(* Title: HOL/Algebra/Group.thy Author: Clemens Ballarin, started 4 February 2003 Based on work by Florian Kammueller, L C Paulson and Markus Wenzel. With additional contributions from Martin Baillon and Paulo EmÃlio de Vilhena. *) 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) one_unique: assumes "u ∈ carrier G" and "⋀x. x ∈ carrier G ⟹ u ⊗ x = x" shows "u = 𝟭" using assms(2)[OF one_closed] r_one[OF assms(1)] by simp 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 [simp, intro]: 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 by simp (metis m_assoc m_closed) 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 (simp add: Units_def m_inv_def) by (metis (mono_tags, lifting) inv_unique the_equality) 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, simp) by (metis (mono_tags, lifting) inv_unique the_equality) lemma (in monoid) Units_r_inv [simp]: "x ∈ Units G ==> x ⊗ inv x = 𝟭" by (metis (full_types) Units_closed Units_inv_closed Units_l_inv Units_r_inv_ex inv_unique) lemma (in monoid) inv_one [simp]: "inv 𝟭 = 𝟭" by (metis Units_one_closed Units_r_inv l_one monoid.Units_inv_closed monoid_axioms) 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_comm: "x ∈ carrier G ⟹ (x [^] (n::nat)) ⊗ (x [^] (m :: nat)) = (x [^] m) ⊗ (x [^] n)" using nat_pow_mult[of x n m] nat_pow_mult[of x m n] by (simp add: add.commute) lemma (in monoid) nat_pow_Suc2: "x ∈ carrier G ⟹ x [^] (Suc n) = x ⊗ (x [^] n)" using nat_pow_mult[of x 1 n] Suc_eq_plus1[of n] by (metis One_nat_def Suc_eq_plus1_left l_one nat.rec(1) nat_pow_Suc nat_pow_def) 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) lemma (in monoid) nat_pow_consistent: "x [^] (n :: nat) = x [^]⇘_{(G ⦇ carrier := H ⦈)⇙}n" unfolding nat_pow_def by simp (* 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" by simp subsection ‹Cancellation Laws and Basic Properties› lemma (in group) r_inv [simp]: "x ∈ carrier G ==> x ⊗ inv x = 𝟭" by simp lemma (in group) right_cancel [simp]: "[| x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==> (y ⊗ x = z ⊗ x) = (y = z)" by (metis inv_closed m_assoc r_inv r_one) 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" using inv_unique r_inv by blast (* 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: assumes "x ∈ carrier G" shows "x [^] (i + j::int) = x [^] i ⊗ x [^] j" proof - have [simp]: "-i - j = -j - i" by simp show ?thesis by (auto simp add: assms int_pow_def2 inv_solve_left inv_solve_right nat_add_distrib [symmetric] nat_pow_mult ) qed lemma (in group) nat_pow_inv: assumes "x ∈ carrier G" shows "(inv x) [^] (i :: nat) = inv (x [^] i)" proof (induction i) case 0 thus ?case by simp next case (Suc i) have "(inv x) [^] Suc i = ((inv x) [^] i) ⊗ inv x" by simp also have " ... = (inv (x [^] i)) ⊗ inv x" by (simp add: Suc.IH Suc.prems) also have " ... = inv (x ⊗ (x [^] i))" by (simp add: assms inv_mult_group) also have " ... = inv (x [^] (Suc i))" using assms nat_pow_Suc2 by auto finally show ?case . qed lemma (in group) int_pow_inv: "x ∈ carrier G ⟹ (inv x) [^] (i :: int) = inv (x [^] i)" by (simp add: nat_pow_inv int_pow_def2) lemma (in group) int_pow_pow: assumes "x ∈ carrier G" shows "(x [^] (n :: int)) [^] (m :: int) = x [^] (n * m :: int)" proof (cases) assume n_ge: "n ≥ 0" thus ?thesis proof (cases) assume m_ge: "m ≥ 0" thus ?thesis using n_ge nat_pow_pow[OF assms, of "nat n" "nat m"] int_pow_def2 by (simp add: mult_less_0_iff nat_mult_distrib) next assume m_lt: "¬ m ≥ 0" with n_ge show ?thesis apply (simp add: int_pow_def2 mult_less_0_iff) by (metis assms mult_minus_right n_ge nat_mult_distrib nat_pow_pow) qed next assume n_lt: "¬ n ≥ 0" thus ?thesis proof (cases) assume m_ge: "m ≥ 0" have "inv x [^] (nat m * nat (- n)) = inv x [^] nat (- (m * n))" by (metis (full_types) m_ge mult_minus_right nat_mult_distrib) with m_ge n_lt show ?thesis by (simp add: int_pow_def2 mult_less_0_iff assms mult.commute nat_pow_inv nat_pow_pow) next assume m_lt: "¬ m ≥ 0" thus ?thesis using n_lt by (auto simp: int_pow_def2 mult_less_0_iff assms nat_mult_distrib_neg nat_pow_inv nat_pow_pow) qed 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) (*Following subsection contributed by Martin Baillon*) subsection ‹Submonoids› locale submonoid = 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" lemma (in submonoid) is_submonoid: "submonoid H G" by (rule submonoid_axioms) lemma (in submonoid) mem_carrier [simp]: "x ∈ H ⟹ x ∈ carrier G" using subset by blast lemma (in submonoid) submonoid_is_monoid [intro]: assumes "monoid G" shows "monoid (G⦇carrier := H⦈)" proof - interpret monoid G by fact show ?thesis by (simp add: monoid_def m_assoc) qed lemma submonoid_nonempty: "~ submonoid {} G" by (blast dest: submonoid.one_closed) lemma (in submonoid) finite_monoid_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_submonoid a have "submonoid {} G" by simp with submonoid_nonempty show ?thesis by contradiction qed lemma (in monoid) monoid_incl_imp_submonoid : assumes "H ⊆ carrier G" and "monoid (G⦇carrier := H⦈)" shows "submonoid H G" proof (intro submonoid.intro[OF assms(1)]) have ab_eq : "⋀ a b. a ∈ H ⟹ b ∈ H ⟹ a ⊗⇘_{G⦇carrier := H⦈⇙}b = a ⊗ b" using assms by simp have "⋀a b. a ∈ H ⟹ b ∈ H ⟹ a ⊗ b ∈ carrier (G⦇carrier := H⦈) " using assms ab_eq unfolding group_def using monoid.m_closed by fastforce thus "⋀a b. a ∈ H ⟹ b ∈ H ⟹ a ⊗ b ∈ H" by simp show "𝟭 ∈ H " using monoid.one_closed[OF assms(2)] assms by simp qed lemma (in monoid) inv_unique': assumes "x ∈ carrier G" "y ∈ carrier G" shows "⟦ x ⊗ y = 𝟭; y ⊗ x = 𝟭 ⟧ ⟹ y = inv x" proof - assume "x ⊗ y = 𝟭" and l_inv: "y ⊗ x = 𝟭" hence unit: "x ∈ Units G" using assms unfolding Units_def by auto show "y = inv x" using inv_unique[OF l_inv Units_r_inv[OF unit] assms Units_inv_closed[OF unit]] . qed lemma (in monoid) m_inv_monoid_consistent: (* contributed by Paulo *) assumes "x ∈ Units (G ⦇ carrier := H ⦈)" and "submonoid H G" shows "inv⇘_{(G ⦇ carrier := H ⦈)⇙}x = inv x" proof - have monoid: "monoid (G ⦇ carrier := H ⦈)" using submonoid.submonoid_is_monoid[OF assms(2) monoid_axioms] . obtain y where y: "y ∈ H" "x ⊗ y = 𝟭" "y ⊗ x = 𝟭" using assms(1) unfolding Units_def by auto have x: "x ∈ H" and in_carrier: "x ∈ carrier G" "y ∈ carrier G" using y(1) submonoid.subset[OF assms(2)] assms(1) unfolding Units_def by auto show ?thesis using monoid.inv_unique'[OF monoid, of x y] x y using inv_unique'[OF in_carrier y(2-3)] by auto qed 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 (in subgroup) subgroup_is_group [intro]: assumes "group G" shows "group (G⦇carrier := H⦈)" proof - interpret group G by fact have "Group.monoid (G⦇carrier := H⦈)" by (simp add: monoid_axioms submonoid.intro submonoid.submonoid_is_monoid subset) then show ?thesis by (rule monoid.group_l_invI) (auto intro: l_inv mem_carrier) qed lemma subgroup_is_submonoid: assumes "subgroup H G" shows "submonoid H G" using assms by (auto intro: submonoid.intro simp add: subgroup_def) lemma (in group) subgroup_Units: assumes "subgroup H G" shows "H ⊆ Units (G ⦇ carrier := H ⦈)" using group.Units[OF subgroup.subgroup_is_group[OF assms group_axioms]] by simp lemma (in group) m_inv_consistent: assumes "subgroup H G" "x ∈ H" shows "inv⇘_{(G ⦇ carrier := H ⦈)⇙}x = inv x" using assms m_inv_monoid_consistent[OF _ subgroup_is_submonoid] subgroup_Units[of H] by auto lemma (in group) int_pow_consistent: (* by Paulo *) assumes "subgroup H G" "x ∈ H" shows "x [^] (n :: int) = x [^]⇘_{(G ⦇ carrier := H ⦈)⇙}n" proof (cases) assume ge: "n ≥ 0" hence "x [^] n = x [^] (nat n)" using int_pow_def2 by auto also have " ... = x [^]⇘_{(G ⦇ carrier := H ⦈)⇙}(nat n)" using nat_pow_consistent by simp also have " ... = x [^]⇘_{(G ⦇ carrier := H ⦈)⇙}n" using group.int_pow_def2[OF subgroup.subgroup_is_group[OF assms(1) is_group]] ge by auto finally show ?thesis . next assume "¬ n ≥ 0" hence lt: "n < 0" by simp hence "x [^] n = inv (x [^] (nat (- n)))" using int_pow_def2 by auto also have " ... = (inv x) [^] (nat (- n))" by (metis assms nat_pow_inv subgroup.mem_carrier) also have " ... = (inv⇘_{(G ⦇ carrier := H ⦈)⇙}x) [^]⇘_{(G ⦇ carrier := H ⦈)⇙}(nat (- n))" using m_inv_consistent[OF assms] nat_pow_consistent by auto also have " ... = inv⇘_{(G ⦇ carrier := H ⦈)⇙}(x [^]⇘_{(G ⦇ carrier := H ⦈)⇙}(nat (- n)))" using group.nat_pow_inv[OF subgroup.subgroup_is_group[OF assms(1) is_group]] assms(2) by auto also have " ... = x [^]⇘_{(G ⦇ carrier := H ⦈)⇙}n" using group.int_pow_def2[OF subgroup.subgroup_is_group[OF assms(1) is_group]] lt by auto finally show ?thesis . 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 lemma (in group) subgroupE: assumes "subgroup H G" shows "H ⊆ carrier G" and "H ≠ {}" and "⋀a. a ∈ H ⟹ inv a ∈ H" and "⋀a b. ⟦ a ∈ H; b ∈ H ⟧ ⟹ a ⊗ b ∈ H" using assms unfolding subgroup_def[of H G] by auto 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" using subset one_closed card_gt_0_iff finite_subset by blast (*Following 3 lemmas contributed by Martin Baillon*) lemma (in subgroup) subgroup_is_submonoid : "submonoid H G" by (simp add: submonoid.intro subset) lemma (in group) submonoid_subgroupI : assumes "submonoid H G" and "⋀a. a ∈ H ⟹ inv a ∈ H" shows "subgroup H G" by (metis assms subgroup_def submonoid_def) lemma (in group) group_incl_imp_subgroup: assumes "H ⊆ carrier G" and "group (G⦇carrier := H⦈)" shows "subgroup H G" proof (intro submonoid_subgroupI[OF monoid_incl_imp_submonoid[OF assms(1)]]) show "monoid (G⦇carrier := H⦈)" using group_def assms by blast have ab_eq : "⋀ a b. a ∈ H ⟹ b ∈ H ⟹ a ⊗⇘_{G⦇carrier := H⦈⇙}b = a ⊗ b" using assms by simp fix a assume aH : "a ∈ H" have " inv⇘_{G⦇carrier := H⦈⇙}a ∈ carrier G" using assms aH group.inv_closed[OF assms(2)] by auto moreover have "𝟭⇘_{G⦇carrier := H⦈⇙}= 𝟭" using assms monoid.one_closed ab_eq one_def by simp hence "a ⊗⇘_{G⦇carrier := H⦈⇙}inv⇘_{G⦇carrier := H⦈⇙}a= 𝟭" using assms ab_eq aH group.r_inv[OF assms(2)] by simp hence "a ⊗ inv⇘_{G⦇carrier := H⦈⇙}a= 𝟭" using aH assms group.inv_closed[OF assms(2)] ab_eq by simp ultimately have "inv⇘_{G⦇carrier := H⦈⇙}a = inv a" by (metis aH assms(1) contra_subsetD group.inv_inv is_group local.inv_equality) moreover have "inv⇘_{G⦇carrier := H⦈⇙}a ∈ H" using aH group.inv_closed[OF assms(2)] by auto ultimately show "inv a ∈ H" by auto qed 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 DirProd_assoc: "(G ×× H ×× I) = (G ×× (H ×× I))" by auto 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 lemma DirProd_subgroups : assumes "group G" and "subgroup H G" and "group K" and "subgroup I K" shows "subgroup (H × I) (G ×× K)" proof (intro group.group_incl_imp_subgroup[OF DirProd_group[OF assms(1)assms(3)]]) have "H ⊆ carrier G" "I ⊆ carrier K" using subgroup.subset assms apply blast+. thus "(H × I) ⊆ carrier (G ×× K)" unfolding DirProd_def by auto have "Group.group ((G⦇carrier := H⦈) ×× (K⦇carrier := I⦈))" using DirProd_group[OF subgroup.subgroup_is_group[OF assms(2)assms(1)] subgroup.subgroup_is_group[OF assms(4)assms(3)]]. moreover have "((G⦇carrier := H⦈) ×× (K⦇carrier := I⦈)) = ((G ×× K)⦇carrier := H × I⦈)" unfolding DirProd_def using assms apply simp. ultimately show "Group.group ((G ×× K)⦇carrier := H × I⦈)" by simp 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" where "iso G H = {h. h ∈ hom G H ∧ bij_betw h (carrier G) (carrier H)}" definition is_iso :: "_ ⇒ _ ⇒ bool" (infixr "≅" 60) where "G ≅ H = (iso G H ≠ {})" lemma iso_set_refl: "(λx. x) ∈ iso G G" by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) corollary iso_refl : "G ≅ G" using iso_set_refl unfolding is_iso_def by auto lemma (in group) iso_set_sym: assumes "h ∈ iso G H" shows "inv_into (carrier G) h ∈ iso H G" proof - have h: "h ∈ hom G H" "bij_betw h (carrier G) (carrier H)" using assms by (auto simp add: iso_def bij_betw_inv_into) then have HG: "bij_betw (inv_into (carrier G) h) (carrier H) (carrier G)" by (simp add: bij_betw_inv_into) have "inv_into (carrier G) h ∈ hom H G" unfolding hom_def proof safe show *: "⋀x. x ∈ carrier H ⟹ inv_into (carrier G) h x ∈ carrier G" by (meson HG bij_betwE) show "inv_into (carrier G) h (x ⊗⇘_{H⇙}y) = inv_into (carrier G) h x ⊗ inv_into (carrier G) h y" if "x ∈ carrier H" "y ∈ carrier H" for x y proof (rule inv_into_f_eq) show "inj_on h (carrier G)" using bij_betw_def h(2) by blast show "inv_into (carrier G) h x ⊗ inv_into (carrier G) h y ∈ carrier G" by (simp add: * that) show "h (inv_into (carrier G) h x ⊗ inv_into (carrier G) h y) = x ⊗⇘_{H⇙}y" using h bij_betw_inv_into_right [of h] unfolding hom_def by (simp add: "*" that) qed qed then show ?thesis by (simp add: Group.iso_def bij_betw_inv_into h) qed corollary (in group) iso_sym: "G ≅ H ⟹ H ≅ G" using iso_set_sym unfolding is_iso_def by auto lemma (in group) iso_set_trans: "[|h ∈ iso G H; i ∈ iso H I|] ==> (compose (carrier G) i h) ∈ iso G I" by (auto simp add: iso_def hom_compose bij_betw_compose) corollary (in group) iso_trans: "⟦G ≅ H ; H ≅ I⟧ ⟹ G ≅ I" using iso_set_trans unfolding is_iso_def by blast (* Next four lemmas contributed by Paulo. *) lemma (in monoid) hom_imp_img_monoid: assumes "h ∈ hom G H" shows "monoid (H ⦇ carrier := h ` (carrier G), one := h 𝟭⇘_{G⇙}⦈)" (is "monoid ?h_img") proof (rule monoidI) show "𝟭⇘_{?h_img⇙}∈ carrier ?h_img" by auto next fix x y z assume "x ∈ carrier ?h_img" "y ∈ carrier ?h_img" "z ∈ carrier ?h_img" then obtain g1 g2 g3 where g1: "g1 ∈ carrier G" "x = h g1" and g2: "g2 ∈ carrier G" "y = h g2" and g3: "g3 ∈ carrier G" "z = h g3" using image_iff[where ?f = h and ?A = "carrier G"] by auto have aux_lemma: "⋀a b. ⟦ a ∈ carrier G; b ∈ carrier G ⟧ ⟹ h a ⊗⇘_{(?h_img)⇙}h b = h (a ⊗ b)" using assms unfolding hom_def by auto show "x ⊗⇘_{(?h_img)⇙}𝟭⇘_{(?h_img)⇙}= x" using aux_lemma[OF g1(1) one_closed] g1(2) r_one[OF g1(1)] by simp show "𝟭⇘_{(?h_img)⇙}⊗⇘_{(?h_img)⇙}x = x" using aux_lemma[OF one_closed g1(1)] g1(2) l_one[OF g1(1)] by simp have "x ⊗⇘_{(?h_img)⇙}y = h (g1 ⊗ g2)" using aux_lemma g1 g2 by auto thus "x ⊗⇘_{(?h_img)⇙}y ∈ carrier ?h_img" using g1(1) g2(1) by simp have "(x ⊗⇘_{(?h_img)⇙}y) ⊗⇘_{(?h_img)⇙}z = h ((g1 ⊗ g2) ⊗ g3)" using aux_lemma g1 g2 g3 by auto also have " ... = h (g1 ⊗ (g2 ⊗ g3))" using m_assoc[OF g1(1) g2(1) g3(1)] by simp also have " ... = x ⊗⇘_{(?h_img)⇙}(y ⊗⇘_{(?h_img)⇙}z)" using aux_lemma g1 g2 g3 by auto finally show "(x ⊗⇘_{(?h_img)⇙}y) ⊗⇘_{(?h_img)⇙}z = x ⊗⇘_{(?h_img)⇙}(y ⊗⇘_{(?h_img)⇙}z)" . qed lemma (in group) hom_imp_img_group: assumes "h ∈ hom G H" shows "group (H ⦇ carrier := h ` (carrier G), one := h 𝟭⇘_{G⇙}⦈)" (is "group ?h_img") proof - interpret monoid ?h_img using hom_imp_img_monoid[OF assms] . show ?thesis proof (unfold_locales) show "carrier ?h_img ⊆ Units ?h_img" proof (auto simp add: Units_def) have aux_lemma: "⋀g1 g2. ⟦ g1 ∈ carrier G; g2 ∈ carrier G ⟧ ⟹ h g1 ⊗⇘_{H⇙}h g2 = h (g1 ⊗ g2)" using assms unfolding hom_def by auto fix g1 assume g1: "g1 ∈ carrier G" thus "∃g2 ∈ carrier G. (h g2) ⊗⇘_{H⇙}(h g1) = h 𝟭 ∧ (h g1) ⊗⇘_{H⇙}(h g2) = h 𝟭" using aux_lemma[OF g1 inv_closed[OF g1]] aux_lemma[OF inv_closed[OF g1] g1] inv_closed by auto qed qed qed lemma (in group) iso_imp_group: assumes "G ≅ H" and "monoid H" shows "group H" proof - obtain φ where phi: "φ ∈ iso G H" "inv_into (carrier G) φ ∈ iso H G" using iso_set_sym assms unfolding is_iso_def by blast define ψ where psi_def: "ψ = inv_into (carrier G) φ" have surj: "φ ` (carrier G) = (carrier H)" "ψ ` (carrier H) = (carrier G)" and inj: "inj_on φ (carrier G)" "inj_on ψ (carrier H)" and phi_hom: "⋀g1 g2. ⟦ g1 ∈ carrier G; g2 ∈ carrier G ⟧ ⟹ φ (g1 ⊗ g2) = (φ g1) ⊗⇘_{H⇙}(φ g2)" and psi_hom: "⋀h1 h2. ⟦ h1 ∈ carrier H; h2 ∈ carrier H ⟧ ⟹ ψ (h1 ⊗⇘_{H⇙}h2) = (ψ h1) ⊗ (ψ h2)" using phi psi_def unfolding iso_def bij_betw_def hom_def by auto have phi_one: "φ 𝟭 = 𝟭⇘_{H⇙}" proof - have "(φ 𝟭) ⊗⇘_{H⇙}𝟭⇘_{H⇙}= (φ 𝟭) ⊗⇘_{H⇙}(φ 𝟭)" by (metis assms(2) image_eqI monoid.r_one one_closed phi_hom r_one surj(1)) thus ?thesis by (metis (no_types, hide_lams) Units_eq Units_one_closed assms(2) f_inv_into_f imageI monoid.l_one monoid.one_closed phi_hom psi_def r_one surj) qed have "carrier H ⊆ Units H" proof fix h assume h: "h ∈ carrier H" let ?inv_h = "φ (inv (ψ h))" have "h ⊗⇘_{H⇙}?inv_h = φ (ψ h) ⊗⇘_{H⇙}?inv_h" by (simp add: f_inv_into_f h psi_def surj(1)) also have " ... = φ ((ψ h) ⊗ inv (ψ h))" by (metis h imageI inv_closed phi_hom surj(2)) also have " ... = φ 𝟭" by (simp add: h inv_into_into psi_def surj(1)) finally have 1: "h ⊗⇘_{H⇙}?inv_h = 𝟭⇘_{H⇙}" using phi_one by simp have "?inv_h ⊗⇘_{H⇙}h = ?inv_h ⊗⇘_{H⇙}φ (ψ h)" by (simp add: f_inv_into_f h psi_def surj(1)) also have " ... = φ (inv (ψ h) ⊗ (ψ h))" by (metis h imageI inv_closed phi_hom surj(2)) also have " ... = φ 𝟭" by (simp add: h inv_into_into psi_def surj(1)) finally have 2: "?inv_h ⊗⇘_{H⇙}h = 𝟭⇘_{H⇙}" using phi_one by simp thus "h ∈ Units H" unfolding Units_def using 1 2 h surj by fastforce qed thus ?thesis unfolding group_def group_axioms_def using assms(2) by simp qed corollary (in group) iso_imp_img_group: assumes "h ∈ iso G H" shows "group (H ⦇ one := h 𝟭 ⦈)" proof - let ?h_img = "H ⦇ carrier := h ` (carrier G), one := h 𝟭 ⦈" have "h ∈ iso G ?h_img" using assms unfolding iso_def hom_def bij_betw_def by auto hence "G ≅ ?h_img" unfolding is_iso_def by auto hence "group ?h_img" using iso_imp_group[of ?h_img] hom_imp_img_monoid[of h H] assms unfolding iso_def by simp moreover have "carrier H = carrier ?h_img" using assms unfolding iso_def bij_betw_def by simp hence "H ⦇ one := h 𝟭 ⦈ = ?h_img" by simp ultimately show ?thesis by simp qed lemma DirProd_commute_iso_set: shows "(λ(x,y). (y,x)) ∈ iso (G ×× H) (H ×× G)" by (auto simp add: iso_def hom_def inj_on_def bij_betw_def) corollary DirProd_commute_iso : "(G ×× H) ≅ (H ×× G)" using DirProd_commute_iso_set unfolding is_iso_def by blast lemma DirProd_assoc_iso_set: shows "(λ(x,y,z). (x,(y,z))) ∈ iso (G ×× H ×× I) (G ×× (H ×× I))" by (auto simp add: iso_def hom_def inj_on_def bij_betw_def) lemma (in group) DirProd_iso_set_trans: assumes "g ∈ iso G G2" and "h ∈ iso H I" shows "(λ(x,y). (g x, h y)) ∈ iso (G ×× H) (G2 ×× I)" proof- have "(λ(x,y). (g x, h y)) ∈ hom (G ×× H) (G2 ×× I)" using assms unfolding iso_def hom_def by auto moreover have " inj_on (λ(x,y). (g x, h y)) (carrier (G ×× H))" using assms unfolding iso_def DirProd_def bij_betw_def inj_on_def by auto moreover have "(λ(x, y). (g x, h y)) ` carrier (G ×× H) = carrier (G2 ×× I)" using assms unfolding iso_def bij_betw_def image_def DirProd_def by fastforce ultimately show "(λ(x,y). (g x, h y)) ∈ iso (G ×× H) (G2 ×× I)" unfolding iso_def bij_betw_def by auto qed corollary (in group) DirProd_iso_trans : assumes "G ≅ G2" and "H ≅ I" shows "G ×× H ≅ G2 ×× I" using DirProd_iso_set_trans assms unfolding is_iso_def by blast 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]: assumes "x ∈ carrier G" shows "h (inv x) = inv⇘_{H⇙}(h x)" proof - have "h x ⊗⇘_{H⇙}h (inv x) = h x ⊗⇘_{H⇙}inv⇘_{H⇙}(h x)" using assms by (simp flip: hom_mult) with assms 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 ⟹ (([^]) x) ∈ hom ⦇ carrier = UNIV, mult = (+), one = 0::int ⦈ G " unfolding hom_def by (simp add: int_pow_mult) (* Next six lemmas contributed by Paulo. *) lemma (in group_hom) img_is_subgroup: "subgroup (h ` (carrier G)) H" apply (rule subgroupI) apply (auto simp add: image_subsetI) apply (metis (no_types, hide_lams) G.inv_closed hom_inv image_iff) by (metis G.monoid_axioms hom_mult image_eqI monoid.m_closed) lemma (in group_hom) subgroup_img_is_subgroup: assumes "subgroup I G" shows "subgroup (h ` I) H" proof - have "h ∈ hom (G ⦇ carrier := I ⦈) H" using G.subgroupE[OF assms] subgroup.mem_carrier[OF assms] homh unfolding hom_def by auto hence "group_hom (G ⦇ carrier := I ⦈) H h" using subgroup.subgroup_is_group[OF assms G.is_group] is_group unfolding group_hom_def group_hom_axioms_def by simp thus ?thesis using group_hom.img_is_subgroup[of "G ⦇ carrier := I ⦈" H h] by simp qed lemma (in group_hom) induced_group_hom: assumes "subgroup I G" shows "group_hom (G ⦇ carrier := I ⦈) (H ⦇ carrier := h ` I ⦈) h" proof - have "h ∈ hom (G ⦇ carrier := I ⦈) (H ⦇ carrier := h ` I ⦈)" using homh subgroup.mem_carrier[OF assms] unfolding hom_def by auto thus ?thesis unfolding group_hom_def group_hom_axioms_def using subgroup.subgroup_is_group[OF assms G.is_group] subgroup.subgroup_is_group[OF subgroup_img_is_subgroup[OF assms] is_group] by simp qed lemma (in group) canonical_inj_is_hom: assumes "subgroup H G" shows "group_hom (G ⦇ carrier := H ⦈) G id" unfolding group_hom_def group_hom_axioms_def hom_def using subgroup.subgroup_is_group[OF assms is_group] is_group subgroup.subset[OF assms] by auto lemma (in group_hom) nat_pow_hom: "x ∈ carrier G ⟹ h (x [^] (n :: nat)) = (h x) [^]⇘_{H⇙}n" by (induction n) auto lemma (in group_hom) int_pow_hom: "x ∈ carrier G ⟹ h (x [^] (n :: int)) = (h x) [^]⇘_{H⇙}n" using int_pow_def2 nat_pow_hom by (simp add: G.int_pow_def2) 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) 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) lemma (in comm_monoid) submonoid_is_comm_monoid : assumes "submonoid H G" shows "comm_monoid (G⦇carrier := H⦈)" proof (intro monoid.monoid_comm_monoidI) show "monoid (G⦇carrier := H⦈)" using submonoid.submonoid_is_monoid assms comm_monoid_axioms comm_monoid_def by blast show "⋀x y. x ∈ carrier (G⦇carrier := H⦈) ⟹ y ∈ carrier (G⦇carrier := H⦈) ⟹ x ⊗⇘_{G⦇carrier := H⦈⇙}y = y ⊗⇘_{G⦇carrier := H⦈⇙}x" apply simp using assms comm_monoid_axioms_def submonoid.mem_carrier by (metis m_comm) qed 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 comm_groupE: fixes G (structure) assumes "comm_group G" shows "⋀x y. ⟦ x ∈ carrier G; y ∈ carrier G ⟧ ⟹ x ⊗ y ∈ carrier G" and "𝟭 ∈ carrier G" and "⋀x y z. ⟦ x ∈ carrier G; y ∈ carrier G; z ∈ carrier G ⟧ ⟹ (x ⊗ y) ⊗ z = x ⊗ (y ⊗ z)" and "⋀x y. ⟦ x ∈ carrier G; y ∈ carrier G ⟧ ⟹ x ⊗ y = y ⊗ x" and "⋀x. x ∈ carrier G ⟹ 𝟭 ⊗ x = x" and "⋀x. x ∈ carrier G ⟹ ∃y ∈ carrier G. y ⊗ x = 𝟭" apply (simp_all add: group.axioms assms comm_group.axioms comm_monoid.m_comm comm_monoid.m_ac(1)) by (simp_all add: Group.group.axioms(1) assms comm_group.axioms(2) monoid.m_closed group.r_inv_ex) 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) (* Next three lemmas contributed by Paulo. *) lemma (in comm_monoid) hom_imp_img_comm_monoid: assumes "h ∈ hom G H" shows "comm_monoid (H ⦇ carrier := h ` (carrier G), one := h 𝟭⇘_{G⇙}⦈)" (is "comm_monoid ?h_img") proof (rule monoid.monoid_comm_monoidI) show "monoid ?h_img" using hom_imp_img_monoid[OF assms] . next fix x y assume "x ∈ carrier ?h_img" "y ∈ carrier ?h_img" then obtain g1 g2 where g1: "g1 ∈ carrier G" "x = h g1" and g2: "g2 ∈ carrier G" "y = h g2" by auto have "x ⊗⇘_{(?h_img)⇙}y = h (g1 ⊗ g2)" using g1 g2 assms unfolding hom_def by auto also have " ... = h (g2 ⊗ g1)" using m_comm[OF g1(1) g2(1)] by simp also have " ... = y ⊗⇘_{(?h_img)⇙}x" using g1 g2 assms unfolding hom_def by auto finally show "x ⊗⇘_{(?h_img)⇙}y = y ⊗⇘_{(?h_img)⇙}x" . qed lemma (in comm_group) hom_imp_img_comm_group: assumes "h ∈ hom G H" shows "comm_group (H ⦇ carrier := h ` (carrier G), one := h 𝟭⇘_{G⇙}⦈)" unfolding comm_group_def using hom_imp_img_group[OF assms] hom_imp_img_comm_monoid[OF assms] by simp lemma (in comm_group) iso_imp_img_comm_group: assumes "h ∈ iso G H" shows "comm_group (H ⦇ one := h 𝟭⇘_{G⇙}⦈)" proof - let ?h_img = "H ⦇ carrier := h ` (carrier G), one := h 𝟭 ⦈" have "comm_group ?h_img" using hom_imp_img_comm_group[of h H] assms unfolding iso_def by auto moreover have "carrier H = carrier ?h_img" using assms unfolding iso_def bij_betw_def by simp hence "H ⦇ one := h 𝟭 ⦈ = ?h_img" by simp ultimately show ?thesis by simp qed lemma (in comm_group) iso_imp_comm_group: assumes "G ≅ H" "monoid H" shows "comm_group H" proof - obtain h where h: "h ∈ iso G H" using assms(1) unfolding is_iso_def by auto hence comm_gr: "comm_group (H ⦇ one := h 𝟭 ⦈)" using iso_imp_img_comm_group[of h H] by simp hence "⋀x. x ∈ carrier H ⟹ h 𝟭 ⊗⇘_{H⇙}x = x" using monoid.l_one[of "H ⦇ one := h 𝟭 ⦈"] unfolding comm_group_def comm_monoid_def by simp moreover have "h 𝟭 ∈ carrier H" using h one_closed unfolding iso_def hom_def by auto ultimately have "h 𝟭 = 𝟭⇘_{H⇙}" using monoid.one_unique[OF assms(2), of "h 𝟭"] by simp hence "H = H ⦇ one := h 𝟭 ⦈" by simp thus ?thesis using comm_gr by simp qed (*A subgroup of a subgroup is a subgroup of the group*) lemma (in group) incl_subgroup: assumes "subgroup J G" and "subgroup I (G⦇carrier:=J⦈)" shows "subgroup I G" unfolding subgroup_def proof have H1: "I ⊆ carrier (G⦇carrier:=J⦈)" using assms(2) subgroup.subset by blast also have H2: "...⊆J" by simp also have "...⊆(carrier G)" by (simp add: assms(1) subgroup.subset) finally have H: "I ⊆ carrier G" by simp have "(⋀x y. ⟦x ∈ I ; y ∈ I⟧ ⟹ x ⊗ y ∈ I)" using assms(2) by (auto simp add: subgroup_def) thus "I ⊆ carrier G ∧ (∀x y. x ∈ I ⟶ y ∈ I ⟶ x ⊗ y ∈ I)" using H by blast have K: "𝟭 ∈ I" using assms(2) by (auto simp add: subgroup_def) have "(⋀x. x ∈ I ⟹ inv x ∈ I)" using assms subgroup.m_inv_closed H by (metis H1 H2 m_inv_consistent subsetCE) thus "𝟭 ∈ I ∧ (∀x. x ∈ I ⟶ inv x ∈ I)" using K by blast qed (*A subgroup included in another subgroup is a subgroup of the subgroup*) lemma (in group) subgroup_incl: assumes "subgroup I G" and "subgroup J G" and "I ⊆ J" shows "subgroup I (G ⦇ carrier := J ⦈)" using group.group_incl_imp_subgroup[of "G ⦇ carrier := J ⦈" I] assms(1-2)[THEN subgroup.subgroup_is_group[OF _ group_axioms]] assms(3) by auto 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 = (=), le = (⊆)⦈" 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_mult_equality: "⟦ subgroup H G; h1 ∈ H; h2 ∈ H ⟧ ⟹ h1 ⊗⇘_{G ⦇ carrier := H ⦈⇙}h2 = h1 ⊗ h2" unfolding subgroup_def by simp 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 lemma (in group) subgroups_Inter_pair : assumes "subgroup I G" and "subgroup J G" shows "subgroup (I∩J) G" using subgroups_Inter[ where ?A = "{I,J}"] assms by auto theorem (in group) subgroups_complete_lattice: "complete_lattice ⦇carrier = {H. subgroup H G}, eq = (=), le = (⊆)⦈" (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 subsection‹Jeremy Avigad's @{text"More_Group"} material› text ‹ Show that the units in any monoid give rise to a group. The file Residues.thy provides some infrastructure to use facts about the unit group within the ring locale. › definition units_of :: "('a, 'b) monoid_scheme ⇒ 'a monoid" where "units_of G = ⦇carrier = Units G, Group.monoid.mult = Group.monoid.mult G, one = one G⦈" lemma (in monoid) units_group: "group (units_of G)" proof - have "⋀x y z. ⟦x ∈ Units G; y ∈ Units G; z ∈ Units G⟧ ⟹ x ⊗ y ⊗ z = x ⊗ (y ⊗ z)" by (simp add: Units_closed m_assoc) moreover have "⋀x. x ∈ Units G ⟹ ∃y∈Units G. y ⊗ x = 𝟭" using Units_l_inv by blast ultimately show ?thesis unfolding units_of_def by (force intro!: groupI) qed lemma (in comm_monoid) units_comm_group: "comm_group (units_of G)" proof - have "⋀x y. ⟦x ∈ carrier (units_of G); y ∈ carrier (units_of G)⟧ ⟹ x ⊗⇘_{units_of G⇙}y = y ⊗⇘_{units_of G⇙}x" by (simp add: Units_closed m_comm units_of_def) then show ?thesis by (rule group.group_comm_groupI [OF units_group]) auto qed lemma units_of_carrier: "carrier (units_of G) = Units G" by (auto simp: units_of_def) lemma units_of_mult: "mult (units_of G) = mult G" by (auto simp: units_of_def) lemma units_of_one: "one (units_of G) = one G" by (auto simp: units_of_def) lemma (in monoid) units_of_inv: assumes "x ∈ Units G" shows "m_inv (units_of G) x = m_inv G x" by (simp add: assms group.inv_equality units_group units_of_carrier units_of_mult units_of_one) lemma units_of_units [simp] : "Units (units_of G) = Units G" unfolding units_of_def Units_def by force lemma (in group) surj_const_mult: "a ∈ carrier G ⟹ (λx. a ⊗ x) ` carrier G = carrier G" apply (auto simp add: image_def) by (metis inv_closed inv_solve_left m_closed) lemma (in group) l_cancel_one [simp]: "x ∈ carrier G ⟹ a ∈ carrier G ⟹ x ⊗ a = x ⟷ a = one G" by (metis Units_eq Units_l_cancel monoid.r_one monoid_axioms one_closed) lemma (in group) r_cancel_one [simp]: "x ∈ carrier G ⟹ a ∈ carrier G ⟹ a ⊗ x = x ⟷ a = one G" by (metis monoid.l_one monoid_axioms one_closed right_cancel) lemma (in group) l_cancel_one' [simp]: "x ∈ carrier G ⟹ a ∈ carrier G ⟹ x = x ⊗ a ⟷ a = one G" using l_cancel_one by fastforce lemma (in group) r_cancel_one' [simp]: "x ∈ carrier G ⟹ a ∈ carrier G ⟹ x = a ⊗ x ⟷ a = one G" using r_cancel_one by fastforce end