submonoid.lean

/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import algebra.big_operators
import algebra.free_monoid
import algebra.group.prod
import data.equiv.mul_add

/-!
# Submonoids

This file defines multiplicative and additive submonoids, first in an unbundled form (deprecated)
and then in a bundled form.

We prove submonoids of a monoid form a complete lattice, and results about images and preimages of
submonoids under monoid homomorphisms. For the unbundled submonoids, these theorems use unbundled
monoid homomorphisms (also deprecated), and the bundled versions use bundled monoid homomorphisms.

There are also theorems about the submonoids generated by an element or a subset of a monoid,
defined both inductively and as the infimum of the set of submonoids containing a given
element/subset.

## Implementation notes

Unbundled submonoids will slowly be removed from mathlib.

(Bundled) submonoid inclusion is denoted `≤` rather than `⊆`, although `∈` is defined as
membership of a submonoid's underlying set.

## Tags
submonoid, submonoids, is_submonoid
-/

variables {M : Type*} [monoid M] {s : set M}
variables {A : Type*} [add_monoid A] {t : set A}

/-- `s` is an additive submonoid: a set containing 0 and closed under addition. -/
class is_add_submonoid (s : set A) : Prop :=
(zero_mem : (0:A) ∈ s)
(add_mem {a b} : a ∈ s → b ∈ s → a + b ∈ s)

/-- `s` is a submonoid: a set containing 1 and closed under multiplication. -/
@[to_additive is_add_submonoid]
class is_submonoid (s : set M) : Prop :=
(one_mem : (1:M) ∈ s)
(mul_mem {a b} : a ∈ s → b ∈ s → a * b ∈ s)

lemma additive.is_add_submonoid
  (s : set M) : ∀ [is_submonoid s], @is_add_submonoid (additive M) _ s
| ⟨h₁, h₂⟩ := ⟨h₁, @h₂⟩

theorem additive.is_add_submonoid_iff
  {s : set M} : @is_add_submonoid (additive M) _ s ↔ is_submonoid s :=
⟨λ ⟨h₁, h₂⟩, ⟨h₁, @h₂⟩, λ h, by exactI additive.is_add_submonoid _⟩

lemma multiplicative.is_submonoid
  (s : set A) : ∀ [is_add_submonoid s], @is_submonoid (multiplicative A) _ s
| ⟨h₁, h₂⟩ := ⟨h₁, @h₂⟩

theorem multiplicative.is_submonoid_iff
  {s : set A} : @is_submonoid (multiplicative A) _ s ↔ is_add_submonoid s :=
⟨λ ⟨h₁, h₂⟩, ⟨h₁, @h₂⟩, λ h, by exactI multiplicative.is_submonoid _⟩

/-- The intersection of two submonoids of a monoid `M` is a submonoid of `M`. -/
@[to_additive "The intersection of two `add_submonoid`s of an `add_monoid` `M` is an `add_submonoid` of M."]
instance is_submonoid.inter (s₁ s₂ : set M) [is_submonoid s₁] [is_submonoid s₂] :
  is_submonoid (s₁ ∩ s₂) :=
{ one_mem := ⟨is_submonoid.one_mem, is_submonoid.one_mem⟩,
  mul_mem := λ x y hx hy,
    ⟨is_submonoid.mul_mem hx.1 hy.1, is_submonoid.mul_mem hx.2 hy.2⟩ }

/-- The intersection of an indexed set of submonoids of a monoid `M` is a submonoid of `M`. -/
@[to_additive "The intersection of an indexed set of `add_submonoid`s of an `add_monoid` `M` is an `add_submonoid` of `M`."]
instance is_submonoid.Inter {ι : Sort*} (s : ι → set M) [h : ∀ y : ι, is_submonoid (s y)] :
  is_submonoid (set.Inter s) :=
{ one_mem := set.mem_Inter.2 $ λ y, is_submonoid.one_mem,
  mul_mem := λ x₁ x₂ h₁ h₂, set.mem_Inter.2 $
    λ y, is_submonoid.mul_mem (set.mem_Inter.1 h₁ y) (set.mem_Inter.1 h₂ y) }

/-- The union of an indexed, directed, nonempty set of submonoids of a monoid `M` is a submonoid
    of `M`. -/
@[to_additive is_add_submonoid_Union_of_directed "The union of an indexed, directed, nonempty set of `add_submonoid`s of an `add_monoid` `M` is an `add_submonoid` of `M`. "]
lemma is_submonoid_Union_of_directed {ι : Type*} [hι : nonempty ι]
  (s : ι → set M) [∀ i, is_submonoid (s i)]
  (directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :
  is_submonoid (⋃i, s i) :=
{ one_mem := let ⟨i⟩ := hι in set.mem_Union.2 ⟨i, is_submonoid.one_mem⟩,
  mul_mem := λ a b ha hb,
    let ⟨i, hi⟩ := set.mem_Union.1 ha in
    let ⟨j, hj⟩ := set.mem_Union.1 hb in
    let ⟨k, hk⟩ := directed i j in
    set.mem_Union.2 ⟨k, is_submonoid.mul_mem (hk.1 hi) (hk.2 hj)⟩ }

section powers

/-- The set of natural number powers `1, x, x², ...` of an element `x` of a monoid. -/
def powers (x : M) : set M := {y | ∃ n:ℕ, x^n = y}
/-- The set of natural number multiples `0, x, 2x, ...` of an element `x` of an `add_monoid`. -/
def multiples (x : A) : set A := {y | ∃ n:ℕ, add_monoid.smul n x = y}
attribute [to_additive multiples] powers

/-- 1 is in the set of natural number powers of an element of a monoid. -/
lemma powers.one_mem {x : M} : (1 : M) ∈ powers x := ⟨0, pow_zero _⟩

/-- 0 is in the set of natural number multiples of an element of an `add_monoid`. -/
lemma multiples.zero_mem {x : A} : (0 : A) ∈ multiples x := ⟨0, add_monoid.zero_smul _⟩
attribute [to_additive] powers.one_mem

/-- An element of a monoid is in the set of that element's natural number powers. -/
lemma powers.self_mem {x : M} : x ∈ powers x := ⟨1, pow_one _⟩

/-- An element of an `add_monoid` is in the set of that element's natural number multiples. -/
lemma multiples.self_mem {x : A} : x ∈ multiples x := ⟨1, add_monoid.one_smul _⟩
attribute [to_additive] powers.self_mem

/-- The set of natural number powers of an element of a monoid is closed under multiplication. -/
lemma powers.mul_mem {x y z : M} : (y ∈ powers x) → (z ∈ powers x) → (y * z ∈ powers x) :=
λ ⟨n₁, h₁⟩ ⟨n₂, h₂⟩, ⟨n₁ + n₂, by simp only [pow_add, *]⟩

/-- The set of natural number multiples of an element of an `add_monoid` is closed under
    addition. -/
lemma multiples.add_mem {x y z : A} :
  (y ∈ multiples x) → (z ∈ multiples x) → (y + z ∈ multiples x) :=
@powers.mul_mem (multiplicative A) _ _ _ _
attribute [to_additive] powers.mul_mem

/-- The set of natural number powers of an element of a monoid `M` is a submonoid of `M`. -/
@[to_additive is_add_submonoid "The set of natural number multiples of an element of an `add_monoid` `M` is an `add_submonoid` of `M`."]
instance powers.is_submonoid (x : M) : is_submonoid (powers x) :=
{ one_mem := powers.one_mem,
  mul_mem := λ y z, powers.mul_mem }

/-- A monoid is a submonoid of itself. -/
@[to_additive is_add_submonoid "An `add_monoid` is an `add_submonoid` of itself."]
instance univ.is_submonoid : is_submonoid (@set.univ M) := by split; simp

/-- The preimage of a submonoid under a monoid hom is a submonoid of the domain. -/
@[to_additive is_add_submonoid "The preimage of an `add_submonoid` under an `add_monoid` hom is an `add_submonoid` of the domain."]
instance preimage.is_submonoid {N : Type*} [monoid N] (f : M → N) [is_monoid_hom f]
  (s : set N) [is_submonoid s] : is_submonoid (f ⁻¹' s) :=
{ one_mem := show f 1 ∈ s, by rw is_monoid_hom.map_one f; exact is_submonoid.one_mem,
  mul_mem := λ a b (ha : f a ∈ s) (hb : f b ∈ s),
    show f (a * b) ∈ s, by rw is_monoid_hom.map_mul f; exact is_submonoid.mul_mem ha hb }

/-- The image of a submonoid under a monoid hom is a submonoid of the codomain. -/
@[instance, to_additive is_add_submonoid "The image of an `add_submonoid` under an `add_monoid` hom is an `add_submonoid` of the codomain."]
lemma image.is_submonoid {γ : Type*} [monoid γ] (f : M → γ) [is_monoid_hom f]
  (s : set M) [is_submonoid s] : is_submonoid (f '' s) :=
{ one_mem := ⟨1, is_submonoid.one_mem, is_monoid_hom.map_one f⟩,
  mul_mem := λ a b ⟨x, hx⟩ ⟨y, hy⟩, ⟨x * y, is_submonoid.mul_mem hx.1 hy.1,
    by rw [is_monoid_hom.map_mul f, hx.2, hy.2]⟩ }

/-- The image of a monoid hom is a submonoid of the codomain. -/
@[to_additive is_add_submonoid "The image of an `add_monoid` hom is an `add_submonoid` of the codomain."]
instance range.is_submonoid {γ : Type*} [monoid γ] (f : M → γ) [is_monoid_hom f] :
  is_submonoid (set.range f) :=
by rw ← set.image_univ; apply_instance

/-- Submonoids are closed under natural powers. -/
lemma is_submonoid.pow_mem {a : M} [is_submonoid s] (h : a ∈ s) : ∀ {n : ℕ}, a ^ n ∈ s
| 0 := is_submonoid.one_mem
| (n + 1) := is_submonoid.mul_mem h is_submonoid.pow_mem

/-- An `add_submonoid` is closed under multiplication by naturals. -/
lemma is_add_submonoid.smul_mem {a : A} [is_add_submonoid t] :
  ∀ (h : a ∈ t) {n : ℕ}, add_monoid.smul n a ∈ t :=
@is_submonoid.pow_mem (multiplicative A) _ _ _ (multiplicative.is_submonoid _)
attribute [to_additive smul_mem] is_submonoid.pow_mem

/-- The set of natural number powers of an element of a submonoid is a subset of the submonoid. -/
lemma is_submonoid.power_subset {a : M} [is_submonoid s] (h : a ∈ s) : powers a ⊆ s :=
assume x ⟨n, hx⟩, hx ▸ is_submonoid.pow_mem h

/-- The set of natural number multiples of an element of an `add_submonoid` is a subset of the
    `add_submonoid`. -/
lemma is_add_submonoid.multiple_subset {a : A} [is_add_submonoid t] :
  a ∈ t → multiples a ⊆ t :=
@is_submonoid.power_subset (multiplicative A) _ _ _ (multiplicative.is_submonoid _)
attribute [to_additive multiple_subset] is_submonoid.power_subset

end powers

namespace is_submonoid

/-- The product of a list of elements of a submonoid is an element of the submonoid. -/
@[to_additive "The sum of a list of elements of an `add_submonoid` is an element of the `add_submonoid`."]
lemma list_prod_mem [is_submonoid s] : ∀{l : list M}, (∀x∈l, x ∈ s) → l.prod ∈ s
| []     h := one_mem
| (a::l) h :=
  suffices a * l.prod ∈ s, by simpa,
  have a ∈ s ∧ (∀x∈l, x ∈ s), by simpa using h,
  is_submonoid.mul_mem this.1 (list_prod_mem this.2)

/-- The product of a multiset of elements of a submonoid of a `comm_monoid` is an element of the submonoid. -/
@[to_additive "The sum of a multiset of elements of an `add_submonoid` of an `add_comm_monoid` is an element of the `add_submonoid`. "]
lemma multiset_prod_mem {M} [comm_monoid M] (s : set M) [is_submonoid s] (m : multiset M) :
  (∀a∈m, a ∈ s) → m.prod ∈ s :=
begin
  refine quotient.induction_on m (assume l hl, _),
  rw [multiset.quot_mk_to_coe, multiset.coe_prod],
  exact list_prod_mem hl
end

/-- The product of elements of a submonoid of a `comm_monoid` indexed by a `finset` is an element of the submonoid. -/
@[to_additive "The sum of elements of an `add_submonoid` of an `add_comm_monoid` indexed by a `finset` is an element of the `add_submonoid`."]
lemma finset_prod_mem {M A} [comm_monoid M] (s : set M) [is_submonoid s] (f : A → M) :
  ∀(t : finset A), (∀b∈t, f b ∈ s) → t.prod f ∈ s
| ⟨m, hm⟩ hs :=
  begin
    refine multiset_prod_mem s _ _,
    simp,
    rintros a b hb rfl,
    exact hs _ hb
  end

end is_submonoid

-- TODO: modify `subtype_instance` to produce this definition, then use it here
--  and for `subtype.group`

/-- Submonoids are themselves monoids. -/
@[to_additive add_monoid "An `add_submonoid` is itself an `add_monoid`."]
instance subtype.monoid {s : set M} [is_submonoid s] : monoid s :=
{ one := ⟨1, is_submonoid.one_mem⟩,
  mul := λ x y, ⟨x * y, is_submonoid.mul_mem x.2 y.2⟩,
  mul_one := λ x, subtype.eq $ mul_one x.1,
  one_mul := λ x, subtype.eq $ one_mul x.1,
  mul_assoc := λ x y z, subtype.eq $ mul_assoc x.1 y.1 z.1 }

/-- Submonoids of commutative monoids are themselves commutative monoids. -/
@[to_additive add_comm_monoid "An `add_submonoid` of a commutative `add_monoid` is itself a commutative `add_monoid`. "]
instance subtype.comm_monoid {M} [comm_monoid M] {s : set M} [is_submonoid s] : comm_monoid s :=
{ mul_comm := λ x y, subtype.eq $ mul_comm x.1 y.1,
  .. subtype.monoid }

/-- Submonoids inherit the 1 of the monoid. -/
@[simp, norm_cast, to_additive "An `add_submonoid` inherits the 0 of the `add_monoid`. "]
lemma is_submonoid.coe_one [is_submonoid s] : ((1 : s) : M) = 1 := rfl
attribute [norm_cast] is_add_submonoid.coe_zero

/-- Submonoids inherit the multiplication of the monoid. -/
@[simp, norm_cast, to_additive "An `add_submonoid` inherits the addition of the `add_monoid`. "]
lemma is_submonoid.coe_mul [is_submonoid s] (a b : s) : ((a * b : s) : M) = a * b := rfl
attribute [norm_cast] is_add_submonoid.coe_add

/-- Submonoids inherit the exponentiation by naturals of the monoid. -/
@[simp, norm_cast] lemma is_submonoid.coe_pow [is_submonoid s] (a : s) (n : ℕ) :
  ((a ^ n : s) : M) = a ^ n :=
by induction n; simp [*, pow_succ]

/-- An `add_submonoid` inherits the multiplication by naturals of the `add_monoid`. -/
@[simp, norm_cast] lemma is_add_submonoid.smul_coe {A : Type*} [add_monoid A] {s : set A}
  [is_add_submonoid s] (a : s) (n : ℕ) : ((add_monoid.smul n a : s) : A) = add_monoid.smul n a :=
by {induction n, refl, simp [*, succ_smul]}

attribute [to_additive smul_coe] is_submonoid.coe_pow

/-- The natural injection from a submonoid into the monoid is a monoid hom. -/
@[to_additive is_add_monoid_hom "The natural injection from an `add_submonoid` into the `add_monoid` is an `add_monoid` hom. "]
instance subtype_val.is_monoid_hom [is_submonoid s] : is_monoid_hom (subtype.val : s → M) :=
{ map_one := rfl, map_mul := λ _ _, rfl }

/-- The natural injection from a submonoid into the monoid is a monoid hom. -/
@[to_additive is_add_monoid_hom "The natural injection from an `add_submonoid` into the `add_monoid` is an `add_monoid` hom. "]
instance coe.is_monoid_hom [is_submonoid s] : is_monoid_hom (coe : s → M) :=
subtype_val.is_monoid_hom

/-- Given a monoid hom `f : γ → M` whose image is contained in a submonoid `s`, the induced map
    from `γ` to `s` is a monoid hom. -/
@[to_additive is_add_monoid_hom "Given an `add_monoid` hom `f : γ → M` whose image is contained in an `add_submonoid` s, the induced map from `γ` to `s` is an `add_monoid` hom."]
instance subtype_mk.is_monoid_hom {γ : Type*} [monoid γ] [is_submonoid s] (f : γ → M)
  [is_monoid_hom f] (h : ∀ x, f x ∈ s) : is_monoid_hom (λ x, (⟨f x, h x⟩ : s)) :=
{ map_one := subtype.eq (is_monoid_hom.map_one f),
  map_mul := λ x y, subtype.eq (is_monoid_hom.map_mul f x y) }

/-- Given two submonoids `s` and `t` such that `s ⊆ t`, the natural injection from `s` into `t` is
    a monoid hom. -/
@[to_additive is_add_monoid_hom "Given two `add_submonoid`s `s` and `t` such that `s ⊆ t`, the natural injection from `s` into `t` is an `add_monoid` hom."]
instance set_inclusion.is_monoid_hom (t : set M) [is_submonoid s] [is_submonoid t] (h : s ⊆ t) :
  is_monoid_hom (set.inclusion h) :=
subtype_mk.is_monoid_hom _ _

namespace add_monoid

/-- The inductively defined membership predicate for the submonoid generated by a subset of a
    monoid. -/
inductive in_closure (s : set A) : A → Prop
| basic {a : A} : a ∈ s → in_closure a
| zero : in_closure 0
| add {a b : A} : in_closure a → in_closure b → in_closure (a + b)

end add_monoid

namespace monoid

/-- The inductively defined membership predicate for the `add_submonoid` generated by a subset of an
    add_monoid. -/
inductive in_closure (s : set M) : M → Prop
| basic {a : M} : a ∈ s → in_closure a
| one : in_closure 1
| mul {a b : M} : in_closure a → in_closure b → in_closure (a * b)

attribute [to_additive] monoid.in_closure
attribute [to_additive] monoid.in_closure.one
attribute [to_additive] monoid.in_closure.mul

/-- The inductively defined submonoid generated by a subset of a monoid. -/
@[to_additive "The inductively defined `add_submonoid` genrated by a subset of an `add_monoid`."]
def closure (s : set M) : set M := {a | in_closure s a }

@[to_additive is_add_submonoid]
instance closure.is_submonoid (s : set M) : is_submonoid (closure s) :=
{ one_mem := in_closure.one, mul_mem := assume a b, in_closure.mul }

/-- A subset of a monoid is contained in the submonoid it generates. -/
@[to_additive "A subset of an `add_monoid` is contained in the `add_submonoid` it generates."]
theorem subset_closure {s : set M} : s ⊆ closure s :=
assume a, in_closure.basic

/-- The submonoid generated by a set is contained in any submonoid that contains the set. -/
@[to_additive "The `add_submonoid` generated by a set is contained in any `add_submonoid` that contains the set."]
theorem closure_subset {s t : set M} [is_submonoid t] (h : s ⊆ t) : closure s ⊆ t :=
assume a ha, by induction ha; simp [h _, *, is_submonoid.one_mem, is_submonoid.mul_mem]

/-- Given subsets `t` and `s` of a monoid `M`, if `s ⊆ t`, the submonoid of `M` generated by `s` is
    contained in the submonoid generated by `t`. -/
@[to_additive "Given subsets `t` and `s` of an `add_monoid M`, if `s ⊆ t`, the `add_submonoid` of `M` generated by `s` is contained in the `add_submonoid` generated by `t`."]
theorem closure_mono {s t : set M} (h : s ⊆ t) : closure s ⊆ closure t :=
closure_subset $ set.subset.trans h subset_closure

/-- The submonoid generated by an element of a monoid equals the set of natural number powers of
    the element. -/
@[to_additive "The `add_submonoid` generated by an element of an `add_monoid` equals the set of natural number multiples of the element."]
theorem closure_singleton {x : M} : closure ({x} : set M) = powers x :=
set.eq_of_subset_of_subset (closure_subset $ set.singleton_subset_iff.2 $ powers.self_mem) $
  is_submonoid.power_subset $ set.singleton_subset_iff.1 $ subset_closure

/-- The image under a monoid hom of the submonoid generated by a set equals the submonoid generated
    by the image of the set under the monoid hom. -/
@[to_additive "The image under an `add_monoid` hom of the `add_submonoid` generated by a set equals the `add_submonoid` generated by the image of the set under the `add_monoid` hom."]
lemma image_closure {A : Type*} [monoid A] (f : M → A) [is_monoid_hom f] (s : set M) :
  f '' closure s = closure (f '' s) :=
le_antisymm
  begin
    rintros _ ⟨x, hx, rfl⟩,
    apply in_closure.rec_on hx; intros,
    { solve_by_elim [subset_closure, set.mem_image_of_mem] },
    { rw [is_monoid_hom.map_one f], apply is_submonoid.one_mem },
    { rw [is_monoid_hom.map_mul f], solve_by_elim [is_submonoid.mul_mem] }
  end
  (closure_subset $ set.image_subset _ subset_closure)

/-- Given an element `a` of the submonoid of a monoid `M` generated by a set `s`, there exists a list of
    elements of `s` whose product is `a`. -/
@[to_additive "Given an element `a` of the `add_submonoid` of an `add_monoid M` generated by a set `s`, there exists a list of elements of `s` whose sum is `a`."]
theorem exists_list_of_mem_closure {s : set M} {a : M} (h : a ∈ closure s) :
  (∃l:list M, (∀x∈l, x ∈ s) ∧ l.prod = a) :=
begin
  induction h,
  case in_closure.basic : a ha { existsi ([a]), simp [ha] },
  case in_closure.one { existsi ([]), simp },
  case in_closure.mul : a b _ _ ha hb {
    rcases ha with ⟨la, ha, eqa⟩,
    rcases hb with ⟨lb, hb, eqb⟩,
    existsi (la ++ lb),
    simp [eqa.symm, eqb.symm, or_imp_distrib],
    exact assume a, ⟨ha a, hb a⟩
  }
end

/-- Given sets `s, t` of a commutative monoid `M`, `x ∈ M` is in the submonoid of `M` generated by
    `s ∪ t` iff there exists an element of the submonoid generated by `s` and an element of the
    submonoid generated by `t` whose product is `x`. -/
@[to_additive "Given sets `s, t` of a commutative `add_monoid M`, `x ∈ M` is in the `add_submonoid` of `M` generated by `s ∪ t` iff there exists an element of the `add_submonoid` generated by `s` and an element of the `add_submonoid` generated by `t` whose sum is `x`."]
theorem mem_closure_union_iff {M : Type*} [comm_monoid M] {s t : set M} {x : M} :
  x ∈ closure (s ∪ t) ↔ ∃ y ∈ closure s, ∃ z ∈ closure t, y * z = x :=
⟨λ hx, let ⟨L, HL1, HL2⟩ := exists_list_of_mem_closure hx in HL2 ▸
  list.rec_on L (λ _, ⟨1, is_submonoid.one_mem, 1, is_submonoid.one_mem, mul_one _⟩)
    (λ hd tl ih HL1, let ⟨y, hy, z, hz, hyzx⟩ := ih (list.forall_mem_of_forall_mem_cons HL1) in
      or.cases_on (HL1 hd $ list.mem_cons_self _ _)
        (λ hs, ⟨hd * y, is_submonoid.mul_mem (subset_closure hs) hy, z, hz, by rw [mul_assoc, list.prod_cons, ← hyzx]; refl⟩)
        (λ ht, ⟨y, hy, z * hd, is_submonoid.mul_mem hz (subset_closure ht), by rw [← mul_assoc, list.prod_cons, ← hyzx, mul_comm hd]; refl⟩)) HL1,
λ ⟨y, hy, z, hz, hyzx⟩, hyzx ▸ is_submonoid.mul_mem (closure_mono (set.subset_union_left _ _) hy)
  (closure_mono (set.subset_union_right _ _) hz)⟩

end monoid

/-!
### Bundled submonoids and `add_submonoid`s
-/

/-- A submonoid of a monoid `M` is a subset containing 1 and closed under multiplication. -/
structure submonoid (M : Type*) [monoid M] :=
(carrier : set M)
(one_mem' : (1 : M) ∈ carrier)
(mul_mem' {a b} : a ∈ carrier → b ∈ carrier → a * b ∈ carrier)

/-- An additive submonoid of an additive monoid `M` is a subset containing 0 and
  closed under addition. -/
structure add_submonoid (M : Type*) [add_monoid M] :=
(carrier : set M)
(zero_mem' : (0 : M) ∈ carrier)
(add_mem' {a b} : a ∈ carrier → b ∈ carrier → a + b ∈ carrier)

attribute [to_additive add_submonoid] submonoid

/-- Create a bundled submonoid from a set `s` and `[is_submonoid s]`. -/
@[to_additive "Create a bundled additive submonoid from a set `s` and `[is_add_submonoid s]`."]
def submonoid.of (s : set M) [h : is_submonoid s] : submonoid M := ⟨s, h.1, h.2⟩

/-- Map from submonoids of monoid `M` to `add_submonoid`s of `additive M`. -/
def submonoid.to_add_submonoid {M : Type*} [monoid M] (S : submonoid M) :
  add_submonoid (additive M) :=
{ carrier := S.carrier,
  zero_mem' := S.one_mem',
  add_mem' := S.mul_mem' }

/-- Map from `add_submonoid`s of `additive M` to submonoids of `M`. -/
def submonoid.of_add_submonoid {M : Type*} [monoid M] (S : add_submonoid (additive M)) :
  submonoid M :=
{ carrier := S.carrier,
  one_mem' := S.zero_mem',
  mul_mem' := S.add_mem' }

/-- Map from `add_submonoid`s of `add_monoid M` to submonoids of `multiplicative M`. -/
def add_submonoid.to_submonoid {M : Type*} [add_monoid M] (S : add_submonoid M) :
  submonoid (multiplicative M) :=
{ carrier := S.carrier,
  one_mem' := S.zero_mem',
  mul_mem' := S.add_mem' }

/-- Map from submonoids of `multiplicative M` to `add_submonoid`s of `add_monoid M`. -/
def add_submonoid.of_submonoid {M : Type*} [add_monoid M] (S : submonoid (multiplicative M)) :
  add_submonoid M :=
{ carrier := S.carrier,
  zero_mem' := S.one_mem',
  add_mem' := S.mul_mem' }

/-- Submonoids of monoid `M` are isomorphic to additive submonoids of `additive M`. -/
def submonoid.add_submonoid_equiv (M : Type*) [monoid M] :
submonoid M ≃ add_submonoid (additive M) :=
{ to_fun := submonoid.to_add_submonoid,
  inv_fun := submonoid.of_add_submonoid,
  left_inv := λ x, by cases x; refl,
  right_inv := λ x, by cases x; refl }

namespace submonoid

@[to_additive]
instance : has_coe (submonoid M) (set M) := ⟨submonoid.carrier⟩

@[to_additive]
instance : has_coe_to_sort (submonoid M) := ⟨Type*, λ S, S.carrier⟩

@[to_additive]
instance : has_mem M (submonoid M) := ⟨λ m S, m ∈ (S:set M)⟩

@[simp, to_additive]
lemma mem_carrier {s : submonoid M} {x : M} : x ∈ s.carrier ↔ x ∈ s := iff.rfl

@[simp, norm_cast, to_additive]
lemma mem_coe {S : submonoid M} {m : M} : m ∈ (S : set M) ↔ m ∈ S := iff.rfl

@[simp, norm_cast, to_additive]
lemma coe_coe (s : submonoid M) : ↥(s : set M) = s := rfl

attribute [norm_cast] add_submonoid.mem_coe add_submonoid.coe_coe

@[to_additive]
instance is_submonoid (S : submonoid M) : is_submonoid (S : set M) := ⟨S.2, S.3⟩

end submonoid

@[to_additive]
protected lemma submonoid.exists {s : submonoid M} {p : s → Prop} :
  (∃ x : s, p x) ↔ ∃ x ∈ s, p ⟨x, ‹x ∈ s›⟩ :=
set_coe.exists

@[to_additive]
protected lemma submonoid.forall {s : submonoid M} {p : s → Prop} :
  (∀ x : s, p x) ↔ ∀ x ∈ s, p ⟨x, ‹x ∈ s›⟩ :=
set_coe.forall

namespace submonoid

variables (S : submonoid M)

/-- Two submonoids are equal if the underlying subsets are equal. -/
@[to_additive "Two `add_submonoid`s are equal if the underlying subsets are equal."]
theorem ext' ⦃S T : submonoid M⦄ (h : (S : set M) = T) : S = T :=
by cases S; cases T; congr'

/-- Two submonoids are equal if and only if the underlying subsets are equal. -/
@[to_additive "Two `add_submonoid`s are equal if and only if the underlying subsets are equal."]
protected theorem ext'_iff {S T : submonoid M}  : S = T ↔ (S : set M) = T :=
⟨λ h, h ▸ rfl, λ h, ext' h⟩

/-- Two submonoids are equal if they have the same elements. -/
@[ext, to_additive "Two `add_submonoid`s are equal if they have the same elements."]
theorem ext {S T : submonoid M}
  (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T := ext' $ set.ext h

attribute [ext] add_submonoid.ext

/-- Copy a submonoid replacing `carrier` with a set that is equal to it. -/
@[to_additive "Copy an additive submonoid replacing `carrier` with a set that is equal to it."]
def copy (S : submonoid M) (s : set M) (hs : s = S) : submonoid M :=
{ carrier := s,
  one_mem' := hs.symm ▸ S.one_mem',
  mul_mem' := hs.symm ▸ S.mul_mem' }

@[simp, to_additive] lemma coe_copy {S : submonoid M} {s : set M} (hs : s = S) :
  (S.copy s hs : set M) = s := rfl

@[to_additive] lemma copy_eq {S : submonoid M} {s : set M} (hs : s = S) : S.copy s hs = S := ext' hs

/-- A submonoid contains the monoid's 1. -/
@[to_additive "An `add_submonoid` contains the monoid's 0."]
theorem one_mem : (1 : M) ∈ S := S.one_mem'

/-- A submonoid is closed under multiplication. -/
@[to_additive "An `add_submonoid` is closed under addition."]
theorem mul_mem {x y : M} : x ∈ S → y ∈ S → x * y ∈ S := submonoid.mul_mem' S

/-- Product of a list of elements in a submonoid is in the submonoid. -/
@[to_additive "Sum of a list of elements in an `add_submonoid` is in the `add_submonoid`."]
lemma list_prod_mem : ∀ {l : list M}, (∀x ∈ l, x ∈ S) → l.prod ∈ S
| []     h := S.one_mem
| (a::l) h :=
  suffices a * l.prod ∈ S, by rwa [list.prod_cons],
  have a ∈ S ∧ (∀ x ∈ l, x ∈ S), from list.forall_mem_cons.1 h,
  S.mul_mem this.1 (list_prod_mem this.2)

/-- Product of a multiset of elements in a submonoid of a `comm_monoid` is in the submonoid. -/
@[to_additive "Sum of a multiset of elements in an `add_submonoid` of an `add_comm_monoid` is in the `add_submonoid`."]
lemma multiset_prod_mem {M} [comm_monoid M] (S : submonoid M) (m : multiset M) :
  (∀a ∈ m, a ∈ S) → m.prod ∈ S :=
begin
  refine quotient.induction_on m (assume l hl, _),
  rw [multiset.quot_mk_to_coe, multiset.coe_prod],
  exact S.list_prod_mem hl
end

/-- Product of elements of a submonoid of a `comm_monoid` indexed by a `finset` is in the
    submonoid. -/
@[to_additive "Sum of elements in an `add_submonoid` of an `add_comm_monoid` indexed by a `finset` is in the `add_submonoid`."]
lemma prod_mem {M : Type*} [comm_monoid M] (S : submonoid M)
  {ι : Type*} {t : finset ι} {f : ι → M} (h : ∀c ∈ t, f c ∈ S) :
  t.prod f ∈ S :=
S.multiset_prod_mem (t.1.map f) $ λ x hx, let ⟨i, hi, hix⟩ := multiset.mem_map.1 hx in hix ▸ h i hi

lemma pow_mem {x : M} (hx : x ∈ S) : ∀ n:ℕ, x^n ∈ S
| 0 := S.one_mem
| (n+1) := S.mul_mem hx (pow_mem n)

/-- A submonoid of a monoid inherits a multiplication. -/
@[to_additive "An `add_submonoid` of an `add_monoid` inherits an addition."]
instance has_mul : has_mul S := ⟨λ a b, ⟨a.1 * b.1, S.mul_mem a.2 b.2⟩⟩

/-- A submonoid of a monoid inherits a 1. -/
@[to_additive "An `add_submonoid` of an `add_monoid` inherits a zero."]
instance has_one : has_one S := ⟨⟨_, S.one_mem⟩⟩

@[simp, to_additive] lemma coe_mul (x y : S) : (↑(x * y) : M) = ↑x * ↑y := rfl
@[simp, to_additive] lemma coe_one : ((1 : S) : M) = 1 := rfl

/-- A submonoid of a monoid inherits a monoid structure. -/
@[to_additive to_add_monoid "An `add_submonoid` of an `add_monoid` inherits an `add_monoid` structure."]
instance to_monoid {M : Type*} [monoid M] (S : submonoid M) : monoid S :=
by refine { mul := (*), one := 1, ..}; by simp [mul_assoc]

/-- A submonoid of a `comm_monoid` is a `comm_monoid`. -/
@[to_additive to_add_comm_monoid "An `add_submonoid` of an `add_comm_monoid` is an `add_comm_monoid`."]
instance to_comm_monoid {M} [comm_monoid M] (S : submonoid M) : comm_monoid S :=
{ mul_comm := λ _ _, subtype.eq $ mul_comm _ _, ..S.to_monoid}

/-- The natural monoid hom from a submonoid of monoid `M` to `M`. -/
@[to_additive "The natural monoid hom from an `add_submonoid` of `add_monoid` `M` to `M`."]
def subtype : S →* M := ⟨coe, rfl, λ _ _, rfl⟩

@[simp, to_additive] theorem coe_subtype : ⇑S.subtype = coe := rfl

@[to_additive]
instance : has_le (submonoid M) := ⟨λ S T, ∀ ⦃x⦄, x ∈ S → x ∈ T⟩

@[to_additive]
lemma le_def {S T : submonoid M} : S ≤ T ↔ ∀ ⦃x : M⦄, x ∈ S → x ∈ T := iff.rfl

@[simp, norm_cast, to_additive]
lemma coe_subset_coe {S T : submonoid M} : (S : set M) ⊆ T ↔ S ≤ T := iff.rfl

@[to_additive]
instance : partial_order (submonoid M) :=
{ le := λ S T, ∀ ⦃x⦄, x ∈ S → x ∈ T,
  .. partial_order.lift (coe : submonoid M → set M) ext' infer_instance }

@[simp, norm_cast, to_additive]
lemma coe_ssubset_coe {S T : submonoid M} : (S : set M) ⊂ T ↔ S < T := iff.rfl

attribute [norm_cast]  add_submonoid.coe_subset_coe add_submonoid.coe_ssubset_coe

/-- The submonoid `M` of the monoid `M`. -/
@[to_additive "The `add_submonoid M` of the `add_monoid M`."]
instance : has_top (submonoid M) :=
⟨{ carrier := set.univ,
   one_mem' := set.mem_univ 1,
   mul_mem' := λ _ _ _ _, set.mem_univ _ }⟩

/-- The trivial submonoid `{1}` of an monoid `M`. -/
@[to_additive "The trivial `add_submonoid` `{0}` of an `add_monoid` `M`."]
instance : has_bot (submonoid M) :=
⟨{ carrier := {1},
   one_mem' := set.mem_singleton 1,
   mul_mem' := λ a b ha hb, by { simp only [set.mem_singleton_iff] at *, rw [ha, hb, mul_one] }}⟩

@[to_additive]
instance : inhabited (submonoid M) := ⟨⊥⟩

@[simp, to_additive] lemma mem_bot {x : M} : x ∈ (⊥ : submonoid M) ↔ x = 1 := set.mem_singleton_iff

@[simp, to_additive] lemma mem_top (x : M) : x ∈ (⊤ : submonoid M) := set.mem_univ x

@[simp, to_additive] lemma coe_top : ((⊤ : submonoid M) : set M) = set.univ := rfl

@[simp, to_additive] lemma coe_bot : ((⊥ : submonoid M) : set M) = {1} := rfl

/-- The inf of two submonoids is their intersection. -/
@[to_additive "The inf of two `add_submonoid`s is their intersection."]
instance : has_inf (submonoid M) :=
⟨λ S₁ S₂,
  { carrier := S₁ ∩ S₂,
    one_mem' := ⟨S₁.one_mem, S₂.one_mem⟩,
    mul_mem' := λ _ _ ⟨hx, hx'⟩ ⟨hy, hy'⟩,
      ⟨S₁.mul_mem hx hy, S₂.mul_mem hx' hy'⟩ }⟩

@[simp, to_additive]
lemma coe_inf (p p' : submonoid M) : ((p ⊓ p' : submonoid M) : set M) = p ∩ p' := rfl

@[simp, to_additive]
lemma mem_inf {p p' : submonoid M} {x : M} : x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p' := iff.rfl

@[to_additive]
instance : has_Inf (submonoid M) :=
⟨λ s, {
  carrier := ⋂ t ∈ s, ↑t,
  one_mem' := set.mem_bInter $ λ i h, i.one_mem,
  mul_mem' := λ x y hx hy, set.mem_bInter $ λ i h,
    i.mul_mem (by apply set.mem_bInter_iff.1 hx i h) (by apply set.mem_bInter_iff.1 hy i h) }⟩

@[simp, to_additive]
lemma coe_Inf (S : set (submonoid M)) : ((Inf S : submonoid M) : set M) = ⋂ s ∈ S, ↑s := rfl

@[to_additive]
lemma mem_Inf {S : set (submonoid M)} {x : M} : x ∈ Inf S ↔ ∀ p ∈ S, x ∈ p := set.mem_bInter_iff

@[to_additive]
lemma mem_infi {ι : Sort*} {S : ι → submonoid M} {x : M} : (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i :=
by simp only [infi, mem_Inf, set.forall_range_iff]

@[simp, to_additive]
lemma coe_infi {ι : Sort*} {S : ι → submonoid M} : (↑(⨅ i, S i) : set M) = ⋂ i, S i :=
by simp only [infi, coe_Inf, set.bInter_range]

attribute [norm_cast] coe_Inf coe_infi

/-- Submonoids of a monoid form a complete lattice. -/
@[to_additive "The `add_submonoid`s of an `add_monoid` form a complete lattice."]
instance : complete_lattice (submonoid M) :=
{ le           := (≤),
  lt           := (<),
  bot          := (⊥),
  bot_le       := λ S x hx, (mem_bot.1 hx).symm ▸ S.one_mem,
  top          := (⊤),
  le_top       := λ S x hx, mem_top x,
  inf          := (⊓),
  Inf          := has_Inf.Inf,
  le_inf       := λ a b c ha hb x hx, ⟨ha hx, hb hx⟩,
  inf_le_left  := λ a b x, and.left,
  inf_le_right := λ a b x, and.right,
  .. complete_lattice_of_Inf (submonoid M) $ λ s,
    is_glb.of_image (λ S T, show (S : set M) ≤ T ↔ S ≤ T, from coe_subset_coe) is_glb_binfi }

/-- The `submonoid` generated by a set. -/
@[to_additive "The `add_submonoid` generated by a set"]
def closure (s : set M) : submonoid M := Inf {S | s ⊆ S}

@[to_additive]
lemma mem_closure {x : M} : x ∈ closure s ↔ ∀ S : submonoid M, s ⊆ S → x ∈ S :=
mem_Inf

/-- The submonoid generated by a set includes the set. -/
@[simp, to_additive "The `add_submonoid` generated by a set includes the set."]
lemma subset_closure : s ⊆ closure s := λ x hx, mem_closure.2 $ λ S hS, hS hx

variable {S}
open set

/-- A submonoid `S` includes `closure s` if and only if it includes `s`. -/
@[simp, to_additive "An additive submonoid `S` includes `closure s` if and only if it includes `s`"]
lemma closure_le : closure s ≤ S ↔ s ⊆ S :=
⟨subset.trans subset_closure, λ h, Inf_le h⟩

/-- Submonoid closure of a set is monotone in its argument: if `s ⊆ t`,
then `closure s ≤ closure t`. -/
@[to_additive "Additive submonoid closure of a set is monotone in its argument: if `s ⊆ t`,
then `closure s ≤ closure t`"]
lemma closure_mono ⦃s t : set M⦄ (h : s ⊆ t) : closure s ≤ closure t :=
closure_le.2 $ subset.trans h subset_closure

@[to_additive]
lemma closure_eq_of_le (h₁ : s ⊆ S) (h₂ : S ≤ closure s) : closure s = S :=
le_antisymm (closure_le.2 h₁) h₂

variable (S)

/-- An induction principle for closure membership. If `p` holds for `1` and all elements of `s`, and
is preserved under multiplication, then `p` holds for all elements of the closure of `s`. -/
@[to_additive "An induction principle for additive closure membership. If `p` holds for `0` and all
elements of `s`, and is preserved under addition, then `p` holds for all elements
of the additive closure of `s`."]
lemma closure_induction {p : M → Prop} {x} (h : x ∈ closure s)
  (Hs : ∀ x ∈ s, p x) (H1 : p 1)
  (Hmul : ∀ x y, p x → p y → p (x * y)) : p x :=
(@closure_le _ _ _ ⟨p, H1, Hmul⟩).2 Hs h

attribute [elab_as_eliminator] submonoid.closure_induction add_submonoid.closure_induction

variable (M)

/-- `closure` forms a Galois insertion with the coercion to set. -/
@[to_additive "`closure` forms a Galois insertion with the coercion to set."]
protected def gi : galois_insertion (@closure M _) coe :=
{ choice := λ s _, closure s,
  gc := λ s t, closure_le,
  le_l_u := λ s, subset_closure,
  choice_eq := λ s h, rfl }

variable {M}

/-- Closure of a submonoid `S` equals `S`. -/
@[simp, to_additive "Additive closure of an additive submonoid `S` equals `S`"]
lemma closure_eq : closure (S : set M) = S := (submonoid.gi M).l_u_eq S

@[simp, to_additive] lemma closure_empty : closure (∅ : set M) = ⊥ :=
(submonoid.gi M).gc.l_bot

@[simp, to_additive] lemma closure_univ : closure (univ : set M) = ⊤ :=
@coe_top M _ ▸ closure_eq ⊤

@[to_additive]
lemma closure_union (s t : set M) : closure (s ∪ t) = closure s ⊔ closure t :=
(submonoid.gi M).gc.l_sup

@[to_additive]
lemma closure_Union {ι} (s : ι → set M) : closure (⋃ i, s i) = ⨆ i, closure (s i) :=
(submonoid.gi M).gc.l_supr

@[to_additive]
lemma mem_supr_of_directed {ι} [hι : nonempty ι] {S : ι → submonoid M} (hS : directed (≤) S)
  {x : M} :
  x ∈ (supr S : submonoid M) ↔ ∃ i, x ∈ S i :=
begin
  refine ⟨_, λ ⟨i, hi⟩, (le_def.1 $ le_supr S i) hi⟩,
  suffices : x ∈ closure (⋃ i, (S i : set M)) → ∃ i, x ∈ S i,
    by simpa only [closure_Union, closure_eq (S _)] using this,
  refine (λ hx, closure_induction hx (λ _, mem_Union.1) _ _),
  { exact hι.elim (λ i, ⟨i, (S i).one_mem⟩) },
  { rintros x y ⟨i, hi⟩ ⟨j, hj⟩,
    rcases hS i j with ⟨k, hki, hkj⟩,
    exact ⟨k, (S k).mul_mem (hki hi) (hkj hj)⟩ }
end

@[to_additive]
lemma mem_Sup_of_directed_on {S : set (submonoid M)} (Sne : S.nonempty)
  (hS : directed_on (≤) S) {x : M} :
  x ∈ Sup S ↔ ∃ s ∈ S, x ∈ s :=
begin
  haveI : nonempty S := Sne.to_subtype,
  rw [Sup_eq_supr, supr_subtype', mem_supr_of_directed, subtype.exists],
  exact (directed_on_iff_directed _).1 hS
end

variables {N : Type*} [monoid N] {P : Type*} [monoid P]

/-- The preimage of a submonoid along a monoid homomorphism is a submonoid. -/
@[to_additive "The preimage of an `add_submonoid` along an `add_monoid` homomorphism is an `add_submonoid`."]
def comap (f : M →* N) (S : submonoid N) : submonoid M :=
{ carrier := (f ⁻¹' S),
  one_mem' := show f 1 ∈ S, by rw f.map_one; exact S.one_mem,
  mul_mem' := λ a b ha hb,
    show f (a * b) ∈ S, by rw f.map_mul; exact S.mul_mem ha hb }

@[simp, to_additive]
lemma coe_comap (S : submonoid N) (f : M →* N) : (S.comap f : set M) = f ⁻¹' S := rfl

@[simp, to_additive]
lemma mem_comap {S : submonoid N} {f : M →* N} {x : M} : x ∈ S.comap f ↔ f x ∈ S := iff.rfl

@[to_additive]
lemma comap_comap (S : submonoid P) (g : N →* P) (f : M →* N) :
  (S.comap g).comap f = S.comap (g.comp f) :=
rfl

/-- The image of a submonoid along a monoid homomorphism is a submonoid. -/
@[to_additive "The image of an `add_submonoid` along an `add_monoid` homomorphism is an `add_submonoid`."]
def map (f : M →* N) (S : submonoid M) : submonoid N :=
{ carrier := (f '' S),
  one_mem' := ⟨1, S.one_mem, f.map_one⟩,
  mul_mem' := begin rintros _ _ ⟨x, hx, rfl⟩ ⟨y, hy, rfl⟩, exact ⟨x * y, S.mul_mem hx hy,
    by rw f.map_mul; refl⟩ end }

@[simp, to_additive]
lemma coe_map (f : M →* N) (S : submonoid M) :
  (S.map f : set N) = f '' S := rfl

@[simp, to_additive]
lemma mem_map {f : M →* N} {S : submonoid M} {y : N} :
  y ∈ S.map f ↔ ∃ x ∈ S, f x = y :=
mem_image_iff_bex

@[to_additive]
lemma map_map (g : N →* P) (f : M →* N) : (S.map f).map g = S.map (g.comp f) :=
ext' $ image_image _ _ _

@[to_additive]
lemma map_le_iff_le_comap {f : M →* N} {S : submonoid M} {T : submonoid N} :
  S.map f ≤ T ↔ S ≤ T.comap f :=
image_subset_iff

@[to_additive]
lemma gc_map_comap (f : M →* N) : galois_connection (map f) (comap f) :=
λ S T, map_le_iff_le_comap

@[to_additive]
lemma map_sup (S T : submonoid M) (f : M →* N) : (S ⊔ T).map f = S.map f ⊔ T.map f :=
(gc_map_comap f).l_sup

@[to_additive]
lemma map_supr {ι : Sort*} (f : M →* N) (s : ι → submonoid M) :
  (supr s).map f = ⨆ i, (s i).map f :=
(gc_map_comap f).l_supr

@[to_additive]
lemma comap_inf (S T : submonoid N) (f : M →* N) : (S ⊓ T).comap f = S.comap f ⊓ T.comap f :=
(gc_map_comap f).u_inf

@[to_additive]
lemma comap_infi {ι : Sort*} (f : M →* N) (s : ι → submonoid N) :
  (infi s).comap f = ⨅ i, (s i).comap f :=
(gc_map_comap f).u_infi

@[simp, to_additive] lemma map_bot (f : M →* N) : (⊥ : submonoid M).map f = ⊥ :=
(gc_map_comap f).l_bot

@[simp, to_additive] lemma comap_top (f : M →* N) : (⊤ : submonoid N).comap f = ⊤ :=
(gc_map_comap f).u_top

/-- Given `submonoid`s `s`, `t` of monoids `M`, `N` respectively, `s × t` as a submonoid of `M × N`. -/
@[to_additive prod "Given `add_submonoid`s `s`, `t` of `add_monoid`s `A`, `B` respectively, `s × t` as an `add_submonoid` of `A × B`."]
def prod (s : submonoid M) (t : submonoid N) : submonoid (M × N) :=
{ carrier := (s : set M).prod t,
  one_mem' := ⟨s.one_mem, t.one_mem⟩,
  mul_mem' := λ p q hp hq, ⟨s.mul_mem hp.1 hq.1, t.mul_mem hp.2 hq.2⟩ }

@[to_additive coe_prod]
lemma coe_prod (s : submonoid M) (t : submonoid N) :
 (s.prod t : set (M × N)) = (s : set M).prod (t : set N) :=
rfl

@[to_additive mem_prod]
lemma mem_prod {s : submonoid M} {t : submonoid N} {p : M × N} :
  p ∈ s.prod t ↔ p.1 ∈ s ∧ p.2 ∈ t := iff.rfl

@[to_additive prod_mono]
lemma prod_mono : ((≤) ⇒ (≤) ⇒ (≤)) (@prod M _ N _) (@prod M _ N _) :=
λ s s' hs t t' ht, set.prod_mono hs ht

@[to_additive prod_mono_right]
lemma prod_mono_right (s : submonoid M) : monotone (λ t : submonoid N, s.prod t) :=
prod_mono (le_refl s)

@[to_additive prod_mono_left]
lemma prod_mono_left (t : submonoid N) : monotone (λ s : submonoid M, s.prod t) :=
λ s₁ s₂ hs, prod_mono hs (le_refl t)

@[to_additive prod_top]
lemma prod_top (s : submonoid M) :
  s.prod (⊤ : submonoid N) = s.comap (monoid_hom.fst M N) :=
ext $ λ x, by simp [mem_prod, monoid_hom.coe_fst]

@[to_additive top_prod]
lemma top_prod (s : submonoid N) :
  (⊤ : submonoid M).prod s = s.comap (monoid_hom.snd M N) :=
ext $ λ x, by simp [mem_prod, monoid_hom.coe_snd]

@[simp, to_additive top_prod_top]
lemma top_prod_top : (⊤ : submonoid M).prod (⊤ : submonoid N) = ⊤ :=
(top_prod _).trans $ comap_top _

@[to_additive] lemma bot_prod_bot : (⊥ : submonoid M).prod (⊥ : submonoid N) = ⊥ :=
ext' $ by simp [coe_prod, prod.one_eq_mk]

/-- Product of submonoids is isomorphic to their product as monoids. -/
@[to_additive prod_equiv "Product of additive submonoids is isomorphic to their product
as additive monoids"]
def prod_equiv (s : submonoid M) (t : submonoid N) : s.prod t ≃* s × t :=
{ map_mul' := λ x y, rfl, .. equiv.set.prod ↑s ↑t }

end submonoid

namespace monoid_hom

variables {N : Type*} {P : Type*} [monoid N] [monoid P] (S : submonoid M)

open submonoid

/-- The range of a monoid homomorphism is a submonoid. -/
@[to_additive "The range of an `add_monoid_hom` is an `add_submonoid`."]
def mrange (f : M →* N) : submonoid N := (⊤ : submonoid M).map f

@[simp, to_additive] lemma coe_mrange (f : M →* N) :
  (f.mrange : set N) = set.range f :=
set.image_univ

@[simp, to_additive] lemma mem_mrange {f : M →* N} {y : N} :
  y ∈ f.mrange ↔ ∃ x, f x = y :=
by simp [mrange]

@[to_additive]
lemma map_mrange (g : N →* P) (f : M →* N) : f.mrange.map g = (g.comp f).mrange :=
(⊤ : submonoid M).map_map g f

/-- Restriction of a monoid hom to a submonoid of the domain. -/
@[to_additive "Restriction of an add_monoid hom to an `add_submonoid` of the domain."]
def restrict {N : Type*} [monoid N] (f : M →* N) (S : submonoid M) : S →* N := f.comp S.subtype

@[to_additive]
lemma restrict_apply {N : Type*} [monoid N] (f : M →* N) (x : S) : f.restrict S x = f x := rfl

@[simp, to_additive] lemma restrict_eq {N : Type} [monoid N] (f : M →* N) (x) :
  f.restrict S x = f x := rfl

/-- Restriction of a monoid hom to a submonoid of the codomain. -/
@[to_additive "Restriction of an `add_monoid` hom to an `add_submonoid` of the codomain."]
def cod_restrict (f : M →* N) (S : submonoid N) (h : ∀ x, f x ∈ S) : M →* S :=
{ to_fun := λ n, ⟨f n, h n⟩,
  map_one' := subtype.eq f.map_one,
  map_mul' := λ x y, subtype.eq (f.map_mul x y) }

/-- Restriction of a monoid hom to its range iterpreted as a submonoid. -/
@[to_additive "Restriction of an `add_monoid` hom to its range interpreted as a submonoid."]
def range_restrict {N} [monoid N] (f : M →* N) : M →* f.mrange :=
f.cod_restrict f.mrange $ λ x, ⟨x, submonoid.mem_top x, rfl⟩

@[simp, to_additive]
lemma coe_range_restrict {N} [monoid N] (f : M →* N) (x : M) :
  (f.range_restrict x : N) = f x :=
rfl

@[to_additive]
lemma mrange_top_iff_surjective {N} [monoid N] {f : M →* N} :
  f.mrange = (⊤ : submonoid N) ↔ function.surjective f :=
submonoid.ext'_iff.trans $ iff.trans (by rw [coe_mrange, coe_top]) set.range_iff_surjective

/-- The range of a surjective monoid hom is the whole of the codomain. -/
@[to_additive "The range of a surjective `add_monoid` hom is the whole of the codomain."]
lemma mrange_top_of_surjective {N} [monoid N] (f : M →* N) (hf : function.surjective f) :
  f.mrange = (⊤ : submonoid N) :=
mrange_top_iff_surjective.2 hf

/-- The submonoid of elements `x : M` such that `f x = g x` -/
@[to_additive "The additive submonoid of elements `x : M` such that `f x = g x`"]
def eq_mlocus (f g : M →* N) : submonoid M :=
{ carrier := {x | f x = g x},
  one_mem' := by rw [set.mem_set_of_eq, f.map_one, g.map_one],
  mul_mem' := λ x y (hx : _ = _) (hy : _ = _), by simp [*] }

/-- If two monoid homomorphisms are equal on a set, then they are equal on its submonoid closure. -/
@[to_additive]
lemma eq_on_mclosure {f g : M →* N} {s : set M} (h : set.eq_on f g s) :
  set.eq_on f g (closure s) :=
show closure s ≤ f.eq_mlocus g, from closure_le.2 h

@[to_additive]
lemma eq_of_eq_on_mtop {f g : M →* N} (h : set.eq_on f g (⊤ : submonoid M)) :
  f = g :=
ext $ λ x, h trivial

@[to_additive]
lemma eq_of_eq_on_mdense {s : set M} (hs : closure s = ⊤) {f g : M →* N} (h : s.eq_on f g) :
  f = g :=
eq_of_eq_on_mtop $ hs ▸ eq_on_mclosure h

@[to_additive]
lemma closure_preimage_le (f : M →* N) (s : set N) :
  closure (f ⁻¹' s) ≤ (closure s).comap f :=
closure_le.2 $ λ x hx, mem_coe.2 $ mem_comap.2 $ subset_closure hx

/-- The image under a monoid hom of the submonoid generated by a set equals the submonoid generated
    by the image of the set. -/
@[to_additive "The image under an `add_monoid` hom of the `add_submonoid` generated by a set equals
the `add_submonoid` generated by the image of the set."]
lemma map_mclosure (f : M →* N) (s : set M) :
  (closure s).map f = closure (f '' s) :=
le_antisymm
  (map_le_iff_le_comap.2 $ le_trans (closure_mono $ set.subset_preimage_image _ _)
    (closure_preimage_le _ _))
  (closure_le.2 $ set.image_subset _ subset_closure)

end monoid_hom

namespace free_monoid

variables {α : Type*}

open submonoid

@[to_additive]
theorem closure_range_of : closure (set.range $ @of α) = ⊤ :=
eq_top_iff.2 $ λ x hx, free_monoid.rec_on x (one_mem _) $ λ x xs hxs,
  mul_mem _ (subset_closure $ set.mem_range_self _) hxs

end free_monoid

namespace submonoid

variables {N : Type*} [monoid N]

open monoid_hom lattice

/-- The monoid hom associated to an inclusion of submonoids. -/
@[to_additive "The `add_monoid` hom associated to an inclusion of submonoids."]
def inclusion {S T : submonoid M} (h : S ≤ T) : S →* T :=
S.subtype.cod_restrict _ (λ x, h x.2)

@[simp, to_additive]
lemma range_subtype (s : submonoid M) : s.subtype.mrange = s :=
ext' $ (coe_mrange _).trans $ set.range_coe_subtype s

lemma closure_singleton_eq (x : M) : closure ({x} : set M) = (powers_hom M x).mrange :=
closure_eq_of_le (set.singleton_subset_iff.2 ⟨multiplicative.of_add 1, trivial, pow_one x⟩) $
  λ x ⟨n, _, hn⟩, hn ▸ pow_mem _ (subset_closure $ set.mem_singleton _) _

/-- The submonoid generated by an element of a monoid equals the set of natural number powers of
    the element. -/
lemma mem_closure_singleton {x y : M} : y ∈ closure ({x} : set M) ↔ ∃ n:ℕ, x^n=y :=
by rw [closure_singleton_eq, mem_mrange]; refl

@[to_additive]
lemma closure_eq_mrange (s : set M) : closure s = (free_monoid.lift s M coe).mrange :=
by rw [mrange, ← free_monoid.closure_range_of, map_mclosure,
  ← set.range_comp, free_monoid.lift_comp_of, set.range_coe_subtype]

@[to_additive]
lemma exists_list_of_mem_closure {s : set M} {x : M} (hx : x ∈ closure s) :
  ∃ (l : list M) (hl : ∀ y ∈ l, y ∈ s), l.prod = x :=
begin
  rw [closure_eq_mrange, mem_mrange] at hx,
  rcases hx with ⟨l, hx⟩,
  exact ⟨list.map coe l, λ y hy, let ⟨z, hz, hy⟩ := list.mem_map.1 hy in hy ▸ z.2, hx⟩
end

@[to_additive]
lemma map_inl (s : submonoid M) : s.map (inl M N) = s.prod ⊥ :=
ext $ λ p, ⟨λ ⟨x, hx, hp⟩, hp ▸ ⟨hx, set.mem_singleton 1⟩,
  λ ⟨hps, hp1⟩, ⟨p.1, hps, prod.ext rfl $ (set.eq_of_mem_singleton hp1).symm⟩⟩

@[to_additive]
lemma map_inr (s : submonoid N) : s.map (inr M N) = prod ⊥ s :=
ext $ λ p, ⟨λ ⟨x, hx, hp⟩, hp ▸ ⟨set.mem_singleton 1, hx⟩,
  λ ⟨hp1, hps⟩, ⟨p.2, hps, prod.ext (set.eq_of_mem_singleton hp1).symm rfl⟩⟩

@[to_additive]
lemma range_inl : (inl M N).mrange = prod ⊤ ⊥ := map_inl ⊤

@[to_additive]
lemma range_inr : (inr M N).mrange = prod ⊥ ⊤ := map_inr ⊤

@[to_additive]
lemma range_inl' : (inl M N).mrange = comap (snd M N) ⊥ := range_inl.trans (top_prod _)

@[to_additive]
lemma range_inr' : (inr M N).mrange = comap (fst M N) ⊥ := range_inr.trans (prod_top _)

@[simp, to_additive]
lemma range_fst : (fst M N).mrange = ⊤ :=
(fst M N).mrange_top_of_surjective $ @prod.fst_surjective _ _ ⟨1⟩

@[simp, to_additive]
lemma range_snd : (snd M N).mrange = ⊤ :=
(snd M N).mrange_top_of_surjective $ @prod.snd_surjective _ _ ⟨1⟩

@[simp, to_additive prod_bot_sup_bot_prod]
lemma prod_bot_sup_bot_prod (s : submonoid M) (t : submonoid N) :
  (s.prod ⊥) ⊔ (prod ⊥ t) = s.prod t :=
le_antisymm (sup_le (prod_mono_right s bot_le) (prod_mono_left t bot_le)) $
assume p hp, prod.fst_mul_snd p ▸ mul_mem _
  ((le_sup_left : s.prod ⊥ ≤ s.prod ⊥ ⊔ prod ⊥ t) ⟨hp.1, set.mem_singleton 1⟩)
  ((le_sup_right : prod ⊥ t ≤ s.prod ⊥ ⊔ prod ⊥ t) ⟨set.mem_singleton 1, hp.2⟩)

@[simp, to_additive]
lemma range_inl_sup_range_inr : (inl M N).mrange ⊔ (inr M N).mrange = ⊤ :=
by simp only [range_inl, range_inr, prod_bot_sup_bot_prod, top_prod_top]

end submonoid

namespace submonoid

variables {N : Type*} [comm_monoid N]

open monoid_hom

@[to_additive]
lemma sup_eq_range (s t : submonoid N) : s ⊔ t = (s.subtype.coprod t.subtype).mrange :=
by rw [mrange, ← range_inl_sup_range_inr, map_sup, map_mrange, coprod_comp_inl,
  map_mrange, coprod_comp_inr, range_subtype, range_subtype]

@[to_additive]
lemma mem_sup {s t : submonoid N} {x : N} :
  x ∈ s ⊔ t ↔ ∃ (y ∈ s) (z ∈ t), y * z = x :=
by simp only [sup_eq_range, mem_mrange, coprod_apply, prod.exists, submonoid.exists,
  coe_subtype, subtype.coe_mk]

end submonoid

namespace add_submonoid

open set

lemma smul_mem (S : add_submonoid A) {x : A} (hx : x ∈ S) :
  ∀ n : ℕ, add_monoid.smul n x ∈ S
| 0     := S.zero_mem
| (n+1) := S.add_mem hx (smul_mem n)

lemma closure_singleton_eq (x : A) : closure ({x} : set A) = (multiples_hom A x).mrange :=
closure_eq_of_le (set.singleton_subset_iff.2 ⟨1, trivial, add_monoid.one_smul x⟩) $
  λ x ⟨n, _, hn⟩, hn ▸ smul_mem _ (subset_closure $ set.mem_singleton _) _

/-- The `add_submonoid` generated by an element of an `add_monoid` equals the set of
natural number multiples of the element. -/
lemma mem_closure_singleton {x y : A} : y ∈ closure ({x} : set A) ↔ ∃ n:ℕ, add_monoid.smul n x = y :=
by rw [closure_singleton_eq, add_monoid_hom.mem_mrange]; refl

end add_submonoid


namespace mul_equiv

variables {S T : submonoid M}

/-- Makes the identity isomorphism from a proof two submonoids of a multiplicative
    monoid are equal. -/
@[to_additive add_submonoid_congr "Makes the identity additive isomorphism from a proof two submonoids of an additive monoid are equal."]
def submonoid_congr (h : S = T) : S ≃* T :=
{ map_mul' :=  λ _ _, rfl, ..equiv.set_congr $ submonoid.ext'_iff.1 h }

end mul_equiv