(*  Title:      HOL/Algebra/Group.thy

     Author:     Clemens Ballarin, started 4 February 2003

 Based on work by Florian Kammueller, L C Paulson and Markus Wenzel.

 *)

 theory Group

 imports Lattice "~~/src/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] \<Rightarrow> 'a" (infixl "\<otimes>\<index>" 70)

   one     :: 'a ("\<one>\<index>")

 definition

   m_inv :: "('a, 'b) monoid_scheme => 'a => 'a" ("inv\<index> _" [81] 80)

   where "inv\<^bsub>G\<^esub> x = (THE y. y \<in> carrier G & x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> & y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>)"

 definition

   Units :: "_ => 'a set"

   --{*The set of invertible elements*}

   where "Units G = {y. y \<in> carrier G & (\<exists>x \<in> carrier G. x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> & y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>)}"

 consts

   pow :: "[('a, 'm) monoid_scheme, 'a, 'b::semiring_1] => 'a"  (infixr "'(^')\<index>" 75)

 overloading nat_pow == "pow :: [_, 'a, nat] => 'a"

 begin

   definition "nat_pow G a n = nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) n"

 end

 overloading int_pow == "pow :: [_, 'a, int] => 'a"

 begin

   definition "int_pow G a z =

    (let p = nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a)

     in if z < 0 then inv\<^bsub>G\<^esub> (p (nat (-z))) else p (nat z))"

 end

 locale monoid =

   fixes G (structure)

   assumes m_closed [intro, simp]:

          "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> carrier G"

       and m_assoc:

          "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> 

           \<Longrightarrow> (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"

       and one_closed [intro, simp]: "\<one> \<in> carrier G"

       and l_one [simp]: "x \<in> carrier G \<Longrightarrow> \<one> \<otimes> x = x"

       and r_one [simp]: "x \<in> carrier G \<Longrightarrow> x \<otimes> \<one> = x"

 lemma monoidI:

   fixes G (structure)

   assumes m_closed:

       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"

     and one_closed: "\<one> \<in> carrier G"

     and m_assoc:

       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>

       (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"

     and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"

     and r_one: "!!x. x \<in> carrier G ==> x \<otimes> \<one> = x"

   shows "monoid G"

   by (fast intro!: monoid.intro intro: assms)

 lemma (in monoid) Units_closed [dest]:

   "x \<in> Units G ==> x \<in> carrier G"

   by (unfold Units_def) fast

 lemma (in monoid) inv_unique:

   assumes eq: "y \<otimes> x = \<one>"  "x \<otimes> y' = \<one>"

     and G: "x \<in> carrier G"  "y \<in> carrier G"  "y' \<in> carrier G"

   shows "y = y'"

 proof -

   from G eq have "y = y \<otimes> (x \<otimes> y')" by simp

   also from G have "... = (y \<otimes> x) \<otimes> y'" by (simp add: m_assoc)

   also from G eq have "... = y'" by simp

   finally show ?thesis .

 qed

 lemma (in monoid) Units_m_closed [intro, simp]:

   assumes x: "x \<in> Units G" and y: "y \<in> Units G"

   shows "x \<otimes> y \<in> Units G"

 proof -

   from x obtain x' where x: "x \<in> carrier G" "x' \<in> carrier G" and xinv: "x \<otimes> x' = \<one>" "x' \<otimes> x = \<one>"

     unfolding Units_def by fast

   from y obtain y' where y: "y \<in> carrier G" "y' \<in> carrier G" and yinv: "y \<otimes> y' = \<one>" "y' \<otimes> y = \<one>"

     unfolding Units_def by fast

   from x y xinv yinv have "y' \<otimes> (x' \<otimes> x) \<otimes> y = \<one>" by simp

   moreover from x y xinv yinv have "x \<otimes> (y \<otimes> y') \<otimes> x' = \<one>" by simp

   moreover note x y

   ultimately show ?thesis unfolding Units_def

     -- "Must avoid premature use of @{text hyp_subst_tac}."

     apply (rule_tac CollectI)

     apply (rule)

     apply (fast)

     apply (rule bexI [where x = "y' \<otimes> x'"])

     apply (auto simp: m_assoc)

     done

 qed

 lemma (in monoid) Units_one_closed [intro, simp]:

   "\<one> \<in> Units G"

   by (unfold Units_def) auto

 lemma (in monoid) Units_inv_closed [intro, simp]:

   "x \<in> Units G ==> inv x \<in> carrier G"

   apply (unfold Units_def m_inv_def, auto)

   apply (rule theI2, fast)

    apply (fast intro: inv_unique, fast)

   done

 lemma (in monoid) Units_l_inv_ex:

   "x \<in> Units G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"

   by (unfold Units_def) auto

 lemma (in monoid) Units_r_inv_ex:

   "x \<in> Units G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>"

   by (unfold Units_def) auto

 lemma (in monoid) Units_l_inv [simp]:

   "x \<in> Units G ==> inv x \<otimes> x = \<one>"

   apply (unfold Units_def m_inv_def, auto)

   apply (rule theI2, fast)

    apply (fast intro: inv_unique, fast)

   done

 lemma (in monoid) Units_r_inv [simp]:

   "x \<in> Units G ==> x \<otimes> inv x = \<one>"

   apply (unfold Units_def m_inv_def, auto)

   apply (rule theI2, fast)

    apply (fast intro: inv_unique, fast)

   done

 lemma (in monoid) Units_inv_Units [intro, simp]:

   "x \<in> Units G ==> inv x \<in> Units G"

 proof -

   assume x: "x \<in> Units G"

   show "inv x \<in> 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 \<in> Units G; y \<in> carrier G; z \<in> carrier G |] ==>

    (x \<otimes> y = x \<otimes> z) = (y = z)"

 proof

   assume eq: "x \<otimes> y = x \<otimes> z"

     and G: "x \<in> Units G"  "y \<in> carrier G"  "z \<in> carrier G"

   then have "(inv x \<otimes> x) \<otimes> y = (inv x \<otimes> x) \<otimes> 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 \<in> Units G"  "y \<in> carrier G"  "z \<in> carrier G"

   then show "x \<otimes> y = x \<otimes> z" by simp

 qed

 lemma (in monoid) Units_inv_inv [simp]:

   "x \<in> Units G ==> inv (inv x) = x"

 proof -

   assume x: "x \<in> Units G"

   then have "inv x \<otimes> inv (inv x) = inv x \<otimes> 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 \<in> Units G"  "y \<in> 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 \<otimes> y = \<one>"

     and G: "x \<in> Units G"  "y \<in> Units G"

   shows "y \<otimes> x = \<one>"

 proof -

   from G have "x \<otimes> y \<otimes> x = x \<otimes> \<one>" by (auto simp add: inv Units_closed)

   with G show ?thesis by (simp del: r_one add: m_assoc Units_closed)

 qed

 text {* Power *}

 lemma (in monoid) nat_pow_closed [intro, simp]:

   "x \<in> carrier G ==> x (^) (n::nat) \<in> carrier G"

   by (induct n) (simp_all add: nat_pow_def)

 lemma (in monoid) nat_pow_0 [simp]:

   "x (^) (0::nat) = \<one>"

   by (simp add: nat_pow_def)

 lemma (in monoid) nat_pow_Suc [simp]:

   "x (^) (Suc n) = x (^) n \<otimes> x"

   by (simp add: nat_pow_def)

 lemma (in monoid) nat_pow_one [simp]:

   "\<one> (^) (n::nat) = \<one>"

   by (induct n) simp_all

 lemma (in monoid) nat_pow_mult:

   "x \<in> carrier G ==> x (^) (n::nat) \<otimes> x (^) m = x (^) (n + m)"

   by (induct m) (simp_all add: m_assoc [THEN sym])

 lemma (in monoid) nat_pow_pow:

   "x \<in> carrier G ==> (x (^) n) (^) m = x (^) (n * m::nat)"

   by (induct m) (simp, simp add: nat_pow_mult add_commute)

 (* 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: "group G" by (rule group_axioms)

 theorem groupI:

   fixes G (structure)

   assumes m_closed [simp]:

       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"

     and one_closed [simp]: "\<one> \<in> carrier G"

     and m_assoc:

       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>

       (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"

     and l_one [simp]: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"

     and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"

   shows "group G"

 proof -

   have l_cancel [simp]:

     "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>

     (x \<otimes> y = x \<otimes> z) = (y = z)"

   proof

     fix x y z

     assume eq: "x \<otimes> y = x \<otimes> z"

       and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"

     with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G"

       and l_inv: "x_inv \<otimes> x = \<one>" by fast

     from G eq xG have "(x_inv \<otimes> x) \<otimes> y = (x_inv \<otimes> x) \<otimes> 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 \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"

     then show "x \<otimes> y = x \<otimes> z" by simp

   qed

   have r_one:

     "!!x. x \<in> carrier G ==> x \<otimes> \<one> = x"

   proof -

     fix x

     assume x: "x \<in> carrier G"

     with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G"

       and l_inv: "x_inv \<otimes> x = \<one>" by fast

     from x xG have "x_inv \<otimes> (x \<otimes> \<one>) = x_inv \<otimes> x"

       by (simp add: m_assoc [symmetric] l_inv)

     with x xG show "x \<otimes> \<one> = x" by simp

   qed

   have inv_ex:

     "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>"

   proof -

     fix x

     assume x: "x \<in> carrier G"

     with l_inv_ex obtain y where y: "y \<in> carrier G"

       and l_inv: "y \<otimes> x = \<one>" by fast

     from x y have "y \<otimes> (x \<otimes> y) = y \<otimes> \<one>"

       by (simp add: m_assoc [symmetric] l_inv r_one)

     with x y have r_inv: "x \<otimes> y = \<one>"

       by simp

     from x y show "\<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>"

       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 default (auto simp: r_one m_assoc carrier_subset_Units)

 qed

 lemma (in monoid) group_l_invI:

   assumes l_inv_ex:

     "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"

   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 \<in> carrier G ==> inv x \<in> carrier G"

   using Units_inv_closed by simp

 lemma (in group) l_inv_ex [simp]:

   "x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"

   using Units_l_inv_ex by simp

 lemma (in group) r_inv_ex [simp]:

   "x \<in> carrier G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>"

   using Units_r_inv_ex by simp

 lemma (in group) l_inv [simp]:

   "x \<in> carrier G ==> inv x \<otimes> x = \<one>"

   using Units_l_inv by simp

 subsection {* Cancellation Laws and Basic Properties *}

 lemma (in group) l_cancel [simp]:

   "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>

    (x \<otimes> y = x \<otimes> z) = (y = z)"

   using Units_l_inv by simp

 lemma (in group) r_inv [simp]:

   "x \<in> carrier G ==> x \<otimes> inv x = \<one>"

 proof -

   assume x: "x \<in> carrier G"

   then have "inv x \<otimes> (x \<otimes> inv x) = inv x \<otimes> \<one>"

     by (simp add: m_assoc [symmetric])

   with x show ?thesis by (simp del: r_one)

 qed

 lemma (in group) r_cancel [simp]:

   "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>

    (y \<otimes> x = z \<otimes> x) = (y = z)"

 proof

   assume eq: "y \<otimes> x = z \<otimes> x"

     and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"

   then have "y \<otimes> (x \<otimes> inv x) = z \<otimes> (x \<otimes> inv x)"

     by (simp add: m_assoc [symmetric] del: r_inv Units_r_inv)

   with G show "y = z" by simp

 next

   assume eq: "y = z"

     and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"

   then show "y \<otimes> x = z \<otimes> x" by simp

 qed

 lemma (in group) inv_one [simp]:

   "inv \<one> = \<one>"

 proof -

   have "inv \<one> = \<one> \<otimes> (inv \<one>)" by (simp del: r_inv Units_r_inv)

   moreover have "... = \<one>" by simp

   finally show ?thesis .

 qed

 lemma (in group) inv_inv [simp]:

   "x \<in> 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 \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv y \<otimes> inv x"

 proof -

   assume G: "x \<in> carrier G"  "y \<in> carrier G"

   then have "inv (x \<otimes> y) \<otimes> (x \<otimes> y) = (inv y \<otimes> inv x) \<otimes> (x \<otimes> 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 \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>"

   by (rule Units_inv_comm) auto

 lemma (in group) inv_equality:

      "[|y \<otimes> x = \<one>; x \<in> carrier G; y \<in> carrier G|] ==> inv x = y"

 apply (simp add: m_inv_def)

 apply (rule the_equality)

  apply (simp add: inv_comm [of y x])

 apply (rule r_cancel [THEN iffD1], auto)

 done

 text {* Power *}

 lemma (in group) int_pow_def2:

   "a (^) (z::int) = (if z < 0 then inv (a (^) (nat (-z))) else a (^) (nat z))"

   by (simp add: int_pow_def nat_pow_def Let_def)

 lemma (in group) int_pow_0 [simp]:

   "x (^) (0::int) = \<one>"

   by (simp add: int_pow_def2)

 lemma (in group) int_pow_one [simp]:

   "\<one> (^) (z::int) = \<one>"

   by (simp add: int_pow_def2)

 subsection {* Subgroups *}

 locale subgroup =

   fixes H and G (structure)

   assumes subset: "H \<subseteq> carrier G"

     and m_closed [intro, simp]: "\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> H"

     and one_closed [simp]: "\<one> \<in> H"

     and m_inv_closed [intro,simp]: "x \<in> H \<Longrightarrow> inv x \<in> 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 \<in> H \<Longrightarrow> x \<in> carrier G"

   using subset by blast

 lemma subgroup_imp_subset:

   "subgroup H G \<Longrightarrow> H \<subseteq> carrier G"

   by (rule subgroup.subset)

 lemma (in subgroup) subgroup_is_group [intro]:

   assumes "group G"

   shows "group (G\<lparr>carrier := H\<rparr>)"

 proof -

   interpret group G by fact

   show ?thesis

     apply (rule monoid.group_l_invI)

     apply (unfold_locales) [1]

     apply (auto intro: m_assoc l_inv mem_carrier)

     done

 qed

 text {*

   Since @{term H} is nonempty, it contains some element @{term x}.  Since

   it is closed under inverse, it contains @{text "inv x"}.  Since

   it is closed under product, it contains @{text "x \<otimes> inv x = \<one>"}.

 *}

 lemma (in group) one_in_subset:

   "[| H \<subseteq> carrier G; H \<noteq> {}; \<forall>a \<in> H. inv a \<in> H; \<forall>a\<in>H. \<forall>b\<in>H. a \<otimes> b \<in> H |]

    ==> \<one> \<in> H"

 by force

 text {* A characterization of subgroups: closed, non-empty subset. *}

 lemma (in group) subgroupI:

   assumes subset: "H \<subseteq> carrier G" and non_empty: "H \<noteq> {}"

     and inv: "!!a. a \<in> H \<Longrightarrow> inv a \<in> H"

     and mult: "!!a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<otimes> b \<in> H"

   shows "subgroup H G"

 proof (simp add: subgroup_def assms)

   show "\<one> \<in> H" by (rule one_in_subset) (auto simp only: assms)

 qed

 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"

 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_subgroup a have "subgroup {} G" by simp

   with subgroup_nonempty show ?thesis by contradiction

 qed

 (*

 lemma (in monoid) Units_subgroup:

   "subgroup (Units G) G"

 *)

 subsection {* Direct Products *}

 definition

   DirProd :: "_ \<Rightarrow> _ \<Rightarrow> ('a \<times> 'b) monoid" (infixr "\<times>\<times>" 80) where

   "G \<times>\<times> H =

     \<lparr>carrier = carrier G \<times> carrier H,

      mult = (\<lambda>(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')),

      one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)\<rparr>"

 lemma DirProd_monoid:

   assumes "monoid G" and "monoid H"

   shows "monoid (G \<times>\<times> 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 \<times>\<times> 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 \<times>\<times> H) = carrier G \<times> carrier H"

   by (simp add: DirProd_def)

 lemma one_DirProd [simp]:

      "\<one>\<^bsub>G \<times>\<times> H\<^esub> = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)"

   by (simp add: DirProd_def)

 lemma mult_DirProd [simp]:

      "(g, h) \<otimes>\<^bsub>(G \<times>\<times> H)\<^esub> (g', h') = (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')"

   by (simp add: DirProd_def)

 lemma inv_DirProd [simp]:

   assumes "group G" and "group H"

   assumes g: "g \<in> carrier G"

       and h: "h \<in> carrier H"

   shows "m_inv (G \<times>\<times> H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)"

 proof -

   interpret G: group G by fact

   interpret H: group H by fact

   interpret Prod: group "G \<times>\<times> H"

     by (auto intro: DirProd_group group.intro group.axioms assms)

   show ?thesis by (simp add: Prod.inv_equality g h)

 qed

 subsection {* Homomorphisms and Isomorphisms *}

 definition

   hom :: "_ => _ => ('a => 'b) set" where

   "hom G H =

     {h. h \<in> carrier G -> carrier H &

       (\<forall>x \<in> carrier G. \<forall>y \<in> carrier G. h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y)}"

 lemma (in group) hom_compose:

   "[|h \<in> hom G H; i \<in> hom H I|] ==> compose (carrier G) i h \<in> hom G I"

 by (fastforce simp add: hom_def compose_def)

 definition

   iso :: "_ => _ => ('a => 'b) set" (infixr "\<cong>" 60)

   where "G \<cong> H = {h. h \<in> hom G H & bij_betw h (carrier G) (carrier H)}"

 lemma iso_refl: "(%x. x) \<in> G \<cong> G"

 by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def)

 lemma (in group) iso_sym:

      "h \<in> G \<cong> H \<Longrightarrow> inv_into (carrier G) h \<in> H \<cong> G"

 apply (simp add: iso_def bij_betw_inv_into) 

 apply (subgoal_tac "inv_into (carrier G) h \<in> carrier H \<rightarrow> carrier G") 

  prefer 2 apply (simp add: bij_betw_imp_funcset [OF bij_betw_inv_into]) 

 apply (simp add: hom_def bij_betw_def inv_into_f_eq f_inv_into_f Pi_def)

 done

 lemma (in group) iso_trans: 

      "[|h \<in> G \<cong> H; i \<in> H \<cong> I|] ==> (compose (carrier G) i h) \<in> G \<cong> I"

 by (auto simp add: iso_def hom_compose bij_betw_compose)

 lemma DirProd_commute_iso:

   shows "(\<lambda>(x,y). (y,x)) \<in> (G \<times>\<times> H) \<cong> (H \<times>\<times> G)"

 by (auto simp add: iso_def hom_def inj_on_def bij_betw_def)

 lemma DirProd_assoc_iso:

   shows "(\<lambda>(x,y,z). (x,(y,z))) \<in> (G \<times>\<times> H \<times>\<times> I) \<cong> (G \<times>\<times> (H \<times>\<times> I))"

 by (auto simp add: iso_def hom_def inj_on_def bij_betw_def)

 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: "h \<in> hom G H"

 lemma (in group_hom) hom_mult [simp]:

   "[| x \<in> carrier G; y \<in> carrier G |] ==> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"

 proof -

   assume "x \<in> carrier G" "y \<in> carrier G"

   with homh [unfolded hom_def] show ?thesis by simp

 qed

 lemma (in group_hom) hom_closed [simp]:

   "x \<in> carrier G ==> h x \<in> carrier H"

 proof -

   assume "x \<in> carrier G"

   with homh [unfolded hom_def] show ?thesis by auto

 qed

 lemma (in group_hom) one_closed [simp]:

   "h \<one> \<in> carrier H"

   by simp

 lemma (in group_hom) hom_one [simp]:

   "h \<one> = \<one>\<^bsub>H\<^esub>"

 proof -

   have "h \<one> \<otimes>\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = h \<one> \<otimes>\<^bsub>H\<^esub> h \<one>"

     by (simp add: hom_mult [symmetric] del: hom_mult)

   then show ?thesis by (simp del: r_one)

 qed

 lemma (in group_hom) inv_closed [simp]:

   "x \<in> carrier G ==> h (inv x) \<in> carrier H"

   by simp

 lemma (in group_hom) hom_inv [simp]:

   "x \<in> carrier G ==> h (inv x) = inv\<^bsub>H\<^esub> (h x)"

 proof -

   assume x: "x \<in> carrier G"

   then have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = \<one>\<^bsub>H\<^esub>"

     by (simp add: hom_mult [symmetric] del: hom_mult)

   also from x have "... = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)"

     by (simp add: hom_mult [symmetric] del: hom_mult)

   finally have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" .

   with x show ?thesis by (simp del: H.r_inv H.Units_r_inv)

 qed

 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: "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x"

 lemma (in comm_monoid) m_lcomm:

   "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow>

    x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"

 proof -

   assume xyz: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"

   from xyz have "x \<otimes> (y \<otimes> z) = (x \<otimes> y) \<otimes> z" by (simp add: m_assoc)

   also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm)

   also from xyz have "... = y \<otimes> (x \<otimes> 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 \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"

     and one_closed: "\<one> \<in> carrier G"

     and m_assoc:

       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>

       (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"

     and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"

     and m_comm:

       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> 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 \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"

   shows "comm_monoid G"

   by (rule comm_monoidI) (auto intro: m_assoc m_comm)

 (*lemma (in comm_monoid) r_one [simp]:

   "x \<in> carrier G ==> x \<otimes> \<one> = x"

 proof -

   assume G: "x \<in> carrier G"

   then have "x \<otimes> \<one> = \<one> \<otimes> x" by (simp add: m_comm)

   also from G have "... = x" by simp

   finally show ?thesis .

 qed*)

 lemma (in comm_monoid) nat_pow_distr:

   "[| x \<in> carrier G; y \<in> carrier G |] ==>

   (x \<otimes> y) (^) (n::nat) = x (^) n \<otimes> y (^) n"

   by (induct n) (simp, simp add: m_ac)

 locale comm_group = comm_monoid + group

 lemma (in group) group_comm_groupI:

   assumes m_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==>

       x \<otimes> y = y \<otimes> x"

   shows "comm_group G"

   by default (simp_all add: m_comm)

 lemma comm_groupI:

   fixes G (structure)

   assumes m_closed:

       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"

     and one_closed: "\<one> \<in> carrier G"

     and m_assoc:

       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>

       (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"

     and m_comm:

       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"

     and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"

     and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"

   shows "comm_group G"

   by (fast intro: group.group_comm_groupI groupI assms)

 lemma (in comm_group) inv_mult:

   "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv x \<otimes> inv y"

   by (simp add: m_ac inv_mult_group)

 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 = op =, le = op \<subseteq> |)"

   by default 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) is_monoid [intro, simp]:

   "monoid G"

   by (auto intro: monoid.intro m_assoc) 

 lemma (in group) subgroup_inv_equality:

   "[| subgroup H G; x \<in> H |] ==> m_inv (G (| carrier := H |)) x = inv x"

 apply (rule_tac inv_equality [THEN sym])

   apply (rule group.l_inv [OF subgroup_imp_group, simplified], assumption+)

  apply (rule subsetD [OF subgroup.subset], assumption+)

 apply (rule subsetD [OF subgroup.subset], assumption)

 apply (rule_tac group.inv_closed [OF subgroup_imp_group, simplified], assumption+)

 done

 theorem (in group) subgroups_Inter:

   assumes subgr: "(!!H. H \<in> A ==> subgroup H G)"

     and not_empty: "A ~= {}"

   shows "subgroup (\<Inter>A) G"

 proof (rule subgroupI)

   from subgr [THEN subgroup.subset] and not_empty

   show "\<Inter>A \<subseteq> carrier G" by blast

 next

   from subgr [THEN subgroup.one_closed]

   show "\<Inter>A ~= {}" by blast

 next

   fix x assume "x \<in> \<Inter>A"

   with subgr [THEN subgroup.m_inv_closed]

   show "inv x \<in> \<Inter>A" by blast

 next

   fix x y assume "x \<in> \<Inter>A" "y \<in> \<Inter>A"

   with subgr [THEN subgroup.m_closed]

   show "x \<otimes> y \<in> \<Inter>A" by blast

 qed

 theorem (in group) subgroups_complete_lattice:

   "complete_lattice (| carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq> |)"

     (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 "\<exists>G. greatest ?L G (carrier ?L)" ..

 next

   fix A

   assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}"

   then have Int_subgroup: "subgroup (\<Inter>A) G"

     by (fastforce intro: subgroups_Inter)

   have "greatest ?L (\<Inter>A) (Lower ?L A)" (is "greatest _ ?Int _")

   proof (rule greatest_LowerI)

     fix H

     assume H: "H \<in> 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 \<subseteq> H" by fastforce

     then show "le ?L ?Int H" by simp

   next

     fix H

     assume H: "H \<in> Lower ?L A"

     with L Int_subgroup show "le ?L H ?Int"

       by (fastforce simp: Lower_def intro: Inter_greatest)

   next

     show "A \<subseteq> carrier ?L" by (rule L)

   next

     show "?Int \<in> carrier ?L" by simp (rule Int_subgroup)

   qed

   then show "\<exists>I. greatest ?L I (Lower ?L A)" ..

 qed

 end