src/HOL/Algebra/Group.thy
 author ballarin Tue, 04 Jul 2006 14:47:01 +0200 changeset 19984 29bb4659f80a parent 19981 c0f124a0d385 child 20318 0e0ea63fe768 permissions -rw-r--r--
Method intro_locales replaced by intro_locales and unfold_locales.

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

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

header {* Groups *}

theory Group imports FuncSet Lattice begin

section {* Monoids and Groups *}

text {*
Definitions follow \cite{Jacobson:1985}.
*}

subsection {* Definitions *}

record 'a monoid =  "'a partial_object" +
mult    :: "['a, 'a] \<Rightarrow> 'a" (infixl "\<otimes>\<index>" 70)
one     :: 'a ("\<one>\<index>")

constdefs (structure G)
m_inv :: "('a, 'b) monoid_scheme => 'a => 'a" ("inv\<index> _" [81] 80)
"inv x == (THE y. y \<in> carrier G & x \<otimes> y = \<one> & y \<otimes> x = \<one>)"

Units :: "_ => 'a set"
--{*The set of invertible elements*}
"Units G == {y. y \<in> carrier G & (\<exists>x \<in> carrier G. x \<otimes> y = \<one> & y \<otimes> x = \<one>)}"

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

defs (overloaded)
nat_pow_def: "pow G a n == nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) n"
int_pow_def: "pow G a z ==
let p = nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a)
in if neg z then inv\<^bsub>G\<^esub> (p (nat (-z))) else p (nat z)"

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: prems)

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_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:
"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:
"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)
with G show "y = z" by (simp add: Units_l_inv)
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 add: Units_l_inv Units_r_inv)
with x show ?thesis by (simp add: Units_closed)
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)

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" .

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 (fast intro!: group.intro monoid.intro group_axioms.intro
carrier_subset_Units intro: prems r_one)
qed

lemma (in monoid) monoid_groupI:
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] l_inv)
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)
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)
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 l_inv) (simp add: m_assoc [symmetric])
with G show ?thesis by (simp del: 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 neg z 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"

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]:
includes group G
shows "group (G\<lparr>carrier := H\<rparr>)"
by (rule groupI) (auto intro: m_assoc l_inv mem_carrier)

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 simp add: l_inv)

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 prems)
show "\<one> \<in> H" by (rule one_in_subset) (auto simp only: prems)
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)" "~ 0 < card H"
then have "finite H" by (blast intro: finite_subset [OF subset])
with prems 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 *}

constdefs
DirProd :: "_ \<Rightarrow> _ \<Rightarrow> ('a \<times> 'b) monoid"  (infixr "\<times>\<times>" 80)
"G \<times>\<times> H \<equiv> \<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:
includes monoid G + monoid H
shows "monoid (G \<times>\<times> H)"
proof -
from prems
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:
includes group G + group H
shows "group (G \<times>\<times> H)"
by (rule groupI)
(auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv
simp add: DirProd_def)

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]:
includes group G + 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)"
apply (rule group.inv_equality [OF DirProd_group])
apply (simp_all add: prems group.l_inv)
done

text{*This alternative proof of the previous result demonstrates interpret.
It uses @{text Prod.inv_equality} (available after @{text interpret})
instead of @{text "group.inv_equality [OF DirProd_group]"}. *}
lemma
includes group G + 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 Prod: group ["G \<times>\<times> H"]
by (auto intro: DirProd_group group.intro group.axioms prems)
show ?thesis by (simp add: Prod.inv_equality g h)
qed

subsection {* Homomorphisms and Isomorphisms *}

constdefs (structure G and H)
hom :: "_ => _ => ('a => 'b) set"
"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 hom_mult:
"[| h \<in> hom G H; x \<in> carrier G; y \<in> carrier G |]
==> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"
by (simp add: hom_def)

lemma hom_closed:
"[| h \<in> hom G H; x \<in> carrier G |] ==> h x \<in> carrier H"
by (auto simp add: hom_def funcset_mem)

lemma (in group) hom_compose:
"[|h \<in> hom G H; i \<in> hom H I|] ==> compose (carrier G) i h \<in> hom G I"
apply (auto simp add: hom_def funcset_compose)
apply (simp add: compose_def funcset_mem)
done

subsection {* Isomorphisms *}

constdefs
iso :: "_ => _ => ('a => 'b) set"  (infixr "\<cong>" 60)
"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 (carrier G) h \<in> H \<cong> G"
apply (simp add: iso_def bij_betw_Inv)
apply (subgoal_tac "Inv (carrier G) h \<in> carrier H \<rightarrow> carrier G")
prefer 2 apply (simp add: bij_betw_imp_funcset [OF bij_betw_Inv])
apply (simp add: hom_def bij_betw_def Inv_f_eq funcset_mem f_Inv_f)
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 Pi_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 Pi_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 = group G + group H + var h +
assumes homh: "h \<in> hom G H"
notes hom_mult [simp] = hom_mult [OF homh]
and hom_closed [simp] = hom_closed [OF homh]

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)
qed

subsection {* Commutative Structures *}

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

subsection {* Definition *}

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: prems 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 unfold_locales (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 prems)

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 {* 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}, le = op \<subseteq> |)"
by (rule partial_order.intro) 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 (rule subgroup.subgroup_is_group)

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}, 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 (fastsimp intro: subgroups_Inter)
have "greatest ?L (\<Inter>A) (Lower ?L A)"
(is "greatest ?L ?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 fastsimp
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 (fastsimp 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