# Theory Group

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theory Group
imports Lattice FuncSet
(*  Title:      HOL/Algebra/Group.thy    Author:     Clemens Ballarin, started 4 February 2003Based on work by Florian Kammueller, L C Paulson and Markus Wenzel.*)theory Groupimports Lattice "~~/src/HOL/Library/FuncSet"beginsection {* 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"endoverloading 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))"endlocale 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) fastlemma (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 .qedlemma (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)    doneqedlemma (in monoid) Units_one_closed [intro, simp]:  "\<one> ∈ Units G"  by (unfold Units_def) autolemma (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)  donelemma (in monoid) Units_l_inv_ex:  "x ∈ Units G ==> ∃y ∈ carrier G. y ⊗ x = \<one>"  by (unfold Units_def) autolemma (in monoid) Units_r_inv_ex:  "x ∈ Units G ==> ∃y ∈ carrier G. x ⊗ y = \<one>"  by (unfold Units_def) autolemma (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)  donelemma (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)  donelemma (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])qedlemma (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 simpnext  assume eq: "y = z"    and G: "x ∈ Units G"  "y ∈ carrier G"  "z ∈ carrier G"  then show "x ⊗ y = x ⊗ z" by simpqedlemma (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)qedlemma (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 simpqedlemma (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)qedtext {* 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_alllemma (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)qedlemma (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 fastnext  show "carrier G <= Units G" by (rule Units)qedlemma (in group) inv_closed [intro, simp]:  "x ∈ carrier G ==> inv x ∈ carrier G"  using Units_inv_closed by simplemma (in group) l_inv_ex [simp]:  "x ∈ carrier G ==> ∃y ∈ carrier G. y ⊗ x = \<one>"  using Units_l_inv_ex by simplemma (in group) r_inv_ex [simp]:  "x ∈ carrier G ==> ∃y ∈ carrier G. x ⊗ y = \<one>"  using Units_r_inv_ex by simplemma (in group) l_inv [simp]:  "x ∈ carrier G ==> inv x ⊗ x = \<one>"  using Units_l_inv by simpsubsection {* 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 simplemma (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)qedlemma (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 simpnext  assume eq: "y = z"    and G: "x ∈ carrier G"  "y ∈ carrier G"  "z ∈ carrier G"  then show "y ⊗ x = z ⊗ x" by simpqedlemma (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 .qedlemma (in group) inv_inv [simp]:  "x ∈ carrier G ==> inv (inv x) = x"  using Units_inv_inv by simplemma (in group) inv_inj:  "inj_on (m_inv G) (carrier G)"  using inv_inj_on_Units by simplemma (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)qedlemma (in group) inv_comm:  "[| x ⊗ y = \<one>; x ∈ carrier G; y ∈ carrier G |] ==> y ⊗ x = \<one>"  by (rule Units_inv_comm) autolemma (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)donetext {* 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 blastlemma 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)    doneqedtext {*  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 forcetext {* 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)qeddeclare 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 contradictionqed(*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) qedtext{*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)qedlemma 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)qedsubsection {* 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)donelemma (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 simpqedlemma (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 autoqedlemma (in group_hom) one_closed [simp]:  "h \<one> ∈ carrier H"  by simplemma (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)qedlemma (in group_hom) inv_closed [simp]:  "x ∈ carrier G ==> h (inv x) ∈ carrier H"  by simplemma (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)qedsubsection {* 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 .qedlemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcommlemma 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 + grouplemma (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_alllemma (in group) subgroup_self:  "subgroup (carrier G) G"  by (rule subgroupI) autolemma (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+)donetheorem (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 blastnext  from subgr [THEN subgroup.one_closed]  show "\<Inter>A ~= {}" by blastnext  fix x assume "x ∈ \<Inter>A"  with subgr [THEN subgroup.m_inv_closed]  show "inv x ∈ \<Inter>A" by blastnext  fix x y assume "x ∈ \<Inter>A" "y ∈ \<Inter>A"  with subgr [THEN subgroup.m_closed]  show "x ⊗ y ∈ \<Inter>A" by blastqedtheorem (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)" ..qedend