src/HOL/Algebra/Coset.thy
author wenzelm
Fri Nov 17 02:20:03 2006 +0100 (2006-11-17)
changeset 21404 eb85850d3eb7
parent 20318 0e0ea63fe768
child 23350 50c5b0912a0c
permissions -rw-r--r--
more robust syntax for definition/abbreviation/notation;
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(*  Title:      HOL/Algebra/Coset.thy
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    ID:         $Id$
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    Author:     Florian Kammueller, with new proofs by L C Paulson, and
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                Stephan Hohe
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*)
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theory Coset imports Group Exponent begin
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section {*Cosets and Quotient Groups*}
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constdefs (structure G)
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  r_coset    :: "[_, 'a set, 'a] \<Rightarrow> 'a set"    (infixl "#>\<index>" 60)
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  "H #> a \<equiv> \<Union>h\<in>H. {h \<otimes> a}"
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  l_coset    :: "[_, 'a, 'a set] \<Rightarrow> 'a set"    (infixl "<#\<index>" 60)
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  "a <# H \<equiv> \<Union>h\<in>H. {a \<otimes> h}"
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  RCOSETS  :: "[_, 'a set] \<Rightarrow> ('a set)set"   ("rcosets\<index> _" [81] 80)
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  "rcosets H \<equiv> \<Union>a\<in>carrier G. {H #> a}"
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  set_mult  :: "[_, 'a set ,'a set] \<Rightarrow> 'a set" (infixl "<#>\<index>" 60)
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  "H <#> K \<equiv> \<Union>h\<in>H. \<Union>k\<in>K. {h \<otimes> k}"
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  SET_INV :: "[_,'a set] \<Rightarrow> 'a set"  ("set'_inv\<index> _" [81] 80)
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  "set_inv H \<equiv> \<Union>h\<in>H. {inv h}"
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locale normal = subgroup + group +
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  assumes coset_eq: "(\<forall>x \<in> carrier G. H #> x = x <# H)"
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abbreviation
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  normal_rel :: "['a set, ('a, 'b) monoid_scheme] \<Rightarrow> bool"  (infixl "\<lhd>" 60) where
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  "H \<lhd> G \<equiv> normal H G"
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subsection {*Basic Properties of Cosets*}
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lemma (in group) coset_mult_assoc:
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     "[| M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G |]
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      ==> (M #> g) #> h = M #> (g \<otimes> h)"
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by (force simp add: r_coset_def m_assoc)
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lemma (in group) coset_mult_one [simp]: "M \<subseteq> carrier G ==> M #> \<one> = M"
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by (force simp add: r_coset_def)
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lemma (in group) coset_mult_inv1:
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     "[| M #> (x \<otimes> (inv y)) = M;  x \<in> carrier G ; y \<in> carrier G;
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         M \<subseteq> carrier G |] ==> M #> x = M #> y"
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apply (erule subst [of concl: "%z. M #> x = z #> y"])
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apply (simp add: coset_mult_assoc m_assoc)
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done
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lemma (in group) coset_mult_inv2:
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     "[| M #> x = M #> y;  x \<in> carrier G;  y \<in> carrier G;  M \<subseteq> carrier G |]
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      ==> M #> (x \<otimes> (inv y)) = M "
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apply (simp add: coset_mult_assoc [symmetric])
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apply (simp add: coset_mult_assoc)
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done
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lemma (in group) coset_join1:
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     "[| H #> x = H;  x \<in> carrier G;  subgroup H G |] ==> x \<in> H"
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apply (erule subst)
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apply (simp add: r_coset_def)
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apply (blast intro: l_one subgroup.one_closed sym)
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done
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lemma (in group) solve_equation:
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    "\<lbrakk>subgroup H G; x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> \<exists>h\<in>H. y = h \<otimes> x"
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apply (rule bexI [of _ "y \<otimes> (inv x)"])
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apply (auto simp add: subgroup.m_closed subgroup.m_inv_closed m_assoc
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                      subgroup.subset [THEN subsetD])
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done
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lemma (in group) repr_independence:
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     "\<lbrakk>y \<in> H #> x;  x \<in> carrier G; subgroup H G\<rbrakk> \<Longrightarrow> H #> x = H #> y"
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by (auto simp add: r_coset_def m_assoc [symmetric]
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                   subgroup.subset [THEN subsetD]
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                   subgroup.m_closed solve_equation)
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lemma (in group) coset_join2:
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     "\<lbrakk>x \<in> carrier G;  subgroup H G;  x\<in>H\<rbrakk> \<Longrightarrow> H #> x = H"
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  --{*Alternative proof is to put @{term "x=\<one>"} in @{text repr_independence}.*}
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by (force simp add: subgroup.m_closed r_coset_def solve_equation)
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lemma (in monoid) r_coset_subset_G:
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     "[| H \<subseteq> carrier G; x \<in> carrier G |] ==> H #> x \<subseteq> carrier G"
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by (auto simp add: r_coset_def)
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lemma (in group) rcosI:
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     "[| h \<in> H; H \<subseteq> carrier G; x \<in> carrier G|] ==> h \<otimes> x \<in> H #> x"
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by (auto simp add: r_coset_def)
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lemma (in group) rcosetsI:
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     "\<lbrakk>H \<subseteq> carrier G; x \<in> carrier G\<rbrakk> \<Longrightarrow> H #> x \<in> rcosets H"
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by (auto simp add: RCOSETS_def)
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text{*Really needed?*}
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lemma (in group) transpose_inv:
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     "[| x \<otimes> y = z;  x \<in> carrier G;  y \<in> carrier G;  z \<in> carrier G |]
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      ==> (inv x) \<otimes> z = y"
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by (force simp add: m_assoc [symmetric])
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lemma (in group) rcos_self: "[| x \<in> carrier G; subgroup H G |] ==> x \<in> H #> x"
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apply (simp add: r_coset_def)
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apply (blast intro: sym l_one subgroup.subset [THEN subsetD]
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                    subgroup.one_closed)
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done
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text {* Opposite of @{thm [locale=group,source] "repr_independence"} *}
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lemma (in group) repr_independenceD:
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  includes subgroup H G
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  assumes ycarr: "y \<in> carrier G"
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      and repr:  "H #> x = H #> y"
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  shows "y \<in> H #> x"
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  by (subst repr, intro rcos_self)
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text {* Elements of a right coset are in the carrier *}
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lemma (in subgroup) elemrcos_carrier:
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  includes group
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  assumes acarr: "a \<in> carrier G"
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    and a': "a' \<in> H #> a"
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  shows "a' \<in> carrier G"
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proof -
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  from subset and acarr
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  have "H #> a \<subseteq> carrier G" by (rule r_coset_subset_G)
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  from this and a'
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  show "a' \<in> carrier G"
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    by fast
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qed
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lemma (in subgroup) rcos_const:
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  includes group
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  assumes hH: "h \<in> H"
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  shows "H #> h = H"
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  apply (unfold r_coset_def)
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  apply rule apply rule
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  apply clarsimp
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  apply (intro subgroup.m_closed)
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  apply assumption+
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  apply rule
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  apply simp
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proof -
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  fix h'
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  assume h'H: "h' \<in> H"
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  note carr = hH[THEN mem_carrier] h'H[THEN mem_carrier]
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  from carr
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  have a: "h' = (h' \<otimes> inv h) \<otimes> h" by (simp add: m_assoc)
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  from h'H hH
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  have "h' \<otimes> inv h \<in> H" by simp
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  from this and a
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  show "\<exists>x\<in>H. h' = x \<otimes> h" by fast
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qed
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text {* Step one for lemma @{text "rcos_module"} *}
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lemma (in subgroup) rcos_module_imp:
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  includes group
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  assumes xcarr: "x \<in> carrier G"
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      and x'cos: "x' \<in> H #> x"
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  shows "(x' \<otimes> inv x) \<in> H"
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proof -
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  from xcarr x'cos
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      have x'carr: "x' \<in> carrier G"
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      by (rule elemrcos_carrier[OF is_group])
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  from xcarr
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      have ixcarr: "inv x \<in> carrier G"
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      by simp
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  from x'cos
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      have "\<exists>h\<in>H. x' = h \<otimes> x"
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      unfolding r_coset_def
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      by fast
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  from this
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      obtain h
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        where hH: "h \<in> H"
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        and x': "x' = h \<otimes> x"
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      by auto
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  from hH and subset
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      have hcarr: "h \<in> carrier G" by fast
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  note carr = xcarr x'carr hcarr
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  from x' and carr
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      have "x' \<otimes> (inv x) = (h \<otimes> x) \<otimes> (inv x)" by fast
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  also from carr
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      have "\<dots> = h \<otimes> (x \<otimes> inv x)" by (simp add: m_assoc)
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  also from carr
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      have "\<dots> = h \<otimes> \<one>" by simp
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  also from carr
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      have "\<dots> = h" by simp
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  finally
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      have "x' \<otimes> (inv x) = h" by simp
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  from hH this
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      show "x' \<otimes> (inv x) \<in> H" by simp
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qed
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text {* Step two for lemma @{text "rcos_module"} *}
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lemma (in subgroup) rcos_module_rev:
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  includes group
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  assumes carr: "x \<in> carrier G" "x' \<in> carrier G"
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      and xixH: "(x' \<otimes> inv x) \<in> H"
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  shows "x' \<in> H #> x"
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proof -
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  from xixH
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      have "\<exists>h\<in>H. x' \<otimes> (inv x) = h" by fast
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  from this
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      obtain h
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        where hH: "h \<in> H"
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        and hsym: "x' \<otimes> (inv x) = h"
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      by fast
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  from hH subset have hcarr: "h \<in> carrier G" by simp
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  note carr = carr hcarr
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  from hsym[symmetric] have "h \<otimes> x = x' \<otimes> (inv x) \<otimes> x" by fast
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  also from carr
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      have "\<dots> = x' \<otimes> ((inv x) \<otimes> x)" by (simp add: m_assoc)
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  also from carr
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      have "\<dots> = x' \<otimes> \<one>" by (simp add: l_inv)
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  also from carr
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      have "\<dots> = x'" by simp
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  finally
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      have "h \<otimes> x = x'" by simp
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  from this[symmetric] and hH
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      show "x' \<in> H #> x"
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      unfolding r_coset_def
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      by fast
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qed
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text {* Module property of right cosets *}
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lemma (in subgroup) rcos_module:
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  includes group
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  assumes carr: "x \<in> carrier G" "x' \<in> carrier G"
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  shows "(x' \<in> H #> x) = (x' \<otimes> inv x \<in> H)"
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proof
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  assume "x' \<in> H #> x"
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  from this and carr
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      show "x' \<otimes> inv x \<in> H"
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      by (intro rcos_module_imp[OF is_group])
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next
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  assume "x' \<otimes> inv x \<in> H"
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  from this and carr
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      show "x' \<in> H #> x"
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      by (intro rcos_module_rev[OF is_group])
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qed
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text {* Right cosets are subsets of the carrier. *} 
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lemma (in subgroup) rcosets_carrier:
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  includes group
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  assumes XH: "X \<in> rcosets H"
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  shows "X \<subseteq> carrier G"
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proof -
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  from XH have "\<exists>x\<in> carrier G. X = H #> x"
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      unfolding RCOSETS_def
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      by fast
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  from this
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      obtain x
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        where xcarr: "x\<in> carrier G"
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        and X: "X = H #> x"
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      by fast
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  from subset and xcarr
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      show "X \<subseteq> carrier G"
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      unfolding X
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      by (rule r_coset_subset_G)
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qed
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text {* Multiplication of general subsets *}
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lemma (in monoid) set_mult_closed:
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  assumes Acarr: "A \<subseteq> carrier G"
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      and Bcarr: "B \<subseteq> carrier G"
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  shows "A <#> B \<subseteq> carrier G"
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apply rule apply (simp add: set_mult_def, clarsimp)
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proof -
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  fix a b
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  assume "a \<in> A"
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  from this and Acarr
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      have acarr: "a \<in> carrier G" by fast
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  assume "b \<in> B"
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  from this and Bcarr
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      have bcarr: "b \<in> carrier G" by fast
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  from acarr bcarr
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      show "a \<otimes> b \<in> carrier G" by (rule m_closed)
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qed
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lemma (in comm_group) mult_subgroups:
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  assumes subH: "subgroup H G"
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      and subK: "subgroup K G"
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  shows "subgroup (H <#> K) G"
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apply (rule subgroup.intro)
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   apply (intro set_mult_closed subgroup.subset[OF subH] subgroup.subset[OF subK])
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  apply (simp add: set_mult_def) apply clarsimp defer 1
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  apply (simp add: set_mult_def) defer 1
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  apply (simp add: set_mult_def, clarsimp) defer 1
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proof -
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  fix ha hb ka kb
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  assume haH: "ha \<in> H" and hbH: "hb \<in> H" and kaK: "ka \<in> K" and kbK: "kb \<in> K"
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  note carr = haH[THEN subgroup.mem_carrier[OF subH]] hbH[THEN subgroup.mem_carrier[OF subH]]
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              kaK[THEN subgroup.mem_carrier[OF subK]] kbK[THEN subgroup.mem_carrier[OF subK]]
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  from carr
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      have "(ha \<otimes> ka) \<otimes> (hb \<otimes> kb) = ha \<otimes> (ka \<otimes> hb) \<otimes> kb" by (simp add: m_assoc)
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  also from carr
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      have "\<dots> = ha \<otimes> (hb \<otimes> ka) \<otimes> kb" by (simp add: m_comm)
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  also from carr
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      have "\<dots> = (ha \<otimes> hb) \<otimes> (ka \<otimes> kb)" by (simp add: m_assoc)
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  finally
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      have eq: "(ha \<otimes> ka) \<otimes> (hb \<otimes> kb) = (ha \<otimes> hb) \<otimes> (ka \<otimes> kb)" .
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  from haH hbH have hH: "ha \<otimes> hb \<in> H" by (simp add: subgroup.m_closed[OF subH])
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  from kaK kbK have kK: "ka \<otimes> kb \<in> K" by (simp add: subgroup.m_closed[OF subK])
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  from hH and kK and eq
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      show "\<exists>h'\<in>H. \<exists>k'\<in>K. (ha \<otimes> ka) \<otimes> (hb \<otimes> kb) = h' \<otimes> k'" by fast
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next
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  have "\<one> = \<one> \<otimes> \<one>" by simp
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  from subgroup.one_closed[OF subH] subgroup.one_closed[OF subK] this
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      show "\<exists>h\<in>H. \<exists>k\<in>K. \<one> = h \<otimes> k" by fast
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next
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  fix h k
ballarin@20318
   316
  assume hH: "h \<in> H"
ballarin@20318
   317
     and kK: "k \<in> K"
ballarin@20318
   318
ballarin@20318
   319
  from hH[THEN subgroup.mem_carrier[OF subH]] kK[THEN subgroup.mem_carrier[OF subK]]
ballarin@20318
   320
      have "inv (h \<otimes> k) = inv h \<otimes> inv k" by (simp add: inv_mult_group m_comm)
ballarin@20318
   321
ballarin@20318
   322
  from subgroup.m_inv_closed[OF subH hH] and subgroup.m_inv_closed[OF subK kK] and this
ballarin@20318
   323
      show "\<exists>ha\<in>H. \<exists>ka\<in>K. inv (h \<otimes> k) = ha \<otimes> ka" by fast
ballarin@20318
   324
qed
ballarin@20318
   325
ballarin@20318
   326
lemma (in subgroup) lcos_module_rev:
ballarin@20318
   327
  includes group
ballarin@20318
   328
  assumes carr: "x \<in> carrier G" "x' \<in> carrier G"
ballarin@20318
   329
      and xixH: "(inv x \<otimes> x') \<in> H"
ballarin@20318
   330
  shows "x' \<in> x <# H"
ballarin@20318
   331
proof -
ballarin@20318
   332
  from xixH
ballarin@20318
   333
      have "\<exists>h\<in>H. (inv x) \<otimes> x' = h" by fast
ballarin@20318
   334
  from this
ballarin@20318
   335
      obtain h
ballarin@20318
   336
        where hH: "h \<in> H"
ballarin@20318
   337
        and hsym: "(inv x) \<otimes> x' = h"
ballarin@20318
   338
      by fast
ballarin@20318
   339
ballarin@20318
   340
  from hH subset have hcarr: "h \<in> carrier G" by simp
ballarin@20318
   341
  note carr = carr hcarr
ballarin@20318
   342
  from hsym[symmetric] have "x \<otimes> h = x \<otimes> ((inv x) \<otimes> x')" by fast
ballarin@20318
   343
  also from carr
ballarin@20318
   344
      have "\<dots> = (x \<otimes> (inv x)) \<otimes> x'" by (simp add: m_assoc[symmetric])
ballarin@20318
   345
  also from carr
ballarin@20318
   346
      have "\<dots> = \<one> \<otimes> x'" by simp
ballarin@20318
   347
  also from carr
ballarin@20318
   348
      have "\<dots> = x'" by simp
ballarin@20318
   349
  finally
ballarin@20318
   350
      have "x \<otimes> h = x'" by simp
ballarin@20318
   351
ballarin@20318
   352
  from this[symmetric] and hH
ballarin@20318
   353
      show "x' \<in> x <# H"
ballarin@20318
   354
      unfolding l_coset_def
ballarin@20318
   355
      by fast
ballarin@20318
   356
qed
ballarin@20318
   357
paulson@13870
   358
wenzelm@14666
   359
subsection {* Normal subgroups *}
paulson@13870
   360
paulson@14963
   361
lemma normal_imp_subgroup: "H \<lhd> G \<Longrightarrow> subgroup H G"
paulson@14963
   362
  by (simp add: normal_def subgroup_def)
paulson@13870
   363
paulson@14963
   364
lemma (in group) normalI: 
paulson@14963
   365
  "subgroup H G \<Longrightarrow> (\<forall>x \<in> carrier G. H #> x = x <# H) \<Longrightarrow> H \<lhd> G";
paulson@14963
   366
  by (simp add: normal_def normal_axioms_def prems) 
paulson@14963
   367
paulson@14963
   368
lemma (in normal) inv_op_closed1:
paulson@14963
   369
     "\<lbrakk>x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> (inv x) \<otimes> h \<otimes> x \<in> H"
paulson@14963
   370
apply (insert coset_eq) 
paulson@14963
   371
apply (auto simp add: l_coset_def r_coset_def)
wenzelm@14666
   372
apply (drule bspec, assumption)
paulson@13870
   373
apply (drule equalityD1 [THEN subsetD], blast, clarify)
paulson@14963
   374
apply (simp add: m_assoc)
paulson@14963
   375
apply (simp add: m_assoc [symmetric])
paulson@13870
   376
done
paulson@13870
   377
paulson@14963
   378
lemma (in normal) inv_op_closed2:
paulson@14963
   379
     "\<lbrakk>x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> h \<otimes> (inv x) \<in> H"
paulson@14963
   380
apply (subgoal_tac "inv (inv x) \<otimes> h \<otimes> (inv x) \<in> H") 
paulson@14963
   381
apply (simp add: ); 
paulson@14963
   382
apply (blast intro: inv_op_closed1) 
paulson@13870
   383
done
paulson@13870
   384
paulson@14747
   385
text{*Alternative characterization of normal subgroups*}
paulson@14747
   386
lemma (in group) normal_inv_iff:
paulson@14747
   387
     "(N \<lhd> G) = 
paulson@14747
   388
      (subgroup N G & (\<forall>x \<in> carrier G. \<forall>h \<in> N. x \<otimes> h \<otimes> (inv x) \<in> N))"
paulson@14747
   389
      (is "_ = ?rhs")
paulson@14747
   390
proof
paulson@14747
   391
  assume N: "N \<lhd> G"
paulson@14747
   392
  show ?rhs
paulson@14963
   393
    by (blast intro: N normal.inv_op_closed2 normal_imp_subgroup) 
paulson@14747
   394
next
paulson@14747
   395
  assume ?rhs
paulson@14747
   396
  hence sg: "subgroup N G" 
paulson@14963
   397
    and closed: "\<And>x. x\<in>carrier G \<Longrightarrow> \<forall>h\<in>N. x \<otimes> h \<otimes> inv x \<in> N" by auto
paulson@14747
   398
  hence sb: "N \<subseteq> carrier G" by (simp add: subgroup.subset) 
paulson@14747
   399
  show "N \<lhd> G"
paulson@14963
   400
  proof (intro normalI [OF sg], simp add: l_coset_def r_coset_def, clarify)
paulson@14747
   401
    fix x
paulson@14747
   402
    assume x: "x \<in> carrier G"
nipkow@15120
   403
    show "(\<Union>h\<in>N. {h \<otimes> x}) = (\<Union>h\<in>N. {x \<otimes> h})"
paulson@14747
   404
    proof
nipkow@15120
   405
      show "(\<Union>h\<in>N. {h \<otimes> x}) \<subseteq> (\<Union>h\<in>N. {x \<otimes> h})"
paulson@14747
   406
      proof clarify
paulson@14747
   407
        fix n
paulson@14747
   408
        assume n: "n \<in> N" 
nipkow@15120
   409
        show "n \<otimes> x \<in> (\<Union>h\<in>N. {x \<otimes> h})"
paulson@14747
   410
        proof 
paulson@14963
   411
          from closed [of "inv x"]
paulson@14963
   412
          show "inv x \<otimes> n \<otimes> x \<in> N" by (simp add: x n)
paulson@14963
   413
          show "n \<otimes> x \<in> {x \<otimes> (inv x \<otimes> n \<otimes> x)}"
paulson@14747
   414
            by (simp add: x n m_assoc [symmetric] sb [THEN subsetD])
paulson@14747
   415
        qed
paulson@14747
   416
      qed
paulson@14747
   417
    next
nipkow@15120
   418
      show "(\<Union>h\<in>N. {x \<otimes> h}) \<subseteq> (\<Union>h\<in>N. {h \<otimes> x})"
paulson@14747
   419
      proof clarify
paulson@14747
   420
        fix n
paulson@14747
   421
        assume n: "n \<in> N" 
nipkow@15120
   422
        show "x \<otimes> n \<in> (\<Union>h\<in>N. {h \<otimes> x})"
paulson@14747
   423
        proof 
paulson@14963
   424
          show "x \<otimes> n \<otimes> inv x \<in> N" by (simp add: x n closed)
paulson@14963
   425
          show "x \<otimes> n \<in> {x \<otimes> n \<otimes> inv x \<otimes> x}"
paulson@14747
   426
            by (simp add: x n m_assoc sb [THEN subsetD])
paulson@14747
   427
        qed
paulson@14747
   428
      qed
paulson@14747
   429
    qed
paulson@14747
   430
  qed
paulson@14747
   431
qed
paulson@13870
   432
paulson@14963
   433
paulson@14803
   434
subsection{*More Properties of Cosets*}
paulson@14803
   435
paulson@14747
   436
lemma (in group) lcos_m_assoc:
paulson@14747
   437
     "[| M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G |]
paulson@14747
   438
      ==> g <# (h <# M) = (g \<otimes> h) <# M"
paulson@14747
   439
by (force simp add: l_coset_def m_assoc)
paulson@13870
   440
paulson@14747
   441
lemma (in group) lcos_mult_one: "M \<subseteq> carrier G ==> \<one> <# M = M"
paulson@14747
   442
by (force simp add: l_coset_def)
paulson@13870
   443
paulson@14747
   444
lemma (in group) l_coset_subset_G:
paulson@14747
   445
     "[| H \<subseteq> carrier G; x \<in> carrier G |] ==> x <# H \<subseteq> carrier G"
paulson@14747
   446
by (auto simp add: l_coset_def subsetD)
paulson@14747
   447
paulson@14747
   448
lemma (in group) l_coset_swap:
paulson@14963
   449
     "\<lbrakk>y \<in> x <# H;  x \<in> carrier G;  subgroup H G\<rbrakk> \<Longrightarrow> x \<in> y <# H"
paulson@14963
   450
proof (simp add: l_coset_def)
paulson@14963
   451
  assume "\<exists>h\<in>H. y = x \<otimes> h"
wenzelm@14666
   452
    and x: "x \<in> carrier G"
paulson@14530
   453
    and sb: "subgroup H G"
paulson@14530
   454
  then obtain h' where h': "h' \<in> H & x \<otimes> h' = y" by blast
paulson@14963
   455
  show "\<exists>h\<in>H. x = y \<otimes> h"
paulson@14530
   456
  proof
paulson@14963
   457
    show "x = y \<otimes> inv h'" using h' x sb
paulson@14530
   458
      by (auto simp add: m_assoc subgroup.subset [THEN subsetD])
paulson@14530
   459
    show "inv h' \<in> H" using h' sb
paulson@14530
   460
      by (auto simp add: subgroup.subset [THEN subsetD] subgroup.m_inv_closed)
paulson@14530
   461
  qed
paulson@14530
   462
qed
paulson@14530
   463
paulson@14747
   464
lemma (in group) l_coset_carrier:
paulson@14530
   465
     "[| y \<in> x <# H;  x \<in> carrier G;  subgroup H G |] ==> y \<in> carrier G"
paulson@14747
   466
by (auto simp add: l_coset_def m_assoc
paulson@14530
   467
                   subgroup.subset [THEN subsetD] subgroup.m_closed)
paulson@14530
   468
paulson@14747
   469
lemma (in group) l_repr_imp_subset:
wenzelm@14666
   470
  assumes y: "y \<in> x <# H" and x: "x \<in> carrier G" and sb: "subgroup H G"
paulson@14530
   471
  shows "y <# H \<subseteq> x <# H"
paulson@14530
   472
proof -
paulson@14530
   473
  from y
paulson@14747
   474
  obtain h' where "h' \<in> H" "x \<otimes> h' = y" by (auto simp add: l_coset_def)
paulson@14530
   475
  thus ?thesis using x sb
paulson@14747
   476
    by (auto simp add: l_coset_def m_assoc
paulson@14530
   477
                       subgroup.subset [THEN subsetD] subgroup.m_closed)
paulson@14530
   478
qed
paulson@14530
   479
paulson@14747
   480
lemma (in group) l_repr_independence:
wenzelm@14666
   481
  assumes y: "y \<in> x <# H" and x: "x \<in> carrier G" and sb: "subgroup H G"
paulson@14530
   482
  shows "x <# H = y <# H"
wenzelm@14666
   483
proof
paulson@14530
   484
  show "x <# H \<subseteq> y <# H"
wenzelm@14666
   485
    by (rule l_repr_imp_subset,
paulson@14530
   486
        (blast intro: l_coset_swap l_coset_carrier y x sb)+)
wenzelm@14666
   487
  show "y <# H \<subseteq> x <# H" by (rule l_repr_imp_subset [OF y x sb])
paulson@14530
   488
qed
paulson@13870
   489
paulson@14747
   490
lemma (in group) setmult_subset_G:
paulson@14963
   491
     "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G\<rbrakk> \<Longrightarrow> H <#> K \<subseteq> carrier G"
paulson@14963
   492
by (auto simp add: set_mult_def subsetD)
paulson@13870
   493
paulson@14963
   494
lemma (in group) subgroup_mult_id: "subgroup H G \<Longrightarrow> H <#> H = H"
paulson@14963
   495
apply (auto simp add: subgroup.m_closed set_mult_def Sigma_def image_def)
paulson@13870
   496
apply (rule_tac x = x in bexI)
paulson@13870
   497
apply (rule bexI [of _ "\<one>"])
wenzelm@14666
   498
apply (auto simp add: subgroup.m_closed subgroup.one_closed
paulson@13870
   499
                      r_one subgroup.subset [THEN subsetD])
paulson@13870
   500
done
paulson@13870
   501
paulson@13870
   502
ballarin@20318
   503
subsubsection {* Set of Inverses of an @{text r_coset}. *}
wenzelm@14666
   504
paulson@14963
   505
lemma (in normal) rcos_inv:
paulson@14963
   506
  assumes x:     "x \<in> carrier G"
paulson@14963
   507
  shows "set_inv (H #> x) = H #> (inv x)" 
paulson@14963
   508
proof (simp add: r_coset_def SET_INV_def x inv_mult_group, safe)
paulson@14963
   509
  fix h
paulson@14963
   510
  assume "h \<in> H"
nipkow@15120
   511
  show "inv x \<otimes> inv h \<in> (\<Union>j\<in>H. {j \<otimes> inv x})"
paulson@14963
   512
  proof
paulson@14963
   513
    show "inv x \<otimes> inv h \<otimes> x \<in> H"
paulson@14963
   514
      by (simp add: inv_op_closed1 prems)
paulson@14963
   515
    show "inv x \<otimes> inv h \<in> {inv x \<otimes> inv h \<otimes> x \<otimes> inv x}"
paulson@14963
   516
      by (simp add: prems m_assoc)
paulson@14963
   517
  qed
paulson@14963
   518
next
paulson@14963
   519
  fix h
paulson@14963
   520
  assume "h \<in> H"
nipkow@15120
   521
  show "h \<otimes> inv x \<in> (\<Union>j\<in>H. {inv x \<otimes> inv j})"
paulson@14963
   522
  proof
paulson@14963
   523
    show "x \<otimes> inv h \<otimes> inv x \<in> H"
paulson@14963
   524
      by (simp add: inv_op_closed2 prems)
paulson@14963
   525
    show "h \<otimes> inv x \<in> {inv x \<otimes> inv (x \<otimes> inv h \<otimes> inv x)}"
paulson@14963
   526
      by (simp add: prems m_assoc [symmetric] inv_mult_group)
paulson@13870
   527
  qed
paulson@13870
   528
qed
paulson@13870
   529
paulson@13870
   530
paulson@14803
   531
subsubsection {*Theorems for @{text "<#>"} with @{text "#>"} or @{text "<#"}.*}
wenzelm@14666
   532
paulson@14747
   533
lemma (in group) setmult_rcos_assoc:
paulson@14963
   534
     "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk>
paulson@14963
   535
      \<Longrightarrow> H <#> (K #> x) = (H <#> K) #> x"
paulson@14963
   536
by (force simp add: r_coset_def set_mult_def m_assoc)
paulson@13870
   537
paulson@14747
   538
lemma (in group) rcos_assoc_lcos:
paulson@14963
   539
     "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk>
paulson@14963
   540
      \<Longrightarrow> (H #> x) <#> K = H <#> (x <# K)"
paulson@14963
   541
by (force simp add: r_coset_def l_coset_def set_mult_def m_assoc)
paulson@13870
   542
paulson@14963
   543
lemma (in normal) rcos_mult_step1:
paulson@14963
   544
     "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
paulson@14963
   545
      \<Longrightarrow> (H #> x) <#> (H #> y) = (H <#> (x <# H)) #> y"
paulson@14963
   546
by (simp add: setmult_rcos_assoc subset
paulson@13870
   547
              r_coset_subset_G l_coset_subset_G rcos_assoc_lcos)
paulson@13870
   548
paulson@14963
   549
lemma (in normal) rcos_mult_step2:
paulson@14963
   550
     "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
paulson@14963
   551
      \<Longrightarrow> (H <#> (x <# H)) #> y = (H <#> (H #> x)) #> y"
paulson@14963
   552
by (insert coset_eq, simp add: normal_def)
paulson@13870
   553
paulson@14963
   554
lemma (in normal) rcos_mult_step3:
paulson@14963
   555
     "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
paulson@14963
   556
      \<Longrightarrow> (H <#> (H #> x)) #> y = H #> (x \<otimes> y)"
paulson@14963
   557
by (simp add: setmult_rcos_assoc coset_mult_assoc
ballarin@19931
   558
              subgroup_mult_id normal.axioms subset prems)
paulson@13870
   559
paulson@14963
   560
lemma (in normal) rcos_sum:
paulson@14963
   561
     "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
paulson@14963
   562
      \<Longrightarrow> (H #> x) <#> (H #> y) = H #> (x \<otimes> y)"
paulson@13870
   563
by (simp add: rcos_mult_step1 rcos_mult_step2 rcos_mult_step3)
paulson@13870
   564
paulson@14963
   565
lemma (in normal) rcosets_mult_eq: "M \<in> rcosets H \<Longrightarrow> H <#> M = M"
wenzelm@14666
   566
  -- {* generalizes @{text subgroup_mult_id} *}
paulson@14963
   567
  by (auto simp add: RCOSETS_def subset
ballarin@19931
   568
        setmult_rcos_assoc subgroup_mult_id normal.axioms prems)
paulson@14963
   569
paulson@14963
   570
paulson@14963
   571
subsubsection{*An Equivalence Relation*}
paulson@14963
   572
paulson@14963
   573
constdefs (structure G)
paulson@14963
   574
  r_congruent :: "[('a,'b)monoid_scheme, 'a set] \<Rightarrow> ('a*'a)set"
paulson@14963
   575
                  ("rcong\<index> _")
paulson@14963
   576
   "rcong H \<equiv> {(x,y). x \<in> carrier G & y \<in> carrier G & inv x \<otimes> y \<in> H}"
paulson@14963
   577
paulson@14963
   578
paulson@14963
   579
lemma (in subgroup) equiv_rcong:
paulson@14963
   580
   includes group G
paulson@14963
   581
   shows "equiv (carrier G) (rcong H)"
paulson@14963
   582
proof (intro equiv.intro)
paulson@14963
   583
  show "refl (carrier G) (rcong H)"
paulson@14963
   584
    by (auto simp add: r_congruent_def refl_def) 
paulson@14963
   585
next
paulson@14963
   586
  show "sym (rcong H)"
paulson@14963
   587
  proof (simp add: r_congruent_def sym_def, clarify)
paulson@14963
   588
    fix x y
paulson@14963
   589
    assume [simp]: "x \<in> carrier G" "y \<in> carrier G" 
paulson@14963
   590
       and "inv x \<otimes> y \<in> H"
paulson@14963
   591
    hence "inv (inv x \<otimes> y) \<in> H" by (simp add: m_inv_closed) 
paulson@14963
   592
    thus "inv y \<otimes> x \<in> H" by (simp add: inv_mult_group)
paulson@14963
   593
  qed
paulson@14963
   594
next
paulson@14963
   595
  show "trans (rcong H)"
paulson@14963
   596
  proof (simp add: r_congruent_def trans_def, clarify)
paulson@14963
   597
    fix x y z
paulson@14963
   598
    assume [simp]: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
paulson@14963
   599
       and "inv x \<otimes> y \<in> H" and "inv y \<otimes> z \<in> H"
paulson@14963
   600
    hence "(inv x \<otimes> y) \<otimes> (inv y \<otimes> z) \<in> H" by simp
paulson@14963
   601
    hence "inv x \<otimes> (y \<otimes> inv y) \<otimes> z \<in> H" by (simp add: m_assoc del: r_inv) 
paulson@14963
   602
    thus "inv x \<otimes> z \<in> H" by simp
paulson@14963
   603
  qed
paulson@14963
   604
qed
paulson@14963
   605
paulson@14963
   606
text{*Equivalence classes of @{text rcong} correspond to left cosets.
paulson@14963
   607
  Was there a mistake in the definitions? I'd have expected them to
paulson@14963
   608
  correspond to right cosets.*}
paulson@14963
   609
paulson@14963
   610
(* CB: This is correct, but subtle.
paulson@14963
   611
   We call H #> a the right coset of a relative to H.  According to
paulson@14963
   612
   Jacobson, this is what the majority of group theory literature does.
paulson@14963
   613
   He then defines the notion of congruence relation ~ over monoids as
paulson@14963
   614
   equivalence relation with a ~ a' & b ~ b' \<Longrightarrow> a*b ~ a'*b'.
paulson@14963
   615
   Our notion of right congruence induced by K: rcong K appears only in
paulson@14963
   616
   the context where K is a normal subgroup.  Jacobson doesn't name it.
paulson@14963
   617
   But in this context left and right cosets are identical.
paulson@14963
   618
*)
paulson@14963
   619
paulson@14963
   620
lemma (in subgroup) l_coset_eq_rcong:
paulson@14963
   621
  includes group G
paulson@14963
   622
  assumes a: "a \<in> carrier G"
paulson@14963
   623
  shows "a <# H = rcong H `` {a}"
paulson@14963
   624
by (force simp add: r_congruent_def l_coset_def m_assoc [symmetric] a ) 
paulson@13870
   625
paulson@13870
   626
ballarin@20318
   627
subsubsection{*Two Distinct Right Cosets are Disjoint*}
paulson@14803
   628
paulson@14803
   629
lemma (in group) rcos_equation:
paulson@14963
   630
  includes subgroup H G
paulson@14963
   631
  shows
paulson@14963
   632
     "\<lbrakk>ha \<otimes> a = h \<otimes> b; a \<in> carrier G;  b \<in> carrier G;  
paulson@14963
   633
        h \<in> H;  ha \<in> H;  hb \<in> H\<rbrakk>
paulson@14963
   634
      \<Longrightarrow> hb \<otimes> a \<in> (\<Union>h\<in>H. {h \<otimes> b})"
paulson@14963
   635
apply (rule UN_I [of "hb \<otimes> ((inv ha) \<otimes> h)"])
paulson@14963
   636
apply (simp add: ); 
paulson@14963
   637
apply (simp add: m_assoc transpose_inv)
paulson@14803
   638
done
paulson@14803
   639
paulson@14803
   640
lemma (in group) rcos_disjoint:
paulson@14963
   641
  includes subgroup H G
paulson@14963
   642
  shows "\<lbrakk>a \<in> rcosets H; b \<in> rcosets H; a\<noteq>b\<rbrakk> \<Longrightarrow> a \<inter> b = {}"
paulson@14963
   643
apply (simp add: RCOSETS_def r_coset_def)
paulson@14963
   644
apply (blast intro: rcos_equation prems sym)
paulson@14803
   645
done
paulson@14803
   646
ballarin@20318
   647
subsection {* Further lemmas for @{text "r_congruent"} *}
ballarin@20318
   648
ballarin@20318
   649
text {* The relation is a congruence *}
ballarin@20318
   650
ballarin@20318
   651
lemma (in normal) congruent_rcong:
ballarin@20318
   652
  shows "congruent2 (rcong H) (rcong H) (\<lambda>a b. a \<otimes> b <# H)"
ballarin@20318
   653
proof (intro congruent2I[of "carrier G" _ "carrier G" _] equiv_rcong is_group)
ballarin@20318
   654
  fix a b c
ballarin@20318
   655
  assume abrcong: "(a, b) \<in> rcong H"
ballarin@20318
   656
    and ccarr: "c \<in> carrier G"
ballarin@20318
   657
ballarin@20318
   658
  from abrcong
ballarin@20318
   659
      have acarr: "a \<in> carrier G"
ballarin@20318
   660
        and bcarr: "b \<in> carrier G"
ballarin@20318
   661
        and abH: "inv a \<otimes> b \<in> H"
ballarin@20318
   662
      unfolding r_congruent_def
ballarin@20318
   663
      by fast+
ballarin@20318
   664
ballarin@20318
   665
  note carr = acarr bcarr ccarr
ballarin@20318
   666
ballarin@20318
   667
  from ccarr and abH
ballarin@20318
   668
      have "inv c \<otimes> (inv a \<otimes> b) \<otimes> c \<in> H" by (rule inv_op_closed1)
ballarin@20318
   669
  moreover
ballarin@20318
   670
      from carr and inv_closed
ballarin@20318
   671
      have "inv c \<otimes> (inv a \<otimes> b) \<otimes> c = (inv c \<otimes> inv a) \<otimes> (b \<otimes> c)" 
ballarin@20318
   672
      by (force cong: m_assoc)
ballarin@20318
   673
  moreover 
ballarin@20318
   674
      from carr and inv_closed
ballarin@20318
   675
      have "\<dots> = (inv (a \<otimes> c)) \<otimes> (b \<otimes> c)"
ballarin@20318
   676
      by (simp add: inv_mult_group)
ballarin@20318
   677
  ultimately
ballarin@20318
   678
      have "(inv (a \<otimes> c)) \<otimes> (b \<otimes> c) \<in> H" by simp
ballarin@20318
   679
  from carr and this
ballarin@20318
   680
     have "(b \<otimes> c) \<in> (a \<otimes> c) <# H"
ballarin@20318
   681
     by (simp add: lcos_module_rev[OF is_group])
ballarin@20318
   682
  from carr and this and is_subgroup
ballarin@20318
   683
     show "(a \<otimes> c) <# H = (b \<otimes> c) <# H" by (intro l_repr_independence, simp+)
ballarin@20318
   684
next
ballarin@20318
   685
  fix a b c
ballarin@20318
   686
  assume abrcong: "(a, b) \<in> rcong H"
ballarin@20318
   687
    and ccarr: "c \<in> carrier G"
ballarin@20318
   688
ballarin@20318
   689
  from ccarr have "c \<in> Units G" by (simp add: Units_eq)
ballarin@20318
   690
  hence cinvc_one: "inv c \<otimes> c = \<one>" by (rule Units_l_inv)
ballarin@20318
   691
ballarin@20318
   692
  from abrcong
ballarin@20318
   693
      have acarr: "a \<in> carrier G"
ballarin@20318
   694
       and bcarr: "b \<in> carrier G"
ballarin@20318
   695
       and abH: "inv a \<otimes> b \<in> H"
ballarin@20318
   696
      by (unfold r_congruent_def, fast+)
ballarin@20318
   697
ballarin@20318
   698
  note carr = acarr bcarr ccarr
ballarin@20318
   699
ballarin@20318
   700
  from carr and inv_closed
ballarin@20318
   701
     have "inv a \<otimes> b = inv a \<otimes> (\<one> \<otimes> b)" by simp
ballarin@20318
   702
  also from carr and inv_closed
ballarin@20318
   703
      have "\<dots> = inv a \<otimes> (inv c \<otimes> c) \<otimes> b" by simp
ballarin@20318
   704
  also from carr and inv_closed
ballarin@20318
   705
      have "\<dots> = (inv a \<otimes> inv c) \<otimes> (c \<otimes> b)" by (force cong: m_assoc)
ballarin@20318
   706
  also from carr and inv_closed
ballarin@20318
   707
      have "\<dots> = inv (c \<otimes> a) \<otimes> (c \<otimes> b)" by (simp add: inv_mult_group)
ballarin@20318
   708
  finally
ballarin@20318
   709
      have "inv a \<otimes> b = inv (c \<otimes> a) \<otimes> (c \<otimes> b)" .
ballarin@20318
   710
  from abH and this
ballarin@20318
   711
      have "inv (c \<otimes> a) \<otimes> (c \<otimes> b) \<in> H" by simp
ballarin@20318
   712
ballarin@20318
   713
  from carr and this
ballarin@20318
   714
     have "(c \<otimes> b) \<in> (c \<otimes> a) <# H"
ballarin@20318
   715
     by (simp add: lcos_module_rev[OF is_group])
ballarin@20318
   716
  from carr and this and is_subgroup
ballarin@20318
   717
     show "(c \<otimes> a) <# H = (c \<otimes> b) <# H" by (intro l_repr_independence, simp+)
ballarin@20318
   718
qed
ballarin@20318
   719
paulson@14803
   720
paulson@14803
   721
subsection {*Order of a Group and Lagrange's Theorem*}
paulson@14803
   722
paulson@14803
   723
constdefs
paulson@14963
   724
  order :: "('a, 'b) monoid_scheme \<Rightarrow> nat"
paulson@14963
   725
  "order S \<equiv> card (carrier S)"
paulson@13870
   726
paulson@14963
   727
lemma (in group) rcos_self:
paulson@14963
   728
  includes subgroup
paulson@14963
   729
  shows "x \<in> carrier G \<Longrightarrow> x \<in> H #> x"
paulson@14963
   730
apply (simp add: r_coset_def)
paulson@14963
   731
apply (rule_tac x="\<one>" in bexI) 
paulson@14963
   732
apply (auto simp add: ); 
paulson@14963
   733
done
paulson@14963
   734
paulson@14963
   735
lemma (in group) rcosets_part_G:
paulson@14963
   736
  includes subgroup
paulson@14963
   737
  shows "\<Union>(rcosets H) = carrier G"
paulson@13870
   738
apply (rule equalityI)
paulson@14963
   739
 apply (force simp add: RCOSETS_def r_coset_def)
paulson@14963
   740
apply (auto simp add: RCOSETS_def intro: rcos_self prems)
paulson@13870
   741
done
paulson@13870
   742
paulson@14747
   743
lemma (in group) cosets_finite:
paulson@14963
   744
     "\<lbrakk>c \<in> rcosets H;  H \<subseteq> carrier G;  finite (carrier G)\<rbrakk> \<Longrightarrow> finite c"
paulson@14963
   745
apply (auto simp add: RCOSETS_def)
paulson@14963
   746
apply (simp add: r_coset_subset_G [THEN finite_subset])
paulson@13870
   747
done
paulson@13870
   748
paulson@14747
   749
text{*The next two lemmas support the proof of @{text card_cosets_equal}.*}
paulson@14747
   750
lemma (in group) inj_on_f:
paulson@14963
   751
    "\<lbrakk>H \<subseteq> carrier G;  a \<in> carrier G\<rbrakk> \<Longrightarrow> inj_on (\<lambda>y. y \<otimes> inv a) (H #> a)"
paulson@13870
   752
apply (rule inj_onI)
paulson@13870
   753
apply (subgoal_tac "x \<in> carrier G & y \<in> carrier G")
paulson@13870
   754
 prefer 2 apply (blast intro: r_coset_subset_G [THEN subsetD])
paulson@13870
   755
apply (simp add: subsetD)
paulson@13870
   756
done
paulson@13870
   757
paulson@14747
   758
lemma (in group) inj_on_g:
paulson@14963
   759
    "\<lbrakk>H \<subseteq> carrier G;  a \<in> carrier G\<rbrakk> \<Longrightarrow> inj_on (\<lambda>y. y \<otimes> a) H"
paulson@13870
   760
by (force simp add: inj_on_def subsetD)
paulson@13870
   761
paulson@14747
   762
lemma (in group) card_cosets_equal:
paulson@14963
   763
     "\<lbrakk>c \<in> rcosets H;  H \<subseteq> carrier G; finite(carrier G)\<rbrakk>
paulson@14963
   764
      \<Longrightarrow> card c = card H"
paulson@14963
   765
apply (auto simp add: RCOSETS_def)
paulson@13870
   766
apply (rule card_bij_eq)
wenzelm@14666
   767
     apply (rule inj_on_f, assumption+)
paulson@14747
   768
    apply (force simp add: m_assoc subsetD r_coset_def)
wenzelm@14666
   769
   apply (rule inj_on_g, assumption+)
paulson@14747
   770
  apply (force simp add: m_assoc subsetD r_coset_def)
paulson@13870
   771
 txt{*The sets @{term "H #> a"} and @{term "H"} are finite.*}
paulson@13870
   772
 apply (simp add: r_coset_subset_G [THEN finite_subset])
paulson@13870
   773
apply (blast intro: finite_subset)
paulson@13870
   774
done
paulson@13870
   775
paulson@14963
   776
lemma (in group) rcosets_subset_PowG:
paulson@14963
   777
     "subgroup H G  \<Longrightarrow> rcosets H \<subseteq> Pow(carrier G)"
paulson@14963
   778
apply (simp add: RCOSETS_def)
paulson@13870
   779
apply (blast dest: r_coset_subset_G subgroup.subset)
paulson@13870
   780
done
paulson@13870
   781
paulson@14803
   782
paulson@14803
   783
theorem (in group) lagrange:
paulson@14963
   784
     "\<lbrakk>finite(carrier G); subgroup H G\<rbrakk>
paulson@14963
   785
      \<Longrightarrow> card(rcosets H) * card(H) = order(G)"
paulson@14963
   786
apply (simp (no_asm_simp) add: order_def rcosets_part_G [symmetric])
paulson@14803
   787
apply (subst mult_commute)
paulson@14803
   788
apply (rule card_partition)
paulson@14963
   789
   apply (simp add: rcosets_subset_PowG [THEN finite_subset])
paulson@14963
   790
  apply (simp add: rcosets_part_G)
paulson@14803
   791
 apply (simp add: card_cosets_equal subgroup.subset)
paulson@14803
   792
apply (simp add: rcos_disjoint)
paulson@14803
   793
done
paulson@14803
   794
paulson@14803
   795
paulson@14747
   796
subsection {*Quotient Groups: Factorization of a Group*}
paulson@13870
   797
paulson@13870
   798
constdefs
paulson@14963
   799
  FactGroup :: "[('a,'b) monoid_scheme, 'a set] \<Rightarrow> ('a set) monoid"
paulson@14803
   800
     (infixl "Mod" 65)
paulson@14747
   801
    --{*Actually defined for groups rather than monoids*}
paulson@14963
   802
  "FactGroup G H \<equiv>
paulson@14963
   803
    \<lparr>carrier = rcosets\<^bsub>G\<^esub> H, mult = set_mult G, one = H\<rparr>"
paulson@14747
   804
paulson@14963
   805
lemma (in normal) setmult_closed:
paulson@14963
   806
     "\<lbrakk>K1 \<in> rcosets H; K2 \<in> rcosets H\<rbrakk> \<Longrightarrow> K1 <#> K2 \<in> rcosets H"
paulson@14963
   807
by (auto simp add: rcos_sum RCOSETS_def)
paulson@13870
   808
paulson@14963
   809
lemma (in normal) setinv_closed:
paulson@14963
   810
     "K \<in> rcosets H \<Longrightarrow> set_inv K \<in> rcosets H"
paulson@14963
   811
by (auto simp add: rcos_inv RCOSETS_def)
ballarin@13889
   812
paulson@14963
   813
lemma (in normal) rcosets_assoc:
paulson@14963
   814
     "\<lbrakk>M1 \<in> rcosets H; M2 \<in> rcosets H; M3 \<in> rcosets H\<rbrakk>
paulson@14963
   815
      \<Longrightarrow> M1 <#> M2 <#> M3 = M1 <#> (M2 <#> M3)"
paulson@14963
   816
by (auto simp add: RCOSETS_def rcos_sum m_assoc)
paulson@13870
   817
paulson@14963
   818
lemma (in subgroup) subgroup_in_rcosets:
paulson@14963
   819
  includes group G
paulson@14963
   820
  shows "H \<in> rcosets H"
ballarin@13889
   821
proof -
paulson@14963
   822
  have "H #> \<one> = H"
paulson@14963
   823
    by (rule coset_join2, auto)
ballarin@13889
   824
  then show ?thesis
paulson@14963
   825
    by (auto simp add: RCOSETS_def)
ballarin@13889
   826
qed
ballarin@13889
   827
paulson@14963
   828
lemma (in normal) rcosets_inv_mult_group_eq:
paulson@14963
   829
     "M \<in> rcosets H \<Longrightarrow> set_inv M <#> M = H"
ballarin@19931
   830
by (auto simp add: RCOSETS_def rcos_inv rcos_sum subgroup.subset normal.axioms prems)
ballarin@13889
   831
paulson@14963
   832
theorem (in normal) factorgroup_is_group:
paulson@14963
   833
  "group (G Mod H)"
wenzelm@14666
   834
apply (simp add: FactGroup_def)
ballarin@13936
   835
apply (rule groupI)
paulson@14747
   836
    apply (simp add: setmult_closed)
paulson@14963
   837
   apply (simp add: normal_imp_subgroup subgroup_in_rcosets [OF is_group])
paulson@14963
   838
  apply (simp add: restrictI setmult_closed rcosets_assoc)
ballarin@13889
   839
 apply (simp add: normal_imp_subgroup
paulson@14963
   840
                  subgroup_in_rcosets rcosets_mult_eq)
paulson@14963
   841
apply (auto dest: rcosets_inv_mult_group_eq simp add: setinv_closed)
ballarin@13889
   842
done
ballarin@13889
   843
paulson@14803
   844
lemma mult_FactGroup [simp]: "X \<otimes>\<^bsub>(G Mod H)\<^esub> X' = X <#>\<^bsub>G\<^esub> X'"
paulson@14803
   845
  by (simp add: FactGroup_def) 
paulson@14803
   846
paulson@14963
   847
lemma (in normal) inv_FactGroup:
paulson@14963
   848
     "X \<in> carrier (G Mod H) \<Longrightarrow> inv\<^bsub>G Mod H\<^esub> X = set_inv X"
paulson@14747
   849
apply (rule group.inv_equality [OF factorgroup_is_group]) 
paulson@14963
   850
apply (simp_all add: FactGroup_def setinv_closed rcosets_inv_mult_group_eq)
paulson@14747
   851
done
paulson@14747
   852
paulson@14747
   853
text{*The coset map is a homomorphism from @{term G} to the quotient group
paulson@14963
   854
  @{term "G Mod H"}*}
paulson@14963
   855
lemma (in normal) r_coset_hom_Mod:
paulson@14963
   856
  "(\<lambda>a. H #> a) \<in> hom G (G Mod H)"
paulson@14963
   857
  by (auto simp add: FactGroup_def RCOSETS_def Pi_def hom_def rcos_sum)
paulson@14747
   858
paulson@14963
   859
 
paulson@14963
   860
subsection{*The First Isomorphism Theorem*}
paulson@14803
   861
paulson@14963
   862
text{*The quotient by the kernel of a homomorphism is isomorphic to the 
paulson@14963
   863
  range of that homomorphism.*}
paulson@14803
   864
paulson@14803
   865
constdefs
paulson@14963
   866
  kernel :: "('a, 'm) monoid_scheme \<Rightarrow> ('b, 'n) monoid_scheme \<Rightarrow> 
paulson@14963
   867
             ('a \<Rightarrow> 'b) \<Rightarrow> 'a set" 
paulson@14803
   868
    --{*the kernel of a homomorphism*}
paulson@14963
   869
  "kernel G H h \<equiv> {x. x \<in> carrier G & h x = \<one>\<^bsub>H\<^esub>}";
paulson@14803
   870
paulson@14803
   871
lemma (in group_hom) subgroup_kernel: "subgroup (kernel G H h) G"
paulson@14963
   872
apply (rule subgroup.intro) 
paulson@14803
   873
apply (auto simp add: kernel_def group.intro prems) 
paulson@14803
   874
done
paulson@14803
   875
paulson@14803
   876
text{*The kernel of a homomorphism is a normal subgroup*}
paulson@14963
   877
lemma (in group_hom) normal_kernel: "(kernel G H h) \<lhd> G"
ballarin@19931
   878
apply (simp add: G.normal_inv_iff subgroup_kernel)
ballarin@19931
   879
apply (simp add: kernel_def)
paulson@14803
   880
done
paulson@14803
   881
paulson@14803
   882
lemma (in group_hom) FactGroup_nonempty:
paulson@14803
   883
  assumes X: "X \<in> carrier (G Mod kernel G H h)"
paulson@14803
   884
  shows "X \<noteq> {}"
paulson@14803
   885
proof -
paulson@14803
   886
  from X
paulson@14803
   887
  obtain g where "g \<in> carrier G" 
paulson@14803
   888
             and "X = kernel G H h #> g"
paulson@14963
   889
    by (auto simp add: FactGroup_def RCOSETS_def)
paulson@14803
   890
  thus ?thesis 
paulson@14963
   891
   by (auto simp add: kernel_def r_coset_def image_def intro: hom_one)
paulson@14803
   892
qed
paulson@14803
   893
paulson@14803
   894
paulson@14803
   895
lemma (in group_hom) FactGroup_contents_mem:
paulson@14803
   896
  assumes X: "X \<in> carrier (G Mod (kernel G H h))"
paulson@14803
   897
  shows "contents (h`X) \<in> carrier H"
paulson@14803
   898
proof -
paulson@14803
   899
  from X
paulson@14803
   900
  obtain g where g: "g \<in> carrier G" 
paulson@14803
   901
             and "X = kernel G H h #> g"
paulson@14963
   902
    by (auto simp add: FactGroup_def RCOSETS_def)
paulson@14963
   903
  hence "h ` X = {h g}" by (auto simp add: kernel_def r_coset_def image_def g)
paulson@14803
   904
  thus ?thesis by (auto simp add: g)
paulson@14803
   905
qed
paulson@14803
   906
paulson@14803
   907
lemma (in group_hom) FactGroup_hom:
paulson@14963
   908
     "(\<lambda>X. contents (h`X)) \<in> hom (G Mod (kernel G H h)) H"
paulson@14963
   909
apply (simp add: hom_def FactGroup_contents_mem  normal.factorgroup_is_group [OF normal_kernel] group.axioms monoid.m_closed)  
paulson@14803
   910
proof (simp add: hom_def funcsetI FactGroup_contents_mem, intro ballI) 
paulson@14803
   911
  fix X and X'
paulson@14803
   912
  assume X:  "X  \<in> carrier (G Mod kernel G H h)"
paulson@14803
   913
     and X': "X' \<in> carrier (G Mod kernel G H h)"
paulson@14803
   914
  then
paulson@14803
   915
  obtain g and g'
paulson@14803
   916
           where "g \<in> carrier G" and "g' \<in> carrier G" 
paulson@14803
   917
             and "X = kernel G H h #> g" and "X' = kernel G H h #> g'"
paulson@14963
   918
    by (auto simp add: FactGroup_def RCOSETS_def)
paulson@14803
   919
  hence all: "\<forall>x\<in>X. h x = h g" "\<forall>x\<in>X'. h x = h g'" 
paulson@14803
   920
    and Xsub: "X \<subseteq> carrier G" and X'sub: "X' \<subseteq> carrier G"
paulson@14803
   921
    by (force simp add: kernel_def r_coset_def image_def)+
paulson@14803
   922
  hence "h ` (X <#> X') = {h g \<otimes>\<^bsub>H\<^esub> h g'}" using X X'
paulson@14803
   923
    by (auto dest!: FactGroup_nonempty
paulson@14803
   924
             simp add: set_mult_def image_eq_UN 
paulson@14803
   925
                       subsetD [OF Xsub] subsetD [OF X'sub]) 
paulson@14803
   926
  thus "contents (h ` (X <#> X')) = contents (h ` X) \<otimes>\<^bsub>H\<^esub> contents (h ` X')"
paulson@14803
   927
    by (simp add: all image_eq_UN FactGroup_nonempty X X')  
paulson@14803
   928
qed
paulson@14803
   929
paulson@14963
   930
paulson@14803
   931
text{*Lemma for the following injectivity result*}
paulson@14803
   932
lemma (in group_hom) FactGroup_subset:
paulson@14963
   933
     "\<lbrakk>g \<in> carrier G; g' \<in> carrier G; h g = h g'\<rbrakk>
paulson@14963
   934
      \<Longrightarrow>  kernel G H h #> g \<subseteq> kernel G H h #> g'"
paulson@14803
   935
apply (clarsimp simp add: kernel_def r_coset_def image_def);
paulson@14803
   936
apply (rename_tac y)  
paulson@14803
   937
apply (rule_tac x="y \<otimes> g \<otimes> inv g'" in exI) 
paulson@14803
   938
apply (simp add: G.m_assoc); 
paulson@14803
   939
done
paulson@14803
   940
paulson@14803
   941
lemma (in group_hom) FactGroup_inj_on:
paulson@14803
   942
     "inj_on (\<lambda>X. contents (h ` X)) (carrier (G Mod kernel G H h))"
paulson@14803
   943
proof (simp add: inj_on_def, clarify) 
paulson@14803
   944
  fix X and X'
paulson@14803
   945
  assume X:  "X  \<in> carrier (G Mod kernel G H h)"
paulson@14803
   946
     and X': "X' \<in> carrier (G Mod kernel G H h)"
paulson@14803
   947
  then
paulson@14803
   948
  obtain g and g'
paulson@14803
   949
           where gX: "g \<in> carrier G"  "g' \<in> carrier G" 
paulson@14803
   950
              "X = kernel G H h #> g" "X' = kernel G H h #> g'"
paulson@14963
   951
    by (auto simp add: FactGroup_def RCOSETS_def)
paulson@14803
   952
  hence all: "\<forall>x\<in>X. h x = h g" "\<forall>x\<in>X'. h x = h g'" 
paulson@14803
   953
    by (force simp add: kernel_def r_coset_def image_def)+
paulson@14803
   954
  assume "contents (h ` X) = contents (h ` X')"
paulson@14803
   955
  hence h: "h g = h g'"
paulson@14803
   956
    by (simp add: image_eq_UN all FactGroup_nonempty X X') 
paulson@14803
   957
  show "X=X'" by (rule equalityI) (simp_all add: FactGroup_subset h gX) 
paulson@14803
   958
qed
paulson@14803
   959
paulson@14803
   960
text{*If the homomorphism @{term h} is onto @{term H}, then so is the
paulson@14803
   961
homomorphism from the quotient group*}
paulson@14803
   962
lemma (in group_hom) FactGroup_onto:
paulson@14803
   963
  assumes h: "h ` carrier G = carrier H"
paulson@14803
   964
  shows "(\<lambda>X. contents (h ` X)) ` carrier (G Mod kernel G H h) = carrier H"
paulson@14803
   965
proof
paulson@14803
   966
  show "(\<lambda>X. contents (h ` X)) ` carrier (G Mod kernel G H h) \<subseteq> carrier H"
paulson@14803
   967
    by (auto simp add: FactGroup_contents_mem)
paulson@14803
   968
  show "carrier H \<subseteq> (\<lambda>X. contents (h ` X)) ` carrier (G Mod kernel G H h)"
paulson@14803
   969
  proof
paulson@14803
   970
    fix y
paulson@14803
   971
    assume y: "y \<in> carrier H"
paulson@14803
   972
    with h obtain g where g: "g \<in> carrier G" "h g = y"
paulson@14803
   973
      by (blast elim: equalityE); 
nipkow@15120
   974
    hence "(\<Union>x\<in>kernel G H h #> g. {h x}) = {y}" 
paulson@14803
   975
      by (auto simp add: y kernel_def r_coset_def) 
paulson@14803
   976
    with g show "y \<in> (\<lambda>X. contents (h ` X)) ` carrier (G Mod kernel G H h)" 
paulson@14963
   977
      by (auto intro!: bexI simp add: FactGroup_def RCOSETS_def image_eq_UN)
paulson@14803
   978
  qed
paulson@14803
   979
qed
paulson@14803
   980
paulson@14803
   981
paulson@14803
   982
text{*If @{term h} is a homomorphism from @{term G} onto @{term H}, then the
paulson@14803
   983
 quotient group @{term "G Mod (kernel G H h)"} is isomorphic to @{term H}.*}
paulson@14803
   984
theorem (in group_hom) FactGroup_iso:
paulson@14803
   985
  "h ` carrier G = carrier H
paulson@14963
   986
   \<Longrightarrow> (\<lambda>X. contents (h`X)) \<in> (G Mod (kernel G H h)) \<cong> H"
paulson@14803
   987
by (simp add: iso_def FactGroup_hom FactGroup_inj_on bij_betw_def 
paulson@14803
   988
              FactGroup_onto) 
paulson@14803
   989
paulson@14963
   990
paulson@13870
   991
end