [semimodule α β] [semimodule α γ] : semimodule α (β × γ) :=
{ smul_add  := assume a p₁ p₂, mk.inj_iff.mpr ⟨smul_add _ _ _, smul_add _ _ _⟩,
  add_smul  := assume a p₁ p₂, mk.inj_iff.mpr ⟨add_smul _ _ _, add_smul _ _ _⟩,
  mul_smul  := assume a₁ a₂ p, mk.inj_iff.mpr ⟨mul_smul _ _ _, mul_smul _ _ _⟩,
  one_smul  := assume ⟨b, c⟩, mk.inj_iff.mpr ⟨one_smul _ _, one_smul _ _⟩,
  zero_smul := assume ⟨b, c⟩, mk.inj_iff.mpr ⟨zero_smul _ _, zero_smul _ _⟩,
  smul_zero := assume a, mk.inj_iff.mpr ⟨smul_zero _, smul_zero _⟩,
  .. prod.has_scalar }

instance {r : ring α} [add_comm_group β] [add_comm_group γ]
  [module α β] [module α γ] : module α (β × γ) := {}

section substructures
variables (s : set α) (t : set β)

@[to_additive is_add_submonoid]
instance [monoid α] [monoid β] [is_submonoid s] [is_submonoid t] :
  is_submonoid (s.prod t) :=
{ one_mem := by rw set.mem_prod; split; apply is_submonoid.one_mem,
  mul_mem := by intros; rw set.mem_prod at *; split; apply is_submonoid.mul_mem; tauto }

@[to_additive prod.is_add_subgroup.prod]
instance is_subgroup.prod [group α] [group β] [is_subgroup s] [is_subgroup t] :
  is_subgroup (s.prod t) :=
{ inv_mem := by intros; rw set.mem_prod at *; split; apply is_subgroup.inv_mem; tauto,
  .. prod.is_submonoid s t }

instance is_subring.prod [ring α] [ring β] [is_subring s] [is_subring t] :
  is_subring (s.prod t) :=
{ .. prod.is_submonoid s t, .. prod.is_add_subgroup.prod s t }

end substructures

end prod

namespace submonoid

/-- Given submonoids `s, t` of monoids `α, β` respectively, `s × t` as a submonoid of `α × β`. -/
@[to_additive prod "Given `add_submonoids` `s, t` of `add_monoids` `α, β` respectively, `s × t` as an `add_submonoid` of `α × β`."]
def prod {α : Type*} {β : Type*} [monoid α] [monoid β] (s : submonoid α) (t : submonoid β) :
  submonoid (α × β) :=
{ carrier := (s : set α).prod t,
  one_mem' := ⟨s.one_mem, t.one_mem⟩,
  mul_mem' := λ _ _ h1 h2, ⟨s.mul_mem h1.1 h2.1, t.mul_mem h1.2 h2.2⟩ }

end submonoid

namespace finset

@[to_additive prod_mk_sum]
lemma prod_mk_prod {α β γ : Type*} [comm_monoid α] [comm_monoid β] (s : finset γ)
  (f : γ → α) (g : γ → β) : (s.prod f, s.prod g) = s.prod (λ x, (f x, g x)) :=
by haveI := classical.dec_eq γ; exact
finset.induction_on s rfl (by simp [prod.ext_iff] {contextual := tt})

end finset