(* 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] => 'a" (infixl "⊗\<index>" 70)

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

definition

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

where "inv⇘_{G⇙}x = (THE y. y ∈ carrier G & x ⊗⇘_{G⇙}y = \<one>⇘_{G⇙}& y ⊗⇘_{G⇙}x = \<one>⇘_{G⇙})"

definition

Units :: "_ => 'a set"

--{*The set of invertible elements*}

where "Units G = {y. y ∈ carrier G & (∃x ∈ carrier G. x ⊗⇘_{G⇙}y = \<one>⇘_{G⇙}& y ⊗⇘_{G⇙}x = \<one>⇘_{G⇙})}"

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>⇘_{G⇙}(%u b. b ⊗⇘_{G⇙}a) n"

end

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

begin

definition "int_pow G a z =

(let p = nat_rec \<one>⇘_{G⇙}(%u b. b ⊗⇘_{G⇙}a)

in if z < 0 then inv⇘_{G⇙}(p (nat (-z))) else p (nat z))"

end

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]: "\<one> ∈ carrier G"

and l_one [simp]: "x ∈ carrier G ==> \<one> ⊗ x = x"

and r_one [simp]: "x ∈ carrier G ==> x ⊗ \<one> = x"

lemma monoidI:

fixes G (structure)

assumes m_closed:

"!!x y. [| x ∈ carrier G; y ∈ carrier G |] ==> x ⊗ y ∈ carrier G"

and one_closed: "\<one> ∈ 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 ==> \<one> ⊗ x = x"

and r_one: "!!x. x ∈ carrier G ==> x ⊗ \<one> = 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) inv_unique:

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

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 [intro, simp]:

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' = \<one>" "x' ⊗ x = \<one>"

unfolding Units_def by fast

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

unfolding Units_def by fast

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

moreover from x y xinv yinv have "x ⊗ (y ⊗ y') ⊗ 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' ⊗ x'"])

apply (auto simp: m_assoc)

done

qed

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

"\<one> ∈ Units G"

by (unfold Units_def) auto

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

"x ∈ Units G ==> inv x ∈ 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 ∈ Units G ==> ∃y ∈ carrier G. y ⊗ x = \<one>"

by (unfold Units_def) auto

lemma (in monoid) Units_r_inv_ex:

"x ∈ Units G ==> ∃y ∈ carrier G. x ⊗ y = \<one>"

by (unfold Units_def) auto

lemma (in monoid) Units_l_inv [simp]:

"x ∈ Units G ==> inv x ⊗ 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 ∈ Units G ==> x ⊗ 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 ∈ 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 = \<one>"

and G: "x ∈ Units G" "y ∈ Units G"

shows "y ⊗ x = \<one>"

proof -

from G have "x ⊗ y ⊗ x = x ⊗ \<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 ∈ 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) = \<one>"

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]:

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

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_pow:

"x ∈ 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 ∈ carrier G; y ∈ carrier G |] ==> x ⊗ y ∈ carrier G"

and one_closed [simp]: "\<one> ∈ 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 ==> \<one> ⊗ x = x"

and l_inv_ex: "!!x. x ∈ carrier G ==> ∃y ∈ carrier G. y ⊗ x = \<one>"

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 = \<one>" 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 ⊗ \<one> = 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 = \<one>" by fast

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

by (simp add: m_assoc [symmetric] l_inv)

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

qed

have inv_ex:

"!!x. x ∈ carrier G ==> ∃y ∈ carrier G. y ⊗ x = \<one> & x ⊗ y = \<one>"

proof -

fix x

assume x: "x ∈ carrier G"

with l_inv_ex obtain y where y: "y ∈ carrier G"

and l_inv: "y ⊗ x = \<one>" by fast

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

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

with x y have r_inv: "x ⊗ y = \<one>"

by simp

from x y show "∃y ∈ carrier G. y ⊗ x = \<one> & x ⊗ 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 ∈ carrier G ==> ∃y ∈ carrier G. y ⊗ 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 ∈ 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 = \<one>"

using Units_l_inv_ex by simp

lemma (in group) r_inv_ex [simp]:

"x ∈ carrier G ==> ∃y ∈ carrier G. x ⊗ y = \<one>"

using Units_r_inv_ex by simp

lemma (in group) l_inv [simp]:

"x ∈ carrier G ==> inv x ⊗ x = \<one>"

using Units_l_inv by simp

subsection {* Cancellation Laws and Basic Properties *}

lemma (in group) l_cancel [simp]:

"[| x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==>

(x ⊗ y = x ⊗ z) = (y = z)"

using Units_l_inv by simp

lemma (in group) r_inv [simp]:

"x ∈ carrier G ==> x ⊗ inv x = \<one>"

proof -

assume x: "x ∈ carrier G"

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

by (simp add: m_assoc [symmetric])

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

qed

lemma (in group) r_cancel [simp]:

"[| x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==>

(y ⊗ x = z ⊗ x) = (y = z)"

proof

assume eq: "y ⊗ x = z ⊗ x"

and G: "x ∈ carrier G" "y ∈ carrier G" "z ∈ carrier G"

then have "y ⊗ (x ⊗ inv x) = z ⊗ (x ⊗ 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 ∈ carrier G" "y ∈ carrier G" "z ∈ carrier G"

then show "y ⊗ x = z ⊗ x" by simp

qed

lemma (in group) inv_one [simp]:

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

proof -

have "inv \<one> = \<one> ⊗ (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 ∈ 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 = \<one>; x ∈ carrier G; y ∈ carrier G |] ==> y ⊗ x = \<one>"

by (rule Units_inv_comm) auto

lemma (in group) inv_equality:

"[|y ⊗ x = \<one>; x ∈ carrier G; y ∈ 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 ⊆ carrier G"

and m_closed [intro, simp]: "[|x ∈ H; y ∈ H|] ==> x ⊗ y ∈ H"

and one_closed [simp]: "\<one> ∈ 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 subgroup_imp_subset:

"subgroup H G ==> H ⊆ carrier G"

by (rule subgroup.subset)

lemma (in subgroup) subgroup_is_group [intro]:

assumes "group G"

shows "group (G(|carrier := H|)),)"

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 ⊗ inv x = \<one>"}.

*}

lemma (in group) one_in_subset:

"[| H ⊆ carrier G; H ≠ {}; ∀a ∈ H. inv a ∈ H; ∀a∈H. ∀b∈H. a ⊗ b ∈ H |]

==> \<one> ∈ 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 "\<one> ∈ 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 :: "_ => _ => ('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 = (\<one>⇘_{G⇙}, \<one>⇘_{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]:

"\<one>⇘_{G ×× H⇙}= (\<one>⇘_{G⇙}, \<one>⇘_{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 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

subsection {* Homomorphisms 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 (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)

definition

iso :: "_ => _ => ('a => 'b) set" (infixr "≅" 60)

where "G ≅ H = {h. h ∈ hom G H & bij_betw h (carrier G) (carrier H)}"

lemma iso_refl: "(%x. x) ∈ G ≅ G"

by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def)

lemma (in group) iso_sym:

"h ∈ G ≅ H ==> inv_into (carrier G) h ∈ H ≅ G"

apply (simp add: iso_def bij_betw_inv_into)

apply (subgoal_tac "inv_into (carrier G) h ∈ carrier H -> 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 ∈ G ≅ H; i ∈ H ≅ I|] ==> (compose (carrier G) i h) ∈ G ≅ I"

by (auto simp add: iso_def hom_compose bij_betw_compose)

lemma DirProd_commute_iso:

shows "(λ(x,y). (y,x)) ∈ (G ×× H) ≅ (H ×× G)"

by (auto simp add: iso_def hom_def inj_on_def bij_betw_def)

lemma DirProd_assoc_iso:

shows "(λ(x,y,z). (x,(y,z))) ∈ (G ×× H ×× I) ≅ (G ×× (H ×× 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 ∈ hom G H"

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 [simp]:

"h \<one> ∈ carrier H"

by simp

lemma (in group_hom) hom_one [simp]:

"h \<one> = \<one>⇘_{H⇙}"

proof -

have "h \<one> ⊗⇘_{H⇙}\<one>⇘_{H⇙}= h \<one> ⊗⇘_{H⇙}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 ∈ carrier G ==> h (inv x) ∈ carrier H"

by simp

lemma (in group_hom) hom_inv [simp]:

"x ∈ carrier G ==> h (inv x) = inv⇘_{H⇙}(h x)"

proof -

assume x: "x ∈ carrier G"

then have "h x ⊗⇘_{H⇙}h (inv x) = \<one>⇘_{H⇙}"

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

also from x have "... = h x ⊗⇘_{H⇙}inv⇘_{H⇙}(h x)"

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

finally have "h x ⊗⇘_{H⇙}h (inv x) = h x ⊗⇘_{H⇙}inv⇘_{H⇙}(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: "[|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: "\<one> ∈ 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 ==> \<one> ⊗ 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) r_one [simp]:

"x ∈ carrier G ==> x ⊗ \<one> = x"

proof -

assume G: "x ∈ carrier G"

then have "x ⊗ \<one> = \<one> ⊗ 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 ∈ carrier G; y ∈ carrier G |] ==>

(x ⊗ y) (^) (n::nat) = x (^) n ⊗ 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 ∈ carrier G; y ∈ carrier G |] ==>

x ⊗ y = y ⊗ x"

shows "comm_group G"

by default (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: "\<one> ∈ 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 ==> \<one> ⊗ x = x"

and l_inv_ex: "!!x. x ∈ carrier G ==> ∃y ∈ carrier G. y ⊗ x = \<one>"

shows "comm_group G"

by (fast intro: group.group_comm_groupI groupI assms)

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)

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 ⊆ |)"

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 ∈ 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 ∈ 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 ⊆ carrier G" by blast

next

from subgr [THEN subgroup.one_closed]

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

next

fix x assume "x ∈ \<Inter>A"

with subgr [THEN subgroup.m_inv_closed]

show "inv x ∈ \<Inter>A" by blast

next

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

with subgr [THEN subgroup.m_closed]

show "x ⊗ y ∈ \<Inter>A" by blast

qed

theorem (in group) subgroups_complete_lattice:

"complete_lattice (| carrier = {H. subgroup H G}, eq = op =, le = op ⊆ |)"

(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 (\<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 ∈ 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

end