Theory Group

(*  Title:      HOL/Algebra/Group.thy
    Author:     Clemens Ballarin, started 4 February 2003

Based on work by Florian Kammueller, L C Paulson and Markus Wenzel.
With additional contributions from Martin Baillon and Paulo Emílio de Vilhena.
*)

theory Group
imports Complete_Lattice "HOL-Library.FuncSet"
begin

section ‹Monoids and Groups›

subsection ‹Definitions›

text ‹
  Definitions follow \<^cite>‹"Jacobson:1985"›.
›

record 'a monoid =  "'a partial_object" +
  mult    :: "['a, 'a] ⇒ 'a" (infixl ‹⊗ı› 70)
  one     :: 'a (‹𝟭ı›)

definition m_inv :: "('a, 'b) monoid_scheme => 'a => 'a"
  where "m_inv G x = (THE y. y ∈ carrier G ∧ x ⊗⇘G⇙ y = 𝟭⇘G⇙ ∧ y ⊗⇘G⇙ x = 𝟭⇘G⇙)"

open_bundle m_inv_syntax
begin
notation m_inv  (‹(‹open_block notation=‹prefix inv››invı _)› [81] 80)
end

definition
  Units :: "_ => 'a set"
  ― ‹The set of invertible elements›
  where "Units G = {y. y ∈ carrier G ∧ (∃x ∈ carrier G. x ⊗⇘G⇙ y = 𝟭⇘G⇙ ∧ y ⊗⇘G⇙ x = 𝟭⇘G⇙)}"

locale monoid =
  fixes G (structure)
  assumes m_closed [intro, simp]:
         "⟦x ∈ carrier G; y ∈ carrier G⟧ ⟹ x ⊗ y ∈ carrier G"
      and m_assoc:
         "⟦x ∈ carrier G; y ∈ carrier G; z ∈ carrier G⟧
          ⟹ (x ⊗ y) ⊗ z = x ⊗ (y ⊗ z)"
      and one_closed [intro, simp]: "𝟭 ∈ carrier G"
      and l_one [simp]: "x ∈ carrier G ⟹ 𝟭 ⊗ x = x"
      and r_one [simp]: "x ∈ carrier G ⟹ x ⊗ 𝟭 = x"

lemma monoidI:
  fixes G (structure)
  assumes m_closed:
      "!!x y. [| x ∈ carrier G; y ∈ carrier G |] ==> x ⊗ y ∈ carrier G"
    and one_closed: "𝟭 ∈ carrier G"
    and m_assoc:
      "!!x y z. [| x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==>
      (x ⊗ y) ⊗ z = x ⊗ (y ⊗ z)"
    and l_one: "!!x. x ∈ carrier G ==> 𝟭 ⊗ x = x"
    and r_one: "!!x. x ∈ carrier G ==> x ⊗ 𝟭 = x"
  shows "monoid G"
  by (fast intro!: monoid.intro intro: assms)

lemma (in monoid) Units_closed [dest]:
  "x ∈ Units G ==> x ∈ carrier G"
  by (unfold Units_def) fast

lemma (in monoid) one_unique:
  assumes "u ∈ carrier G"
    and "⋀x. x ∈ carrier G ⟹ u ⊗ x = x"
  shows "u = 𝟭"
  using assms(2)[OF one_closed] r_one[OF assms(1)] by simp

lemma (in monoid) inv_unique:
  assumes eq: "y ⊗ x = 𝟭"  "x ⊗ y' = 𝟭"
    and G: "x ∈ carrier G"  "y ∈ carrier G"  "y' ∈ carrier G"
  shows "y = y'"
proof -
  from G eq have "y = y ⊗ (x ⊗ y')" by simp
  also from G have "... = (y ⊗ x) ⊗ y'" by (simp add: m_assoc)
  also from G eq have "... = y'" by simp
  finally show ?thesis .
qed

lemma (in monoid) Units_m_closed [simp, intro]:
  assumes x: "x ∈ Units G" and y: "y ∈ Units G"
  shows "x ⊗ y ∈ Units G"
proof -
  from x obtain x' where x: "x ∈ carrier G" "x' ∈ carrier G" and xinv: "x ⊗ x' = 𝟭" "x' ⊗ x = 𝟭"
    unfolding Units_def by fast
  from y obtain y' where y: "y ∈ carrier G" "y' ∈ carrier G" and yinv: "y ⊗ y' = 𝟭" "y' ⊗ y = 𝟭"
    unfolding Units_def by fast
  from x y xinv yinv have "y' ⊗ (x' ⊗ x) ⊗ y = 𝟭" by simp
  moreover from x y xinv yinv have "x ⊗ (y ⊗ y') ⊗ x' = 𝟭" by simp
  moreover note x y
  ultimately show ?thesis unfolding Units_def
    by simp (metis m_assoc m_closed)
qed

lemma (in monoid) Units_one_closed [intro, simp]:
  "𝟭 ∈ Units G"
  by (unfold Units_def) auto

lemma (in monoid) Units_inv_closed [intro, simp]:
  "x ∈ Units G ==> inv x ∈ carrier G"
  apply (simp add: Units_def m_inv_def)
  by (metis (mono_tags, lifting) inv_unique the_equality)

lemma (in monoid) Units_l_inv_ex:
  "x ∈ Units G ==> ∃y ∈ carrier G. y ⊗ x = 𝟭"
  by (unfold Units_def) auto

lemma (in monoid) Units_r_inv_ex:
  "x ∈ Units G ==> ∃y ∈ carrier G. x ⊗ y = 𝟭"
  by (unfold Units_def) auto

lemma (in monoid) Units_l_inv [simp]:
  "x ∈ Units G ==> inv x ⊗ x = 𝟭"
  apply (unfold Units_def m_inv_def, simp)
  by (metis (mono_tags, lifting) inv_unique the_equality)

lemma (in monoid) Units_r_inv [simp]:
  "x ∈ Units G ==> x ⊗ inv x = 𝟭"
  by (metis (full_types) Units_closed Units_inv_closed Units_l_inv Units_r_inv_ex inv_unique)

lemma (in monoid) inv_one [simp]:
  "inv 𝟭 = 𝟭"
  by (metis Units_one_closed Units_r_inv l_one monoid.Units_inv_closed monoid_axioms)

lemma (in monoid) Units_inv_Units [intro, simp]:
  "x ∈ Units G ==> inv x ∈ Units G"
proof -
  assume x: "x ∈ Units G"
  show "inv x ∈ Units G"
    by (auto simp add: Units_def
      intro: Units_l_inv Units_r_inv x Units_closed [OF x])
qed

lemma (in monoid) Units_l_cancel [simp]:
  "[| x ∈ Units G; y ∈ carrier G; z ∈ carrier G |] ==>
   (x ⊗ y = x ⊗ z) = (y = z)"
proof
  assume eq: "x ⊗ y = x ⊗ z"
    and G: "x ∈ Units G"  "y ∈ carrier G"  "z ∈ carrier G"
  then have "(inv x ⊗ x) ⊗ y = (inv x ⊗ x) ⊗ z"
    by (simp add: m_assoc Units_closed del: Units_l_inv)
  with G show "y = z" by simp
next
  assume eq: "y = z"
    and G: "x ∈ Units G"  "y ∈ carrier G"  "z ∈ carrier G"
  then show "x ⊗ y = x ⊗ z" by simp
qed

lemma (in monoid) Units_inv_inv [simp]:
  "x ∈ Units G ==> inv (inv x) = x"
proof -
  assume x: "x ∈ Units G"
  then have "inv x ⊗ inv (inv x) = inv x ⊗ x" by simp
  with x show ?thesis by (simp add: Units_closed del: Units_l_inv Units_r_inv)
qed

lemma (in monoid) inv_inj_on_Units:
  "inj_on (m_inv G) (Units G)"
proof (rule inj_onI)
  fix x y
  assume G: "x ∈ Units G"  "y ∈ Units G" and eq: "inv x = inv y"
  then have "inv (inv x) = inv (inv y)" by simp
  with G show "x = y" by simp
qed

lemma (in monoid) Units_inv_comm:
  assumes inv: "x ⊗ y = 𝟭"
    and G: "x ∈ Units G"  "y ∈ Units G"
  shows "y ⊗ x = 𝟭"
proof -
  from G have "x ⊗ y ⊗ x = x ⊗ 𝟭" by (auto simp add: inv Units_closed)
  with G show ?thesis by (simp del: r_one add: m_assoc Units_closed)
qed

lemma (in monoid) carrier_not_empty: "carrier G ≠ {}"
by auto

(* Jacobson defines submonoid here. *)
(* Jacobson defines the order of a monoid here. *)


subsection ‹Groups›

text ‹
  A group is a monoid all of whose elements are invertible.
›

locale group = monoid +
  assumes Units: "carrier G <= Units G"

lemma (in group) is_group [iff]: "group G" by (rule group_axioms)

lemma (in group) is_monoid [iff]: "monoid G"
  by (rule monoid_axioms)

theorem groupI:
  fixes G (structure)
  assumes m_closed [simp]:
      "!!x y. [| x ∈ carrier G; y ∈ carrier G |] ==> x ⊗ y ∈ carrier G"
    and one_closed [simp]: "𝟭 ∈ carrier G"
    and m_assoc:
      "!!x y z. [| x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==>
      (x ⊗ y) ⊗ z = x ⊗ (y ⊗ z)"
    and l_one [simp]: "!!x. x ∈ carrier G ==> 𝟭 ⊗ x = x"
    and l_inv_ex: "!!x. x ∈ carrier G ==> ∃y ∈ carrier G. y ⊗ x = 𝟭"
  shows "group G"
proof -
  have l_cancel [simp]:
    "!!x y z. [| x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==>
    (x ⊗ y = x ⊗ z) = (y = z)"
  proof
    fix x y z
    assume eq: "x ⊗ y = x ⊗ z"
      and G: "x ∈ carrier G"  "y ∈ carrier G"  "z ∈ carrier G"
    with l_inv_ex obtain x_inv where xG: "x_inv ∈ carrier G"
      and l_inv: "x_inv ⊗ x = 𝟭" by fast
    from G eq xG have "(x_inv ⊗ x) ⊗ y = (x_inv ⊗ x) ⊗ z"
      by (simp add: m_assoc)
    with G show "y = z" by (simp add: l_inv)
  next
    fix x y z
    assume eq: "y = z"
      and G: "x ∈ carrier G"  "y ∈ carrier G"  "z ∈ carrier G"
    then show "x ⊗ y = x ⊗ z" by simp
  qed
  have r_one:
    "!!x. x ∈ carrier G ==> x ⊗ 𝟭 = x"
  proof -
    fix x
    assume x: "x ∈ carrier G"
    with l_inv_ex obtain x_inv where xG: "x_inv ∈ carrier G"
      and l_inv: "x_inv ⊗ x = 𝟭" by fast
    from x xG have "x_inv ⊗ (x ⊗ 𝟭) = x_inv ⊗ x"
      by (simp add: m_assoc [symmetric] l_inv)
    with x xG show "x ⊗ 𝟭 = x" by simp
  qed
  have inv_ex:
    "⋀x. x ∈ carrier G ⟹ ∃y ∈ carrier G. y ⊗ x = 𝟭 ∧ x ⊗ y = 𝟭"
  proof -
    fix x
    assume x: "x ∈ carrier G"
    with l_inv_ex obtain y where y: "y ∈ carrier G"
      and l_inv: "y ⊗ x = 𝟭" by fast
    from x y have "y ⊗ (x ⊗ y) = y ⊗ 𝟭"
      by (simp add: m_assoc [symmetric] l_inv r_one)
    with x y have r_inv: "x ⊗ y = 𝟭"
      by simp
    from x y show "∃y ∈ carrier G. y ⊗ x = 𝟭 ∧ x ⊗ y = 𝟭"
      by (fast intro: l_inv r_inv)
  qed
  then have carrier_subset_Units: "carrier G ⊆ Units G"
    by (unfold Units_def) fast
  show ?thesis
    by standard (auto simp: r_one m_assoc carrier_subset_Units)
qed

lemma (in monoid) group_l_invI:
  assumes l_inv_ex:
    "!!x. x ∈ carrier G ==> ∃y ∈ carrier G. y ⊗ x = 𝟭"
  shows "group G"
  by (rule groupI) (auto intro: m_assoc l_inv_ex)

lemma (in group) Units_eq [simp]:
  "Units G = carrier G"
proof
  show "Units G ⊆ carrier G" by fast
next
  show "carrier G ⊆ Units G" by (rule Units)
qed

lemma (in group) inv_closed [intro, simp]:
  "x ∈ carrier G ==> inv x ∈ carrier G"
  using Units_inv_closed by simp

lemma (in group) l_inv_ex [simp]:
  "x ∈ carrier G ==> ∃y ∈ carrier G. y ⊗ x = 𝟭"
  using Units_l_inv_ex by simp

lemma (in group) r_inv_ex [simp]:
  "x ∈ carrier G ==> ∃y ∈ carrier G. x ⊗ y = 𝟭"
  using Units_r_inv_ex by simp

lemma (in group) l_inv [simp]:
  "x ∈ carrier G ==> inv x ⊗ x = 𝟭"
  by simp


subsection ‹Cancellation Laws and Basic Properties›

lemma (in group) inv_eq_1_iff [simp]:
  assumes "x ∈ carrier G" shows "inv⇘G⇙ x = 𝟭⇘G⇙ ⟷ x = 𝟭⇘G⇙"
proof -
  have "x = 𝟭" if "inv x = 𝟭"
  proof -
    have "inv x ⊗ x = 𝟭"
      using assms l_inv by blast
    then show "x = 𝟭"
      using that assms by simp
  qed
  then show ?thesis
    by auto
qed

lemma (in group) r_inv [simp]:
  "x ∈ carrier G ==> x ⊗ inv x = 𝟭"
  by simp

lemma (in group) right_cancel [simp]:
  "[| x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==>
   (y ⊗ x = z ⊗ x) = (y = z)"
  by (metis inv_closed m_assoc r_inv r_one)

lemma (in group) inv_inv [simp]:
  "x ∈ carrier G ==> inv (inv x) = x"
  using Units_inv_inv by simp

lemma (in group) inv_inj:
  "inj_on (m_inv G) (carrier G)"
  using inv_inj_on_Units by simp

lemma (in group) inv_mult_group:
  "[| x ∈ carrier G; y ∈ carrier G |] ==> inv (x ⊗ y) = inv y ⊗ inv x"
proof -
  assume G: "x ∈ carrier G"  "y ∈ carrier G"
  then have "inv (x ⊗ y) ⊗ (x ⊗ y) = (inv y ⊗ inv x) ⊗ (x ⊗ y)"
    by (simp add: m_assoc) (simp add: m_assoc [symmetric])
  with G show ?thesis by (simp del: l_inv Units_l_inv)
qed

lemma (in group) inv_comm:
  "[| x ⊗ y = 𝟭; x ∈ carrier G; y ∈ carrier G |] ==> y ⊗ x = 𝟭"
  by (rule Units_inv_comm) auto

lemma (in group) inv_equality:
     "[|y ⊗ x = 𝟭; x ∈ carrier G; y ∈ carrier G|] ==> inv x = y"
  using inv_unique r_inv by blast

lemma (in group) inv_solve_left:
  "⟦ a ∈ carrier G; b ∈ carrier G; c ∈ carrier G ⟧ ⟹ a = inv b ⊗ c ⟷ c = b ⊗ a"
  by (metis inv_equality l_inv_ex l_one m_assoc r_inv)

lemma (in group) inv_solve_left':
  "⟦ a ∈ carrier G; b ∈ carrier G; c ∈ carrier G ⟧ ⟹ inv b ⊗ c = a ⟷ c = b ⊗ a"
  by (metis inv_equality l_inv_ex l_one m_assoc r_inv)

lemma (in group) inv_solve_right:
  "⟦ a ∈ carrier G; b ∈ carrier G; c ∈ carrier G ⟧ ⟹ a = b ⊗ inv c ⟷ b = a ⊗ c"
  by (metis inv_equality l_inv_ex l_one m_assoc r_inv)

lemma (in group) inv_solve_right':
  "⟦a ∈ carrier G; b ∈ carrier G; c ∈ carrier G⟧ ⟹ b ⊗ inv c = a ⟷ b = a ⊗ c"
  by (auto simp: m_assoc)
  

subsection ‹Power›

consts
  pow :: "[('a, 'm) monoid_scheme, 'a, 'b::semiring_1] => 'a"  (infixr ‹[^]ı› 75)

overloading nat_pow == "pow :: [_, 'a, nat] => 'a"
begin
  definition "nat_pow G a n = rec_nat 𝟭⇘G⇙ (%u b. b ⊗⇘G⇙ a) n"
end

lemma (in monoid) nat_pow_closed [intro, simp]:
  "x ∈ carrier G ==> x [^] (n::nat) ∈ carrier G"
  by (induct n) (simp_all add: nat_pow_def)

lemma (in monoid) nat_pow_0 [simp]:
  "x [^] (0::nat) = 𝟭"
  by (simp add: nat_pow_def)

lemma (in monoid) nat_pow_Suc [simp]:
  "x [^] (Suc n) = x [^] n ⊗ x"
  by (simp add: nat_pow_def)

lemma (in monoid) nat_pow_one [simp]:
  "𝟭 [^] (n::nat) = 𝟭"
  by (induct n) simp_all

lemma (in monoid) nat_pow_mult:
  "x ∈ carrier G ==> x [^] (n::nat) ⊗ x [^] m = x [^] (n + m)"
  by (induct m) (simp_all add: m_assoc [THEN sym])

lemma (in monoid) nat_pow_comm:
  "x ∈ carrier G ⟹ (x [^] (n::nat)) ⊗ (x [^] (m :: nat)) = (x [^] m) ⊗ (x [^] n)"
  using nat_pow_mult[of x n m] nat_pow_mult[of x m n] by (simp add: add.commute)

lemma (in monoid) nat_pow_Suc2:
  "x ∈ carrier G ⟹ x [^] (Suc n) = x ⊗ (x [^] n)"
  using nat_pow_mult[of x 1 n] Suc_eq_plus1[of n]
  by (metis One_nat_def Suc_eq_plus1_left l_one nat.rec(1) nat_pow_Suc nat_pow_def)

lemma (in monoid) nat_pow_pow:
  "x ∈ carrier G ==> (x [^] n) [^] m = x [^] (n * m::nat)"
  by (induct m) (simp, simp add: nat_pow_mult add.commute)

lemma (in monoid) nat_pow_consistent:
  "x [^] (n :: nat) = x [^]⇘(G ⦇ carrier := H ⦈)⇙ n"
  unfolding nat_pow_def by simp

lemma nat_pow_0 [simp]: "x [^]⇘G⇙ (0::nat) = 𝟭⇘G⇙"
  by (simp add: nat_pow_def)

lemma nat_pow_Suc [simp]: "x [^]⇘G⇙ (Suc n) = (x [^]⇘G⇙ n)⊗⇘G⇙ x"
  by (simp add: nat_pow_def)

lemma (in group) nat_pow_inv:
  assumes "x ∈ carrier G" shows "(inv x) [^] (i :: nat) = inv (x [^] i)"
proof (induction i)
  case 0 thus ?case by simp
next
  case (Suc i)
  have "(inv x) [^] Suc i = ((inv x) [^] i) ⊗ inv x"
    by simp
  also have " ... = (inv (x [^] i)) ⊗ inv x"
    by (simp add: Suc.IH Suc.prems)
  also have " ... = inv (x ⊗ (x [^] i))"
    by (simp add: assms inv_mult_group)
  also have " ... = inv (x [^] (Suc i))"
    using assms nat_pow_Suc2 by auto
  finally show ?case .
qed

overloading int_pow == "pow :: [_, 'a, int] => 'a"
begin
  definition "int_pow G a z =
   (let p = rec_nat 𝟭⇘G⇙ (%u b. b ⊗⇘G⇙ a)
    in if z < 0 then inv⇘G⇙ (p (nat (-z))) else p (nat z))"
end

lemma int_pow_int: "x [^]⇘G⇙ (int n) = x [^]⇘G⇙ n"
  by(simp add: int_pow_def nat_pow_def)

lemma pow_nat:
  assumes "i≥0"
  shows "x [^]⇘G⇙ nat i = x [^]⇘G⇙ i"
proof (cases i rule: int_cases)
  case (nonneg n)
  then show ?thesis
    by (simp add: int_pow_int)
next
  case (neg n)
  then show ?thesis
    using assms by linarith
qed

lemma int_pow_0 [simp]: "x [^]⇘G⇙ (0::int) = 𝟭⇘G⇙"
  by (simp add: int_pow_def)

lemma int_pow_def2: "a [^]⇘G⇙ z =
   (if z < 0 then inv⇘G⇙ (a [^]⇘G⇙ (nat (-z))) else a [^]⇘G⇙ (nat z))"
  by (simp add: int_pow_def nat_pow_def)

lemma (in group) int_pow_one [simp]:
  "𝟭 [^] (z::int) = 𝟭"
  by (simp add: int_pow_def2)

lemma (in group) int_pow_closed [intro, simp]:
  "x ∈ carrier G ==> x [^] (i::int) ∈ carrier G"
  by (simp add: int_pow_def2)

lemma (in group) int_pow_1 [simp]:
  "x ∈ carrier G ⟹ x [^] (1::int) = x"
  by (simp add: int_pow_def2)

lemma (in group) int_pow_neg:
  "x ∈ carrier G ⟹ x [^] (-i::int) = inv (x [^] i)"
  by (simp add: int_pow_def2)

lemma (in group) int_pow_neg_int: "x ∈ carrier G ⟹ x [^] -(int n) = inv (x [^] n)"
  by (simp add: int_pow_neg int_pow_int)

lemma (in group) int_pow_mult:
  assumes "x ∈ carrier G" shows "x [^] (i + j::int) = x [^] i ⊗ x [^] j"
proof -
  have [simp]: "-i - j = -j - i" by simp
  show ?thesis
    by (auto simp: assms int_pow_def2 inv_solve_left inv_solve_right nat_add_distrib [symmetric] nat_pow_mult)
qed

lemma (in group) int_pow_inv:
  "x ∈ carrier G ⟹ (inv x) [^] (i :: int) = inv (x [^] i)"
  by (metis int_pow_def2 nat_pow_inv)

lemma (in group) int_pow_pow:
  assumes "x ∈ carrier G"
  shows "(x [^] (n :: int)) [^] (m :: int) = x [^] (n * m :: int)"
proof (cases)
  assume n_ge: "n ≥ 0" thus ?thesis
  proof (cases)
    assume m_ge: "m ≥ 0" thus ?thesis
      using n_ge nat_pow_pow[OF assms, of "nat n" "nat m"] int_pow_def2 [where G=G]
      by (simp add: mult_less_0_iff nat_mult_distrib)
  next
    assume m_lt: "¬ m ≥ 0" 
    with n_ge show ?thesis
      apply (simp add: int_pow_def2 mult_less_0_iff)
      by (metis assms mult_minus_right n_ge nat_mult_distrib nat_pow_pow)
  qed
next
  assume n_lt: "¬ n ≥ 0" thus ?thesis
  proof (cases)
    assume m_ge: "m ≥ 0" 
    have "inv x [^] (nat m * nat (- n)) = inv x [^] nat (- (m * n))"
      by (metis (full_types) m_ge mult_minus_right nat_mult_distrib)
    with m_ge n_lt show ?thesis
      by (simp add: int_pow_def2 mult_less_0_iff assms mult.commute nat_pow_inv nat_pow_pow)
  next
    assume m_lt: "¬ m ≥ 0" thus ?thesis
      using n_lt by (auto simp: int_pow_def2 mult_less_0_iff assms nat_mult_distrib_neg nat_pow_inv nat_pow_pow)
  qed
qed

lemma (in group) int_pow_diff:
  "x ∈ carrier G ⟹ x [^] (n - m :: int) = x [^] n ⊗ inv (x [^] m)"
  by(simp only: diff_conv_add_uminus int_pow_mult int_pow_neg)

lemma (in group) inj_on_multc: "c ∈ carrier G ⟹ inj_on (λx. x ⊗ c) (carrier G)"
  by(simp add: inj_on_def)

lemma (in group) inj_on_cmult: "c ∈ carrier G ⟹ inj_on (λx. c ⊗ x) (carrier G)"
  by(simp add: inj_on_def)


lemma (in monoid) group_commutes_pow:
  fixes n::nat
  shows "⟦x ⊗ y = y ⊗ x; x ∈ carrier G; y ∈ carrier G⟧ ⟹ x [^] n ⊗ y = y ⊗ x [^] n"
  apply (induction n, auto)
  by (metis m_assoc nat_pow_closed)

lemma (in monoid) pow_mult_distrib:
  assumes eq: "x ⊗ y = y ⊗ x" and xy: "x ∈ carrier G" "y ∈ carrier G"
  shows "(x ⊗ y) [^] (n::nat) = x [^] n ⊗ y [^] n"
proof (induct n)
  case (Suc n)
  have "x ⊗ (y [^] n ⊗ y) = y [^] n ⊗ x ⊗ y"
    by (simp add: eq group_commutes_pow m_assoc xy)
  then show ?case
    using assms Suc.hyps m_assoc by auto
qed auto

lemma (in group) int_pow_mult_distrib:
  assumes eq: "x ⊗ y = y ⊗ x" and xy: "x ∈ carrier G" "y ∈ carrier G"
  shows "(x ⊗ y) [^] (i::int) = x [^] i ⊗ y [^] i"
proof (cases i rule: int_cases)
  case (nonneg n)
  then show ?thesis
    by (metis eq int_pow_int pow_mult_distrib xy)
next
  case (neg n)
  then show ?thesis
    unfolding neg
    apply (simp add: xy int_pow_neg_int del: of_nat_Suc)
    by (metis eq inv_mult_group local.nat_pow_Suc nat_pow_closed pow_mult_distrib xy)
qed

lemma (in group) pow_eq_div2:
  fixes m n :: nat
  assumes x_car: "x ∈ carrier G"
  assumes pow_eq: "x [^] m = x [^] n"
  shows "x [^] (m - n) = 𝟭"
proof (cases "m < n")
  case False
  have "𝟭 ⊗ x [^] m = x [^] m" by (simp add: x_car)
  also have "… = x [^] (m - n) ⊗ x [^] n"
    using False by (simp add: nat_pow_mult x_car)
  also have "… = x [^] (m - n) ⊗ x [^] m"
    by (simp add: pow_eq)
  finally show ?thesis
    by (metis nat_pow_closed one_closed right_cancel x_car)
qed simp

subsection ‹Submonoids›

locale submonoid = ✐‹contributor ‹Martin Baillon››
  fixes H and G (structure)
  assumes subset: "H ⊆ carrier G"
    and m_closed [intro, simp]: "⟦x ∈ H; y ∈ H⟧ ⟹ x ⊗ y ∈ H"
    and one_closed [simp]: "𝟭 ∈ H"

lemma (in submonoid) is_submonoid: ✐‹contributor ‹Martin Baillon››
  "submonoid H G" by (rule submonoid_axioms)

lemma (in submonoid) mem_carrier [simp]: ✐‹contributor ‹Martin Baillon››
  "x ∈ H ⟹ x ∈ carrier G"
  using subset by blast

lemma (in submonoid) submonoid_is_monoid [intro]: ✐‹contributor ‹Martin Baillon››
  assumes "monoid G"
  shows "monoid (G⦇carrier := H⦈)"
proof -
  interpret monoid G by fact
  show ?thesis
    by (simp add: monoid_def m_assoc)
qed

lemma submonoid_nonempty: ✐‹contributor ‹Martin Baillon››
  "~ submonoid {} G"
  by (blast dest: submonoid.one_closed)

lemma (in submonoid) finite_monoid_imp_card_positive: ✐‹contributor ‹Martin Baillon››
  "finite (carrier G) ==> 0 < card H"
proof (rule classical)
  assume "finite (carrier G)" and a: "~ 0 < card H"
  then have "finite H" by (blast intro: finite_subset [OF subset])
  with is_submonoid a have "submonoid {} G" by simp
  with submonoid_nonempty show ?thesis by contradiction
qed


lemma (in monoid) monoid_incl_imp_submonoid : ✐‹contributor ‹Martin Baillon››
  assumes "H ⊆ carrier G"
and "monoid (G⦇carrier := H⦈)"
shows "submonoid H G"
proof (intro submonoid.intro[OF assms(1)])
  have ab_eq : "⋀ a b. a ∈ H ⟹ b ∈ H ⟹ a ⊗⇘G⦇carrier := H⦈⇙ b = a ⊗ b" using assms by simp
  have "⋀a b. a ∈ H ⟹ b ∈ H ⟹ a ⊗ b ∈ carrier (G⦇carrier := H⦈) "
    using assms ab_eq unfolding group_def using monoid.m_closed by fastforce
  thus "⋀a b. a ∈ H ⟹ b ∈ H ⟹ a ⊗ b ∈ H" by simp
  show "𝟭 ∈ H " using monoid.one_closed[OF assms(2)] assms by simp
qed

lemma (in monoid) inv_unique': ✐‹contributor ‹Martin Baillon››
  assumes "x ∈ carrier G" "y ∈ carrier G"
  shows "⟦ x ⊗ y = 𝟭; y ⊗ x = 𝟭 ⟧ ⟹ y = inv x"
proof -
  assume "x ⊗ y = 𝟭" and l_inv: "y ⊗ x = 𝟭"
  hence unit: "x ∈ Units G"
    using assms unfolding Units_def by auto
  show "y = inv x"
    using inv_unique[OF l_inv Units_r_inv[OF unit] assms Units_inv_closed[OF unit]] .
qed

lemma (in monoid) m_inv_monoid_consistent: ✐‹contributor ‹Paulo Emílio de Vilhena››
  assumes "x ∈ Units (G ⦇ carrier := H ⦈)" and "submonoid H G"
  shows "inv⇘(G ⦇ carrier := H ⦈)⇙ x = inv x"
proof -
  have monoid: "monoid (G ⦇ carrier := H ⦈)"
    using submonoid.submonoid_is_monoid[OF assms(2) monoid_axioms] .
  obtain y where y: "y ∈ H" "x ⊗ y = 𝟭" "y ⊗ x = 𝟭"
    using assms(1) unfolding Units_def by auto
  have x: "x ∈ H" and in_carrier: "x ∈ carrier G" "y ∈ carrier G"
    using y(1) submonoid.subset[OF assms(2)] assms(1) unfolding Units_def by auto
  show ?thesis
    using monoid.inv_unique'[OF monoid, of x y] x y
    using inv_unique'[OF in_carrier y(2-3)] by auto
qed

subsection ‹Subgroups›

locale subgroup =
  fixes H and G (structure)
  assumes subset: "H ⊆ carrier G"
    and m_closed [intro, simp]: "⟦x ∈ H; y ∈ H⟧ ⟹ x ⊗ y ∈ H"
    and one_closed [simp]: "𝟭 ∈ H"
    and m_inv_closed [intro,simp]: "x ∈ H ⟹ inv x ∈ H"

lemma (in subgroup) is_subgroup:
  "subgroup H G" by (rule subgroup_axioms)

declare (in subgroup) group.intro [intro]

lemma (in subgroup) mem_carrier [simp]:
  "x ∈ H ⟹ x ∈ carrier G"
  using subset by blast

lemma (in subgroup) subgroup_is_group [intro]:
  assumes "group G"
  shows "group (G⦇carrier := H⦈)"
proof -
  interpret group G by fact
  have "Group.monoid (G⦇carrier := H⦈)"
    by (simp add: monoid_axioms submonoid.intro submonoid.submonoid_is_monoid subset)
  then show ?thesis
    by (rule monoid.group_l_invI) (auto intro: l_inv mem_carrier)
qed

lemma (in group) triv_subgroup: "subgroup {𝟭} G"
  by (auto simp: subgroup_def)

lemma subgroup_is_submonoid:
  assumes "subgroup H G" shows "submonoid H G"
  using assms by (auto intro: submonoid.intro simp add: subgroup_def)

lemma (in group) subgroup_Units:
  assumes "subgroup H G" shows "H ⊆ Units (G ⦇ carrier := H ⦈)"
  using group.Units[OF subgroup.subgroup_is_group[OF assms group_axioms]] by simp

lemma (in group) m_inv_consistent [simp]:
  assumes "subgroup H G" "x ∈ H"
  shows "inv⇘(G ⦇ carrier := H ⦈)⇙ x = inv x"
  using assms m_inv_monoid_consistent[OF _ subgroup_is_submonoid] subgroup_Units[of H] by auto

lemma (in group) int_pow_consistent: ✐‹contributor ‹Paulo Emílio de Vilhena››
  assumes "subgroup H G" "x ∈ H"
  shows "x [^] (n :: int) = x [^]⇘(G ⦇ carrier := H ⦈)⇙ n"
proof (cases)
  assume ge: "n ≥ 0"
  hence "x [^] n = x [^] (nat n)"
    using int_pow_def2 [of G] by auto
  also have " ... = x [^]⇘(G ⦇ carrier := H ⦈)⇙ (nat n)"
    using nat_pow_consistent by simp
  also have " ... = x [^]⇘(G ⦇ carrier := H ⦈)⇙ n"
    by (metis ge int_nat_eq int_pow_int)
  finally show ?thesis .
next
  assume "¬ n ≥ 0" hence lt: "n < 0" by simp
  hence "x [^] n = inv (x [^] (nat (- n)))"
    using int_pow_def2 [of G] by auto
  also have " ... = (inv x) [^] (nat (- n))"
    by (metis assms nat_pow_inv subgroup.mem_carrier)
  also have " ... = (inv⇘(G ⦇ carrier := H ⦈)⇙ x) [^]⇘(G ⦇ carrier := H ⦈)⇙ (nat (- n))"
    using m_inv_consistent[OF assms] nat_pow_consistent by auto
  also have " ... = inv⇘(G ⦇ carrier := H ⦈)⇙ (x [^]⇘(G ⦇ carrier := H ⦈)⇙ (nat (- n)))"
    using group.nat_pow_inv[OF subgroup.subgroup_is_group[OF assms(1) is_group]] assms(2) by auto
  also have " ... = x [^]⇘(G ⦇ carrier := H ⦈)⇙ n"
    by (simp add: int_pow_def2 lt)
  finally show ?thesis .
qed

text ‹
  Since \<^term>‹H› is nonempty, it contains some element \<^term>‹x›.  Since
  it is closed under inverse, it contains ‹inv x›.  Since
  it is closed under product, it contains ‹x ⊗ inv x = 𝟭›.
›

lemma (in group) one_in_subset:
  "⟦H ⊆ carrier G; H ≠ {}; ∀a ∈ H. inv a ∈ H; ∀a∈H. ∀b∈H. a ⊗ b ∈ H⟧
   ⟹ 𝟭 ∈ H"
by force

text ‹A characterization of subgroups: closed, non-empty subset.›

lemma (in group) subgroupI:
  assumes subset: "H ⊆ carrier G" and non_empty: "H ≠ {}"
    and inv: "!!a. a ∈ H ⟹ inv a ∈ H"
    and mult: "!!a b. ⟦a ∈ H; b ∈ H⟧ ⟹ a ⊗ b ∈ H"
  shows "subgroup H G"
proof (simp add: subgroup_def assms)
  show "𝟭 ∈ H" by (rule one_in_subset) (auto simp only: assms)
qed

lemma (in group) subgroupE:
  assumes "subgroup H G"
  shows "H ⊆ carrier G"
    and "H ≠ {}"
    and "⋀a. a ∈ H ⟹ inv a ∈ H"
    and "⋀a b. ⟦ a ∈ H; b ∈ H ⟧ ⟹ a ⊗ b ∈ H"
  using assms unfolding subgroup_def[of H G] by auto

declare monoid.one_closed [iff] group.inv_closed [simp]
  monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp]

lemma subgroup_nonempty:
  "¬ subgroup {} G"
  by (blast dest: subgroup.one_closed)

lemma (in subgroup) finite_imp_card_positive: "finite (carrier G) ⟹ 0 < card H"
  using subset one_closed card_gt_0_iff finite_subset by blast

lemma (in subgroup) subgroup_is_submonoid : ✐‹contributor ‹Martin Baillon››
  "submonoid H G"
  by (simp add: submonoid.intro subset)

lemma (in group) submonoid_subgroupI : ✐‹contributor ‹Martin Baillon››
  assumes "submonoid H G"
    and "⋀a. a ∈ H ⟹ inv a ∈ H"
  shows "subgroup H G"
  by (metis assms subgroup_def submonoid_def)

lemma (in group) group_incl_imp_subgroup: ✐‹contributor ‹Martin Baillon››
  assumes "H ⊆ carrier G"
    and "group (G⦇carrier := H⦈)"
  shows "subgroup H G"
proof (intro submonoid_subgroupI[OF monoid_incl_imp_submonoid[OF assms(1)]])
  show "monoid (G⦇carrier := H⦈)" using group_def assms by blast
  have ab_eq : "⋀ a b. a ∈ H ⟹ b ∈ H ⟹ a ⊗⇘G⦇carrier := H⦈⇙ b = a ⊗ b" using assms by simp
  fix a  assume aH : "a ∈ H"
  have " inv⇘G⦇carrier := H⦈⇙ a ∈ carrier G"
    using assms aH group.inv_closed[OF assms(2)] by auto
  moreover have "𝟭⇘G⦇carrier := H⦈⇙ = 𝟭" using assms monoid.one_closed ab_eq one_def by simp
  hence "a ⊗⇘G⦇carrier := H⦈⇙ inv⇘G⦇carrier := H⦈⇙ a= 𝟭"
    using assms ab_eq aH  group.r_inv[OF assms(2)] by simp
  hence "a ⊗ inv⇘G⦇carrier := H⦈⇙ a= 𝟭"
    using aH assms group.inv_closed[OF assms(2)] ab_eq by simp
  ultimately have "inv⇘G⦇carrier := H⦈⇙ a = inv a"
    by (metis aH assms(1) contra_subsetD group.inv_inv is_group local.inv_equality)
  moreover have "inv⇘G⦇carrier := H⦈⇙ a ∈ H" 
    using aH group.inv_closed[OF assms(2)] by auto
  ultimately show "inv a ∈ H" by auto
qed


subsection ‹Direct Products›

definition
  DirProd :: "_ ⇒ _ ⇒ ('a × 'b) monoid" (infixr ‹××› 80) where
  "G ×× H =
    ⦇carrier = carrier G × carrier H,
     mult = (λ(g, h) (g', h'). (g ⊗⇘G⇙ g', h ⊗⇘H⇙ h')),
     one = (𝟭⇘G⇙, 𝟭⇘H⇙)⦈"

lemma DirProd_monoid:
  assumes "monoid G" and "monoid H"
  shows "monoid (G ×× H)"
proof -
  interpret G: monoid G by fact
  interpret H: monoid H by fact
  from assms
  show ?thesis by (unfold monoid_def DirProd_def, auto)
qed


text‹Does not use the previous result because it's easier just to use auto.›
lemma DirProd_group:
  assumes "group G" and "group H"
  shows "group (G ×× H)"
proof -
  interpret G: group G by fact
  interpret H: group H by fact
  show ?thesis by (rule groupI)
     (auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv
           simp add: DirProd_def)
qed

lemma carrier_DirProd [simp]: "carrier (G ×× H) = carrier G × carrier H"
  by (simp add: DirProd_def)

lemma one_DirProd [simp]: "𝟭⇘G ×× H⇙ = (𝟭⇘G⇙, 𝟭⇘H⇙)"
  by (simp add: DirProd_def)

lemma mult_DirProd [simp]: "(g, h) ⊗⇘(G ×× H)⇙ (g', h') = (g ⊗⇘G⇙ g', h ⊗⇘H⇙ h')"
  by (simp add: DirProd_def)

lemma mult_DirProd': "x ⊗⇘(G ×× H)⇙ y = (fst x ⊗⇘G⇙ fst y, snd x ⊗⇘H⇙ snd y)"
  by (subst mult_DirProd [symmetric]) simp

lemma DirProd_assoc: "(G ×× H ×× I) = (G ×× (H ×× I))"
  by auto

lemma inv_DirProd [simp]:
  assumes "group G" and "group H"
  assumes g: "g ∈ carrier G"
      and h: "h ∈ carrier H"
  shows "m_inv (G ×× H) (g, h) = (inv⇘G⇙ g, inv⇘H⇙ h)"
proof -
  interpret G: group G by fact
  interpret H: group H by fact
  interpret Prod: group "G ×× H"
    by (auto intro: DirProd_group group.intro group.axioms assms)
  show ?thesis by (simp add: Prod.inv_equality g h)
qed

lemma DirProd_subgroups :
  assumes "group G"
    and "subgroup H G"
    and "group K"
    and "subgroup I K"
  shows "subgroup (H × I) (G ×× K)"
proof (intro group.group_incl_imp_subgroup[OF DirProd_group[OF assms(1)assms(3)]])
  have "H ⊆ carrier G" "I ⊆ carrier K" using subgroup.subset assms by blast+
  thus "(H × I) ⊆ carrier (G ×× K)" unfolding DirProd_def by auto
  have "Group.group ((G⦇carrier := H⦈) ×× (K⦇carrier := I⦈))"
    using DirProd_group[OF subgroup.subgroup_is_group[OF assms(2)assms(1)]
        subgroup.subgroup_is_group[OF assms(4)assms(3)]].
  moreover have "((G⦇carrier := H⦈) ×× (K⦇carrier := I⦈)) = ((G ×× K)⦇carrier := H × I⦈)"
    unfolding DirProd_def using assms by simp
  ultimately show "Group.group ((G ×× K)⦇carrier := H × I⦈)" by simp
qed

subsection ‹Homomorphisms (mono and epi) and Isomorphisms›

definition
  hom :: "_ => _ => ('a => 'b) set" where
  "hom G H =
    {h. h ∈ carrier G → carrier H ∧
      (∀x ∈ carrier G. ∀y ∈ carrier G. h (x ⊗⇘G⇙ y) = h x ⊗⇘H⇙ h y)}"

lemma homI:
  "⟦⋀x. x ∈ carrier G ⟹ h x ∈ carrier H;
    ⋀x y. ⟦x ∈ carrier G; y ∈ carrier G⟧ ⟹ h (x ⊗⇘G⇙ y) = h x ⊗⇘H⇙ h y⟧ ⟹ h ∈ hom G H"
  by (auto simp: hom_def)

lemma hom_carrier: "h ∈ hom G H ⟹ h ` carrier G ⊆ carrier H"
  by (auto simp: hom_def)

lemma hom_in_carrier: "⟦h ∈ hom G H; x ∈ carrier G⟧ ⟹ h x ∈ carrier H"
  by (auto simp: hom_def)

lemma hom_compose:
  "⟦ f ∈ hom G H; g ∈ hom H I ⟧ ⟹ g ∘ f ∈ hom G I"
  unfolding hom_def by (auto simp add: Pi_iff)

lemma (in group) hom_restrict:
  assumes "h ∈ hom G H" and "⋀g. g ∈ carrier G ⟹ h g = t g" shows "t ∈ hom G H"
  using assms unfolding hom_def by (auto simp add: Pi_iff)

lemma (in group) hom_compose:
  "[|h ∈ hom G H; i ∈ hom H I|] ==> compose (carrier G) i h ∈ hom G I"
by (fastforce simp add: hom_def compose_def)

lemma (in group) restrict_hom_iff [simp]:
  "(λx. if x ∈ carrier G then f x else g x) ∈ hom G H ⟷ f ∈ hom G H"
  by (simp add: hom_def Pi_iff)

definition iso :: "_ => _ => ('a => 'b) set"
  where "iso G H = {h. h ∈ hom G H ∧ bij_betw h (carrier G) (carrier H)}"

definition is_iso :: "_ ⇒ _ ⇒ bool" (infixr ‹≅› 60)
  where "G ≅ H = (iso G H  ≠ {})"

definition mon where "mon G H = {f ∈ hom G H. inj_on f (carrier G)}"

definition epi where "epi G H = {f ∈ hom G H. f ` (carrier G) = carrier H}"

lemma isoI:
  "⟦h ∈ hom G H; bij_betw h (carrier G) (carrier H)⟧ ⟹ h ∈ iso G H"
  by (auto simp: iso_def)

lemma is_isoI: "h ∈ iso G H ⟹ G ≅ H"
  using is_iso_def by auto

lemma epi_iff_subset:
   "f ∈ epi G G' ⟷ f ∈ hom G G' ∧ carrier G' ⊆ f ` carrier G"
  by (auto simp: epi_def hom_def)

lemma iso_iff_mon_epi: "f ∈ iso G H ⟷ f ∈ mon G H ∧ f ∈ epi G H"
  by (auto simp: iso_def mon_def epi_def bij_betw_def)

lemma iso_set_refl: "(λx. x) ∈ iso G G"
  by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def)

lemma id_iso: "id ∈ iso G G"
  by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def)

corollary iso_refl [simp]: "G ≅ G"
  using iso_set_refl unfolding is_iso_def by auto

lemma iso_iff:
   "h ∈ iso G H ⟷ h ∈ hom G H ∧ h ` (carrier G) = carrier H ∧ inj_on h (carrier G)"
  by (auto simp: iso_def hom_def bij_betw_def)

lemma iso_imp_homomorphism:
   "h ∈ iso G H ⟹ h ∈ hom G H"
  by (simp add: iso_iff)

lemma trivial_hom:
   "group H ⟹ (λx. one H) ∈ hom G H"
  by (auto simp: hom_def Group.group_def)

lemma (in group) hom_eq:
  assumes "f ∈ hom G H" "⋀x. x ∈ carrier G ⟹ f' x = f x"
  shows "f' ∈ hom G H"
  using assms by (auto simp: hom_def)

lemma (in group) iso_eq:
  assumes "f ∈ iso G H" "⋀x. x ∈ carrier G ⟹ f' x = f x"
  shows "f' ∈ iso G H"
  using assms  by (fastforce simp: iso_def inj_on_def bij_betw_def hom_eq image_iff)

lemma (in group) iso_set_sym:
  assumes "h ∈ iso G H"
  shows "inv_into (carrier G) h ∈ iso H G"
proof -
  have h: "h ∈ hom G H" "bij_betw h (carrier G) (carrier H)"
    using assms by (auto simp add: iso_def bij_betw_inv_into)
  then have HG: "bij_betw (inv_into (carrier G) h) (carrier H) (carrier G)"
    by (simp add: bij_betw_inv_into)
  have "inv_into (carrier G) h ∈ hom H G"
    unfolding hom_def
  proof safe
    show *: "⋀x. x ∈ carrier H ⟹ inv_into (carrier G) h x ∈ carrier G"
      by (meson HG bij_betwE)
    show "inv_into (carrier G) h (x ⊗⇘H⇙ y) = inv_into (carrier G) h x ⊗ inv_into (carrier G) h y"
      if "x ∈ carrier H" "y ∈ carrier H" for x y
    proof (rule inv_into_f_eq)
      show "inj_on h (carrier G)"
        using bij_betw_def h(2) by blast
      show "inv_into (carrier G) h x ⊗ inv_into (carrier G) h y ∈ carrier G"
        by (simp add: * that)
      show "h (inv_into (carrier G) h x ⊗ inv_into (carrier G) h y) = x ⊗⇘H⇙ y"
        using h bij_betw_inv_into_right [of h] unfolding hom_def by (simp add: "*" that)
    qed
  qed
  then show ?thesis
    by (simp add: Group.iso_def bij_betw_inv_into h)
qed

corollary (in group) iso_sym: "G ≅ H ⟹ H ≅ G"
  using iso_set_sym unfolding is_iso_def by auto

lemma iso_set_trans:
  "⟦h ∈ Group.iso G H; i ∈ Group.iso H I⟧ ⟹ i ∘ h ∈ Group.iso G I"
  by (force simp: iso_def hom_compose intro: bij_betw_trans)

corollary iso_trans [trans]: "⟦G ≅ H ; H ≅ I⟧ ⟹ G ≅ I"
  using iso_set_trans unfolding is_iso_def by blast

lemma iso_same_card: "G ≅ H ⟹ card (carrier G) = card (carrier H)"
  using bij_betw_same_card  unfolding is_iso_def iso_def by auto

lemma iso_finite: "G ≅ H ⟹ finite(carrier G) ⟷ finite(carrier H)"
  by (auto simp: is_iso_def iso_def bij_betw_finite)

lemma mon_compose:
   "⟦f ∈ mon G H; g ∈ mon H K⟧ ⟹ (g ∘ f) ∈ mon G K"
  by (auto simp: mon_def intro: hom_compose comp_inj_on inj_on_subset [OF _ hom_carrier])

lemma mon_compose_rev:
   "⟦f ∈ hom G H; g ∈ hom H K; (g ∘ f) ∈ mon G K⟧ ⟹ f ∈ mon G H"
  using inj_on_imageI2 by (auto simp: mon_def)

lemma epi_compose:
   "⟦f ∈ epi G H; g ∈ epi H K⟧ ⟹ (g ∘ f) ∈ epi G K"
  using hom_compose by (force simp: epi_def hom_compose simp flip: image_image)

lemma epi_compose_rev:
   "⟦f ∈ hom G H; g ∈ hom H K; (g ∘ f) ∈ epi G K⟧ ⟹ g ∈ epi H K"
  by (fastforce simp: epi_def hom_def Pi_iff image_def set_eq_iff)

lemma iso_compose_rev:
   "⟦f ∈ hom G H; g ∈ hom H K; (g ∘ f) ∈ iso G K⟧ ⟹ f ∈ mon G H ∧ g ∈ epi H K"
  unfolding iso_iff_mon_epi using mon_compose_rev epi_compose_rev by blast

lemma epi_iso_compose_rev:
  assumes "f ∈ epi G H" "g ∈ hom H K" "(g ∘ f) ∈ iso G K"
  shows "f ∈ iso G H ∧ g ∈ iso H K"
proof
  show "f ∈ iso G H"
    by (metis (no_types, lifting) assms epi_def iso_compose_rev iso_iff_mon_epi mem_Collect_eq)
  then have "f ∈ hom G H ∧ bij_betw f (carrier G) (carrier H)"
    using Group.iso_def ‹f ∈ Group.iso G H› by blast
  then have "bij_betw g (carrier H) (carrier K)"
    using Group.iso_def assms(3) bij_betw_comp_iff by blast
  then show "g ∈ iso H K"
    using Group.iso_def assms(2) by blast
qed

lemma mon_left_invertible:
   "⟦f ∈ hom G H; ⋀x. x ∈ carrier G ⟹ g(f x) = x⟧ ⟹ f ∈ mon G H"
  by (simp add: mon_def inj_on_def) metis

lemma epi_right_invertible:
   "⟦g ∈ hom H G; f ∈ carrier G → carrier H; ⋀x. x ∈ carrier G ⟹ g(f x) = x⟧ ⟹ g ∈ epi H G"
  by (force simp: Pi_iff epi_iff_subset image_subset_iff_funcset subset_iff)

lemma (in monoid) hom_imp_img_monoid: ✐‹contributor ‹Paulo Emílio de Vilhena››
  assumes "h ∈ hom G H"
  shows "monoid (H ⦇ carrier := h ` (carrier G), one := h 𝟭⇘G⇙ ⦈)" (is "monoid ?h_img")
proof (rule monoidI)
  show "𝟭⇘?h_img⇙ ∈ carrier ?h_img"
    by auto
next
  fix x y z assume "x ∈ carrier ?h_img" "y ∈ carrier ?h_img" "z ∈ carrier ?h_img"
  then obtain g1 g2 g3
    where g1: "g1 ∈ carrier G" "x = h g1"
      and g2: "g2 ∈ carrier G" "y = h g2"
      and g3: "g3 ∈ carrier G" "z = h g3"
    using image_iff[where ?f = h and ?A = "carrier G"] by auto
  have aux_lemma:
    "⋀a b. ⟦ a ∈ carrier G; b ∈ carrier G ⟧ ⟹ h a ⊗⇘(?h_img)⇙ h b = h (a ⊗ b)"
    using assms unfolding hom_def by auto

  show "x ⊗⇘(?h_img)⇙ 𝟭⇘(?h_img)⇙ = x"
    using aux_lemma[OF g1(1) one_closed] g1(2) r_one[OF g1(1)] by simp

  show "𝟭⇘(?h_img)⇙ ⊗⇘(?h_img)⇙ x = x"
    using aux_lemma[OF one_closed g1(1)] g1(2) l_one[OF g1(1)] by simp

  have "x ⊗⇘(?h_img)⇙ y = h (g1 ⊗ g2)"
    using aux_lemma g1 g2 by auto
  thus "x ⊗⇘(?h_img)⇙ y ∈ carrier ?h_img"
    using g1(1) g2(1) by simp

  have "(x ⊗⇘(?h_img)⇙ y) ⊗⇘(?h_img)⇙ z = h ((g1 ⊗ g2) ⊗ g3)"
    using aux_lemma g1 g2 g3 by auto
  also have " ... = h (g1 ⊗ (g2 ⊗ g3))"
    using m_assoc[OF g1(1) g2(1) g3(1)] by simp
  also have " ... = x ⊗⇘(?h_img)⇙ (y ⊗⇘(?h_img)⇙ z)"
    using aux_lemma g1 g2 g3 by auto
  finally show "(x ⊗⇘(?h_img)⇙ y) ⊗⇘(?h_img)⇙ z = x ⊗⇘(?h_img)⇙ (y ⊗⇘(?h_img)⇙ z)" .
qed

lemma (in group) hom_imp_img_group: ✐‹contributor ‹Paulo Emílio de Vilhena››
  assumes "h ∈ hom G H"
  shows "group (H ⦇ carrier := h ` (carrier G), one := h 𝟭⇘G⇙ ⦈)" (is "group ?h_img")
proof -
  interpret monoid ?h_img
    using hom_imp_img_monoid[OF assms] .

  show ?thesis
  proof (unfold_locales)
    show "carrier ?h_img ⊆ Units ?h_img"
    proof (auto simp add: Units_def)
      have aux_lemma:
        "⋀g1 g2. ⟦ g1 ∈ carrier G; g2 ∈ carrier G ⟧ ⟹ h g1 ⊗⇘H⇙ h g2 = h (g1 ⊗ g2)"
        using assms unfolding hom_def by auto

      fix g1 assume g1: "g1 ∈ carrier G"
      thus "∃g2 ∈ carrier G. (h g2) ⊗⇘H⇙ (h g1) = h 𝟭 ∧ (h g1) ⊗⇘H⇙ (h g2) = h 𝟭"
        using aux_lemma[OF g1 inv_closed[OF g1]]
              aux_lemma[OF inv_closed[OF g1] g1]
              inv_closed by auto
    qed
  qed
qed

lemma (in group) iso_imp_group: ✐‹contributor ‹Paulo Emílio de Vilhena››
  assumes "G ≅ H" and "monoid H"
  shows "group H"
proof -
  obtain φ where phi: "φ ∈ iso G H" "inv_into (carrier G) φ ∈ iso H G"
    using iso_set_sym assms unfolding is_iso_def by blast
  define ψ where psi_def: "ψ = inv_into (carrier G) φ"

  have surj: "φ ` (carrier G) = (carrier H)" "ψ ` (carrier H) = (carrier G)"
   and inj: "inj_on φ (carrier G)" "inj_on ψ (carrier H)"
   and phi_hom: "⋀g1 g2. ⟦ g1 ∈ carrier G; g2 ∈ carrier G ⟧ ⟹ φ (g1 ⊗ g2) = (φ g1) ⊗⇘H⇙ (φ g2)"
   and psi_hom: "⋀h1 h2. ⟦ h1 ∈ carrier H; h2 ∈ carrier H ⟧ ⟹ ψ (h1 ⊗⇘H⇙ h2) = (ψ h1) ⊗ (ψ h2)"
   using phi psi_def unfolding iso_def bij_betw_def hom_def by auto

  have phi_one: "φ 𝟭 = 𝟭⇘H⇙"
  proof -
    have "(φ 𝟭) ⊗⇘H⇙ 𝟭⇘H⇙ = (φ 𝟭) ⊗⇘H⇙ (φ 𝟭)"
      by (metis assms(2) image_eqI monoid.r_one one_closed phi_hom r_one surj(1))
    thus ?thesis
      by (metis (no_types, opaque_lifting) Units_eq Units_one_closed assms(2) f_inv_into_f imageI
          monoid.l_one monoid.one_closed phi_hom psi_def r_one surj)
  qed

  have "carrier H ⊆ Units H"
  proof
    fix h assume h: "h ∈ carrier H"
    let ?inv_h = "φ (inv (ψ h))"
    have "h ⊗⇘H⇙ ?inv_h = φ (ψ h) ⊗⇘H⇙ ?inv_h"
      by (simp add: f_inv_into_f h psi_def surj(1))
    also have " ... = φ ((ψ h) ⊗ inv (ψ h))"
      by (metis h imageI inv_closed phi_hom surj(2))
    also have " ... = φ 𝟭"
      by (simp add: h inv_into_into psi_def surj(1))
    finally have 1: "h ⊗⇘H⇙ ?inv_h = 𝟭⇘H⇙"
      using phi_one by simp

    have "?inv_h ⊗⇘H⇙ h = ?inv_h ⊗⇘H⇙ φ (ψ h)"
      by (simp add: f_inv_into_f h psi_def surj(1))
    also have " ... = φ (inv (ψ h) ⊗ (ψ h))"
      by (metis h imageI inv_closed phi_hom surj(2))
    also have " ... = φ 𝟭"
      by (simp add: h inv_into_into psi_def surj(1))
    finally have 2: "?inv_h ⊗⇘H⇙ h = 𝟭⇘H⇙"
      using phi_one by simp

    thus "h ∈ Units H" unfolding Units_def using 1 2 h surj by fastforce
  qed
  thus ?thesis unfolding group_def group_axioms_def using assms(2) by simp
qed

corollary (in group) iso_imp_img_group: ✐‹contributor ‹Paulo Emílio de Vilhena››
  assumes "h ∈ iso G H"
  shows "group (H ⦇ one := h 𝟭 ⦈)"
proof -
  let ?h_img = "H ⦇ carrier := h ` (carrier G), one := h 𝟭 ⦈"
  have "h ∈ iso G ?h_img"
    using assms unfolding iso_def hom_def bij_betw_def by auto
  hence "G ≅ ?h_img"
    unfolding is_iso_def by auto
  hence "group ?h_img"
    using iso_imp_group[of ?h_img] hom_imp_img_monoid[of h H] assms unfolding iso_def by simp
  moreover have "carrier H = carrier ?h_img"
    using assms unfolding iso_def bij_betw_def by simp
  hence "H ⦇ one := h 𝟭 ⦈ = ?h_img"
    by simp
  ultimately show ?thesis by simp
qed

subsubsection ‹HOL Light's concept of an isomorphism pair›

definition group_isomorphisms
  where
 "group_isomorphisms G H f g ≡
        f ∈ hom G H ∧ g ∈ hom H G ∧
        (∀x ∈ carrier G. g(f x) = x) ∧
        (∀y ∈ carrier H. f(g y) = y)"

lemma group_isomorphisms_sym: "group_isomorphisms G H f g ⟹ group_isomorphisms H G g f"
  by (auto simp: group_isomorphisms_def)

lemma group_isomorphisms_imp_iso: "group_isomorphisms G H f g ⟹ f ∈ iso G H"
by (auto simp: iso_def inj_on_def image_def group_isomorphisms_def hom_def bij_betw_def Pi_iff, metis+)

lemma (in group) iso_iff_group_isomorphisms:
  "f ∈ iso G H ⟷ (∃g. group_isomorphisms G H f g)"
proof safe
  show "∃g. group_isomorphisms G H f g" if "f ∈ Group.iso G H"
    unfolding group_isomorphisms_def
  proof (intro exI conjI)
    let ?g = "inv_into (carrier G) f"
    show "∀x∈carrier G. ?g (f x) = x"
      by (metis (no_types, lifting) Group.iso_def bij_betw_inv_into_left mem_Collect_eq that)
    show "∀y∈carrier H. f (?g y) = y"
      by (metis (no_types, lifting) Group.iso_def bij_betw_inv_into_right mem_Collect_eq that)
  qed (use Group.iso_def iso_set_sym that in ‹blast+›)
next
  fix g
  assume "group_isomorphisms G H f g"
  then show "f ∈ Group.iso G H"
    by (auto simp: iso_def group_isomorphisms_def hom_in_carrier intro: bij_betw_byWitness)
qed


subsubsection ‹Involving direct products›

lemma DirProd_commute_iso_set:
  shows "(λ(x,y). (y,x)) ∈ iso (G ×× H) (H ×× G)"
  by (auto simp add: iso_def hom_def inj_on_def bij_betw_def)

corollary DirProd_commute_iso :
"(G ×× H) ≅ (H ×× G)"
  using DirProd_commute_iso_set unfolding is_iso_def by blast

lemma DirProd_assoc_iso_set:
  shows "(λ(x,y,z). (x,(y,z))) ∈ iso (G ×× H ×× I) (G ×× (H ×× I))"
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def)

lemma (in group) DirProd_iso_set_trans:
  assumes "g ∈ iso G G2"
    and "h ∈ iso H I"
  shows "(λ(x,y). (g x, h y)) ∈ iso (G ×× H) (G2 ×× I)"
proof-
  have "(λ(x,y). (g x, h y)) ∈ hom (G ×× H) (G2 ×× I)"
    using assms unfolding iso_def hom_def by auto
  moreover have " inj_on (λ(x,y). (g x, h y)) (carrier (G ×× H))"
    using assms unfolding iso_def DirProd_def bij_betw_def inj_on_def by auto
  moreover have "(λ(x, y). (g x, h y)) ` carrier (G ×× H) = carrier (G2 ×× I)"
    using assms unfolding iso_def bij_betw_def image_def DirProd_def by fastforce
  ultimately show "(λ(x,y). (g x, h y)) ∈ iso (G ×× H) (G2 ×× I)"
    unfolding iso_def bij_betw_def by auto
qed

corollary (in group) DirProd_iso_trans :
  assumes "G ≅ G2" and "H ≅ I"
  shows "G ×× H ≅ G2 ×× I"
  using DirProd_iso_set_trans assms unfolding is_iso_def by blast

lemma hom_pairwise: "f ∈ hom G (DirProd H K) ⟷ (fst ∘ f) ∈ hom G H ∧ (snd ∘ f) ∈ hom G K"
  apply (auto simp: hom_def mult_DirProd' dest: Pi_mem)
   apply (metis Product_Type.mem_Times_iff comp_eq_dest_lhs funcset_mem)
  by (metis mult_DirProd prod.collapse)

lemma hom_paired:
   "(λx. (f x,g x)) ∈ hom G (DirProd H K) ⟷ f ∈ hom G H ∧ g ∈ hom G K"
  by (simp add: hom_pairwise o_def)

lemma hom_paired2:
  assumes "group G" "group H"
  shows "(λ(x,y). (f x,g y)) ∈ hom (DirProd G H) (DirProd G' H') ⟷ f ∈ hom G G' ∧ g ∈ hom H H'"
  using assms
  by (fastforce simp: hom_def Pi_def dest!: group.is_monoid)

lemma iso_paired2:
  assumes "group G" "group H"
  shows "(λ(x,y). (f x,g y)) ∈ iso (DirProd G H) (DirProd G' H') ⟷ f ∈ iso G G' ∧ g ∈ iso H H'"
  using assms
  by (fastforce simp add: iso_def inj_on_def bij_betw_def hom_paired2 image_paired_Times
      times_eq_iff group_def monoid.carrier_not_empty)

lemma hom_of_fst:
  assumes "group H"
  shows "(f ∘ fst) ∈ hom (DirProd G H) K ⟷ f ∈ hom G K"
proof -
  interpret group H
    by (rule assms)
  show ?thesis
    using one_closed by (auto simp: hom_def Pi_def)
qed

lemma hom_of_snd:
  assumes "group G"
  shows "(f ∘ snd) ∈ hom (DirProd G H) K ⟷ f ∈ hom H K"
proof -
  interpret group G
    by (rule assms)
  show ?thesis
    using one_closed by (auto simp: hom_def Pi_def)
qed


subsection‹The locale for a homomorphism between two groups›

text‹Basis for homomorphism proofs: we assume two groups \<^term>‹G› and
  \<^term>‹H›, with a homomorphism \<^term>‹h› between them›
locale group_hom = G?: group G + H?: group H for G (structure) and H (structure) +
  fixes h
  assumes homh [simp]: "h ∈ hom G H"

declare group_hom.homh [simp]

lemma (in group_hom) hom_mult [simp]:
  "[| x ∈ carrier G; y ∈ carrier G |] ==> h (x ⊗⇘G⇙ y) = h x ⊗⇘H⇙ h y"
proof -
  assume "x ∈ carrier G" "y ∈ carrier G"
  with homh [unfolded hom_def] show ?thesis by simp
qed

lemma (in group_hom) hom_closed [simp]:
  "x ∈ carrier G ==> h x ∈ carrier H"
proof -
  assume "x ∈ carrier G"
  with homh [unfolded hom_def] show ?thesis by auto
qed

lemma (in group_hom) one_closed: "h 𝟭 ∈ carrier H"
  by simp

lemma (in group_hom) hom_one [simp]: "h 𝟭 = 𝟭⇘H⇙"
proof -
  have "h 𝟭 ⊗⇘H⇙ 𝟭⇘H⇙ = h 𝟭 ⊗⇘H⇙ h 𝟭"
    by (simp add: hom_mult [symmetric] del: hom_mult)
  then show ?thesis
    by (metis H.Units_eq H.Units_l_cancel H.one_closed local.one_closed)
qed

lemma hom_one:
  assumes "h ∈ hom G H" "group G" "group H"
  shows "h (one G) = one H"
  apply (rule group_hom.hom_one)
  by (simp add: assms group_hom_axioms_def group_hom_def)

lemma hom_mult:
  "⟦h ∈ hom G H; x ∈ carrier G; y ∈ carrier G⟧ ⟹ h (x ⊗⇘G⇙ y) = h x ⊗⇘H⇙ h y"
  by (auto simp: hom_def)

lemma (in group_hom) inv_closed [simp]:
  "x ∈ carrier G ==> h (inv x) ∈ carrier H"
  by simp

lemma (in group_hom) hom_inv [simp]:
  assumes "x ∈ carrier G" shows "h (inv x) = inv⇘H⇙ (h x)"
proof -
  have "h x ⊗⇘H⇙ h (inv x) = h x ⊗⇘H⇙ inv⇘H⇙ (h x)" 
    using assms by (simp flip: hom_mult)
  with assms show ?thesis by (simp del: H.r_inv H.Units_r_inv)
qed

lemma (in group) int_pow_is_hom: ✐‹contributor ‹Joachim Breitner››
  "x ∈ carrier G ⟹ (([^]) x) ∈ hom ⦇ carrier = UNIV, mult = (+), one = 0::int ⦈ G "
  unfolding hom_def by (simp add: int_pow_mult)

lemma (in group_hom) img_is_subgroup: "subgroup (h ` (carrier G)) H" ✐‹contributor ‹Paulo Emílio de Vilhena››
  apply (rule subgroupI)
  apply (auto simp add: image_subsetI)
  apply (metis G.inv_closed hom_inv image_iff)
  by (metis G.monoid_axioms hom_mult image_eqI monoid.m_closed)

lemma (in group_hom) subgroup_img_is_subgroup: ✐‹contributor ‹Paulo Emílio de Vilhena››
  assumes "subgroup I G"
  shows "subgroup (h ` I) H"
proof -
  have "h ∈ hom (G ⦇ carrier := I ⦈) H"
    using G.subgroupE[OF assms] subgroup.mem_carrier[OF assms] homh
    unfolding hom_def by auto
  hence "group_hom (G ⦇ carrier := I ⦈) H h"
    using subgroup.subgroup_is_group[OF assms G.is_group] is_group
    unfolding group_hom_def group_hom_axioms_def by simp
  thus ?thesis
    using group_hom.img_is_subgroup[of "G ⦇ carrier := I ⦈" H h] by simp
qed

lemma (in subgroup) iso_subgroup: ✐‹contributor ‹Jakob von Raumer››
  assumes "group G" "group F"
  assumes "φ ∈ iso G F"
  shows "subgroup (φ ` H) F"
  by (metis assms Group.iso_iff group_hom.intro group_hom_axioms_def group_hom.subgroup_img_is_subgroup subgroup_axioms)

lemma (in group_hom) induced_group_hom: ✐‹contributor ‹Paulo Emílio de Vilhena››
  assumes "subgroup I G"
  shows "group_hom (G ⦇ carrier := I ⦈) (H ⦇ carrier := h ` I ⦈) h"
proof -
  have "h ∈ hom (G ⦇ carrier := I ⦈) (H ⦇ carrier := h ` I ⦈)"
    using homh subgroup.mem_carrier[OF assms] unfolding hom_def by auto
  thus ?thesis
    unfolding group_hom_def group_hom_axioms_def
    using subgroup.subgroup_is_group[OF assms G.is_group]
          subgroup.subgroup_is_group[OF subgroup_img_is_subgroup[OF assms] is_group] by simp
qed

text ‹An isomorphism restricts to an isomorphism of subgroups.›

lemma iso_restrict:
  assumes "φ ∈ iso G F"
  assumes groups: "group G" "group F"
  assumes HG: "subgroup H G"
  shows "(restrict φ H) ∈ iso (G⦇carrier := H⦈) (F⦇carrier := φ ` H⦈)"
proof -
  have "⋀x y. ⟦x ∈ H; y ∈ H; x ⊗⇘G⇙ y ∈ H⟧ ⟹ φ (x ⊗⇘G⇙ y) = φ x ⊗⇘F⇙ φ y"
    by (meson assms hom_mult iso_imp_homomorphism subgroup.mem_carrier)
  moreover have "⋀x y. ⟦x ∈ H; y ∈ H; x ⊗⇘G⇙ y ∉ H⟧ ⟹ φ x ⊗⇘F⇙ φ y = undefined"
    by (simp add: HG subgroup.m_closed)
  moreover have "⋀x y. ⟦x ∈ H; y ∈ H; φ x = φ y⟧ ⟹ x = y"
    by (smt (verit, ccfv_SIG) assms group.iso_iff_group_isomorphisms group_isomorphisms_def subgroup.mem_carrier)
  ultimately show ?thesis
    by (auto simp: iso_def hom_def bij_betw_def inj_on_def)
qed

lemma (in group) canonical_inj_is_hom: ✐‹contributor ‹Paulo Emílio de Vilhena››
  assumes "subgroup H G"
  shows "group_hom (G ⦇ carrier := H ⦈) G id"
  unfolding group_hom_def group_hom_axioms_def hom_def
  using subgroup.subgroup_is_group[OF assms is_group]
        is_group subgroup.subset[OF assms] by auto

lemma (in group_hom) hom_nat_pow: ✐‹contributor ‹Paulo Emílio de Vilhena››
  "x ∈ carrier G ⟹ h (x [^] (n :: nat)) = (h x) [^]⇘H⇙ n"
  by (induction n) auto

lemma (in group_hom) hom_int_pow: ✐‹contributor ‹Paulo Emílio de Vilhena››
  "x ∈ carrier G ⟹ h (x [^] (n :: int)) = (h x) [^]⇘H⇙ n"
  using hom_nat_pow by (simp add: int_pow_def2)

lemma hom_nat_pow:
  "⟦h ∈ hom G H; x ∈ carrier G; group G; group H⟧ ⟹ h (x [^]⇘G⇙ (n :: nat)) = (h x) [^]⇘H⇙ n"
  by (simp add: group_hom.hom_nat_pow group_hom_axioms_def group_hom_def)

lemma hom_int_pow:
  "⟦h ∈ hom G H; x ∈ carrier G; group G; group H⟧ ⟹ h (x [^]⇘G⇙ (n :: int)) = (h x) [^]⇘H⇙ n"
  by (simp add: group_hom.hom_int_pow group_hom_axioms.intro group_hom_def)

subsection ‹Commutative Structures›

text ‹
  Naming convention: multiplicative structures that are commutative
  are called \emph{commutative}, additive structures are called
  \emph{Abelian}.
›

locale comm_monoid = monoid +
  assumes m_comm: "⟦x ∈ carrier G; y ∈ carrier G⟧ ⟹ x ⊗ y = y ⊗ x"

lemma (in comm_monoid) m_lcomm:
  "⟦x ∈ carrier G; y ∈ carrier G; z ∈ carrier G⟧ ⟹
   x ⊗ (y ⊗ z) = y ⊗ (x ⊗ z)"
proof -
  assume xyz: "x ∈ carrier G"  "y ∈ carrier G"  "z ∈ carrier G"
  from xyz have "x ⊗ (y ⊗ z) = (x ⊗ y) ⊗ z" by (simp add: m_assoc)
  also from xyz have "... = (y ⊗ x) ⊗ z" by (simp add: m_comm)
  also from xyz have "... = y ⊗ (x ⊗ z)" by (simp add: m_assoc)
  finally show ?thesis .
qed

lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm

lemma comm_monoidI:
  fixes G (structure)
  assumes m_closed:
      "!!x y. [| x ∈ carrier G; y ∈ carrier G |] ==> x ⊗ y ∈ carrier G"
    and one_closed: "𝟭 ∈ carrier G"
    and m_assoc:
      "!!x y z. [| x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==>
      (x ⊗ y) ⊗ z = x ⊗ (y ⊗ z)"
    and l_one: "!!x. x ∈ carrier G ==> 𝟭 ⊗ x = x"
    and m_comm:
      "!!x y. [| x ∈ carrier G; y ∈ carrier G |] ==> x ⊗ y = y ⊗ x"
  shows "comm_monoid G"
  using l_one
    by (auto intro!: comm_monoid.intro comm_monoid_axioms.intro monoid.intro
             intro: assms simp: m_closed one_closed m_comm)

lemma (in monoid) monoid_comm_monoidI:
  assumes m_comm:
      "!!x y. [| x ∈ carrier G; y ∈ carrier G |] ==> x ⊗ y = y ⊗ x"
  shows "comm_monoid G"
  by (rule comm_monoidI) (auto intro: m_assoc m_comm)

lemma (in comm_monoid) submonoid_is_comm_monoid :
  assumes "submonoid H G"
  shows "comm_monoid (G⦇carrier := H⦈)"
proof (intro monoid.monoid_comm_monoidI)
  show "monoid (G⦇carrier := H⦈)"
    using submonoid.submonoid_is_monoid assms comm_monoid_axioms comm_monoid_def by blast
  show "⋀x y. x ∈ carrier (G⦇carrier := H⦈) ⟹ y ∈ carrier (G⦇carrier := H⦈)
        ⟹ x ⊗⇘G⦇carrier := H⦈⇙ y = y ⊗⇘G⦇carrier := H⦈⇙ x" 
    by simp (meson assms m_comm submonoid.mem_carrier)
qed

locale comm_group = comm_monoid + group

lemma (in group) group_comm_groupI:
  assumes m_comm: "!!x y. [| x ∈ carrier G; y ∈ carrier G |] ==> x ⊗ y = y ⊗ x"
  shows "comm_group G"
  by standard (simp_all add: m_comm)

lemma comm_groupI:
  fixes G (structure)
  assumes m_closed:
      "!!x y. [| x ∈ carrier G; y ∈ carrier G |] ==> x ⊗ y ∈ carrier G"
    and one_closed: "𝟭 ∈ carrier G"
    and m_assoc:
      "!!x y z. [| x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==>
      (x ⊗ y) ⊗ z = x ⊗ (y ⊗ z)"
    and m_comm:
      "!!x y. [| x ∈ carrier G; y ∈ carrier G |] ==> x ⊗ y = y ⊗ x"
    and l_one: "!!x. x ∈ carrier G ==> 𝟭 ⊗ x = x"
    and l_inv_ex: "!!x. x ∈ carrier G ==> ∃y ∈ carrier G. y ⊗ x = 𝟭"
  shows "comm_group G"
  by (fast intro: group.group_comm_groupI groupI assms)

lemma comm_groupE:
  fixes G (structure)
  assumes "comm_group G"
  shows "⋀x y. ⟦ x ∈ carrier G; y ∈ carrier G ⟧ ⟹ x ⊗ y ∈ carrier G"
    and "𝟭 ∈ carrier G"
    and "⋀x y z. ⟦ x ∈ carrier G; y ∈ carrier G; z ∈ carrier G ⟧ ⟹ (x ⊗ y) ⊗ z = x ⊗ (y ⊗ z)"
    and "⋀x y. ⟦ x ∈ carrier G; y ∈ carrier G ⟧ ⟹ x ⊗ y = y ⊗ x"
    and "⋀x. x ∈ carrier G ⟹ 𝟭 ⊗ x = x"
    and "⋀x. x ∈ carrier G ⟹ ∃y ∈ carrier G. y ⊗ x = 𝟭"
  apply (simp_all add: group.axioms assms comm_group.axioms comm_monoid.m_comm comm_monoid.m_ac(1))
  by (simp_all add: Group.group.axioms(1) assms comm_group.axioms(2) monoid.m_closed group.r_inv_ex)

lemma (in comm_group) inv_mult:
  "[| x ∈ carrier G; y ∈ carrier G |] ==> inv (x ⊗ y) = inv x ⊗ inv y"
  by (simp add: m_ac inv_mult_group)

lemma (in comm_monoid) nat_pow_distrib:
  fixes n::nat
  assumes "x ∈ carrier G" "y ∈ carrier G"
  shows "(x ⊗ y) [^] n = x [^] n ⊗ y [^] n"
  by (simp add: assms pow_mult_distrib m_comm)

lemma (in comm_group) int_pow_distrib:
  assumes "x ∈ carrier G" "y ∈ carrier G"
  shows "(x ⊗ y) [^] (i::int) = x [^] i ⊗ y [^] i"
  by (simp add: assms int_pow_mult_distrib m_comm)

lemma (in comm_monoid) hom_imp_img_comm_monoid: ✐‹contributor ‹Paulo Emílio de Vilhena››
  assumes "h ∈ hom G H"
  shows "comm_monoid (H ⦇ carrier := h ` (carrier G), one := h 𝟭⇘G⇙ ⦈)" (is "comm_monoid ?h_img")
proof (rule monoid.monoid_comm_monoidI)
  show "monoid ?h_img"
    using hom_imp_img_monoid[OF assms] .
next
  fix x y assume "x ∈ carrier ?h_img" "y ∈ carrier ?h_img"
  then obtain g1 g2
    where g1: "g1 ∈ carrier G" "x = h g1"
      and g2: "g2 ∈ carrier G" "y = h g2"
    by auto
  have "x ⊗⇘(?h_img)⇙ y = h (g1 ⊗ g2)"
    using g1 g2 assms unfolding hom_def by auto
  also have " ... = h (g2 ⊗ g1)"
    using m_comm[OF g1(1) g2(1)] by simp
  also have " ... = y ⊗⇘(?h_img)⇙ x"
    using g1 g2 assms unfolding hom_def by auto
  finally show "x ⊗⇘(?h_img)⇙ y = y ⊗⇘(?h_img)⇙ x" .
qed

lemma (in comm_group) hom_group_mult:
  assumes "f ∈ hom H G" "g ∈ hom H G"
 shows "(λx. f x ⊗⇘G⇙ g x) ∈ hom H G"
    using assms by (auto simp: hom_def Pi_def m_ac)

lemma (in comm_group) hom_imp_img_comm_group: ✐‹contributor ‹Paulo Emílio de Vilhena››
  assumes "h ∈ hom G H"
  shows "comm_group (H ⦇ carrier := h ` (carrier G), one := h 𝟭⇘G⇙ ⦈)"
  unfolding comm_group_def
  using hom_imp_img_group[OF assms] hom_imp_img_comm_monoid[OF assms] by simp

lemma (in comm_group) iso_imp_img_comm_group: ✐‹contributor ‹Paulo Emílio de Vilhena››
  assumes "h ∈ iso G H"
  shows "comm_group (H ⦇ one := h 𝟭⇘G⇙ ⦈)"
proof -
  let ?h_img = "H ⦇ carrier := h ` (carrier G), one := h 𝟭 ⦈"
  have "comm_group ?h_img"
    using hom_imp_img_comm_group[of h H] assms unfolding iso_def by auto
  moreover have "carrier H = carrier ?h_img"
    using assms unfolding iso_def bij_betw_def by simp
  hence "H ⦇ one := h 𝟭 ⦈ = ?h_img"
    by simp
  ultimately show ?thesis by simp
qed

lemma (in comm_group) iso_imp_comm_group: ✐‹contributor ‹Paulo Emílio de Vilhena››
  assumes "G ≅ H" "monoid H"
  shows "comm_group H"
proof -
  obtain h where h: "h ∈ iso G H"
    using assms(1) unfolding is_iso_def by auto
  hence comm_gr: "comm_group (H ⦇ one := h 𝟭 ⦈)"
    using iso_imp_img_comm_group[of h H] by simp
  hence "⋀x. x ∈ carrier H ⟹ h 𝟭 ⊗⇘H⇙ x = x"
    using monoid.l_one[of "H ⦇ one := h 𝟭 ⦈"] unfolding comm_group_def comm_monoid_def by simp
  moreover have "h 𝟭 ∈ carrier H"
    using h one_closed unfolding iso_def hom_def by auto
  ultimately have "h 𝟭 = 𝟭⇘H⇙"
    using monoid.one_unique[OF assms(2), of "h 𝟭"] by simp
  hence "H = H ⦇ one := h 𝟭 ⦈"
    by simp
  thus ?thesis
    using comm_gr by simp
qed

(*A subgroup of a subgroup is a subgroup of the group*)
lemma (in group) incl_subgroup:
  assumes "subgroup J G"
    and "subgroup I (G⦇carrier:=J⦈)"
  shows "subgroup I G" unfolding subgroup_def
proof
  have H1: "I ⊆ carrier (G⦇carrier:=J⦈)" using assms(2) subgroup.subset by blast
  also have H2: "...⊆J" by simp
  also  have "...⊆(carrier G)"  by (simp add: assms(1) subgroup.subset)
  finally have H: "I ⊆ carrier G" by simp
  have "(⋀x y. ⟦x ∈ I ; y ∈ I⟧ ⟹ x ⊗ y ∈ I)" using assms(2) by (auto simp add: subgroup_def)
  thus  "I ⊆ carrier G ∧ (∀x y. x ∈ I ⟶ y ∈ I ⟶ x ⊗ y ∈ I)"  using H by blast
  have K: "𝟭 ∈ I" using assms(2) by (auto simp add: subgroup_def)
  have "(⋀x. x ∈ I ⟹ inv x ∈ I)" using assms  subgroup.m_inv_closed H
    by (metis H1 H2 m_inv_consistent subsetCE)
  thus "𝟭 ∈ I ∧ (∀x. x ∈ I ⟶ inv x ∈ I)" using K by blast
qed

(*A subgroup included in another subgroup is a subgroup of the subgroup*)
lemma (in group) subgroup_incl:
  assumes "subgroup I G" and "subgroup J G" and "I ⊆ J"
  shows "subgroup I (G ⦇ carrier := J ⦈)"
  using group.group_incl_imp_subgroup[of "G ⦇ carrier := J ⦈" I]
        assms(1-2)[THEN subgroup.subgroup_is_group[OF _ group_axioms]] assms(3) by auto


subsection ‹The Lattice of Subgroups of a Group›

text_raw ‹\label{sec:subgroup-lattice}›

theorem (in group) subgroups_partial_order:
  "partial_order ⦇carrier = {H. subgroup H G}, eq = (=), le = (⊆)⦈"
  by standard simp_all

lemma (in group) subgroup_self:
  "subgroup (carrier G) G"
  by (rule subgroupI) auto

lemma (in group) subgroup_imp_group:
  "subgroup H G ==> group (G⦇carrier := H⦈)"
  by (erule subgroup.subgroup_is_group) (rule group_axioms)

lemma (in group) subgroup_mult_equality:
  "⟦ subgroup H G; h1 ∈ H; h2 ∈ H ⟧ ⟹  h1 ⊗⇘G ⦇ carrier := H ⦈⇙ h2 = h1 ⊗ h2"
  unfolding subgroup_def by simp

theorem (in group) subgroups_Inter:
  assumes subgr: "(⋀H. H ∈ A ⟹ subgroup H G)"
    and not_empty: "A ≠ {}"
  shows "subgroup (⋂A) G"
proof (rule subgroupI)
  from subgr [THEN subgroup.subset] and not_empty
  show "⋂A ⊆ carrier G" by blast
next
  from subgr [THEN subgroup.one_closed]
  show "⋂A ≠ {}" by blast
next
  fix x assume "x ∈ ⋂A"
  with subgr [THEN subgroup.m_inv_closed]
  show "inv x ∈ ⋂A" by blast
next
  fix x y assume "x ∈ ⋂A" "y ∈ ⋂A"
  with subgr [THEN subgroup.m_closed]
  show "x ⊗ y ∈ ⋂A" by blast
qed

lemma (in group) subgroups_Inter_pair :
  assumes "subgroup I G" "subgroup J G" shows "subgroup (I∩J) G" 
  using subgroups_Inter[ where ?A = "{I,J}"] assms by auto

theorem (in group) subgroups_complete_lattice:
  "complete_lattice ⦇carrier = {H. subgroup H G}, eq = (=), le = (⊆)⦈"
    (is "complete_lattice ?L")
proof (rule partial_order.complete_lattice_criterion1)
  show "partial_order ?L" by (rule subgroups_partial_order)
next
  have "greatest ?L (carrier G) (carrier ?L)"
    by (unfold greatest_def) (simp add: subgroup.subset subgroup_self)
  then show "∃G. greatest ?L G (carrier ?L)" ..
next
  fix A
  assume L: "A ⊆ carrier ?L" and non_empty: "A ≠ {}"
  then have Int_subgroup: "subgroup (⋂A) G"
    by (fastforce intro: subgroups_Inter)
  have "greatest ?L (⋂A) (Lower ?L A)" (is "greatest _ ?Int _")
  proof (rule greatest_LowerI)
    fix H
    assume H: "H ∈ A"
    with L have subgroupH: "subgroup H G" by auto
    from subgroupH have groupH: "group (G ⦇carrier := H⦈)" (is "group ?H")
      by (rule subgroup_imp_group)
    from groupH have monoidH: "monoid ?H"
      by (rule group.is_monoid)
    from H have Int_subset: "?Int ⊆ H" by fastforce
    then show "le ?L ?Int H" by simp
  next
    fix H
    assume H: "H ∈ Lower ?L A"
    with L Int_subgroup show "le ?L H ?Int"
      by (fastforce simp: Lower_def intro: Inter_greatest)
  next
    show "A ⊆ carrier ?L" by (rule L)
  next
    show "?Int ∈ carrier ?L" by simp (rule Int_subgroup)
  qed
  then show "∃I. greatest ?L I (Lower ?L A)" ..
qed

subsection‹The units in any monoid give rise to a group›

text ‹Thanks to Jeremy Avigad. The file Residues.thy provides some infrastructure to use
  facts about the unit group within the ring locale.
›

definition units_of :: "('a, 'b) monoid_scheme ⇒ 'a monoid"
  where "units_of G =
    ⦇carrier = Units G, Group.monoid.mult = Group.monoid.mult G, one  = one G⦈"

lemma (in monoid) units_group: "group (units_of G)"
proof -
  have "⋀x y z. ⟦x ∈ Units G; y ∈ Units G; z ∈ Units G⟧ ⟹ x ⊗ y ⊗ z = x ⊗ (y ⊗ z)"
    by (simp add: Units_closed m_assoc)
  moreover have "⋀x. x ∈ Units G ⟹ ∃y∈Units G. y ⊗ x = 𝟭"
    using Units_l_inv by blast
  ultimately show ?thesis
    unfolding units_of_def
    by (force intro!: groupI)
qed

lemma (in comm_monoid) units_comm_group: "comm_group (units_of G)"
proof -
  have "⋀x y. ⟦x ∈ carrier (units_of G); y ∈ carrier (units_of G)⟧
              ⟹ x ⊗⇘units_of G⇙ y = y ⊗⇘units_of G⇙ x"
    by (simp add: Units_closed m_comm units_of_def)
  then show ?thesis
    by (rule group.group_comm_groupI [OF units_group]) auto
qed

lemma units_of_carrier: "carrier (units_of G) = Units G"
  by (auto simp: units_of_def)

lemma units_of_mult: "mult (units_of G) = mult G"
  by (auto simp: units_of_def)

lemma units_of_one: "one (units_of G) = one G"
  by (auto simp: units_of_def)

lemma (in monoid) units_of_inv:
  assumes "x ∈ Units G"
  shows "m_inv (units_of G) x = m_inv G x"
  by (simp add: assms group.inv_equality units_group units_of_carrier units_of_mult units_of_one)

lemma units_of_units [simp] : "Units (units_of G) = Units G"
  unfolding units_of_def Units_def by force

lemma (in group) surj_const_mult: "a ∈ carrier G ⟹ (λx. a ⊗ x) ` carrier G = carrier G"
  apply (auto simp add: image_def)
  by (metis inv_closed inv_solve_left m_closed)

lemma (in group) l_cancel_one [simp]: "x ∈ carrier G ⟹ a ∈ carrier G ⟹ x ⊗ a = x ⟷ a = one G"
  by (metis Units_eq Units_l_cancel monoid.r_one monoid_axioms one_closed)

lemma (in group) r_cancel_one [simp]: "x ∈ carrier G ⟹ a ∈ carrier G ⟹ a ⊗ x = x ⟷ a = one G"
  by (metis monoid.l_one monoid_axioms one_closed right_cancel)

lemma (in group) l_cancel_one' [simp]: "x ∈ carrier G ⟹ a ∈ carrier G ⟹ x = x ⊗ a ⟷ a = one G"
  using l_cancel_one by fastforce

lemma (in group) r_cancel_one' [simp]: "x ∈ carrier G ⟹ a ∈ carrier G ⟹ x = a ⊗ x ⟷ a = one G"
  using r_cancel_one by fastforce

declare pow_nat [simp] (*causes looping if added above, especially with int_pow_def2*)

end