src/HOL/Algebra/AbelCoset.thy
author haftmann
Mon Mar 01 13:40:23 2010 +0100 (2010-03-01)
changeset 35416 d8d7d1b785af
parent 30729 461ee3e49ad3
child 35847 19f1f7066917
permissions -rw-r--r--
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
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(*
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  Title:     HOL/Algebra/AbelCoset.thy
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  Author:    Stephan Hohe, TU Muenchen
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*)
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theory AbelCoset
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imports Coset Ring
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begin
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subsection {* More Lifting from Groups to Abelian Groups *}
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subsubsection {* Definitions *}
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text {* Hiding @{text "<+>"} from @{theory Sum_Type} until I come
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  up with better syntax here *}
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no_notation Plus (infixr "<+>" 65)
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constdefs (structure G)
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  a_r_coset    :: "[_, 'a set, 'a] \<Rightarrow> 'a set"    (infixl "+>\<index>" 60)
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  "a_r_coset G \<equiv> r_coset \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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  a_l_coset    :: "[_, 'a, 'a set] \<Rightarrow> 'a set"    (infixl "<+\<index>" 60)
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  "a_l_coset G \<equiv> l_coset \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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  A_RCOSETS  :: "[_, 'a set] \<Rightarrow> ('a set)set"   ("a'_rcosets\<index> _" [81] 80)
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  "A_RCOSETS G H \<equiv> RCOSETS \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> H"
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  set_add  :: "[_, 'a set ,'a set] \<Rightarrow> 'a set" (infixl "<+>\<index>" 60)
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  "set_add G \<equiv> set_mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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  A_SET_INV :: "[_,'a set] \<Rightarrow> 'a set"  ("a'_set'_inv\<index> _" [81] 80)
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  "A_SET_INV G H \<equiv> SET_INV \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> H"
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constdefs (structure G)
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  a_r_congruent :: "[('a,'b)ring_scheme, 'a set] \<Rightarrow> ('a*'a)set"
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                  ("racong\<index> _")
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   "a_r_congruent G \<equiv> r_congruent \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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definition A_FactGroup :: "[('a,'b) ring_scheme, 'a set] \<Rightarrow> ('a set) monoid" (infixl "A'_Mod" 65) where
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    --{*Actually defined for groups rather than monoids*}
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  "A_FactGroup G H \<equiv> FactGroup \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> H"
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definition a_kernel :: "('a, 'm) ring_scheme \<Rightarrow> ('b, 'n) ring_scheme \<Rightarrow> 
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             ('a \<Rightarrow> 'b) \<Rightarrow> 'a set" where 
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    --{*the kernel of a homomorphism (additive)*}
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  "a_kernel G H h \<equiv> kernel \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>
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                              \<lparr>carrier = carrier H, mult = add H, one = zero H\<rparr> h"
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locale abelian_group_hom = G: abelian_group G + H: abelian_group H
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    for G (structure) and H (structure) +
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  fixes h
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  assumes a_group_hom: "group_hom (| carrier = carrier G, mult = add G, one = zero G |)
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                                  (| carrier = carrier H, mult = add H, one = zero H |) h"
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lemmas a_r_coset_defs =
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  a_r_coset_def r_coset_def
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lemma a_r_coset_def':
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  fixes G (structure)
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  shows "H +> a \<equiv> \<Union>h\<in>H. {h \<oplus> a}"
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unfolding a_r_coset_defs
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by simp
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lemmas a_l_coset_defs =
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  a_l_coset_def l_coset_def
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lemma a_l_coset_def':
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  fixes G (structure)
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  shows "a <+ H \<equiv> \<Union>h\<in>H. {a \<oplus> h}"
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unfolding a_l_coset_defs
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by simp
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lemmas A_RCOSETS_defs =
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  A_RCOSETS_def RCOSETS_def
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lemma A_RCOSETS_def':
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  fixes G (structure)
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  shows "a_rcosets H \<equiv> \<Union>a\<in>carrier G. {H +> a}"
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unfolding A_RCOSETS_defs
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by (fold a_r_coset_def, simp)
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lemmas set_add_defs =
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  set_add_def set_mult_def
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lemma set_add_def':
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  fixes G (structure)
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  shows "H <+> K \<equiv> \<Union>h\<in>H. \<Union>k\<in>K. {h \<oplus> k}"
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unfolding set_add_defs
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by simp
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lemmas A_SET_INV_defs =
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  A_SET_INV_def SET_INV_def
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lemma A_SET_INV_def':
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  fixes G (structure)
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  shows "a_set_inv H \<equiv> \<Union>h\<in>H. {\<ominus> h}"
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unfolding A_SET_INV_defs
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by (fold a_inv_def)
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subsubsection {* Cosets *}
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lemma (in abelian_group) a_coset_add_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 \<oplus> h)"
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by (rule group.coset_mult_assoc [OF a_group,
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    folded a_r_coset_def, simplified monoid_record_simps])
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lemma (in abelian_group) a_coset_add_zero [simp]:
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  "M \<subseteq> carrier G ==> M +> \<zero> = M"
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by (rule group.coset_mult_one [OF a_group,
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    folded a_r_coset_def, simplified monoid_record_simps])
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lemma (in abelian_group) a_coset_add_inv1:
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     "[| M +> (x \<oplus> (\<ominus> 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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by (rule group.coset_mult_inv1 [OF a_group,
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    folded a_r_coset_def a_inv_def, simplified monoid_record_simps])
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lemma (in abelian_group) a_coset_add_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 \<oplus> (\<ominus> y)) = M"
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by (rule group.coset_mult_inv2 [OF a_group,
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    folded a_r_coset_def a_inv_def, simplified monoid_record_simps])
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lemma (in abelian_group) a_coset_join1:
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     "[| H +> x = H;  x \<in> carrier G;  subgroup H (|carrier = carrier G, mult = add G, one = zero G|) |] ==> x \<in> H"
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by (rule group.coset_join1 [OF a_group,
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    folded a_r_coset_def, simplified monoid_record_simps])
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lemma (in abelian_group) a_solve_equation:
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    "\<lbrakk>subgroup H (|carrier = carrier G, mult = add G, one = zero G|); x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> \<exists>h\<in>H. y = h \<oplus> x"
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by (rule group.solve_equation [OF a_group,
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    folded a_r_coset_def, simplified monoid_record_simps])
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lemma (in abelian_group) a_repr_independence:
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     "\<lbrakk>y \<in> H +> x;  x \<in> carrier G; subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> \<rbrakk> \<Longrightarrow> H +> x = H +> y"
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by (rule group.repr_independence [OF a_group,
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    folded a_r_coset_def, simplified monoid_record_simps])
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lemma (in abelian_group) a_coset_join2:
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     "\<lbrakk>x \<in> carrier G;  subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>; x\<in>H\<rbrakk> \<Longrightarrow> H +> x = H"
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by (rule group.coset_join2 [OF a_group,
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    folded a_r_coset_def, simplified monoid_record_simps])
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lemma (in abelian_monoid) a_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 (rule monoid.r_coset_subset_G [OF a_monoid,
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    folded a_r_coset_def, simplified monoid_record_simps])
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lemma (in abelian_group) a_rcosI:
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     "[| h \<in> H; H \<subseteq> carrier G; x \<in> carrier G|] ==> h \<oplus> x \<in> H +> x"
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by (rule group.rcosI [OF a_group,
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    folded a_r_coset_def, simplified monoid_record_simps])
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lemma (in abelian_group) a_rcosetsI:
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     "\<lbrakk>H \<subseteq> carrier G; x \<in> carrier G\<rbrakk> \<Longrightarrow> H +> x \<in> a_rcosets H"
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by (rule group.rcosetsI [OF a_group,
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    folded a_r_coset_def A_RCOSETS_def, simplified monoid_record_simps])
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text{*Really needed?*}
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lemma (in abelian_group) a_transpose_inv:
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     "[| x \<oplus> y = z;  x \<in> carrier G;  y \<in> carrier G;  z \<in> carrier G |]
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      ==> (\<ominus> x) \<oplus> z = y"
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by (rule group.transpose_inv [OF a_group,
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    folded a_r_coset_def a_inv_def, simplified monoid_record_simps])
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(*
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--"duplicate"
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lemma (in abelian_group) a_rcos_self:
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     "[| x \<in> carrier G; subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> |] ==> x \<in> H +> x"
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by (rule group.rcos_self [OF a_group,
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    folded a_r_coset_def, simplified monoid_record_simps])
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*)
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subsubsection {* Subgroups *}
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locale additive_subgroup =
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  fixes H and G (structure)
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  assumes a_subgroup: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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lemma (in additive_subgroup) is_additive_subgroup:
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  shows "additive_subgroup H G"
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by (rule additive_subgroup_axioms)
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lemma additive_subgroupI:
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  fixes G (structure)
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  assumes a_subgroup: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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  shows "additive_subgroup H G"
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by (rule additive_subgroup.intro) (rule a_subgroup)
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lemma (in additive_subgroup) a_subset:
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     "H \<subseteq> carrier G"
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by (rule subgroup.subset[OF a_subgroup,
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    simplified monoid_record_simps])
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lemma (in additive_subgroup) a_closed [intro, simp]:
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     "\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<oplus> y \<in> H"
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by (rule subgroup.m_closed[OF a_subgroup,
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    simplified monoid_record_simps])
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lemma (in additive_subgroup) zero_closed [simp]:
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     "\<zero> \<in> H"
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by (rule subgroup.one_closed[OF a_subgroup,
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    simplified monoid_record_simps])
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lemma (in additive_subgroup) a_inv_closed [intro,simp]:
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     "x \<in> H \<Longrightarrow> \<ominus> x \<in> H"
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by (rule subgroup.m_inv_closed[OF a_subgroup,
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    folded a_inv_def, simplified monoid_record_simps])
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subsubsection {* Additive subgroups are normal *}
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text {* Every subgroup of an @{text "abelian_group"} is normal *}
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locale abelian_subgroup = additive_subgroup + abelian_group G +
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  assumes a_normal: "normal H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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lemma (in abelian_subgroup) is_abelian_subgroup:
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  shows "abelian_subgroup H G"
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by (rule abelian_subgroup_axioms)
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lemma abelian_subgroupI:
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  assumes a_normal: "normal H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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      and a_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<oplus>\<^bsub>G\<^esub> y = y \<oplus>\<^bsub>G\<^esub> x"
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  shows "abelian_subgroup H G"
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proof -
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  interpret normal "H" "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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  by (rule a_normal)
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  show "abelian_subgroup H G"
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  proof qed (simp add: a_comm)
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qed
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lemma abelian_subgroupI2:
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  fixes G (structure)
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  assumes a_comm_group: "comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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      and a_subgroup: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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  shows "abelian_subgroup H G"
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proof -
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  interpret comm_group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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  by (rule a_comm_group)
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  interpret subgroup "H" "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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  by (rule a_subgroup)
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  show "abelian_subgroup H G"
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  apply unfold_locales
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  proof (simp add: r_coset_def l_coset_def, clarsimp)
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    fix x
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    assume xcarr: "x \<in> carrier G"
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    from a_subgroup
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        have Hcarr: "H \<subseteq> carrier G" by (unfold subgroup_def, simp)
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    from xcarr Hcarr
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        show "(\<Union>h\<in>H. {h \<oplus>\<^bsub>G\<^esub> x}) = (\<Union>h\<in>H. {x \<oplus>\<^bsub>G\<^esub> h})"
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        using m_comm[simplified]
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        by fast
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  qed
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qed
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lemma abelian_subgroupI3:
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  fixes G (structure)
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  assumes asg: "additive_subgroup H G"
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      and ag: "abelian_group G"
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  shows "abelian_subgroup H G"
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apply (rule abelian_subgroupI2)
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 apply (rule abelian_group.a_comm_group[OF ag])
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apply (rule additive_subgroup.a_subgroup[OF asg])
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done
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lemma (in abelian_subgroup) a_coset_eq:
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     "(\<forall>x \<in> carrier G. H +> x = x <+ H)"
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by (rule normal.coset_eq[OF a_normal,
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    folded a_r_coset_def a_l_coset_def, simplified monoid_record_simps])
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lemma (in abelian_subgroup) a_inv_op_closed1:
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  shows "\<lbrakk>x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> (\<ominus> x) \<oplus> h \<oplus> x \<in> H"
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by (rule normal.inv_op_closed1 [OF a_normal,
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    folded a_inv_def, simplified monoid_record_simps])
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lemma (in abelian_subgroup) a_inv_op_closed2:
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  shows "\<lbrakk>x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> x \<oplus> h \<oplus> (\<ominus> x) \<in> H"
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by (rule normal.inv_op_closed2 [OF a_normal,
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    folded a_inv_def, simplified monoid_record_simps])
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text{*Alternative characterization of normal subgroups*}
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lemma (in abelian_group) a_normal_inv_iff:
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     "(N \<lhd> \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>) = 
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      (subgroup N \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> & (\<forall>x \<in> carrier G. \<forall>h \<in> N. x \<oplus> h \<oplus> (\<ominus> x) \<in> N))"
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      (is "_ = ?rhs")
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by (rule group.normal_inv_iff [OF a_group,
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    folded a_inv_def, simplified monoid_record_simps])
ballarin@20318
   296
ballarin@20318
   297
lemma (in abelian_group) a_lcos_m_assoc:
ballarin@20318
   298
     "[| M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G |]
ballarin@20318
   299
      ==> g <+ (h <+ M) = (g \<oplus> h) <+ M"
ballarin@20318
   300
by (rule group.lcos_m_assoc [OF a_group,
ballarin@20318
   301
    folded a_l_coset_def, simplified monoid_record_simps])
ballarin@20318
   302
ballarin@20318
   303
lemma (in abelian_group) a_lcos_mult_one:
ballarin@20318
   304
     "M \<subseteq> carrier G ==> \<zero> <+ M = M"
ballarin@20318
   305
by (rule group.lcos_mult_one [OF a_group,
ballarin@20318
   306
    folded a_l_coset_def, simplified monoid_record_simps])
ballarin@20318
   307
ballarin@20318
   308
ballarin@20318
   309
lemma (in abelian_group) a_l_coset_subset_G:
ballarin@20318
   310
     "[| H \<subseteq> carrier G; x \<in> carrier G |] ==> x <+ H \<subseteq> carrier G"
ballarin@20318
   311
by (rule group.l_coset_subset_G [OF a_group,
ballarin@20318
   312
    folded a_l_coset_def, simplified monoid_record_simps])
ballarin@20318
   313
ballarin@20318
   314
ballarin@20318
   315
lemma (in abelian_group) a_l_coset_swap:
ballarin@20318
   316
     "\<lbrakk>y \<in> x <+ H;  x \<in> carrier G;  subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>\<rbrakk> \<Longrightarrow> x \<in> y <+ H"
ballarin@20318
   317
by (rule group.l_coset_swap [OF a_group,
ballarin@20318
   318
    folded a_l_coset_def, simplified monoid_record_simps])
ballarin@20318
   319
ballarin@20318
   320
lemma (in abelian_group) a_l_coset_carrier:
ballarin@20318
   321
     "[| y \<in> x <+ H;  x \<in> carrier G;  subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> |] ==> y \<in> carrier G"
ballarin@20318
   322
by (rule group.l_coset_carrier [OF a_group,
ballarin@20318
   323
    folded a_l_coset_def, simplified monoid_record_simps])
ballarin@20318
   324
ballarin@20318
   325
lemma (in abelian_group) a_l_repr_imp_subset:
ballarin@20318
   326
  assumes y: "y \<in> x <+ H" and x: "x \<in> carrier G" and sb: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
ballarin@20318
   327
  shows "y <+ H \<subseteq> x <+ H"
wenzelm@23350
   328
apply (rule group.l_repr_imp_subset [OF a_group,
ballarin@20318
   329
    folded a_l_coset_def, simplified monoid_record_simps])
wenzelm@23350
   330
apply (rule y)
wenzelm@23350
   331
apply (rule x)
wenzelm@23350
   332
apply (rule sb)
wenzelm@23350
   333
done
ballarin@20318
   334
ballarin@20318
   335
lemma (in abelian_group) a_l_repr_independence:
ballarin@20318
   336
  assumes y: "y \<in> x <+ H" and x: "x \<in> carrier G" and sb: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
ballarin@20318
   337
  shows "x <+ H = y <+ H"
wenzelm@23350
   338
apply (rule group.l_repr_independence [OF a_group,
ballarin@20318
   339
    folded a_l_coset_def, simplified monoid_record_simps])
wenzelm@23350
   340
apply (rule y)
wenzelm@23350
   341
apply (rule x)
wenzelm@23350
   342
apply (rule sb)
wenzelm@23350
   343
done
ballarin@20318
   344
ballarin@20318
   345
lemma (in abelian_group) setadd_subset_G:
ballarin@20318
   346
     "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G\<rbrakk> \<Longrightarrow> H <+> K \<subseteq> carrier G"
ballarin@20318
   347
by (rule group.setmult_subset_G [OF a_group,
ballarin@20318
   348
    folded set_add_def, simplified monoid_record_simps])
ballarin@20318
   349
ballarin@20318
   350
lemma (in abelian_group) subgroup_add_id: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> \<Longrightarrow> H <+> H = H"
ballarin@20318
   351
by (rule group.subgroup_mult_id [OF a_group,
ballarin@20318
   352
    folded set_add_def, simplified monoid_record_simps])
ballarin@20318
   353
ballarin@20318
   354
lemma (in abelian_subgroup) a_rcos_inv:
ballarin@20318
   355
  assumes x:     "x \<in> carrier G"
ballarin@20318
   356
  shows "a_set_inv (H +> x) = H +> (\<ominus> x)" 
ballarin@20318
   357
by (rule normal.rcos_inv [OF a_normal,
wenzelm@23350
   358
  folded a_r_coset_def a_inv_def A_SET_INV_def, simplified monoid_record_simps]) (rule x)
ballarin@20318
   359
ballarin@20318
   360
lemma (in abelian_group) a_setmult_rcos_assoc:
ballarin@20318
   361
     "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk>
ballarin@20318
   362
      \<Longrightarrow> H <+> (K +> x) = (H <+> K) +> x"
ballarin@20318
   363
by (rule group.setmult_rcos_assoc [OF a_group,
ballarin@20318
   364
    folded set_add_def a_r_coset_def, simplified monoid_record_simps])
ballarin@20318
   365
ballarin@20318
   366
lemma (in abelian_group) a_rcos_assoc_lcos:
ballarin@20318
   367
     "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk>
ballarin@20318
   368
      \<Longrightarrow> (H +> x) <+> K = H <+> (x <+ K)"
ballarin@20318
   369
by (rule group.rcos_assoc_lcos [OF a_group,
ballarin@20318
   370
     folded set_add_def a_r_coset_def a_l_coset_def, simplified monoid_record_simps])
ballarin@20318
   371
ballarin@20318
   372
lemma (in abelian_subgroup) a_rcos_sum:
ballarin@20318
   373
     "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
ballarin@20318
   374
      \<Longrightarrow> (H +> x) <+> (H +> y) = H +> (x \<oplus> y)"
ballarin@20318
   375
by (rule normal.rcos_sum [OF a_normal,
ballarin@20318
   376
    folded set_add_def a_r_coset_def, simplified monoid_record_simps])
ballarin@20318
   377
ballarin@20318
   378
lemma (in abelian_subgroup) rcosets_add_eq:
ballarin@20318
   379
  "M \<in> a_rcosets H \<Longrightarrow> H <+> M = M"
ballarin@20318
   380
  -- {* generalizes @{text subgroup_mult_id} *}
ballarin@20318
   381
by (rule normal.rcosets_mult_eq [OF a_normal,
ballarin@20318
   382
    folded set_add_def A_RCOSETS_def, simplified monoid_record_simps])
ballarin@20318
   383
ballarin@20318
   384
ballarin@27717
   385
subsubsection {* Congruence Relation *}
ballarin@20318
   386
ballarin@20318
   387
lemma (in abelian_subgroup) a_equiv_rcong:
ballarin@20318
   388
   shows "equiv (carrier G) (racong H)"
ballarin@20318
   389
by (rule subgroup.equiv_rcong [OF a_subgroup a_group,
ballarin@20318
   390
    folded a_r_congruent_def, simplified monoid_record_simps])
ballarin@20318
   391
ballarin@20318
   392
lemma (in abelian_subgroup) a_l_coset_eq_rcong:
ballarin@20318
   393
  assumes a: "a \<in> carrier G"
ballarin@20318
   394
  shows "a <+ H = racong H `` {a}"
ballarin@20318
   395
by (rule subgroup.l_coset_eq_rcong [OF a_subgroup a_group,
wenzelm@23350
   396
    folded a_r_congruent_def a_l_coset_def, simplified monoid_record_simps]) (rule a)
ballarin@20318
   397
ballarin@20318
   398
lemma (in abelian_subgroup) a_rcos_equation:
ballarin@20318
   399
  shows
ballarin@20318
   400
     "\<lbrakk>ha \<oplus> a = h \<oplus> b; a \<in> carrier G;  b \<in> carrier G;  
ballarin@20318
   401
        h \<in> H;  ha \<in> H;  hb \<in> H\<rbrakk>
ballarin@20318
   402
      \<Longrightarrow> hb \<oplus> a \<in> (\<Union>h\<in>H. {h \<oplus> b})"
ballarin@20318
   403
by (rule group.rcos_equation [OF a_group a_subgroup,
ballarin@20318
   404
    folded a_r_congruent_def a_l_coset_def, simplified monoid_record_simps])
ballarin@20318
   405
ballarin@20318
   406
lemma (in abelian_subgroup) a_rcos_disjoint:
ballarin@20318
   407
  shows "\<lbrakk>a \<in> a_rcosets H; b \<in> a_rcosets H; a\<noteq>b\<rbrakk> \<Longrightarrow> a \<inter> b = {}"
ballarin@20318
   408
by (rule group.rcos_disjoint [OF a_group a_subgroup,
ballarin@20318
   409
    folded A_RCOSETS_def, simplified monoid_record_simps])
ballarin@20318
   410
ballarin@20318
   411
lemma (in abelian_subgroup) a_rcos_self:
ballarin@20318
   412
  shows "x \<in> carrier G \<Longrightarrow> x \<in> H +> x"
wenzelm@26310
   413
by (rule group.rcos_self [OF a_group _ a_subgroup,
ballarin@20318
   414
    folded a_r_coset_def, simplified monoid_record_simps])
ballarin@20318
   415
ballarin@20318
   416
lemma (in abelian_subgroup) a_rcosets_part_G:
ballarin@20318
   417
  shows "\<Union>(a_rcosets H) = carrier G"
ballarin@20318
   418
by (rule group.rcosets_part_G [OF a_group a_subgroup,
ballarin@20318
   419
    folded A_RCOSETS_def, simplified monoid_record_simps])
ballarin@20318
   420
ballarin@20318
   421
lemma (in abelian_subgroup) a_cosets_finite:
ballarin@20318
   422
     "\<lbrakk>c \<in> a_rcosets H;  H \<subseteq> carrier G;  finite (carrier G)\<rbrakk> \<Longrightarrow> finite c"
ballarin@20318
   423
by (rule group.cosets_finite [OF a_group,
ballarin@20318
   424
    folded A_RCOSETS_def, simplified monoid_record_simps])
ballarin@20318
   425
ballarin@20318
   426
lemma (in abelian_group) a_card_cosets_equal:
ballarin@20318
   427
     "\<lbrakk>c \<in> a_rcosets H;  H \<subseteq> carrier G; finite(carrier G)\<rbrakk>
ballarin@20318
   428
      \<Longrightarrow> card c = card H"
ballarin@20318
   429
by (rule group.card_cosets_equal [OF a_group,
ballarin@20318
   430
    folded A_RCOSETS_def, simplified monoid_record_simps])
ballarin@20318
   431
ballarin@20318
   432
lemma (in abelian_group) rcosets_subset_PowG:
ballarin@20318
   433
     "additive_subgroup H G  \<Longrightarrow> a_rcosets H \<subseteq> Pow(carrier G)"
ballarin@20318
   434
by (rule group.rcosets_subset_PowG [OF a_group,
ballarin@20318
   435
    folded A_RCOSETS_def, simplified monoid_record_simps],
ballarin@20318
   436
    rule additive_subgroup.a_subgroup)
ballarin@20318
   437
ballarin@20318
   438
theorem (in abelian_group) a_lagrange:
ballarin@20318
   439
     "\<lbrakk>finite(carrier G); additive_subgroup H G\<rbrakk>
ballarin@20318
   440
      \<Longrightarrow> card(a_rcosets H) * card(H) = order(G)"
ballarin@20318
   441
by (rule group.lagrange [OF a_group,
ballarin@20318
   442
    folded A_RCOSETS_def, simplified monoid_record_simps order_def, folded order_def])
ballarin@20318
   443
    (fast intro!: additive_subgroup.a_subgroup)+
ballarin@20318
   444
ballarin@20318
   445
ballarin@27717
   446
subsubsection {* Factorization *}
ballarin@20318
   447
ballarin@20318
   448
lemmas A_FactGroup_defs = A_FactGroup_def FactGroup_def
ballarin@20318
   449
ballarin@20318
   450
lemma A_FactGroup_def':
ballarin@27611
   451
  fixes G (structure)
ballarin@20318
   452
  shows "G A_Mod H \<equiv> \<lparr>carrier = a_rcosets\<^bsub>G\<^esub> H, mult = set_add G, one = H\<rparr>"
ballarin@20318
   453
unfolding A_FactGroup_defs
ballarin@20318
   454
by (fold A_RCOSETS_def set_add_def)
ballarin@20318
   455
ballarin@20318
   456
ballarin@20318
   457
lemma (in abelian_subgroup) a_setmult_closed:
ballarin@20318
   458
     "\<lbrakk>K1 \<in> a_rcosets H; K2 \<in> a_rcosets H\<rbrakk> \<Longrightarrow> K1 <+> K2 \<in> a_rcosets H"
ballarin@20318
   459
by (rule normal.setmult_closed [OF a_normal,
ballarin@20318
   460
    folded A_RCOSETS_def set_add_def, simplified monoid_record_simps])
ballarin@20318
   461
ballarin@20318
   462
lemma (in abelian_subgroup) a_setinv_closed:
ballarin@20318
   463
     "K \<in> a_rcosets H \<Longrightarrow> a_set_inv K \<in> a_rcosets H"
ballarin@20318
   464
by (rule normal.setinv_closed [OF a_normal,
ballarin@20318
   465
    folded A_RCOSETS_def A_SET_INV_def, simplified monoid_record_simps])
ballarin@20318
   466
ballarin@20318
   467
lemma (in abelian_subgroup) a_rcosets_assoc:
ballarin@20318
   468
     "\<lbrakk>M1 \<in> a_rcosets H; M2 \<in> a_rcosets H; M3 \<in> a_rcosets H\<rbrakk>
ballarin@20318
   469
      \<Longrightarrow> M1 <+> M2 <+> M3 = M1 <+> (M2 <+> M3)"
ballarin@20318
   470
by (rule normal.rcosets_assoc [OF a_normal,
ballarin@20318
   471
    folded A_RCOSETS_def set_add_def, simplified monoid_record_simps])
ballarin@20318
   472
ballarin@20318
   473
lemma (in abelian_subgroup) a_subgroup_in_rcosets:
ballarin@20318
   474
     "H \<in> a_rcosets H"
ballarin@20318
   475
by (rule subgroup.subgroup_in_rcosets [OF a_subgroup a_group,
ballarin@20318
   476
    folded A_RCOSETS_def, simplified monoid_record_simps])
ballarin@20318
   477
ballarin@20318
   478
lemma (in abelian_subgroup) a_rcosets_inv_mult_group_eq:
ballarin@20318
   479
     "M \<in> a_rcosets H \<Longrightarrow> a_set_inv M <+> M = H"
ballarin@20318
   480
by (rule normal.rcosets_inv_mult_group_eq [OF a_normal,
ballarin@20318
   481
    folded A_RCOSETS_def A_SET_INV_def set_add_def, simplified monoid_record_simps])
ballarin@20318
   482
ballarin@20318
   483
theorem (in abelian_subgroup) a_factorgroup_is_group:
ballarin@20318
   484
  "group (G A_Mod H)"
ballarin@20318
   485
by (rule normal.factorgroup_is_group [OF a_normal,
ballarin@20318
   486
    folded A_FactGroup_def, simplified monoid_record_simps])
ballarin@20318
   487
ballarin@20318
   488
text {* Since the Factorization is based on an \emph{abelian} subgroup, is results in 
ballarin@20318
   489
        a commutative group *}
ballarin@20318
   490
theorem (in abelian_subgroup) a_factorgroup_is_comm_group:
ballarin@20318
   491
  "comm_group (G A_Mod H)"
ballarin@20318
   492
apply (intro comm_group.intro comm_monoid.intro) prefer 3
ballarin@20318
   493
  apply (rule a_factorgroup_is_group)
ballarin@20318
   494
 apply (rule group.axioms[OF a_factorgroup_is_group])
ballarin@20318
   495
apply (rule comm_monoid_axioms.intro)
ballarin@20318
   496
apply (unfold A_FactGroup_def FactGroup_def RCOSETS_def, fold set_add_def a_r_coset_def, clarsimp)
ballarin@20318
   497
apply (simp add: a_rcos_sum a_comm)
ballarin@20318
   498
done
ballarin@20318
   499
ballarin@20318
   500
lemma add_A_FactGroup [simp]: "X \<otimes>\<^bsub>(G A_Mod H)\<^esub> X' = X <+>\<^bsub>G\<^esub> X'"
ballarin@20318
   501
by (simp add: A_FactGroup_def set_add_def)
ballarin@20318
   502
ballarin@20318
   503
lemma (in abelian_subgroup) a_inv_FactGroup:
ballarin@20318
   504
     "X \<in> carrier (G A_Mod H) \<Longrightarrow> inv\<^bsub>G A_Mod H\<^esub> X = a_set_inv X"
ballarin@20318
   505
by (rule normal.inv_FactGroup [OF a_normal,
ballarin@20318
   506
    folded A_FactGroup_def A_SET_INV_def, simplified monoid_record_simps])
ballarin@20318
   507
ballarin@20318
   508
text{*The coset map is a homomorphism from @{term G} to the quotient group
ballarin@20318
   509
  @{term "G Mod H"}*}
ballarin@20318
   510
lemma (in abelian_subgroup) a_r_coset_hom_A_Mod:
ballarin@20318
   511
  "(\<lambda>a. H +> a) \<in> hom \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> (G A_Mod H)"
ballarin@20318
   512
by (rule normal.r_coset_hom_Mod [OF a_normal,
ballarin@20318
   513
    folded A_FactGroup_def a_r_coset_def, simplified monoid_record_simps])
ballarin@20318
   514
ballarin@20318
   515
text {* The isomorphism theorems have been omitted from lifting, at
ballarin@20318
   516
  least for now *}
ballarin@20318
   517
ballarin@27717
   518
subsubsection{*The First Isomorphism Theorem*}
ballarin@20318
   519
ballarin@20318
   520
text{*The quotient by the kernel of a homomorphism is isomorphic to the 
ballarin@20318
   521
  range of that homomorphism.*}
ballarin@20318
   522
ballarin@20318
   523
lemmas a_kernel_defs =
ballarin@20318
   524
  a_kernel_def kernel_def
ballarin@20318
   525
ballarin@20318
   526
lemma a_kernel_def':
ballarin@20318
   527
  "a_kernel R S h \<equiv> {x \<in> carrier R. h x = \<zero>\<^bsub>S\<^esub>}"
ballarin@20318
   528
by (rule a_kernel_def[unfolded kernel_def, simplified ring_record_simps])
ballarin@20318
   529
ballarin@20318
   530
ballarin@27717
   531
subsubsection {* Homomorphisms *}
ballarin@20318
   532
ballarin@20318
   533
lemma abelian_group_homI:
ballarin@27611
   534
  assumes "abelian_group G"
ballarin@27611
   535
  assumes "abelian_group H"
ballarin@20318
   536
  assumes a_group_hom: "group_hom (| carrier = carrier G, mult = add G, one = zero G |)
ballarin@20318
   537
                                  (| carrier = carrier H, mult = add H, one = zero H |) h"
ballarin@20318
   538
  shows "abelian_group_hom G H h"
ballarin@27611
   539
proof -
wenzelm@30729
   540
  interpret G: abelian_group G by fact
wenzelm@30729
   541
  interpret H: abelian_group H by fact
ballarin@27611
   542
  show ?thesis apply (intro abelian_group_hom.intro abelian_group_hom_axioms.intro)
ballarin@27611
   543
    apply fact
ballarin@27611
   544
    apply fact
ballarin@27611
   545
    apply (rule a_group_hom)
ballarin@27611
   546
    done
ballarin@27611
   547
qed
ballarin@20318
   548
ballarin@20318
   549
lemma (in abelian_group_hom) is_abelian_group_hom:
ballarin@20318
   550
  "abelian_group_hom G H h"
haftmann@28823
   551
  ..
ballarin@20318
   552
ballarin@20318
   553
lemma (in abelian_group_hom) hom_add [simp]:
ballarin@20318
   554
  "[| x : carrier G; y : carrier G |]
ballarin@20318
   555
        ==> h (x \<oplus>\<^bsub>G\<^esub> y) = h x \<oplus>\<^bsub>H\<^esub> h y"
ballarin@20318
   556
by (rule group_hom.hom_mult[OF a_group_hom,
ballarin@20318
   557
    simplified ring_record_simps])
ballarin@20318
   558
ballarin@20318
   559
lemma (in abelian_group_hom) hom_closed [simp]:
ballarin@20318
   560
  "x \<in> carrier G \<Longrightarrow> h x \<in> carrier H"
ballarin@20318
   561
by (rule group_hom.hom_closed[OF a_group_hom,
ballarin@20318
   562
    simplified ring_record_simps])
ballarin@20318
   563
ballarin@20318
   564
lemma (in abelian_group_hom) zero_closed [simp]:
ballarin@20318
   565
  "h \<zero> \<in> carrier H"
ballarin@20318
   566
by (rule group_hom.one_closed[OF a_group_hom,
ballarin@20318
   567
    simplified ring_record_simps])
ballarin@20318
   568
ballarin@20318
   569
lemma (in abelian_group_hom) hom_zero [simp]:
ballarin@20318
   570
  "h \<zero> = \<zero>\<^bsub>H\<^esub>"
ballarin@20318
   571
by (rule group_hom.hom_one[OF a_group_hom,
ballarin@20318
   572
    simplified ring_record_simps])
ballarin@20318
   573
ballarin@20318
   574
lemma (in abelian_group_hom) a_inv_closed [simp]:
ballarin@20318
   575
  "x \<in> carrier G ==> h (\<ominus>x) \<in> carrier H"
ballarin@20318
   576
by (rule group_hom.inv_closed[OF a_group_hom,
ballarin@20318
   577
    folded a_inv_def, simplified ring_record_simps])
ballarin@20318
   578
ballarin@20318
   579
lemma (in abelian_group_hom) hom_a_inv [simp]:
ballarin@20318
   580
  "x \<in> carrier G ==> h (\<ominus>x) = \<ominus>\<^bsub>H\<^esub> (h x)"
ballarin@20318
   581
by (rule group_hom.hom_inv[OF a_group_hom,
ballarin@20318
   582
    folded a_inv_def, simplified ring_record_simps])
ballarin@20318
   583
ballarin@20318
   584
lemma (in abelian_group_hom) additive_subgroup_a_kernel:
ballarin@20318
   585
  "additive_subgroup (a_kernel G H h) G"
ballarin@20318
   586
apply (rule additive_subgroup.intro)
ballarin@20318
   587
apply (rule group_hom.subgroup_kernel[OF a_group_hom,
ballarin@20318
   588
       folded a_kernel_def, simplified ring_record_simps])
ballarin@20318
   589
done
ballarin@20318
   590
ballarin@20318
   591
text{*The kernel of a homomorphism is an abelian subgroup*}
ballarin@20318
   592
lemma (in abelian_group_hom) abelian_subgroup_a_kernel:
ballarin@20318
   593
  "abelian_subgroup (a_kernel G H h) G"
ballarin@20318
   594
apply (rule abelian_subgroupI)
ballarin@20318
   595
apply (rule group_hom.normal_kernel[OF a_group_hom,
ballarin@20318
   596
       folded a_kernel_def, simplified ring_record_simps])
ballarin@20318
   597
apply (simp add: G.a_comm)
ballarin@20318
   598
done
ballarin@20318
   599
ballarin@20318
   600
lemma (in abelian_group_hom) A_FactGroup_nonempty:
ballarin@20318
   601
  assumes X: "X \<in> carrier (G A_Mod a_kernel G H h)"
ballarin@20318
   602
  shows "X \<noteq> {}"
ballarin@20318
   603
by (rule group_hom.FactGroup_nonempty[OF a_group_hom,
wenzelm@23350
   604
    folded a_kernel_def A_FactGroup_def, simplified ring_record_simps]) (rule X)
ballarin@20318
   605
ballarin@20318
   606
lemma (in abelian_group_hom) FactGroup_contents_mem:
ballarin@20318
   607
  assumes X: "X \<in> carrier (G A_Mod (a_kernel G H h))"
ballarin@20318
   608
  shows "contents (h`X) \<in> carrier H"
ballarin@20318
   609
by (rule group_hom.FactGroup_contents_mem[OF a_group_hom,
wenzelm@23350
   610
    folded a_kernel_def A_FactGroup_def, simplified ring_record_simps]) (rule X)
ballarin@20318
   611
ballarin@20318
   612
lemma (in abelian_group_hom) A_FactGroup_hom:
ballarin@20318
   613
     "(\<lambda>X. contents (h`X)) \<in> hom (G A_Mod (a_kernel G H h))
ballarin@20318
   614
          \<lparr>carrier = carrier H, mult = add H, one = zero H\<rparr>"
ballarin@20318
   615
by (rule group_hom.FactGroup_hom[OF a_group_hom,
ballarin@20318
   616
    folded a_kernel_def A_FactGroup_def, simplified ring_record_simps])
ballarin@20318
   617
ballarin@20318
   618
lemma (in abelian_group_hom) A_FactGroup_inj_on:
ballarin@20318
   619
     "inj_on (\<lambda>X. contents (h ` X)) (carrier (G A_Mod a_kernel G H h))"
ballarin@20318
   620
by (rule group_hom.FactGroup_inj_on[OF a_group_hom,
ballarin@20318
   621
    folded a_kernel_def A_FactGroup_def, simplified ring_record_simps])
ballarin@20318
   622
ballarin@20318
   623
text{*If the homomorphism @{term h} is onto @{term H}, then so is the
ballarin@20318
   624
homomorphism from the quotient group*}
ballarin@20318
   625
lemma (in abelian_group_hom) A_FactGroup_onto:
ballarin@20318
   626
  assumes h: "h ` carrier G = carrier H"
ballarin@20318
   627
  shows "(\<lambda>X. contents (h ` X)) ` carrier (G A_Mod a_kernel G H h) = carrier H"
ballarin@20318
   628
by (rule group_hom.FactGroup_onto[OF a_group_hom,
wenzelm@23350
   629
    folded a_kernel_def A_FactGroup_def, simplified ring_record_simps]) (rule h)
ballarin@20318
   630
ballarin@20318
   631
text{*If @{term h} is a homomorphism from @{term G} onto @{term H}, then the
ballarin@20318
   632
 quotient group @{term "G Mod (kernel G H h)"} is isomorphic to @{term H}.*}
ballarin@20318
   633
theorem (in abelian_group_hom) A_FactGroup_iso:
ballarin@20318
   634
  "h ` carrier G = carrier H
ballarin@20318
   635
   \<Longrightarrow> (\<lambda>X. contents (h`X)) \<in> (G A_Mod (a_kernel G H h)) \<cong>
ballarin@20318
   636
          (| carrier = carrier H, mult = add H, one = zero H |)"
ballarin@20318
   637
by (rule group_hom.FactGroup_iso[OF a_group_hom,
ballarin@20318
   638
    folded a_kernel_def A_FactGroup_def, simplified ring_record_simps])
ballarin@20318
   639
ballarin@27717
   640
subsubsection {* Cosets *}
ballarin@20318
   641
ballarin@20318
   642
text {* Not eveything from \texttt{CosetExt.thy} is lifted here. *}
ballarin@20318
   643
ballarin@20318
   644
lemma (in additive_subgroup) a_Hcarr [simp]:
ballarin@20318
   645
  assumes hH: "h \<in> H"
ballarin@20318
   646
  shows "h \<in> carrier G"
ballarin@20318
   647
by (rule subgroup.mem_carrier [OF a_subgroup,
wenzelm@23350
   648
    simplified monoid_record_simps]) (rule hH)
ballarin@20318
   649
ballarin@20318
   650
ballarin@20318
   651
lemma (in abelian_subgroup) a_elemrcos_carrier:
ballarin@20318
   652
  assumes acarr: "a \<in> carrier G"
ballarin@20318
   653
      and a': "a' \<in> H +> a"
ballarin@20318
   654
  shows "a' \<in> carrier G"
ballarin@20318
   655
by (rule subgroup.elemrcos_carrier [OF a_subgroup a_group,
wenzelm@23350
   656
    folded a_r_coset_def, simplified monoid_record_simps]) (rule acarr, rule a')
ballarin@20318
   657
ballarin@20318
   658
lemma (in abelian_subgroup) a_rcos_const:
ballarin@20318
   659
  assumes hH: "h \<in> H"
ballarin@20318
   660
  shows "H +> h = H"
ballarin@20318
   661
by (rule subgroup.rcos_const [OF a_subgroup a_group,
wenzelm@23350
   662
    folded a_r_coset_def, simplified monoid_record_simps]) (rule hH)
ballarin@20318
   663
ballarin@20318
   664
lemma (in abelian_subgroup) a_rcos_module_imp:
ballarin@20318
   665
  assumes xcarr: "x \<in> carrier G"
ballarin@20318
   666
      and x'cos: "x' \<in> H +> x"
ballarin@20318
   667
  shows "(x' \<oplus> \<ominus>x) \<in> H"
ballarin@20318
   668
by (rule subgroup.rcos_module_imp [OF a_subgroup a_group,
wenzelm@23350
   669
    folded a_r_coset_def a_inv_def, simplified monoid_record_simps]) (rule xcarr, rule x'cos)
ballarin@20318
   670
ballarin@20318
   671
lemma (in abelian_subgroup) a_rcos_module_rev:
wenzelm@23350
   672
  assumes "x \<in> carrier G" "x' \<in> carrier G"
wenzelm@23350
   673
      and "(x' \<oplus> \<ominus>x) \<in> H"
ballarin@20318
   674
  shows "x' \<in> H +> x"
wenzelm@23350
   675
using assms
ballarin@20318
   676
by (rule subgroup.rcos_module_rev [OF a_subgroup a_group,
ballarin@20318
   677
    folded a_r_coset_def a_inv_def, simplified monoid_record_simps])
ballarin@20318
   678
ballarin@20318
   679
lemma (in abelian_subgroup) a_rcos_module:
wenzelm@23350
   680
  assumes "x \<in> carrier G" "x' \<in> carrier G"
ballarin@20318
   681
  shows "(x' \<in> H +> x) = (x' \<oplus> \<ominus>x \<in> H)"
wenzelm@23350
   682
using assms
ballarin@20318
   683
by (rule subgroup.rcos_module [OF a_subgroup a_group,
ballarin@20318
   684
    folded a_r_coset_def a_inv_def, simplified monoid_record_simps])
ballarin@20318
   685
ballarin@20318
   686
--"variant"
ballarin@20318
   687
lemma (in abelian_subgroup) a_rcos_module_minus:
ballarin@27611
   688
  assumes "ring G"
ballarin@20318
   689
  assumes carr: "x \<in> carrier G" "x' \<in> carrier G"
ballarin@20318
   690
  shows "(x' \<in> H +> x) = (x' \<ominus> x \<in> H)"
ballarin@20318
   691
proof -
wenzelm@30729
   692
  interpret G: ring G by fact
ballarin@20318
   693
  from carr
wenzelm@23350
   694
  have "(x' \<in> H +> x) = (x' \<oplus> \<ominus>x \<in> H)" by (rule a_rcos_module)
wenzelm@23350
   695
  with carr
wenzelm@23350
   696
  show "(x' \<in> H +> x) = (x' \<ominus> x \<in> H)"
wenzelm@23350
   697
    by (simp add: minus_eq)
ballarin@20318
   698
qed
ballarin@20318
   699
ballarin@20318
   700
lemma (in abelian_subgroup) a_repr_independence':
wenzelm@23463
   701
  assumes y: "y \<in> H +> x"
wenzelm@23463
   702
      and xcarr: "x \<in> carrier G"
ballarin@20318
   703
  shows "H +> x = H +> y"
wenzelm@23463
   704
  apply (rule a_repr_independence)
wenzelm@23463
   705
    apply (rule y)
wenzelm@23463
   706
   apply (rule xcarr)
wenzelm@23463
   707
  apply (rule a_subgroup)
wenzelm@23463
   708
  done
ballarin@20318
   709
ballarin@20318
   710
lemma (in abelian_subgroup) a_repr_independenceD:
ballarin@20318
   711
  assumes ycarr: "y \<in> carrier G"
ballarin@20318
   712
      and repr:  "H +> x = H +> y"
ballarin@20318
   713
  shows "y \<in> H +> x"
ballarin@20318
   714
by (rule group.repr_independenceD [OF a_group a_subgroup,
wenzelm@23383
   715
    folded a_r_coset_def, simplified monoid_record_simps]) (rule ycarr, rule repr)
ballarin@20318
   716
ballarin@20318
   717
ballarin@20318
   718
lemma (in abelian_subgroup) a_rcosets_carrier:
ballarin@20318
   719
  "X \<in> a_rcosets H \<Longrightarrow> X \<subseteq> carrier G"
ballarin@20318
   720
by (rule subgroup.rcosets_carrier [OF a_subgroup a_group,
ballarin@20318
   721
    folded A_RCOSETS_def, simplified monoid_record_simps])
ballarin@20318
   722
ballarin@20318
   723
ballarin@20318
   724
ballarin@27717
   725
subsubsection {* Addition of Subgroups *}
ballarin@20318
   726
ballarin@20318
   727
lemma (in abelian_monoid) set_add_closed:
ballarin@20318
   728
  assumes Acarr: "A \<subseteq> carrier G"
ballarin@20318
   729
      and Bcarr: "B \<subseteq> carrier G"
ballarin@20318
   730
  shows "A <+> B \<subseteq> carrier G"
ballarin@20318
   731
by (rule monoid.set_mult_closed [OF a_monoid,
wenzelm@23383
   732
    folded set_add_def, simplified monoid_record_simps]) (rule Acarr, rule Bcarr)
ballarin@20318
   733
ballarin@20318
   734
lemma (in abelian_group) add_additive_subgroups:
ballarin@20318
   735
  assumes subH: "additive_subgroup H G"
ballarin@20318
   736
      and subK: "additive_subgroup K G"
ballarin@20318
   737
  shows "additive_subgroup (H <+> K) G"
ballarin@20318
   738
apply (rule additive_subgroup.intro)
ballarin@20318
   739
apply (unfold set_add_def)
ballarin@20318
   740
apply (intro comm_group.mult_subgroups)
ballarin@20318
   741
  apply (rule a_comm_group)
ballarin@20318
   742
 apply (rule additive_subgroup.a_subgroup[OF subH])
ballarin@20318
   743
apply (rule additive_subgroup.a_subgroup[OF subK])
ballarin@20318
   744
done
ballarin@20318
   745
ballarin@20318
   746
end