(* 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 Complete_Lattice "HOL-Library.FuncSet" begin section ‹Monoids and Groups› subsection ‹Definitions› text ‹ Definitions follow @{cite "Jacobson:1985"}. › record 'a monoid = "'a partial_object" + mult :: "['a, 'a] ⇒ 'a" (infixl "⊗ı" 70) one :: 'a ("𝟭ı") definition m_inv :: "('a, 'b) monoid_scheme => 'a => 'a" ("invı _" [81] 80) where "inv⇘_{G⇙}x = (THE y. y ∈ carrier G & x ⊗⇘_{G⇙}y = 𝟭⇘_{G⇙}& y ⊗⇘_{G⇙}x = 𝟭⇘_{G⇙})" definition Units :: "_ => 'a set" ―‹The set of invertible elements› where "Units G = {y. y ∈ carrier G & (∃x ∈ carrier G. x ⊗⇘_{G⇙}y = 𝟭⇘_{G⇙}& y ⊗⇘_{G⇙}x = 𝟭⇘_{G⇙})}" consts pow :: "[('a, 'm) monoid_scheme, 'a, 'b::semiring_1] => 'a" (infixr "'(^')ı" 75) overloading nat_pow == "pow :: [_, 'a, nat] => 'a" begin definition "nat_pow G a n = rec_nat 𝟭⇘_{G⇙}(%u b. b ⊗⇘_{G⇙}a) n" end overloading int_pow == "pow :: [_, 'a, int] => 'a" begin definition "int_pow G a z = (let p = rec_nat 𝟭⇘_{G⇙}(%u b. b ⊗⇘_{G⇙}a) in if z < 0 then inv⇘_{G⇙}(p (nat (-z))) else p (nat z))" end lemma int_pow_int: "x (^)⇘_{G⇙}(int n) = x (^)⇘_{G⇙}n" by(simp add: int_pow_def nat_pow_def) locale monoid = fixes G (structure) assumes m_closed [intro, simp]: "⟦x ∈ carrier G; y ∈ carrier G⟧ ⟹ x ⊗ y ∈ carrier G" and m_assoc: "⟦x ∈ carrier G; y ∈ carrier G; z ∈ carrier G⟧ ⟹ (x ⊗ y) ⊗ z = x ⊗ (y ⊗ z)" and one_closed [intro, simp]: "𝟭 ∈ carrier G" and l_one [simp]: "x ∈ carrier G ⟹ 𝟭 ⊗ x = x" and r_one [simp]: "x ∈ carrier G ⟹ x ⊗ 𝟭 = x" lemma monoidI: fixes G (structure) assumes m_closed: "!!x y. [| x ∈ carrier G; y ∈ carrier G |] ==> x ⊗ y ∈ carrier G" and one_closed: "𝟭 ∈ carrier G" and m_assoc: "!!x y z. [| x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==> (x ⊗ y) ⊗ z = x ⊗ (y ⊗ z)" and l_one: "!!x. x ∈ carrier G ==> 𝟭 ⊗ x = x" and r_one: "!!x. x ∈ carrier G ==> x ⊗ 𝟭 = x" shows "monoid G" by (fast intro!: monoid.intro intro: assms) lemma (in monoid) Units_closed [dest]: "x ∈ Units G ==> x ∈ carrier G" by (unfold Units_def) fast lemma (in monoid) inv_unique: assumes eq: "y ⊗ x = 𝟭" "x ⊗ y' = 𝟭" and G: "x ∈ carrier G" "y ∈ carrier G" "y' ∈ carrier G" shows "y = y'" proof - from G eq have "y = y ⊗ (x ⊗ y')" by simp also from G have "... = (y ⊗ x) ⊗ y'" by (simp add: m_assoc) also from G eq have "... = y'" by simp finally show ?thesis . qed lemma (in monoid) Units_m_closed [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' = 𝟭" "x' ⊗ x = 𝟭" unfolding Units_def by fast from y obtain y' where y: "y ∈ carrier G" "y' ∈ carrier G" and yinv: "y ⊗ y' = 𝟭" "y' ⊗ y = 𝟭" unfolding Units_def by fast from x y xinv yinv have "y' ⊗ (x' ⊗ x) ⊗ y = 𝟭" by simp moreover from x y xinv yinv have "x ⊗ (y ⊗ y') ⊗ x' = 𝟭" by simp moreover note x y ultimately show ?thesis unfolding Units_def ― "Must avoid premature use of ‹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]: "𝟭 ∈ 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 = 𝟭" by (unfold Units_def) auto lemma (in monoid) Units_r_inv_ex: "x ∈ Units G ==> ∃y ∈ carrier G. x ⊗ y = 𝟭" by (unfold Units_def) auto lemma (in monoid) Units_l_inv [simp]: "x ∈ Units G ==> inv x ⊗ x = 𝟭" apply (unfold Units_def m_inv_def, 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 = 𝟭" 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 = 𝟭" and G: "x ∈ Units G" "y ∈ Units G" shows "y ⊗ x = 𝟭" proof - from G have "x ⊗ y ⊗ x = x ⊗ 𝟭" by (auto simp add: inv Units_closed) with G show ?thesis by (simp del: r_one add: m_assoc Units_closed) qed lemma (in monoid) carrier_not_empty: "carrier G ≠ {}" by auto 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) = 𝟭" by (simp add: nat_pow_def) lemma (in monoid) nat_pow_Suc [simp]: "x (^) (Suc n) = x (^) n ⊗ x" by (simp add: nat_pow_def) lemma (in monoid) nat_pow_one [simp]: "𝟭 (^) (n::nat) = 𝟭" by (induct n) simp_all lemma (in monoid) nat_pow_mult: "x ∈ carrier G ==> x (^) (n::nat) ⊗ x (^) m = x (^) (n + m)" by (induct m) (simp_all add: m_assoc [THEN sym]) lemma (in monoid) nat_pow_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]: "𝟭 ∈ carrier G" and m_assoc: "!!x y z. [| x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==> (x ⊗ y) ⊗ z = x ⊗ (y ⊗ z)" and l_one [simp]: "!!x. x ∈ carrier G ==> 𝟭 ⊗ x = x" and l_inv_ex: "!!x. x ∈ carrier G ==> ∃y ∈ carrier G. y ⊗ x = 𝟭" shows "group G" proof - have l_cancel [simp]: "!!x y z. [| x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==> (x ⊗ y = x ⊗ z) = (y = z)" proof fix x y z assume eq: "x ⊗ y = x ⊗ z" and G: "x ∈ carrier G" "y ∈ carrier G" "z ∈ carrier G" with l_inv_ex obtain x_inv where xG: "x_inv ∈ carrier G" and l_inv: "x_inv ⊗ x = 𝟭" by fast from G eq xG have "(x_inv ⊗ x) ⊗ y = (x_inv ⊗ x) ⊗ z" by (simp add: m_assoc) with G show "y = z" by (simp add: l_inv) next fix x y z assume eq: "y = z" and G: "x ∈ carrier G" "y ∈ carrier G" "z ∈ carrier G" then show "x ⊗ y = x ⊗ z" by simp qed have r_one: "!!x. x ∈ carrier G ==> x ⊗ 𝟭 = x" proof - fix x assume x: "x ∈ carrier G" with l_inv_ex obtain x_inv where xG: "x_inv ∈ carrier G" and l_inv: "x_inv ⊗ x = 𝟭" by fast from x xG have "x_inv ⊗ (x ⊗ 𝟭) = x_inv ⊗ x" by (simp add: m_assoc [symmetric] l_inv) with x xG show "x ⊗ 𝟭 = x" by simp qed have inv_ex: "!!x. x ∈ carrier G ==> ∃y ∈ carrier G. y ⊗ x = 𝟭 & x ⊗ y = 𝟭" proof - fix x assume x: "x ∈ carrier G" with l_inv_ex obtain y where y: "y ∈ carrier G" and l_inv: "y ⊗ x = 𝟭" by fast from x y have "y ⊗ (x ⊗ y) = y ⊗ 𝟭" by (simp add: m_assoc [symmetric] l_inv r_one) with x y have r_inv: "x ⊗ y = 𝟭" by simp from x y show "∃y ∈ carrier G. y ⊗ x = 𝟭 & x ⊗ y = 𝟭" by (fast intro: l_inv r_inv) qed then have carrier_subset_Units: "carrier G <= Units G" by (unfold Units_def) fast show ?thesis by standard (auto simp: r_one m_assoc carrier_subset_Units) qed lemma (in monoid) group_l_invI: assumes l_inv_ex: "!!x. x ∈ carrier G ==> ∃y ∈ carrier G. y ⊗ x = 𝟭" shows "group G" by (rule groupI) (auto intro: m_assoc l_inv_ex) lemma (in group) Units_eq [simp]: "Units G = carrier G" proof show "Units G <= carrier G" by fast next show "carrier G <= Units G" by (rule Units) qed lemma (in group) inv_closed [intro, simp]: "x ∈ carrier G ==> inv x ∈ carrier G" using Units_inv_closed by simp lemma (in group) l_inv_ex [simp]: "x ∈ carrier G ==> ∃y ∈ carrier G. y ⊗ x = 𝟭" using Units_l_inv_ex by simp lemma (in group) r_inv_ex [simp]: "x ∈ carrier G ==> ∃y ∈ carrier G. x ⊗ y = 𝟭" using Units_r_inv_ex by simp lemma (in group) l_inv [simp]: "x ∈ carrier G ==> inv x ⊗ x = 𝟭" 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 = 𝟭" proof - assume x: "x ∈ carrier G" then have "inv x ⊗ (x ⊗ inv x) = inv x ⊗ 𝟭" 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 𝟭 = 𝟭" proof - have "inv 𝟭 = 𝟭 ⊗ (inv 𝟭)" by (simp del: r_inv Units_r_inv) moreover have "... = 𝟭" 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 = 𝟭; x ∈ carrier G; y ∈ carrier G |] ==> y ⊗ x = 𝟭" by (rule Units_inv_comm) auto lemma (in group) inv_equality: "[|y ⊗ x = 𝟭; x ∈ carrier G; y ∈ carrier G|] ==> inv x = y" 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 (* Contributed by Joachim Breitner *) lemma (in group) inv_solve_left: "⟦ a ∈ carrier G; b ∈ carrier G; c ∈ carrier G ⟧ ⟹ a = inv b ⊗ c ⟷ c = b ⊗ a" by (metis inv_equality l_inv_ex l_one m_assoc r_inv) lemma (in group) inv_solve_right: "⟦ a ∈ carrier G; b ∈ carrier G; c ∈ carrier G ⟧ ⟹ a = b ⊗ inv c ⟷ b = a ⊗ c" by (metis inv_equality l_inv_ex l_one m_assoc r_inv) 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) = 𝟭" by (simp add: int_pow_def2) lemma (in group) int_pow_one [simp]: "𝟭 (^) (z::int) = 𝟭" by (simp add: int_pow_def2) (* The following are contributed by Joachim Breitner *) lemma (in group) int_pow_closed [intro, simp]: "x ∈ carrier G ==> x (^) (i::int) ∈ carrier G" by (simp add: int_pow_def2) lemma (in group) int_pow_1 [simp]: "x ∈ carrier G ⟹ x (^) (1::int) = x" by (simp add: int_pow_def2) lemma (in group) int_pow_neg: "x ∈ carrier G ⟹ x (^) (-i::int) = inv (x (^) i)" by (simp add: int_pow_def2) lemma (in group) int_pow_mult: "x ∈ carrier G ⟹ x (^) (i + j::int) = x (^) i ⊗ x (^) j" proof - have [simp]: "-i - j = -j - i" by simp assume "x : carrier G" then show ?thesis by (auto simp add: int_pow_def2 inv_solve_left inv_solve_right nat_add_distrib [symmetric] nat_pow_mult ) qed lemma (in group) int_pow_diff: "x ∈ carrier G ⟹ x (^) (n - m :: int) = x (^) n ⊗ inv (x (^) m)" by(simp only: diff_conv_add_uminus int_pow_mult int_pow_neg) lemma (in group) inj_on_multc: "c ∈ carrier G ⟹ inj_on (λx. x ⊗ c) (carrier G)" by(simp add: inj_on_def) lemma (in group) inj_on_cmult: "c ∈ carrier G ⟹ inj_on (λx. c ⊗ x) (carrier G)" by(simp add: inj_on_def) subsection ‹Subgroups› locale subgroup = fixes H and G (structure) assumes subset: "H ⊆ carrier G" and m_closed [intro, simp]: "⟦x ∈ H; y ∈ H⟧ ⟹ x ⊗ y ∈ H" and one_closed [simp]: "𝟭 ∈ H" and m_inv_closed [intro,simp]: "x ∈ H ⟹ inv x ∈ H" lemma (in subgroup) is_subgroup: "subgroup H G" by (rule subgroup_axioms) declare (in subgroup) group.intro [intro] lemma (in subgroup) mem_carrier [simp]: "x ∈ H ⟹ x ∈ carrier G" using subset by blast lemma 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 ‹inv x›. Since it is closed under product, it contains ‹x ⊗ inv x = 𝟭›. › lemma (in group) one_in_subset: "[| H ⊆ carrier G; H ≠ {}; ∀a ∈ H. inv a ∈ H; ∀a∈H. ∀b∈H. a ⊗ b ∈ H |] ==> 𝟭 ∈ H" by force text ‹A characterization of subgroups: closed, non-empty subset.› lemma (in group) subgroupI: assumes subset: "H ⊆ carrier G" and non_empty: "H ≠ {}" and inv: "!!a. a ∈ H ⟹ inv a ∈ H" and mult: "!!a b. ⟦a ∈ H; b ∈ H⟧ ⟹ a ⊗ b ∈ H" shows "subgroup H G" proof (simp add: subgroup_def assms) show "𝟭 ∈ H" by (rule one_in_subset) (auto simp only: assms) qed 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 = (𝟭⇘_{G⇙}, 𝟭⇘_{H⇙})⦈" lemma DirProd_monoid: assumes "monoid G" and "monoid H" shows "monoid (G ×× H)" proof - interpret G: monoid G by fact interpret H: monoid H by fact from assms show ?thesis by (unfold monoid_def DirProd_def, auto) qed text‹Does not use the previous result because it's easier just to use auto.› lemma DirProd_group: assumes "group G" and "group H" shows "group (G ×× H)" proof - interpret G: group G by fact interpret H: group H by fact show ?thesis by (rule groupI) (auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv simp add: DirProd_def) qed lemma carrier_DirProd [simp]: "carrier (G ×× H) = carrier G × carrier H" by (simp add: DirProd_def) lemma one_DirProd [simp]: "𝟭⇘_{G ×× H⇙}= (𝟭⇘_{G⇙}, 𝟭⇘_{H⇙})" by (simp add: DirProd_def) lemma mult_DirProd [simp]: "(g, h) ⊗⇘_{(G ×× H)⇙}(g', h') = (g ⊗⇘_{G⇙}g', h ⊗⇘_{H⇙}h')" by (simp add: DirProd_def) lemma 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 𝟭 ∈ carrier H" by simp lemma (in group_hom) hom_one [simp]: "h 𝟭 = 𝟭⇘_{H⇙}" proof - have "h 𝟭 ⊗⇘_{H⇙}𝟭⇘_{H⇙}= h 𝟭 ⊗⇘_{H⇙}h 𝟭" by (simp add: hom_mult [symmetric] del: hom_mult) then show ?thesis by (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) = 𝟭⇘_{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 (* Contributed by Joachim Breitner *) lemma (in group) int_pow_is_hom: "x ∈ carrier G ⟹ (op(^) x) ∈ hom ⦇ carrier = UNIV, mult = op +, one = 0::int ⦈ G " unfolding hom_def by (simp add: int_pow_mult) subsection ‹Commutative Structures› text ‹ Naming convention: multiplicative structures that are commutative are called \emph{commutative}, additive structures are called \emph{Abelian}. › locale comm_monoid = monoid + assumes m_comm: "⟦x ∈ carrier G; y ∈ carrier G⟧ ⟹ x ⊗ y = y ⊗ x" lemma (in comm_monoid) m_lcomm: "⟦x ∈ carrier G; y ∈ carrier G; z ∈ carrier G⟧ ⟹ x ⊗ (y ⊗ z) = y ⊗ (x ⊗ z)" proof - assume xyz: "x ∈ carrier G" "y ∈ carrier G" "z ∈ carrier G" from xyz have "x ⊗ (y ⊗ z) = (x ⊗ y) ⊗ z" by (simp add: m_assoc) also from xyz have "... = (y ⊗ x) ⊗ z" by (simp add: m_comm) also from xyz have "... = y ⊗ (x ⊗ z)" by (simp add: m_assoc) finally show ?thesis . qed lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm lemma comm_monoidI: fixes G (structure) assumes m_closed: "!!x y. [| x ∈ carrier G; y ∈ carrier G |] ==> x ⊗ y ∈ carrier G" and one_closed: "𝟭 ∈ carrier G" and m_assoc: "!!x y z. [| x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==> (x ⊗ y) ⊗ z = x ⊗ (y ⊗ z)" and l_one: "!!x. x ∈ carrier G ==> 𝟭 ⊗ x = x" and m_comm: "!!x y. [| x ∈ carrier G; y ∈ carrier G |] ==> x ⊗ y = y ⊗ x" shows "comm_monoid G" using l_one by (auto intro!: comm_monoid.intro comm_monoid_axioms.intro monoid.intro intro: assms simp: m_closed one_closed m_comm) lemma (in monoid) monoid_comm_monoidI: assumes m_comm: "!!x y. [| x ∈ carrier G; y ∈ carrier G |] ==> x ⊗ y = y ⊗ x" shows "comm_monoid G" by (rule comm_monoidI) (auto intro: m_assoc m_comm) (*lemma (in comm_monoid) r_one [simp]: "x ∈ carrier G ==> x ⊗ 𝟭 = x" proof - assume G: "x ∈ carrier G" then have "x ⊗ 𝟭 = 𝟭 ⊗ 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 standard (simp_all add: m_comm) lemma comm_groupI: fixes G (structure) assumes m_closed: "!!x y. [| x ∈ carrier G; y ∈ carrier G |] ==> x ⊗ y ∈ carrier G" and one_closed: "𝟭 ∈ carrier G" and m_assoc: "!!x y z. [| x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==> (x ⊗ y) ⊗ z = x ⊗ (y ⊗ z)" and m_comm: "!!x y. [| x ∈ carrier G; y ∈ carrier G |] ==> x ⊗ y = y ⊗ x" and l_one: "!!x. x ∈ carrier G ==> 𝟭 ⊗ x = x" and l_inv_ex: "!!x. x ∈ carrier G ==> ∃y ∈ carrier G. y ⊗ x = 𝟭" shows "comm_group G" by (fast intro: group.group_comm_groupI groupI assms) lemma (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 standard simp_all lemma (in group) subgroup_self: "subgroup (carrier G) G" by (rule subgroupI) auto lemma (in group) subgroup_imp_group: "subgroup H G ==> group (G⦇carrier := H⦈)" by (erule subgroup.subgroup_is_group) (rule group_axioms) lemma (in group) 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 (⋂A) G" proof (rule subgroupI) from subgr [THEN subgroup.subset] and not_empty show "⋂A ⊆ carrier G" by blast next from subgr [THEN subgroup.one_closed] show "⋂A ~= {}" by blast next fix x assume "x ∈ ⋂A" with subgr [THEN subgroup.m_inv_closed] show "inv x ∈ ⋂A" by blast next fix x y assume "x ∈ ⋂A" "y ∈ ⋂A" with subgr [THEN subgroup.m_closed] show "x ⊗ y ∈ ⋂A" by blast qed 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 (⋂A) G" by (fastforce intro: subgroups_Inter) have "greatest ?L (⋂A) (Lower ?L A)" (is "greatest _ ?Int _") proof (rule greatest_LowerI) fix H assume H: "H ∈ A" with L have subgroupH: "subgroup H G" by auto from subgroupH have groupH: "group (G ⦇carrier := H⦈)" (is "group ?H") by (rule subgroup_imp_group) from groupH have monoidH: "monoid ?H" by (rule group.is_monoid) from H have Int_subset: "?Int ⊆ H" by fastforce then show "le ?L ?Int H" by simp next fix H assume H: "H ∈ Lower ?L A" with L Int_subgroup show "le ?L H ?Int" by (fastforce simp: Lower_def intro: Inter_greatest) next show "A ⊆ carrier ?L" by (rule L) next show "?Int ∈ carrier ?L" by simp (rule Int_subgroup) qed then show "∃I. greatest ?L I (Lower ?L A)" .. qed end