(* Title: HOL/Algebra/Coset.thy ID: $Id: Coset.thy,v 1.24 2008/03/17 21:34:23 wenzelm Exp $ Author: Florian Kammueller, with new proofs by L C Paulson, and Stephan Hohe *) theory Coset imports Group Exponent begin section {*Cosets and Quotient Groups*} constdefs (structure G) r_coset :: "[_, 'a set, 'a] => 'a set" (infixl "#>\<index>" 60) "H #> a ≡ \<Union>h∈H. {h ⊗ a}" l_coset :: "[_, 'a, 'a set] => 'a set" (infixl "<#\<index>" 60) "a <# H ≡ \<Union>h∈H. {a ⊗ h}" RCOSETS :: "[_, 'a set] => ('a set)set" ("rcosets\<index> _" [81] 80) "rcosets H ≡ \<Union>a∈carrier G. {H #> a}" set_mult :: "[_, 'a set ,'a set] => 'a set" (infixl "<#>\<index>" 60) "H <#> K ≡ \<Union>h∈H. \<Union>k∈K. {h ⊗ k}" SET_INV :: "[_,'a set] => 'a set" ("set'_inv\<index> _" [81] 80) "set_inv H ≡ \<Union>h∈H. {inv h}" locale normal = subgroup + group + assumes coset_eq: "(∀x ∈ carrier G. H #> x = x <# H)" abbreviation normal_rel :: "['a set, ('a, 'b) monoid_scheme] => bool" (infixl "\<lhd>" 60) where "H \<lhd> G ≡ normal H G" subsection {*Basic Properties of Cosets*} lemma (in group) coset_mult_assoc: "[| M ⊆ carrier G; g ∈ carrier G; h ∈ carrier G |] ==> (M #> g) #> h = M #> (g ⊗ h)" by (force simp add: r_coset_def m_assoc) lemma (in group) coset_mult_one [simp]: "M ⊆ carrier G ==> M #> \<one> = M" by (force simp add: r_coset_def) lemma (in group) coset_mult_inv1: "[| M #> (x ⊗ (inv y)) = M; x ∈ carrier G ; y ∈ carrier G; M ⊆ carrier G |] ==> M #> x = M #> y" apply (erule subst [of concl: "%z. M #> x = z #> y"]) apply (simp add: coset_mult_assoc m_assoc) done lemma (in group) coset_mult_inv2: "[| M #> x = M #> y; x ∈ carrier G; y ∈ carrier G; M ⊆ carrier G |] ==> M #> (x ⊗ (inv y)) = M " apply (simp add: coset_mult_assoc [symmetric]) apply (simp add: coset_mult_assoc) done lemma (in group) coset_join1: "[| H #> x = H; x ∈ carrier G; subgroup H G |] ==> x ∈ H" apply (erule subst) apply (simp add: r_coset_def) apply (blast intro: l_one subgroup.one_closed sym) done lemma (in group) solve_equation: "[|subgroup H G; x ∈ H; y ∈ H|] ==> ∃h∈H. y = h ⊗ x" apply (rule bexI [of _ "y ⊗ (inv x)"]) apply (auto simp add: subgroup.m_closed subgroup.m_inv_closed m_assoc subgroup.subset [THEN subsetD]) done lemma (in group) repr_independence: "[|y ∈ H #> x; x ∈ carrier G; subgroup H G|] ==> H #> x = H #> y" by (auto simp add: r_coset_def m_assoc [symmetric] subgroup.subset [THEN subsetD] subgroup.m_closed solve_equation) lemma (in group) coset_join2: "[|x ∈ carrier G; subgroup H G; x∈H|] ==> H #> x = H" --{*Alternative proof is to put @{term "x=\<one>"} in @{text repr_independence}.*} by (force simp add: subgroup.m_closed r_coset_def solve_equation) lemma (in monoid) r_coset_subset_G: "[| H ⊆ carrier G; x ∈ carrier G |] ==> H #> x ⊆ carrier G" by (auto simp add: r_coset_def) lemma (in group) rcosI: "[| h ∈ H; H ⊆ carrier G; x ∈ carrier G|] ==> h ⊗ x ∈ H #> x" by (auto simp add: r_coset_def) lemma (in group) rcosetsI: "[|H ⊆ carrier G; x ∈ carrier G|] ==> H #> x ∈ rcosets H" by (auto simp add: RCOSETS_def) text{*Really needed?*} lemma (in group) transpose_inv: "[| x ⊗ y = z; x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==> (inv x) ⊗ z = y" by (force simp add: m_assoc [symmetric]) lemma (in group) rcos_self: "[| x ∈ carrier G; subgroup H G |] ==> x ∈ H #> x" apply (simp add: r_coset_def) apply (blast intro: sym l_one subgroup.subset [THEN subsetD] subgroup.one_closed) done text (in group) {* Opposite of @{thm [source] "repr_independence"} *} lemma (in group) repr_independenceD: includes subgroup H G assumes ycarr: "y ∈ carrier G" and repr: "H #> x = H #> y" shows "y ∈ H #> x" apply (subst repr) apply (intro rcos_self) apply (rule ycarr) apply (rule is_subgroup) done text {* Elements of a right coset are in the carrier *} lemma (in subgroup) elemrcos_carrier: includes group assumes acarr: "a ∈ carrier G" and a': "a' ∈ H #> a" shows "a' ∈ carrier G" proof - from subset and acarr have "H #> a ⊆ carrier G" by (rule r_coset_subset_G) from this and a' show "a' ∈ carrier G" by fast qed lemma (in subgroup) rcos_const: includes group assumes hH: "h ∈ H" shows "H #> h = H" apply (unfold r_coset_def) apply rule apply rule apply clarsimp apply (intro subgroup.m_closed) apply (rule is_subgroup) apply assumption apply (rule hH) apply rule apply simp proof - fix h' assume h'H: "h' ∈ H" note carr = hH[THEN mem_carrier] h'H[THEN mem_carrier] from carr have a: "h' = (h' ⊗ inv h) ⊗ h" by (simp add: m_assoc) from h'H hH have "h' ⊗ inv h ∈ H" by simp from this and a show "∃x∈H. h' = x ⊗ h" by fast qed text {* Step one for lemma @{text "rcos_module"} *} lemma (in subgroup) rcos_module_imp: includes group assumes xcarr: "x ∈ carrier G" and x'cos: "x' ∈ H #> x" shows "(x' ⊗ inv x) ∈ H" proof - from xcarr x'cos have x'carr: "x' ∈ carrier G" by (rule elemrcos_carrier[OF is_group]) from xcarr have ixcarr: "inv x ∈ carrier G" by simp from x'cos have "∃h∈H. x' = h ⊗ x" unfolding r_coset_def by fast from this obtain h where hH: "h ∈ H" and x': "x' = h ⊗ x" by auto from hH and subset have hcarr: "h ∈ carrier G" by fast note carr = xcarr x'carr hcarr from x' and carr have "x' ⊗ (inv x) = (h ⊗ x) ⊗ (inv x)" by fast also from carr have "… = h ⊗ (x ⊗ inv x)" by (simp add: m_assoc) also from carr have "… = h ⊗ \<one>" by simp also from carr have "… = h" by simp finally have "x' ⊗ (inv x) = h" by simp from hH this show "x' ⊗ (inv x) ∈ H" by simp qed text {* Step two for lemma @{text "rcos_module"} *} lemma (in subgroup) rcos_module_rev: includes group assumes carr: "x ∈ carrier G" "x' ∈ carrier G" and xixH: "(x' ⊗ inv x) ∈ H" shows "x' ∈ H #> x" proof - from xixH have "∃h∈H. x' ⊗ (inv x) = h" by fast from this obtain h where hH: "h ∈ H" and hsym: "x' ⊗ (inv x) = h" by fast from hH subset have hcarr: "h ∈ carrier G" by simp note carr = carr hcarr from hsym[symmetric] have "h ⊗ x = x' ⊗ (inv x) ⊗ x" by fast also from carr have "… = x' ⊗ ((inv x) ⊗ x)" by (simp add: m_assoc) also from carr have "… = x' ⊗ \<one>" by (simp add: l_inv) also from carr have "… = x'" by simp finally have "h ⊗ x = x'" by simp from this[symmetric] and hH show "x' ∈ H #> x" unfolding r_coset_def by fast qed text {* Module property of right cosets *} lemma (in subgroup) rcos_module: includes group assumes carr: "x ∈ carrier G" "x' ∈ carrier G" shows "(x' ∈ H #> x) = (x' ⊗ inv x ∈ H)" proof assume "x' ∈ H #> x" from this and carr show "x' ⊗ inv x ∈ H" by (intro rcos_module_imp[OF is_group]) next assume "x' ⊗ inv x ∈ H" from this and carr show "x' ∈ H #> x" by (intro rcos_module_rev[OF is_group]) qed text {* Right cosets are subsets of the carrier. *} lemma (in subgroup) rcosets_carrier: includes group assumes XH: "X ∈ rcosets H" shows "X ⊆ carrier G" proof - from XH have "∃x∈ carrier G. X = H #> x" unfolding RCOSETS_def by fast from this obtain x where xcarr: "x∈ carrier G" and X: "X = H #> x" by fast from subset and xcarr show "X ⊆ carrier G" unfolding X by (rule r_coset_subset_G) qed text {* Multiplication of general subsets *} lemma (in monoid) set_mult_closed: assumes Acarr: "A ⊆ carrier G" and Bcarr: "B ⊆ carrier G" shows "A <#> B ⊆ carrier G" apply rule apply (simp add: set_mult_def, clarsimp) proof - fix a b assume "a ∈ A" from this and Acarr have acarr: "a ∈ carrier G" by fast assume "b ∈ B" from this and Bcarr have bcarr: "b ∈ carrier G" by fast from acarr bcarr show "a ⊗ b ∈ carrier G" by (rule m_closed) qed lemma (in comm_group) mult_subgroups: assumes subH: "subgroup H G" and subK: "subgroup K G" shows "subgroup (H <#> K) G" apply (rule subgroup.intro) apply (intro set_mult_closed subgroup.subset[OF subH] subgroup.subset[OF subK]) apply (simp add: set_mult_def) apply clarsimp defer 1 apply (simp add: set_mult_def) defer 1 apply (simp add: set_mult_def, clarsimp) defer 1 proof - fix ha hb ka kb assume haH: "ha ∈ H" and hbH: "hb ∈ H" and kaK: "ka ∈ K" and kbK: "kb ∈ K" note carr = haH[THEN subgroup.mem_carrier[OF subH]] hbH[THEN subgroup.mem_carrier[OF subH]] kaK[THEN subgroup.mem_carrier[OF subK]] kbK[THEN subgroup.mem_carrier[OF subK]] from carr have "(ha ⊗ ka) ⊗ (hb ⊗ kb) = ha ⊗ (ka ⊗ hb) ⊗ kb" by (simp add: m_assoc) also from carr have "… = ha ⊗ (hb ⊗ ka) ⊗ kb" by (simp add: m_comm) also from carr have "… = (ha ⊗ hb) ⊗ (ka ⊗ kb)" by (simp add: m_assoc) finally have eq: "(ha ⊗ ka) ⊗ (hb ⊗ kb) = (ha ⊗ hb) ⊗ (ka ⊗ kb)" . from haH hbH have hH: "ha ⊗ hb ∈ H" by (simp add: subgroup.m_closed[OF subH]) from kaK kbK have kK: "ka ⊗ kb ∈ K" by (simp add: subgroup.m_closed[OF subK]) from hH and kK and eq show "∃h'∈H. ∃k'∈K. (ha ⊗ ka) ⊗ (hb ⊗ kb) = h' ⊗ k'" by fast next have "\<one> = \<one> ⊗ \<one>" by simp from subgroup.one_closed[OF subH] subgroup.one_closed[OF subK] this show "∃h∈H. ∃k∈K. \<one> = h ⊗ k" by fast next fix h k assume hH: "h ∈ H" and kK: "k ∈ K" from hH[THEN subgroup.mem_carrier[OF subH]] kK[THEN subgroup.mem_carrier[OF subK]] have "inv (h ⊗ k) = inv h ⊗ inv k" by (simp add: inv_mult_group m_comm) from subgroup.m_inv_closed[OF subH hH] and subgroup.m_inv_closed[OF subK kK] and this show "∃ha∈H. ∃ka∈K. inv (h ⊗ k) = ha ⊗ ka" by fast qed lemma (in subgroup) lcos_module_rev: includes group assumes carr: "x ∈ carrier G" "x' ∈ carrier G" and xixH: "(inv x ⊗ x') ∈ H" shows "x' ∈ x <# H" proof - from xixH have "∃h∈H. (inv x) ⊗ x' = h" by fast from this obtain h where hH: "h ∈ H" and hsym: "(inv x) ⊗ x' = h" by fast from hH subset have hcarr: "h ∈ carrier G" by simp note carr = carr hcarr from hsym[symmetric] have "x ⊗ h = x ⊗ ((inv x) ⊗ x')" by fast also from carr have "… = (x ⊗ (inv x)) ⊗ x'" by (simp add: m_assoc[symmetric]) also from carr have "… = \<one> ⊗ x'" by simp also from carr have "… = x'" by simp finally have "x ⊗ h = x'" by simp from this[symmetric] and hH show "x' ∈ x <# H" unfolding l_coset_def by fast qed subsection {* Normal subgroups *} lemma normal_imp_subgroup: "H \<lhd> G ==> subgroup H G" by (simp add: normal_def subgroup_def) lemma (in group) normalI: "subgroup H G ==> (∀x ∈ carrier G. H #> x = x <# H) ==> H \<lhd> G" by (simp add: normal_def normal_axioms_def prems) lemma (in normal) inv_op_closed1: "[|x ∈ carrier G; h ∈ H|] ==> (inv x) ⊗ h ⊗ x ∈ H" apply (insert coset_eq) apply (auto simp add: l_coset_def r_coset_def) apply (drule bspec, assumption) apply (drule equalityD1 [THEN subsetD], blast, clarify) apply (simp add: m_assoc) apply (simp add: m_assoc [symmetric]) done lemma (in normal) inv_op_closed2: "[|x ∈ carrier G; h ∈ H|] ==> x ⊗ h ⊗ (inv x) ∈ H" apply (subgoal_tac "inv (inv x) ⊗ h ⊗ (inv x) ∈ H") apply (simp add: ) apply (blast intro: inv_op_closed1) done text{*Alternative characterization of normal subgroups*} lemma (in group) normal_inv_iff: "(N \<lhd> G) = (subgroup N G & (∀x ∈ carrier G. ∀h ∈ N. x ⊗ h ⊗ (inv x) ∈ N))" (is "_ = ?rhs") proof assume N: "N \<lhd> G" show ?rhs by (blast intro: N normal.inv_op_closed2 normal_imp_subgroup) next assume ?rhs hence sg: "subgroup N G" and closed: "!!x. x∈carrier G ==> ∀h∈N. x ⊗ h ⊗ inv x ∈ N" by auto hence sb: "N ⊆ carrier G" by (simp add: subgroup.subset) show "N \<lhd> G" proof (intro normalI [OF sg], simp add: l_coset_def r_coset_def, clarify) fix x assume x: "x ∈ carrier G" show "(\<Union>h∈N. {h ⊗ x}) = (\<Union>h∈N. {x ⊗ h})" proof show "(\<Union>h∈N. {h ⊗ x}) ⊆ (\<Union>h∈N. {x ⊗ h})" proof clarify fix n assume n: "n ∈ N" show "n ⊗ x ∈ (\<Union>h∈N. {x ⊗ h})" proof from closed [of "inv x"] show "inv x ⊗ n ⊗ x ∈ N" by (simp add: x n) show "n ⊗ x ∈ {x ⊗ (inv x ⊗ n ⊗ x)}" by (simp add: x n m_assoc [symmetric] sb [THEN subsetD]) qed qed next show "(\<Union>h∈N. {x ⊗ h}) ⊆ (\<Union>h∈N. {h ⊗ x})" proof clarify fix n assume n: "n ∈ N" show "x ⊗ n ∈ (\<Union>h∈N. {h ⊗ x})" proof show "x ⊗ n ⊗ inv x ∈ N" by (simp add: x n closed) show "x ⊗ n ∈ {x ⊗ n ⊗ inv x ⊗ x}" by (simp add: x n m_assoc sb [THEN subsetD]) qed qed qed qed qed subsection{*More Properties of Cosets*} lemma (in group) lcos_m_assoc: "[| M ⊆ carrier G; g ∈ carrier G; h ∈ carrier G |] ==> g <# (h <# M) = (g ⊗ h) <# M" by (force simp add: l_coset_def m_assoc) lemma (in group) lcos_mult_one: "M ⊆ carrier G ==> \<one> <# M = M" by (force simp add: l_coset_def) lemma (in group) l_coset_subset_G: "[| H ⊆ carrier G; x ∈ carrier G |] ==> x <# H ⊆ carrier G" by (auto simp add: l_coset_def subsetD) lemma (in group) l_coset_swap: "[|y ∈ x <# H; x ∈ carrier G; subgroup H G|] ==> x ∈ y <# H" proof (simp add: l_coset_def) assume "∃h∈H. y = x ⊗ h" and x: "x ∈ carrier G" and sb: "subgroup H G" then obtain h' where h': "h' ∈ H & x ⊗ h' = y" by blast show "∃h∈H. x = y ⊗ h" proof show "x = y ⊗ inv h'" using h' x sb by (auto simp add: m_assoc subgroup.subset [THEN subsetD]) show "inv h' ∈ H" using h' sb by (auto simp add: subgroup.subset [THEN subsetD] subgroup.m_inv_closed) qed qed lemma (in group) l_coset_carrier: "[| y ∈ x <# H; x ∈ carrier G; subgroup H G |] ==> y ∈ carrier G" by (auto simp add: l_coset_def m_assoc subgroup.subset [THEN subsetD] subgroup.m_closed) lemma (in group) l_repr_imp_subset: assumes y: "y ∈ x <# H" and x: "x ∈ carrier G" and sb: "subgroup H G" shows "y <# H ⊆ x <# H" proof - from y obtain h' where "h' ∈ H" "x ⊗ h' = y" by (auto simp add: l_coset_def) thus ?thesis using x sb by (auto simp add: l_coset_def m_assoc subgroup.subset [THEN subsetD] subgroup.m_closed) qed lemma (in group) l_repr_independence: assumes y: "y ∈ x <# H" and x: "x ∈ carrier G" and sb: "subgroup H G" shows "x <# H = y <# H" proof show "x <# H ⊆ y <# H" by (rule l_repr_imp_subset, (blast intro: l_coset_swap l_coset_carrier y x sb)+) show "y <# H ⊆ x <# H" by (rule l_repr_imp_subset [OF y x sb]) qed lemma (in group) setmult_subset_G: "[|H ⊆ carrier G; K ⊆ carrier G|] ==> H <#> K ⊆ carrier G" by (auto simp add: set_mult_def subsetD) lemma (in group) subgroup_mult_id: "subgroup H G ==> H <#> H = H" apply (auto simp add: subgroup.m_closed set_mult_def Sigma_def image_def) apply (rule_tac x = x in bexI) apply (rule bexI [of _ "\<one>"]) apply (auto simp add: subgroup.m_closed subgroup.one_closed r_one subgroup.subset [THEN subsetD]) done subsubsection {* Set of Inverses of an @{text r_coset}. *} lemma (in normal) rcos_inv: assumes x: "x ∈ carrier G" shows "set_inv (H #> x) = H #> (inv x)" proof (simp add: r_coset_def SET_INV_def x inv_mult_group, safe) fix h assume "h ∈ H" show "inv x ⊗ inv h ∈ (\<Union>j∈H. {j ⊗ inv x})" proof show "inv x ⊗ inv h ⊗ x ∈ H" by (simp add: inv_op_closed1 prems) show "inv x ⊗ inv h ∈ {inv x ⊗ inv h ⊗ x ⊗ inv x}" by (simp add: prems m_assoc) qed next fix h assume "h ∈ H" show "h ⊗ inv x ∈ (\<Union>j∈H. {inv x ⊗ inv j})" proof show "x ⊗ inv h ⊗ inv x ∈ H" by (simp add: inv_op_closed2 prems) show "h ⊗ inv x ∈ {inv x ⊗ inv (x ⊗ inv h ⊗ inv x)}" by (simp add: prems m_assoc [symmetric] inv_mult_group) qed qed subsubsection {*Theorems for @{text "<#>"} with @{text "#>"} or @{text "<#"}.*} lemma (in group) setmult_rcos_assoc: "[|H ⊆ carrier G; K ⊆ carrier G; x ∈ carrier G|] ==> H <#> (K #> x) = (H <#> K) #> x" by (force simp add: r_coset_def set_mult_def m_assoc) lemma (in group) rcos_assoc_lcos: "[|H ⊆ carrier G; K ⊆ carrier G; x ∈ carrier G|] ==> (H #> x) <#> K = H <#> (x <# K)" by (force simp add: r_coset_def l_coset_def set_mult_def m_assoc) lemma (in normal) rcos_mult_step1: "[|x ∈ carrier G; y ∈ carrier G|] ==> (H #> x) <#> (H #> y) = (H <#> (x <# H)) #> y" by (simp add: setmult_rcos_assoc subset r_coset_subset_G l_coset_subset_G rcos_assoc_lcos) lemma (in normal) rcos_mult_step2: "[|x ∈ carrier G; y ∈ carrier G|] ==> (H <#> (x <# H)) #> y = (H <#> (H #> x)) #> y" by (insert coset_eq, simp add: normal_def) lemma (in normal) rcos_mult_step3: "[|x ∈ carrier G; y ∈ carrier G|] ==> (H <#> (H #> x)) #> y = H #> (x ⊗ y)" by (simp add: setmult_rcos_assoc coset_mult_assoc subgroup_mult_id normal.axioms subset prems) lemma (in normal) rcos_sum: "[|x ∈ carrier G; y ∈ carrier G|] ==> (H #> x) <#> (H #> y) = H #> (x ⊗ y)" by (simp add: rcos_mult_step1 rcos_mult_step2 rcos_mult_step3) lemma (in normal) rcosets_mult_eq: "M ∈ rcosets H ==> H <#> M = M" -- {* generalizes @{text subgroup_mult_id} *} by (auto simp add: RCOSETS_def subset setmult_rcos_assoc subgroup_mult_id normal.axioms prems) subsubsection{*An Equivalence Relation*} constdefs (structure G) r_congruent :: "[('a,'b)monoid_scheme, 'a set] => ('a*'a)set" ("rcong\<index> _") "rcong H ≡ {(x,y). x ∈ carrier G & y ∈ carrier G & inv x ⊗ y ∈ H}" lemma (in subgroup) equiv_rcong: includes group G shows "equiv (carrier G) (rcong H)" proof (intro equiv.intro) show "refl (carrier G) (rcong H)" by (auto simp add: r_congruent_def refl_def) next show "sym (rcong H)" proof (simp add: r_congruent_def sym_def, clarify) fix x y assume [simp]: "x ∈ carrier G" "y ∈ carrier G" and "inv x ⊗ y ∈ H" hence "inv (inv x ⊗ y) ∈ H" by (simp add: m_inv_closed) thus "inv y ⊗ x ∈ H" by (simp add: inv_mult_group) qed next show "trans (rcong H)" proof (simp add: r_congruent_def trans_def, clarify) fix x y z assume [simp]: "x ∈ carrier G" "y ∈ carrier G" "z ∈ carrier G" and "inv x ⊗ y ∈ H" and "inv y ⊗ z ∈ H" hence "(inv x ⊗ y) ⊗ (inv y ⊗ z) ∈ H" by simp hence "inv x ⊗ (y ⊗ inv y) ⊗ z ∈ H" by (simp add: m_assoc del: r_inv) thus "inv x ⊗ z ∈ H" by simp qed qed text{*Equivalence classes of @{text rcong} correspond to left cosets. Was there a mistake in the definitions? I'd have expected them to correspond to right cosets.*} (* CB: This is correct, but subtle. We call H #> a the right coset of a relative to H. According to Jacobson, this is what the majority of group theory literature does. He then defines the notion of congruence relation ~ over monoids as equivalence relation with a ~ a' & b ~ b' ==> a*b ~ a'*b'. Our notion of right congruence induced by K: rcong K appears only in the context where K is a normal subgroup. Jacobson doesn't name it. But in this context left and right cosets are identical. *) lemma (in subgroup) l_coset_eq_rcong: includes group G assumes a: "a ∈ carrier G" shows "a <# H = rcong H `` {a}" by (force simp add: r_congruent_def l_coset_def m_assoc [symmetric] a ) subsubsection{*Two Distinct Right Cosets are Disjoint*} lemma (in group) rcos_equation: includes subgroup H G shows "[|ha ⊗ a = h ⊗ b; a ∈ carrier G; b ∈ carrier G; h ∈ H; ha ∈ H; hb ∈ H|] ==> hb ⊗ a ∈ (\<Union>h∈H. {h ⊗ b})" apply (rule UN_I [of "hb ⊗ ((inv ha) ⊗ h)"]) apply (simp add: ) apply (simp add: m_assoc transpose_inv) done lemma (in group) rcos_disjoint: includes subgroup H G shows "[|a ∈ rcosets H; b ∈ rcosets H; a≠b|] ==> a ∩ b = {}" apply (simp add: RCOSETS_def r_coset_def) apply (blast intro: rcos_equation prems sym) done subsection {* Further lemmas for @{text "r_congruent"} *} text {* The relation is a congruence *} lemma (in normal) congruent_rcong: shows "congruent2 (rcong H) (rcong H) (λa b. a ⊗ b <# H)" proof (intro congruent2I[of "carrier G" _ "carrier G" _] equiv_rcong is_group) fix a b c assume abrcong: "(a, b) ∈ rcong H" and ccarr: "c ∈ carrier G" from abrcong have acarr: "a ∈ carrier G" and bcarr: "b ∈ carrier G" and abH: "inv a ⊗ b ∈ H" unfolding r_congruent_def by fast+ note carr = acarr bcarr ccarr from ccarr and abH have "inv c ⊗ (inv a ⊗ b) ⊗ c ∈ H" by (rule inv_op_closed1) moreover from carr and inv_closed have "inv c ⊗ (inv a ⊗ b) ⊗ c = (inv c ⊗ inv a) ⊗ (b ⊗ c)" by (force cong: m_assoc) moreover from carr and inv_closed have "… = (inv (a ⊗ c)) ⊗ (b ⊗ c)" by (simp add: inv_mult_group) ultimately have "(inv (a ⊗ c)) ⊗ (b ⊗ c) ∈ H" by simp from carr and this have "(b ⊗ c) ∈ (a ⊗ c) <# H" by (simp add: lcos_module_rev[OF is_group]) from carr and this and is_subgroup show "(a ⊗ c) <# H = (b ⊗ c) <# H" by (intro l_repr_independence, simp+) next fix a b c assume abrcong: "(a, b) ∈ rcong H" and ccarr: "c ∈ carrier G" from ccarr have "c ∈ Units G" by (simp add: Units_eq) hence cinvc_one: "inv c ⊗ c = \<one>" by (rule Units_l_inv) from abrcong have acarr: "a ∈ carrier G" and bcarr: "b ∈ carrier G" and abH: "inv a ⊗ b ∈ H" by (unfold r_congruent_def, fast+) note carr = acarr bcarr ccarr from carr and inv_closed have "inv a ⊗ b = inv a ⊗ (\<one> ⊗ b)" by simp also from carr and inv_closed have "… = inv a ⊗ (inv c ⊗ c) ⊗ b" by simp also from carr and inv_closed have "… = (inv a ⊗ inv c) ⊗ (c ⊗ b)" by (force cong: m_assoc) also from carr and inv_closed have "… = inv (c ⊗ a) ⊗ (c ⊗ b)" by (simp add: inv_mult_group) finally have "inv a ⊗ b = inv (c ⊗ a) ⊗ (c ⊗ b)" . from abH and this have "inv (c ⊗ a) ⊗ (c ⊗ b) ∈ H" by simp from carr and this have "(c ⊗ b) ∈ (c ⊗ a) <# H" by (simp add: lcos_module_rev[OF is_group]) from carr and this and is_subgroup show "(c ⊗ a) <# H = (c ⊗ b) <# H" by (intro l_repr_independence, simp+) qed subsection {*Order of a Group and Lagrange's Theorem*} constdefs order :: "('a, 'b) monoid_scheme => nat" "order S ≡ card (carrier S)" lemma (in group) rcosets_part_G: includes subgroup shows "\<Union>(rcosets H) = carrier G" apply (rule equalityI) apply (force simp add: RCOSETS_def r_coset_def) apply (auto simp add: RCOSETS_def intro: rcos_self prems) done lemma (in group) cosets_finite: "[|c ∈ rcosets H; H ⊆ carrier G; finite (carrier G)|] ==> finite c" apply (auto simp add: RCOSETS_def) apply (simp add: r_coset_subset_G [THEN finite_subset]) done text{*The next two lemmas support the proof of @{text card_cosets_equal}.*} lemma (in group) inj_on_f: "[|H ⊆ carrier G; a ∈ carrier G|] ==> inj_on (λy. y ⊗ inv a) (H #> a)" apply (rule inj_onI) apply (subgoal_tac "x ∈ carrier G & y ∈ carrier G") prefer 2 apply (blast intro: r_coset_subset_G [THEN subsetD]) apply (simp add: subsetD) done lemma (in group) inj_on_g: "[|H ⊆ carrier G; a ∈ carrier G|] ==> inj_on (λy. y ⊗ a) H" by (force simp add: inj_on_def subsetD) lemma (in group) card_cosets_equal: "[|c ∈ rcosets H; H ⊆ carrier G; finite(carrier G)|] ==> card c = card H" apply (auto simp add: RCOSETS_def) apply (rule card_bij_eq) apply (rule inj_on_f, assumption+) apply (force simp add: m_assoc subsetD r_coset_def) apply (rule inj_on_g, assumption+) apply (force simp add: m_assoc subsetD r_coset_def) txt{*The sets @{term "H #> a"} and @{term "H"} are finite.*} apply (simp add: r_coset_subset_G [THEN finite_subset]) apply (blast intro: finite_subset) done lemma (in group) rcosets_subset_PowG: "subgroup H G ==> rcosets H ⊆ Pow(carrier G)" apply (simp add: RCOSETS_def) apply (blast dest: r_coset_subset_G subgroup.subset) done theorem (in group) lagrange: "[|finite(carrier G); subgroup H G|] ==> card(rcosets H) * card(H) = order(G)" apply (simp (no_asm_simp) add: order_def rcosets_part_G [symmetric]) apply (subst mult_commute) apply (rule card_partition) apply (simp add: rcosets_subset_PowG [THEN finite_subset]) apply (simp add: rcosets_part_G) apply (simp add: card_cosets_equal subgroup.subset) apply (simp add: rcos_disjoint) done subsection {*Quotient Groups: Factorization of a Group*} constdefs FactGroup :: "[('a,'b) monoid_scheme, 'a set] => ('a set) monoid" (infixl "Mod" 65) --{*Actually defined for groups rather than monoids*} "FactGroup G H ≡ (|carrier = rcosetsG H, mult = set_mult G, one = H|))," lemma (in normal) setmult_closed: "[|K1 ∈ rcosets H; K2 ∈ rcosets H|] ==> K1 <#> K2 ∈ rcosets H" by (auto simp add: rcos_sum RCOSETS_def) lemma (in normal) setinv_closed: "K ∈ rcosets H ==> set_inv K ∈ rcosets H" by (auto simp add: rcos_inv RCOSETS_def) lemma (in normal) rcosets_assoc: "[|M1 ∈ rcosets H; M2 ∈ rcosets H; M3 ∈ rcosets H|] ==> M1 <#> M2 <#> M3 = M1 <#> (M2 <#> M3)" by (auto simp add: RCOSETS_def rcos_sum m_assoc) lemma (in subgroup) subgroup_in_rcosets: includes group G shows "H ∈ rcosets H" proof - from _ subgroup_axioms have "H #> \<one> = H" by (rule coset_join2) auto then show ?thesis by (auto simp add: RCOSETS_def) qed lemma (in normal) rcosets_inv_mult_group_eq: "M ∈ rcosets H ==> set_inv M <#> M = H" by (auto simp add: RCOSETS_def rcos_inv rcos_sum subgroup.subset normal.axioms prems) theorem (in normal) factorgroup_is_group: "group (G Mod H)" apply (simp add: FactGroup_def) apply (rule groupI) apply (simp add: setmult_closed) apply (simp add: normal_imp_subgroup subgroup_in_rcosets [OF is_group]) apply (simp add: restrictI setmult_closed rcosets_assoc) apply (simp add: normal_imp_subgroup subgroup_in_rcosets rcosets_mult_eq) apply (auto dest: rcosets_inv_mult_group_eq simp add: setinv_closed) done lemma mult_FactGroup [simp]: "X ⊗(G Mod H) X' = X <#>G X'" by (simp add: FactGroup_def) lemma (in normal) inv_FactGroup: "X ∈ carrier (G Mod H) ==> invG Mod H X = set_inv X" apply (rule group.inv_equality [OF factorgroup_is_group]) apply (simp_all add: FactGroup_def setinv_closed rcosets_inv_mult_group_eq) done text{*The coset map is a homomorphism from @{term G} to the quotient group @{term "G Mod H"}*} lemma (in normal) r_coset_hom_Mod: "(λa. H #> a) ∈ hom G (G Mod H)" by (auto simp add: FactGroup_def RCOSETS_def Pi_def hom_def rcos_sum) subsection{*The First Isomorphism Theorem*} text{*The quotient by the kernel of a homomorphism is isomorphic to the range of that homomorphism.*} constdefs kernel :: "('a, 'm) monoid_scheme => ('b, 'n) monoid_scheme => ('a => 'b) => 'a set" --{*the kernel of a homomorphism*} "kernel G H h ≡ {x. x ∈ carrier G & h x = \<one>H}" lemma (in group_hom) subgroup_kernel: "subgroup (kernel G H h) G" apply (rule subgroup.intro) apply (auto simp add: kernel_def group.intro prems) done text{*The kernel of a homomorphism is a normal subgroup*} lemma (in group_hom) normal_kernel: "(kernel G H h) \<lhd> G" apply (simp add: G.normal_inv_iff subgroup_kernel) apply (simp add: kernel_def) done lemma (in group_hom) FactGroup_nonempty: assumes X: "X ∈ carrier (G Mod kernel G H h)" shows "X ≠ {}" proof - from X obtain g where "g ∈ carrier G" and "X = kernel G H h #> g" by (auto simp add: FactGroup_def RCOSETS_def) thus ?thesis by (auto simp add: kernel_def r_coset_def image_def intro: hom_one) qed lemma (in group_hom) FactGroup_contents_mem: assumes X: "X ∈ carrier (G Mod (kernel G H h))" shows "contents (h`X) ∈ carrier H" proof - from X obtain g where g: "g ∈ carrier G" and "X = kernel G H h #> g" by (auto simp add: FactGroup_def RCOSETS_def) hence "h ` X = {h g}" by (auto simp add: kernel_def r_coset_def image_def g) thus ?thesis by (auto simp add: g) qed lemma (in group_hom) FactGroup_hom: "(λX. contents (h`X)) ∈ hom (G Mod (kernel G H h)) H" apply (simp add: hom_def FactGroup_contents_mem normal.factorgroup_is_group [OF normal_kernel] group.axioms monoid.m_closed) proof (simp add: hom_def funcsetI FactGroup_contents_mem, intro ballI) fix X and X' assume X: "X ∈ carrier (G Mod kernel G H h)" and X': "X' ∈ carrier (G Mod kernel G H h)" then obtain g and g' where "g ∈ carrier G" and "g' ∈ carrier G" and "X = kernel G H h #> g" and "X' = kernel G H h #> g'" by (auto simp add: FactGroup_def RCOSETS_def) hence all: "∀x∈X. h x = h g" "∀x∈X'. h x = h g'" and Xsub: "X ⊆ carrier G" and X'sub: "X' ⊆ carrier G" by (force simp add: kernel_def r_coset_def image_def)+ hence "h ` (X <#> X') = {h g ⊗H h g'}" using X X' by (auto dest!: FactGroup_nonempty simp add: set_mult_def image_eq_UN subsetD [OF Xsub] subsetD [OF X'sub]) thus "contents (h ` (X <#> X')) = contents (h ` X) ⊗H contents (h ` X')" by (simp add: all image_eq_UN FactGroup_nonempty X X') qed text{*Lemma for the following injectivity result*} lemma (in group_hom) FactGroup_subset: "[|g ∈ carrier G; g' ∈ carrier G; h g = h g'|] ==> kernel G H h #> g ⊆ kernel G H h #> g'" apply (clarsimp simp add: kernel_def r_coset_def image_def) apply (rename_tac y) apply (rule_tac x="y ⊗ g ⊗ inv g'" in exI) apply (simp add: G.m_assoc) done lemma (in group_hom) FactGroup_inj_on: "inj_on (λX. contents (h ` X)) (carrier (G Mod kernel G H h))" proof (simp add: inj_on_def, clarify) fix X and X' assume X: "X ∈ carrier (G Mod kernel G H h)" and X': "X' ∈ carrier (G Mod kernel G H h)" then obtain g and g' where gX: "g ∈ carrier G" "g' ∈ carrier G" "X = kernel G H h #> g" "X' = kernel G H h #> g'" by (auto simp add: FactGroup_def RCOSETS_def) hence all: "∀x∈X. h x = h g" "∀x∈X'. h x = h g'" by (force simp add: kernel_def r_coset_def image_def)+ assume "contents (h ` X) = contents (h ` X')" hence h: "h g = h g'" by (simp add: image_eq_UN all FactGroup_nonempty X X') show "X=X'" by (rule equalityI) (simp_all add: FactGroup_subset h gX) qed text{*If the homomorphism @{term h} is onto @{term H}, then so is the homomorphism from the quotient group*} lemma (in group_hom) FactGroup_onto: assumes h: "h ` carrier G = carrier H" shows "(λX. contents (h ` X)) ` carrier (G Mod kernel G H h) = carrier H" proof show "(λX. contents (h ` X)) ` carrier (G Mod kernel G H h) ⊆ carrier H" by (auto simp add: FactGroup_contents_mem) show "carrier H ⊆ (λX. contents (h ` X)) ` carrier (G Mod kernel G H h)" proof fix y assume y: "y ∈ carrier H" with h obtain g where g: "g ∈ carrier G" "h g = y" by (blast elim: equalityE) hence "(\<Union>x∈kernel G H h #> g. {h x}) = {y}" by (auto simp add: y kernel_def r_coset_def) with g show "y ∈ (λX. contents (h ` X)) ` carrier (G Mod kernel G H h)" by (auto intro!: bexI simp add: FactGroup_def RCOSETS_def image_eq_UN) qed qed text{*If @{term h} is a homomorphism from @{term G} onto @{term H}, then the quotient group @{term "G Mod (kernel G H h)"} is isomorphic to @{term H}.*} theorem (in group_hom) FactGroup_iso: "h ` carrier G = carrier H ==> (λX. contents (h`X)) ∈ (G Mod (kernel G H h)) ≅ H" by (simp add: iso_def FactGroup_hom FactGroup_inj_on bij_betw_def FactGroup_onto) end
lemma coset_mult_assoc:
[| M ⊆ carrier G; g ∈ carrier G; h ∈ carrier G |] ==> M #> g #> h = M #> g ⊗ h
lemma coset_mult_one:
M ⊆ carrier G ==> M #> \<one> = M
lemma coset_mult_inv1:
[| M #> x ⊗ inv y = M; x ∈ carrier G; y ∈ carrier G; M ⊆ carrier G |]
==> M #> x = M #> y
lemma coset_mult_inv2:
[| M #> x = M #> y; x ∈ carrier G; y ∈ carrier G; M ⊆ carrier G |]
==> M #> x ⊗ inv y = M
lemma coset_join1:
[| H #> x = H; x ∈ carrier G; subgroup H G |] ==> x ∈ H
lemma solve_equation:
[| subgroup H G; x ∈ H; y ∈ H |] ==> ∃h∈H. y = h ⊗ x
lemma repr_independence:
[| y ∈ H #> x; x ∈ carrier G; subgroup H G |] ==> H #> x = H #> y
lemma coset_join2:
[| x ∈ carrier G; subgroup H G; x ∈ H |] ==> H #> x = H
lemma r_coset_subset_G:
[| H ⊆ carrier G; x ∈ carrier G |] ==> H #> x ⊆ carrier G
lemma rcosI:
[| h ∈ H; H ⊆ carrier G; x ∈ carrier G |] ==> h ⊗ x ∈ H #> x
lemma rcosetsI:
[| H ⊆ carrier G; x ∈ carrier G |] ==> H #> x ∈ rcosets H
lemma transpose_inv:
[| x ⊗ y = z; x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==> inv x ⊗ z = y
lemma rcos_self:
[| x ∈ carrier G; subgroup H G |] ==> x ∈ H #> x
lemma repr_independenceD:
[| subgroup H G; y ∈ carrier G; H #> x = H #> y |] ==> y ∈ H #> x
lemma elemrcos_carrier:
[| group G; a ∈ carrier G; a' ∈ H #> a |] ==> a' ∈ carrier G
lemma rcos_const:
[| group G; h ∈ H |] ==> H #> h = H
lemma rcos_module_imp:
[| group G; x ∈ carrier G; x' ∈ H #> x |] ==> x' ⊗ inv x ∈ H
lemma rcos_module_rev:
[| group G; x ∈ carrier G; x' ∈ carrier G; x' ⊗ inv x ∈ H |] ==> x' ∈ H #> x
lemma rcos_module:
[| group G; x ∈ carrier G; x' ∈ carrier G |]
==> (x' ∈ H #> x) = (x' ⊗ inv x ∈ H)
lemma rcosets_carrier:
[| group G; X ∈ rcosets H |] ==> X ⊆ carrier G
lemma set_mult_closed:
[| A ⊆ carrier G; B ⊆ carrier G |] ==> A <#> B ⊆ carrier G
lemma mult_subgroups:
[| subgroup H G; subgroup K G |] ==> subgroup (H <#> K) G
lemma lcos_module_rev:
[| group G; x ∈ carrier G; x' ∈ carrier G; inv x ⊗ x' ∈ H |] ==> x' ∈ x <# H
lemma normal_imp_subgroup:
H \<lhd> G ==> subgroup H G
lemma normalI:
[| subgroup H G; ∀x∈carrier G. H #> x = x <# H |] ==> H \<lhd> G
lemma inv_op_closed1:
[| x ∈ carrier G; h ∈ H |] ==> inv x ⊗ h ⊗ x ∈ H
lemma inv_op_closed2:
[| x ∈ carrier G; h ∈ H |] ==> x ⊗ h ⊗ inv x ∈ H
lemma normal_inv_iff:
N \<lhd> G = (subgroup N G ∧ (∀x∈carrier G. ∀h∈N. x ⊗ h ⊗ inv x ∈ N))
lemma lcos_m_assoc:
[| M ⊆ carrier G; g ∈ carrier G; h ∈ carrier G |] ==> g <# (h <# M) = g ⊗ h <# M
lemma lcos_mult_one:
M ⊆ carrier G ==> \<one> <# M = M
lemma l_coset_subset_G:
[| H ⊆ carrier G; x ∈ carrier G |] ==> x <# H ⊆ carrier G
lemma l_coset_swap:
[| y ∈ x <# H; x ∈ carrier G; subgroup H G |] ==> x ∈ y <# H
lemma l_coset_carrier:
[| y ∈ x <# H; x ∈ carrier G; subgroup H G |] ==> y ∈ carrier G
lemma l_repr_imp_subset:
[| y ∈ x <# H; x ∈ carrier G; subgroup H G |] ==> y <# H ⊆ x <# H
lemma l_repr_independence:
[| y ∈ x <# H; x ∈ carrier G; subgroup H G |] ==> x <# H = y <# H
lemma setmult_subset_G:
[| H ⊆ carrier G; K ⊆ carrier G |] ==> H <#> K ⊆ carrier G
lemma subgroup_mult_id:
subgroup H G ==> H <#> H = H
lemma rcos_inv:
x ∈ carrier G ==> set_inv (H #> x) = H #> inv x
lemma setmult_rcos_assoc:
[| H ⊆ carrier G; K ⊆ carrier G; x ∈ carrier G |]
==> H <#> (K #> x) = H <#> K #> x
lemma rcos_assoc_lcos:
[| H ⊆ carrier G; K ⊆ carrier G; x ∈ carrier G |]
==> H #> x <#> K = H <#> (x <# K)
lemma rcos_mult_step1:
[| x ∈ carrier G; y ∈ carrier G |] ==> H #> x <#> (H #> y) = H <#> (x <# H) #> y
lemma rcos_mult_step2:
[| x ∈ carrier G; y ∈ carrier G |] ==> H <#> (x <# H) #> y = H <#> (H #> x) #> y
lemma rcos_mult_step3:
[| x ∈ carrier G; y ∈ carrier G |] ==> H <#> (H #> x) #> y = H #> x ⊗ y
lemma rcos_sum:
[| x ∈ carrier G; y ∈ carrier G |] ==> H #> x <#> (H #> y) = H #> x ⊗ y
lemma rcosets_mult_eq:
M ∈ rcosets H ==> H <#> M = M
lemma equiv_rcong:
group G ==> equiv (carrier G) rcong H
lemma l_coset_eq_rcong:
[| group G; a ∈ carrier G |] ==> a <# H = rcong H `` {a}
lemma rcos_equation:
[| subgroup H G; ha ⊗ a = h ⊗ b; a ∈ carrier G; b ∈ carrier G; h ∈ H; ha ∈ H;
hb ∈ H |]
==> hb ⊗ a ∈ (UN h:H. {h ⊗ b})
lemma rcos_disjoint:
[| subgroup H G; a ∈ rcosets H; b ∈ rcosets H; a ≠ b |] ==> a ∩ b = {}
lemma congruent_rcong:
(λa b. a ⊗ b <# H) respects2 rcong H
lemma rcosets_part_G:
subgroup H G ==> Union (rcosets H) = carrier G
lemma cosets_finite:
[| c ∈ rcosets H; H ⊆ carrier G; finite (carrier G) |] ==> finite c
lemma inj_on_f:
[| H ⊆ carrier G; a ∈ carrier G |] ==> inj_on (λy. y ⊗ inv a) (H #> a)
lemma inj_on_g:
[| H ⊆ carrier G; a ∈ carrier G |] ==> inj_on (λy. y ⊗ a) H
lemma card_cosets_equal:
[| c ∈ rcosets H; H ⊆ carrier G; finite (carrier G) |] ==> card c = card H
lemma rcosets_subset_PowG:
subgroup H G ==> rcosets H ⊆ Pow (carrier G)
theorem lagrange:
[| finite (carrier G); subgroup H G |]
==> card (rcosets H) * card H = Coset.order G
lemma setmult_closed:
[| K1.0 ∈ rcosets H; K2.0 ∈ rcosets H |] ==> K1.0 <#> K2.0 ∈ rcosets H
lemma setinv_closed:
K ∈ rcosets H ==> set_inv K ∈ rcosets H
lemma rcosets_assoc:
[| M1.0 ∈ rcosets H; M2.0 ∈ rcosets H; M3.0 ∈ rcosets H |]
==> M1.0 <#> M2.0 <#> M3.0 = M1.0 <#> (M2.0 <#> M3.0)
lemma subgroup_in_rcosets:
group G ==> H ∈ rcosets H
lemma rcosets_inv_mult_group_eq:
M ∈ rcosets H ==> set_inv M <#> M = H
theorem factorgroup_is_group:
group (G Mod H)
lemma mult_FactGroup:
X ⊗G Mod H X' = X <#>G X'
lemma inv_FactGroup:
X ∈ carrier (G Mod H) ==> invG Mod H X = set_inv X
lemma r_coset_hom_Mod:
op #> H ∈ hom G (G Mod H)
lemma subgroup_kernel:
subgroup (kernel G H h) G
lemma normal_kernel:
kernel G H h \<lhd> G
lemma FactGroup_nonempty:
X ∈ carrier (G Mod kernel G H h) ==> X ≠ {}
lemma FactGroup_contents_mem:
X ∈ carrier (G Mod kernel G H h) ==> contents (h ` X) ∈ carrier H
lemma FactGroup_hom:
(λX. contents (h ` X)) ∈ hom (G Mod kernel G H h) H
lemma FactGroup_subset:
[| g ∈ carrier G; g' ∈ carrier G; h g = h g' |]
==> kernel G H h #> g ⊆ kernel G H h #> g'
lemma FactGroup_inj_on:
inj_on (λX. contents (h ` X)) (carrier (G Mod kernel G H h))
lemma FactGroup_onto:
h ` carrier G = carrier H
==> (λX. contents (h ` X)) ` carrier (G Mod kernel G H h) = carrier H
theorem FactGroup_iso:
h ` carrier G = carrier H ==> (λX. contents (h ` X)) ∈ G Mod kernel G H h ≅ H