src/HOL/Algebra/Coset.thy
 author nipkow Fri, 13 Nov 2009 14:14:04 +0100 changeset 33657 a4179bf442d1 parent 32960 69916a850301 child 35416 d8d7d1b785af permissions -rw-r--r--
renamed lemmas "anti_sym" -> "antisym"
```
(*  Title:      HOL/Algebra/Coset.thy
Author:     Florian Kammueller, with new proofs by L C Paulson, and
Stephan Hohe
*)

theory Coset imports Group begin

section {*Cosets and Quotient Groups*}

constdefs (structure G)
r_coset    :: "[_, 'a set, 'a] \<Rightarrow> 'a set"    (infixl "#>\<index>" 60)
"H #> a \<equiv> \<Union>h\<in>H. {h \<otimes> a}"

l_coset    :: "[_, 'a, 'a set] \<Rightarrow> 'a set"    (infixl "<#\<index>" 60)
"a <# H \<equiv> \<Union>h\<in>H. {a \<otimes> h}"

RCOSETS  :: "[_, 'a set] \<Rightarrow> ('a set)set"   ("rcosets\<index> _" [81] 80)
"rcosets H \<equiv> \<Union>a\<in>carrier G. {H #> a}"

set_mult  :: "[_, 'a set ,'a set] \<Rightarrow> 'a set" (infixl "<#>\<index>" 60)
"H <#> K \<equiv> \<Union>h\<in>H. \<Union>k\<in>K. {h \<otimes> k}"

SET_INV :: "[_,'a set] \<Rightarrow> 'a set"  ("set'_inv\<index> _" [81] 80)
"set_inv H \<equiv> \<Union>h\<in>H. {inv h}"

locale normal = subgroup + group +
assumes coset_eq: "(\<forall>x \<in> carrier G. H #> x = x <# H)"

abbreviation
normal_rel :: "['a set, ('a, 'b) monoid_scheme] \<Rightarrow> bool"  (infixl "\<lhd>" 60) where
"H \<lhd> G \<equiv> normal H G"

subsection {*Basic Properties of Cosets*}

lemma (in group) coset_mult_assoc:
"[| M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G |]
==> (M #> g) #> h = M #> (g \<otimes> h)"
by (force simp add: r_coset_def m_assoc)

lemma (in group) coset_mult_one [simp]: "M \<subseteq> carrier G ==> M #> \<one> = M"

lemma (in group) coset_mult_inv1:
"[| M #> (x \<otimes> (inv y)) = M;  x \<in> carrier G ; y \<in> carrier G;
M \<subseteq> carrier G |] ==> M #> x = M #> y"
apply (erule subst [of concl: "%z. M #> x = z #> y"])
done

lemma (in group) coset_mult_inv2:
"[| M #> x = M #> y;  x \<in> carrier G;  y \<in> carrier G;  M \<subseteq> carrier G |]
==> M #> (x \<otimes> (inv y)) = M "
done

lemma (in group) coset_join1:
"[| H #> x = H;  x \<in> carrier G;  subgroup H G |] ==> x \<in> H"
apply (erule subst)
apply (blast intro: l_one subgroup.one_closed sym)
done

lemma (in group) solve_equation:
"\<lbrakk>subgroup H G; x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> \<exists>h\<in>H. y = h \<otimes> x"
apply (rule bexI [of _ "y \<otimes> (inv x)"])
apply (auto simp add: subgroup.m_closed subgroup.m_inv_closed m_assoc
subgroup.subset [THEN subsetD])
done

lemma (in group) repr_independence:
"\<lbrakk>y \<in> H #> x;  x \<in> carrier G; subgroup H G\<rbrakk> \<Longrightarrow> H #> x = H #> y"
by (auto simp add: r_coset_def m_assoc [symmetric]
subgroup.subset [THEN subsetD]
subgroup.m_closed solve_equation)

lemma (in group) coset_join2:
"\<lbrakk>x \<in> carrier G;  subgroup H G;  x\<in>H\<rbrakk> \<Longrightarrow> H #> x = H"
--{*Alternative proof is to put @{term "x=\<one>"} in @{text repr_independence}.*}
by (force simp add: subgroup.m_closed r_coset_def solve_equation)

lemma (in monoid) r_coset_subset_G:
"[| H \<subseteq> carrier G; x \<in> carrier G |] ==> H #> x \<subseteq> carrier G"

lemma (in group) rcosI:
"[| h \<in> H; H \<subseteq> carrier G; x \<in> carrier G|] ==> h \<otimes> x \<in> H #> x"

lemma (in group) rcosetsI:
"\<lbrakk>H \<subseteq> carrier G; x \<in> carrier G\<rbrakk> \<Longrightarrow> H #> x \<in> rcosets H"

text{*Really needed?*}
lemma (in group) transpose_inv:
"[| x \<otimes> y = z;  x \<in> carrier G;  y \<in> carrier G;  z \<in> carrier G |]
==> (inv x) \<otimes> z = y"
by (force simp add: m_assoc [symmetric])

lemma (in group) rcos_self: "[| x \<in> carrier G; subgroup H G |] ==> x \<in> H #> x"
apply (blast intro: sym l_one subgroup.subset [THEN subsetD]
subgroup.one_closed)
done

text (in group) {* Opposite of @{thm [source] "repr_independence"} *}
lemma (in group) repr_independenceD:
assumes "subgroup H G"
assumes ycarr: "y \<in> carrier G"
and repr:  "H #> x = H #> y"
shows "y \<in> H #> x"
proof -
interpret subgroup H G by fact
show ?thesis  apply (subst repr)
apply (intro rcos_self)
apply (rule ycarr)
apply (rule is_subgroup)
done
qed

text {* Elements of a right coset are in the carrier *}
lemma (in subgroup) elemrcos_carrier:
assumes "group G"
assumes acarr: "a \<in> carrier G"
and a': "a' \<in> H #> a"
shows "a' \<in> carrier G"
proof -
interpret group G by fact
from subset and acarr
have "H #> a \<subseteq> carrier G" by (rule r_coset_subset_G)
from this and a'
show "a' \<in> carrier G"
by fast
qed

lemma (in subgroup) rcos_const:
assumes "group G"
assumes hH: "h \<in> H"
shows "H #> h = H"
proof -
interpret group G by fact
show ?thesis apply (unfold r_coset_def)
apply rule
apply rule
apply clarsimp
apply (intro subgroup.m_closed)
apply (rule is_subgroup)
apply assumption
apply (rule hH)
apply rule
apply simp
proof -
fix h'
assume h'H: "h' \<in> H"
note carr = hH[THEN mem_carrier] h'H[THEN mem_carrier]
from carr
have a: "h' = (h' \<otimes> inv h) \<otimes> h" by (simp add: m_assoc)
from h'H hH
have "h' \<otimes> inv h \<in> H" by simp
from this and a
show "\<exists>x\<in>H. h' = x \<otimes> h" by fast
qed
qed

text {* Step one for lemma @{text "rcos_module"} *}
lemma (in subgroup) rcos_module_imp:
assumes "group G"
assumes xcarr: "x \<in> carrier G"
and x'cos: "x' \<in> H #> x"
shows "(x' \<otimes> inv x) \<in> H"
proof -
interpret group G by fact
from xcarr x'cos
have x'carr: "x' \<in> carrier G"
by (rule elemrcos_carrier[OF is_group])
from xcarr
have ixcarr: "inv x \<in> carrier G"
by simp
from x'cos
have "\<exists>h\<in>H. x' = h \<otimes> x"
unfolding r_coset_def
by fast
from this
obtain h
where hH: "h \<in> H"
and x': "x' = h \<otimes> x"
by auto
from hH and subset
have hcarr: "h \<in> carrier G" by fast
note carr = xcarr x'carr hcarr
from x' and carr
have "x' \<otimes> (inv x) = (h \<otimes> x) \<otimes> (inv x)" by fast
also from carr
have "\<dots> = h \<otimes> (x \<otimes> inv x)" by (simp add: m_assoc)
also from carr
have "\<dots> = h \<otimes> \<one>" by simp
also from carr
have "\<dots> = h" by simp
finally
have "x' \<otimes> (inv x) = h" by simp
from hH this
show "x' \<otimes> (inv x) \<in> H" by simp
qed

text {* Step two for lemma @{text "rcos_module"} *}
lemma (in subgroup) rcos_module_rev:
assumes "group G"
assumes carr: "x \<in> carrier G" "x' \<in> carrier G"
and xixH: "(x' \<otimes> inv x) \<in> H"
shows "x' \<in> H #> x"
proof -
interpret group G by fact
from xixH
have "\<exists>h\<in>H. x' \<otimes> (inv x) = h" by fast
from this
obtain h
where hH: "h \<in> H"
and hsym: "x' \<otimes> (inv x) = h"
by fast
from hH subset have hcarr: "h \<in> carrier G" by simp
note carr = carr hcarr
from hsym[symmetric] have "h \<otimes> x = x' \<otimes> (inv x) \<otimes> x" by fast
also from carr
have "\<dots> = x' \<otimes> ((inv x) \<otimes> x)" by (simp add: m_assoc)
also from carr
have "\<dots> = x' \<otimes> \<one>" by (simp add: l_inv)
also from carr
have "\<dots> = x'" by simp
finally
have "h \<otimes> x = x'" by simp
from this[symmetric] and hH
show "x' \<in> H #> x"
unfolding r_coset_def
by fast
qed

text {* Module property of right cosets *}
lemma (in subgroup) rcos_module:
assumes "group G"
assumes carr: "x \<in> carrier G" "x' \<in> carrier G"
shows "(x' \<in> H #> x) = (x' \<otimes> inv x \<in> H)"
proof -
interpret group G by fact
show ?thesis proof  assume "x' \<in> H #> x"
from this and carr
show "x' \<otimes> inv x \<in> H"
by (intro rcos_module_imp[OF is_group])
next
assume "x' \<otimes> inv x \<in> H"
from this and carr
show "x' \<in> H #> x"
by (intro rcos_module_rev[OF is_group])
qed
qed

text {* Right cosets are subsets of the carrier. *}
lemma (in subgroup) rcosets_carrier:
assumes "group G"
assumes XH: "X \<in> rcosets H"
shows "X \<subseteq> carrier G"
proof -
interpret group G by fact
from XH have "\<exists>x\<in> carrier G. X = H #> x"
unfolding RCOSETS_def
by fast
from this
obtain x
where xcarr: "x\<in> carrier G"
and X: "X = H #> x"
by fast
from subset and xcarr
show "X \<subseteq> carrier G"
unfolding X
by (rule r_coset_subset_G)
qed

text {* Multiplication of general subsets *}
lemma (in monoid) set_mult_closed:
assumes Acarr: "A \<subseteq> carrier G"
and Bcarr: "B \<subseteq> carrier G"
shows "A <#> B \<subseteq> carrier G"
apply rule apply (simp add: set_mult_def, clarsimp)
proof -
fix a b
assume "a \<in> A"
from this and Acarr
have acarr: "a \<in> carrier G" by fast

assume "b \<in> B"
from this and Bcarr
have bcarr: "b \<in> carrier G" by fast

from acarr bcarr
show "a \<otimes> b \<in> carrier G" by (rule m_closed)
qed

lemma (in comm_group) mult_subgroups:
assumes subH: "subgroup H G"
and subK: "subgroup K G"
shows "subgroup (H <#> K) G"
apply (rule subgroup.intro)
apply (intro set_mult_closed subgroup.subset[OF subH] subgroup.subset[OF subK])
apply (simp add: set_mult_def) apply clarsimp defer 1
apply (simp add: set_mult_def) defer 1
apply (simp add: set_mult_def, clarsimp) defer 1
proof -
fix ha hb ka kb
assume haH: "ha \<in> H" and hbH: "hb \<in> H" and kaK: "ka \<in> K" and kbK: "kb \<in> K"
note carr = haH[THEN subgroup.mem_carrier[OF subH]] hbH[THEN subgroup.mem_carrier[OF subH]]
kaK[THEN subgroup.mem_carrier[OF subK]] kbK[THEN subgroup.mem_carrier[OF subK]]
from carr
have "(ha \<otimes> ka) \<otimes> (hb \<otimes> kb) = ha \<otimes> (ka \<otimes> hb) \<otimes> kb" by (simp add: m_assoc)
also from carr
have "\<dots> = ha \<otimes> (hb \<otimes> ka) \<otimes> kb" by (simp add: m_comm)
also from carr
have "\<dots> = (ha \<otimes> hb) \<otimes> (ka \<otimes> kb)" by (simp add: m_assoc)
finally
have eq: "(ha \<otimes> ka) \<otimes> (hb \<otimes> kb) = (ha \<otimes> hb) \<otimes> (ka \<otimes> kb)" .

from haH hbH have hH: "ha \<otimes> hb \<in> H" by (simp add: subgroup.m_closed[OF subH])
from kaK kbK have kK: "ka \<otimes> kb \<in> K" by (simp add: subgroup.m_closed[OF subK])

from hH and kK and eq
show "\<exists>h'\<in>H. \<exists>k'\<in>K. (ha \<otimes> ka) \<otimes> (hb \<otimes> kb) = h' \<otimes> k'" by fast
next
have "\<one> = \<one> \<otimes> \<one>" by simp
from subgroup.one_closed[OF subH] subgroup.one_closed[OF subK] this
show "\<exists>h\<in>H. \<exists>k\<in>K. \<one> = h \<otimes> k" by fast
next
fix h k
assume hH: "h \<in> H"
and kK: "k \<in> K"

from hH[THEN subgroup.mem_carrier[OF subH]] kK[THEN subgroup.mem_carrier[OF subK]]
have "inv (h \<otimes> k) = inv h \<otimes> inv k" by (simp add: inv_mult_group m_comm)

from subgroup.m_inv_closed[OF subH hH] and subgroup.m_inv_closed[OF subK kK] and this
show "\<exists>ha\<in>H. \<exists>ka\<in>K. inv (h \<otimes> k) = ha \<otimes> ka" by fast
qed

lemma (in subgroup) lcos_module_rev:
assumes "group G"
assumes carr: "x \<in> carrier G" "x' \<in> carrier G"
and xixH: "(inv x \<otimes> x') \<in> H"
shows "x' \<in> x <# H"
proof -
interpret group G by fact
from xixH
have "\<exists>h\<in>H. (inv x) \<otimes> x' = h" by fast
from this
obtain h
where hH: "h \<in> H"
and hsym: "(inv x) \<otimes> x' = h"
by fast

from hH subset have hcarr: "h \<in> carrier G" by simp
note carr = carr hcarr
from hsym[symmetric] have "x \<otimes> h = x \<otimes> ((inv x) \<otimes> x')" by fast
also from carr
have "\<dots> = (x \<otimes> (inv x)) \<otimes> x'" by (simp add: m_assoc[symmetric])
also from carr
have "\<dots> = \<one> \<otimes> x'" by simp
also from carr
have "\<dots> = x'" by simp
finally
have "x \<otimes> h = x'" by simp

from this[symmetric] and hH
show "x' \<in> x <# H"
unfolding l_coset_def
by fast
qed

subsection {* Normal subgroups *}

lemma normal_imp_subgroup: "H \<lhd> G \<Longrightarrow> subgroup H G"

lemma (in group) normalI:
"subgroup H G \<Longrightarrow> (\<forall>x \<in> carrier G. H #> x = x <# H) \<Longrightarrow> H \<lhd> G"
by (simp add: normal_def normal_axioms_def prems)

lemma (in normal) inv_op_closed1:
"\<lbrakk>x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> (inv x) \<otimes> h \<otimes> x \<in> H"
apply (insert coset_eq)
apply (auto simp add: l_coset_def r_coset_def)
apply (drule bspec, assumption)
apply (drule equalityD1 [THEN subsetD], blast, clarify)
done

lemma (in normal) inv_op_closed2:
"\<lbrakk>x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> h \<otimes> (inv x) \<in> H"
apply (subgoal_tac "inv (inv x) \<otimes> h \<otimes> (inv x) \<in> H")
apply (blast intro: inv_op_closed1)
done

text{*Alternative characterization of normal subgroups*}
lemma (in group) normal_inv_iff:
"(N \<lhd> G) =
(subgroup N G & (\<forall>x \<in> carrier G. \<forall>h \<in> N. x \<otimes> h \<otimes> (inv x) \<in> N))"
(is "_ = ?rhs")
proof
assume N: "N \<lhd> G"
show ?rhs
by (blast intro: N normal.inv_op_closed2 normal_imp_subgroup)
next
assume ?rhs
hence sg: "subgroup N G"
and closed: "\<And>x. x\<in>carrier G \<Longrightarrow> \<forall>h\<in>N. x \<otimes> h \<otimes> inv x \<in> N" by auto
hence sb: "N \<subseteq> carrier G" by (simp add: subgroup.subset)
show "N \<lhd> G"
proof (intro normalI [OF sg], simp add: l_coset_def r_coset_def, clarify)
fix x
assume x: "x \<in> carrier G"
show "(\<Union>h\<in>N. {h \<otimes> x}) = (\<Union>h\<in>N. {x \<otimes> h})"
proof
show "(\<Union>h\<in>N. {h \<otimes> x}) \<subseteq> (\<Union>h\<in>N. {x \<otimes> h})"
proof clarify
fix n
assume n: "n \<in> N"
show "n \<otimes> x \<in> (\<Union>h\<in>N. {x \<otimes> h})"
proof
from closed [of "inv x"]
show "inv x \<otimes> n \<otimes> x \<in> N" by (simp add: x n)
show "n \<otimes> x \<in> {x \<otimes> (inv x \<otimes> n \<otimes> x)}"
by (simp add: x n m_assoc [symmetric] sb [THEN subsetD])
qed
qed
next
show "(\<Union>h\<in>N. {x \<otimes> h}) \<subseteq> (\<Union>h\<in>N. {h \<otimes> x})"
proof clarify
fix n
assume n: "n \<in> N"
show "x \<otimes> n \<in> (\<Union>h\<in>N. {h \<otimes> x})"
proof
show "x \<otimes> n \<otimes> inv x \<in> N" by (simp add: x n closed)
show "x \<otimes> n \<in> {x \<otimes> n \<otimes> inv x \<otimes> x}"
by (simp add: x n m_assoc sb [THEN subsetD])
qed
qed
qed
qed
qed

subsection{*More Properties of Cosets*}

lemma (in group) lcos_m_assoc:
"[| M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G |]
==> g <# (h <# M) = (g \<otimes> h) <# M"
by (force simp add: l_coset_def m_assoc)

lemma (in group) lcos_mult_one: "M \<subseteq> carrier G ==> \<one> <# M = M"

lemma (in group) l_coset_subset_G:
"[| H \<subseteq> carrier G; x \<in> carrier G |] ==> x <# H \<subseteq> carrier G"
by (auto simp add: l_coset_def subsetD)

lemma (in group) l_coset_swap:
"\<lbrakk>y \<in> x <# H;  x \<in> carrier G;  subgroup H G\<rbrakk> \<Longrightarrow> x \<in> y <# H"
assume "\<exists>h\<in>H. y = x \<otimes> h"
and x: "x \<in> carrier G"
and sb: "subgroup H G"
then obtain h' where h': "h' \<in> H & x \<otimes> h' = y" by blast
show "\<exists>h\<in>H. x = y \<otimes> h"
proof
show "x = y \<otimes> inv h'" using h' x sb
by (auto simp add: m_assoc subgroup.subset [THEN subsetD])
show "inv h' \<in> H" using h' sb
by (auto simp add: subgroup.subset [THEN subsetD] subgroup.m_inv_closed)
qed
qed

lemma (in group) l_coset_carrier:
"[| y \<in> x <# H;  x \<in> carrier G;  subgroup H G |] ==> y \<in> carrier G"
by (auto simp add: l_coset_def m_assoc
subgroup.subset [THEN subsetD] subgroup.m_closed)

lemma (in group) l_repr_imp_subset:
assumes y: "y \<in> x <# H" and x: "x \<in> carrier G" and sb: "subgroup H G"
shows "y <# H \<subseteq> x <# H"
proof -
from y
obtain h' where "h' \<in> H" "x \<otimes> h' = y" by (auto simp add: l_coset_def)
thus ?thesis using x sb
by (auto simp add: l_coset_def m_assoc
subgroup.subset [THEN subsetD] subgroup.m_closed)
qed

lemma (in group) l_repr_independence:
assumes y: "y \<in> x <# H" and x: "x \<in> carrier G" and sb: "subgroup H G"
shows "x <# H = y <# H"
proof
show "x <# H \<subseteq> y <# H"
by (rule l_repr_imp_subset,
(blast intro: l_coset_swap l_coset_carrier y x sb)+)
show "y <# H \<subseteq> x <# H" by (rule l_repr_imp_subset [OF y x sb])
qed

lemma (in group) setmult_subset_G:
"\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G\<rbrakk> \<Longrightarrow> H <#> K \<subseteq> carrier G"
by (auto simp add: set_mult_def subsetD)

lemma (in group) subgroup_mult_id: "subgroup H G \<Longrightarrow> H <#> H = H"
apply (auto simp add: subgroup.m_closed set_mult_def Sigma_def image_def)
apply (rule_tac x = x in bexI)
apply (rule bexI [of _ "\<one>"])
apply (auto simp add: subgroup.m_closed subgroup.one_closed
r_one subgroup.subset [THEN subsetD])
done

subsubsection {* Set of Inverses of an @{text r_coset}. *}

lemma (in normal) rcos_inv:
assumes x:     "x \<in> carrier G"
shows "set_inv (H #> x) = H #> (inv x)"
proof (simp add: r_coset_def SET_INV_def x inv_mult_group, safe)
fix h
assume "h \<in> H"
show "inv x \<otimes> inv h \<in> (\<Union>j\<in>H. {j \<otimes> inv x})"
proof
show "inv x \<otimes> inv h \<otimes> x \<in> H"
show "inv x \<otimes> inv h \<in> {inv x \<otimes> inv h \<otimes> x \<otimes> inv x}"
qed
next
fix h
assume "h \<in> H"
show "h \<otimes> inv x \<in> (\<Union>j\<in>H. {inv x \<otimes> inv j})"
proof
show "x \<otimes> inv h \<otimes> inv x \<in> H"
show "h \<otimes> inv x \<in> {inv x \<otimes> inv (x \<otimes> inv h \<otimes> inv x)}"
by (simp add: prems m_assoc [symmetric] inv_mult_group)
qed
qed

subsubsection {*Theorems for @{text "<#>"} with @{text "#>"} or @{text "<#"}.*}

lemma (in group) setmult_rcos_assoc:
"\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk>
\<Longrightarrow> H <#> (K #> x) = (H <#> K) #> x"
by (force simp add: r_coset_def set_mult_def m_assoc)

lemma (in group) rcos_assoc_lcos:
"\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk>
\<Longrightarrow> (H #> x) <#> K = H <#> (x <# K)"
by (force simp add: r_coset_def l_coset_def set_mult_def m_assoc)

lemma (in normal) rcos_mult_step1:
"\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
\<Longrightarrow> (H #> x) <#> (H #> y) = (H <#> (x <# H)) #> y"
r_coset_subset_G l_coset_subset_G rcos_assoc_lcos)

lemma (in normal) rcos_mult_step2:
"\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
\<Longrightarrow> (H <#> (x <# H)) #> y = (H <#> (H #> x)) #> y"
by (insert coset_eq, simp add: normal_def)

lemma (in normal) rcos_mult_step3:
"\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
\<Longrightarrow> (H <#> (H #> x)) #> y = H #> (x \<otimes> y)"
subgroup_mult_id normal.axioms subset prems)

lemma (in normal) rcos_sum:
"\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
\<Longrightarrow> (H #> x) <#> (H #> y) = H #> (x \<otimes> y)"
by (simp add: rcos_mult_step1 rcos_mult_step2 rcos_mult_step3)

lemma (in normal) rcosets_mult_eq: "M \<in> rcosets H \<Longrightarrow> H <#> M = M"
-- {* generalizes @{text subgroup_mult_id} *}
by (auto simp add: RCOSETS_def subset
setmult_rcos_assoc subgroup_mult_id normal.axioms prems)

subsubsection{*An Equivalence Relation*}

constdefs (structure G)
r_congruent :: "[('a,'b)monoid_scheme, 'a set] \<Rightarrow> ('a*'a)set"
("rcong\<index> _")
"rcong H \<equiv> {(x,y). x \<in> carrier G & y \<in> carrier G & inv x \<otimes> y \<in> H}"

lemma (in subgroup) equiv_rcong:
assumes "group G"
shows "equiv (carrier G) (rcong H)"
proof -
interpret group G by fact
show ?thesis
proof (intro equiv.intro)
show "refl_on (carrier G) (rcong H)"
by (auto simp add: r_congruent_def refl_on_def)
next
show "sym (rcong H)"
proof (simp add: r_congruent_def sym_def, clarify)
fix x y
assume [simp]: "x \<in> carrier G" "y \<in> carrier G"
and "inv x \<otimes> y \<in> H"
hence "inv (inv x \<otimes> y) \<in> H" by (simp add: m_inv_closed)
thus "inv y \<otimes> x \<in> H" by (simp add: inv_mult_group)
qed
next
show "trans (rcong H)"
proof (simp add: r_congruent_def trans_def, clarify)
fix x y z
assume [simp]: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
and "inv x \<otimes> y \<in> H" and "inv y \<otimes> z \<in> H"
hence "(inv x \<otimes> y) \<otimes> (inv y \<otimes> z) \<in> H" by simp
hence "inv x \<otimes> (y \<otimes> inv y) \<otimes> z \<in> H"
by (simp add: m_assoc del: r_inv Units_r_inv)
thus "inv x \<otimes> z \<in> H" by simp
qed
qed
qed

text{*Equivalence classes of @{text rcong} correspond to left cosets.
Was there a mistake in the definitions? I'd have expected them to
correspond to right cosets.*}

(* CB: This is correct, but subtle.
We call H #> a the right coset of a relative to H.  According to
Jacobson, this is what the majority of group theory literature does.
He then defines the notion of congruence relation ~ over monoids as
equivalence relation with a ~ a' & b ~ b' \<Longrightarrow> a*b ~ a'*b'.
Our notion of right congruence induced by K: rcong K appears only in
the context where K is a normal subgroup.  Jacobson doesn't name it.
But in this context left and right cosets are identical.
*)

lemma (in subgroup) l_coset_eq_rcong:
assumes "group G"
assumes a: "a \<in> carrier G"
shows "a <# H = rcong H `` {a}"
proof -
interpret group G by fact
show ?thesis by (force simp add: r_congruent_def l_coset_def m_assoc [symmetric] a )
qed

subsubsection{*Two Distinct Right Cosets are Disjoint*}

lemma (in group) rcos_equation:
assumes "subgroup H G"
assumes p: "ha \<otimes> a = h \<otimes> b" "a \<in> carrier G" "b \<in> carrier G" "h \<in> H" "ha \<in> H" "hb \<in> H"
shows "hb \<otimes> a \<in> (\<Union>h\<in>H. {h \<otimes> b})"
proof -
interpret subgroup H G by fact
from p show ?thesis apply (rule_tac UN_I [of "hb \<otimes> ((inv ha) \<otimes> h)"])
done
qed

lemma (in group) rcos_disjoint:
assumes "subgroup H G"
assumes p: "a \<in> rcosets H" "b \<in> rcosets H" "a\<noteq>b"
shows "a \<inter> b = {}"
proof -
interpret subgroup H G by fact
from p show ?thesis apply (simp add: RCOSETS_def r_coset_def)
apply (blast intro: rcos_equation prems sym)
done
qed

subsection {* Further lemmas for @{text "r_congruent"} *}

text {* The relation is a congruence *}

lemma (in normal) congruent_rcong:
shows "congruent2 (rcong H) (rcong H) (\<lambda>a b. a \<otimes> b <# H)"
proof (intro congruent2I[of "carrier G" _ "carrier G" _] equiv_rcong is_group)
fix a b c
assume abrcong: "(a, b) \<in> rcong H"
and ccarr: "c \<in> carrier G"

from abrcong
have acarr: "a \<in> carrier G"
and bcarr: "b \<in> carrier G"
and abH: "inv a \<otimes> b \<in> H"
unfolding r_congruent_def
by fast+

note carr = acarr bcarr ccarr

from ccarr and abH
have "inv c \<otimes> (inv a \<otimes> b) \<otimes> c \<in> H" by (rule inv_op_closed1)
moreover
from carr and inv_closed
have "inv c \<otimes> (inv a \<otimes> b) \<otimes> c = (inv c \<otimes> inv a) \<otimes> (b \<otimes> c)"
by (force cong: m_assoc)
moreover
from carr and inv_closed
have "\<dots> = (inv (a \<otimes> c)) \<otimes> (b \<otimes> c)"
ultimately
have "(inv (a \<otimes> c)) \<otimes> (b \<otimes> c) \<in> H" by simp
from carr and this
have "(b \<otimes> c) \<in> (a \<otimes> c) <# H"
from carr and this and is_subgroup
show "(a \<otimes> c) <# H = (b \<otimes> c) <# H" by (intro l_repr_independence, simp+)
next
fix a b c
assume abrcong: "(a, b) \<in> rcong H"
and ccarr: "c \<in> carrier G"

from ccarr have "c \<in> Units G" by (simp add: Units_eq)
hence cinvc_one: "inv c \<otimes> c = \<one>" by (rule Units_l_inv)

from abrcong
have acarr: "a \<in> carrier G"
and bcarr: "b \<in> carrier G"
and abH: "inv a \<otimes> b \<in> H"
by (unfold r_congruent_def, fast+)

note carr = acarr bcarr ccarr

from carr and inv_closed
have "inv a \<otimes> b = inv a \<otimes> (\<one> \<otimes> b)" by simp
also from carr and inv_closed
have "\<dots> = inv a \<otimes> (inv c \<otimes> c) \<otimes> b" by simp
also from carr and inv_closed
have "\<dots> = (inv a \<otimes> inv c) \<otimes> (c \<otimes> b)" by (force cong: m_assoc)
also from carr and inv_closed
have "\<dots> = inv (c \<otimes> a) \<otimes> (c \<otimes> b)" by (simp add: inv_mult_group)
finally
have "inv a \<otimes> b = inv (c \<otimes> a) \<otimes> (c \<otimes> b)" .
from abH and this
have "inv (c \<otimes> a) \<otimes> (c \<otimes> b) \<in> H" by simp

from carr and this
have "(c \<otimes> b) \<in> (c \<otimes> a) <# H"
from carr and this and is_subgroup
show "(c \<otimes> a) <# H = (c \<otimes> b) <# H" by (intro l_repr_independence, simp+)
qed

subsection {*Order of a Group and Lagrange's Theorem*}

constdefs
order :: "('a, 'b) monoid_scheme \<Rightarrow> nat"
"order S \<equiv> card (carrier S)"

lemma (in group) rcosets_part_G:
assumes "subgroup H G"
shows "\<Union>(rcosets H) = carrier G"
proof -
interpret subgroup H G by fact
show ?thesis
apply (rule equalityI)
apply (force simp add: RCOSETS_def r_coset_def)
apply (auto simp add: RCOSETS_def intro: rcos_self prems)
done
qed

lemma (in group) cosets_finite:
"\<lbrakk>c \<in> rcosets H;  H \<subseteq> carrier G;  finite (carrier G)\<rbrakk> \<Longrightarrow> finite c"
apply (simp add: r_coset_subset_G [THEN finite_subset])
done

text{*The next two lemmas support the proof of @{text card_cosets_equal}.*}
lemma (in group) inj_on_f:
"\<lbrakk>H \<subseteq> carrier G;  a \<in> carrier G\<rbrakk> \<Longrightarrow> inj_on (\<lambda>y. y \<otimes> inv a) (H #> a)"
apply (rule inj_onI)
apply (subgoal_tac "x \<in> carrier G & y \<in> carrier G")
prefer 2 apply (blast intro: r_coset_subset_G [THEN subsetD])
done

lemma (in group) inj_on_g:
"\<lbrakk>H \<subseteq> carrier G;  a \<in> carrier G\<rbrakk> \<Longrightarrow> inj_on (\<lambda>y. y \<otimes> a) H"
by (force simp add: inj_on_def subsetD)

lemma (in group) card_cosets_equal:
"\<lbrakk>c \<in> rcosets H;  H \<subseteq> carrier G; finite(carrier G)\<rbrakk>
\<Longrightarrow> card c = card H"
apply (rule card_bij_eq)
apply (rule inj_on_f, assumption+)
apply (force simp add: m_assoc subsetD r_coset_def)
apply (rule inj_on_g, assumption+)
apply (force simp add: m_assoc subsetD r_coset_def)
txt{*The sets @{term "H #> a"} and @{term "H"} are finite.*}
apply (simp add: r_coset_subset_G [THEN finite_subset])
apply (blast intro: finite_subset)
done

lemma (in group) rcosets_subset_PowG:
"subgroup H G  \<Longrightarrow> rcosets H \<subseteq> Pow(carrier G)"
apply (blast dest: r_coset_subset_G subgroup.subset)
done

theorem (in group) lagrange:
"\<lbrakk>finite(carrier G); subgroup H G\<rbrakk>
\<Longrightarrow> card(rcosets H) * card(H) = order(G)"
apply (simp (no_asm_simp) add: order_def rcosets_part_G [symmetric])
apply (subst mult_commute)
apply (rule card_partition)
apply (simp add: rcosets_subset_PowG [THEN finite_subset])
done

subsection {*Quotient Groups: Factorization of a Group*}

constdefs
FactGroup :: "[('a,'b) monoid_scheme, 'a set] \<Rightarrow> ('a set) monoid"
(infixl "Mod" 65)
--{*Actually defined for groups rather than monoids*}
"FactGroup G H \<equiv>
\<lparr>carrier = rcosets\<^bsub>G\<^esub> H, mult = set_mult G, one = H\<rparr>"

lemma (in normal) setmult_closed:
"\<lbrakk>K1 \<in> rcosets H; K2 \<in> rcosets H\<rbrakk> \<Longrightarrow> K1 <#> K2 \<in> rcosets H"
by (auto simp add: rcos_sum RCOSETS_def)

lemma (in normal) setinv_closed:
"K \<in> rcosets H \<Longrightarrow> set_inv K \<in> rcosets H"
by (auto simp add: rcos_inv RCOSETS_def)

lemma (in normal) rcosets_assoc:
"\<lbrakk>M1 \<in> rcosets H; M2 \<in> rcosets H; M3 \<in> rcosets H\<rbrakk>
\<Longrightarrow> M1 <#> M2 <#> M3 = M1 <#> (M2 <#> M3)"
by (auto simp add: RCOSETS_def rcos_sum m_assoc)

lemma (in subgroup) subgroup_in_rcosets:
assumes "group G"
shows "H \<in> rcosets H"
proof -
interpret group G by fact
from _ subgroup_axioms have "H #> \<one> = H"
by (rule coset_join2) auto
then show ?thesis
qed

lemma (in normal) rcosets_inv_mult_group_eq:
"M \<in> rcosets H \<Longrightarrow> set_inv M <#> M = H"
by (auto simp add: RCOSETS_def rcos_inv rcos_sum subgroup.subset normal.axioms prems)

theorem (in normal) factorgroup_is_group:
"group (G Mod H)"
apply (rule groupI)
apply (simp add: normal_imp_subgroup subgroup_in_rcosets [OF is_group])
apply (simp add: restrictI setmult_closed rcosets_assoc)
subgroup_in_rcosets rcosets_mult_eq)
apply (auto dest: rcosets_inv_mult_group_eq simp add: setinv_closed)
done

lemma mult_FactGroup [simp]: "X \<otimes>\<^bsub>(G Mod H)\<^esub> X' = X <#>\<^bsub>G\<^esub> X'"

lemma (in normal) inv_FactGroup:
"X \<in> carrier (G Mod H) \<Longrightarrow> inv\<^bsub>G Mod H\<^esub> X = set_inv X"
apply (rule group.inv_equality [OF factorgroup_is_group])
apply (simp_all add: FactGroup_def setinv_closed rcosets_inv_mult_group_eq)
done

text{*The coset map is a homomorphism from @{term G} to the quotient group
@{term "G Mod H"}*}
lemma (in normal) r_coset_hom_Mod:
"(\<lambda>a. H #> a) \<in> hom G (G Mod H)"
by (auto simp add: FactGroup_def RCOSETS_def Pi_def hom_def rcos_sum)

subsection{*The First Isomorphism Theorem*}

text{*The quotient by the kernel of a homomorphism is isomorphic to the
range of that homomorphism.*}

constdefs
kernel :: "('a, 'm) monoid_scheme \<Rightarrow> ('b, 'n) monoid_scheme \<Rightarrow>
('a \<Rightarrow> 'b) \<Rightarrow> 'a set"
--{*the kernel of a homomorphism*}
"kernel G H h \<equiv> {x. x \<in> carrier G & h x = \<one>\<^bsub>H\<^esub>}"

lemma (in group_hom) subgroup_kernel: "subgroup (kernel G H h) G"
apply (rule subgroup.intro)
apply (auto simp add: kernel_def group.intro prems)
done

text{*The kernel of a homomorphism is a normal subgroup*}
lemma (in group_hom) normal_kernel: "(kernel G H h) \<lhd> G"
done

lemma (in group_hom) FactGroup_nonempty:
assumes X: "X \<in> carrier (G Mod kernel G H h)"
shows "X \<noteq> {}"
proof -
from X
obtain g where "g \<in> carrier G"
and "X = kernel G H h #> g"
by (auto simp add: FactGroup_def RCOSETS_def)
thus ?thesis
by (auto simp add: kernel_def r_coset_def image_def intro: hom_one)
qed

lemma (in group_hom) FactGroup_contents_mem:
assumes X: "X \<in> carrier (G Mod (kernel G H h))"
shows "contents (h`X) \<in> carrier H"
proof -
from X
obtain g where g: "g \<in> carrier G"
and "X = kernel G H h #> g"
by (auto simp add: FactGroup_def RCOSETS_def)
hence "h ` X = {h g}" by (auto simp add: kernel_def r_coset_def image_def g)
thus ?thesis by (auto simp add: g)
qed

lemma (in group_hom) FactGroup_hom:
"(\<lambda>X. contents (h`X)) \<in> hom (G Mod (kernel G H h)) H"
apply (simp add: hom_def FactGroup_contents_mem  normal.factorgroup_is_group [OF normal_kernel] group.axioms monoid.m_closed)
proof (intro ballI)
fix X and X'
assume X:  "X  \<in> carrier (G Mod kernel G H h)"
and X': "X' \<in> carrier (G Mod kernel G H h)"
then
obtain g and g'
where "g \<in> carrier G" and "g' \<in> carrier G"
and "X = kernel G H h #> g" and "X' = kernel G H h #> g'"
by (auto simp add: FactGroup_def RCOSETS_def)
hence all: "\<forall>x\<in>X. h x = h g" "\<forall>x\<in>X'. h x = h g'"
and Xsub: "X \<subseteq> carrier G" and X'sub: "X' \<subseteq> carrier G"
by (force simp add: kernel_def r_coset_def image_def)+
hence "h ` (X <#> X') = {h g \<otimes>\<^bsub>H\<^esub> h g'}" using X X'
by (auto dest!: FactGroup_nonempty
subsetD [OF Xsub] subsetD [OF X'sub])
thus "contents (h ` (X <#> X')) = contents (h ` X) \<otimes>\<^bsub>H\<^esub> contents (h ` X')"
by (simp add: all image_eq_UN FactGroup_nonempty X X')
qed

text{*Lemma for the following injectivity result*}
lemma (in group_hom) FactGroup_subset:
"\<lbrakk>g \<in> carrier G; g' \<in> carrier G; h g = h g'\<rbrakk>
\<Longrightarrow>  kernel G H h #> g \<subseteq> kernel G H h #> g'"
apply (clarsimp simp add: kernel_def r_coset_def image_def)
apply (rename_tac y)
apply (rule_tac x="y \<otimes> g \<otimes> inv g'" in exI)
done

lemma (in group_hom) FactGroup_inj_on:
"inj_on (\<lambda>X. contents (h ` X)) (carrier (G Mod kernel G H h))"
fix X and X'
assume X:  "X  \<in> carrier (G Mod kernel G H h)"
and X': "X' \<in> carrier (G Mod kernel G H h)"
then
obtain g and g'
where gX: "g \<in> carrier G"  "g' \<in> carrier G"
"X = kernel G H h #> g" "X' = kernel G H h #> g'"
by (auto simp add: FactGroup_def RCOSETS_def)
hence all: "\<forall>x\<in>X. h x = h g" "\<forall>x\<in>X'. h x = h g'"
by (force simp add: kernel_def r_coset_def image_def)+
assume "contents (h ` X) = contents (h ` X')"
hence h: "h g = h g'"
by (simp add: image_eq_UN all FactGroup_nonempty X X')
show "X=X'" by (rule equalityI) (simp_all add: FactGroup_subset h gX)
qed

text{*If the homomorphism @{term h} is onto @{term H}, then so is the
homomorphism from the quotient group*}
lemma (in group_hom) FactGroup_onto:
assumes h: "h ` carrier G = carrier H"
shows "(\<lambda>X. contents (h ` X)) ` carrier (G Mod kernel G H h) = carrier H"
proof
show "(\<lambda>X. contents (h ` X)) ` carrier (G Mod kernel G H h) \<subseteq> carrier H"
show "carrier H \<subseteq> (\<lambda>X. contents (h ` X)) ` carrier (G Mod kernel G H h)"
proof
fix y
assume y: "y \<in> carrier H"
with h obtain g where g: "g \<in> carrier G" "h g = y"
by (blast elim: equalityE)
hence "(\<Union>x\<in>kernel G H h #> g. {h x}) = {y}"
by (auto simp add: y kernel_def r_coset_def)
with g show "y \<in> (\<lambda>X. contents (h ` X)) ` carrier (G Mod kernel G H h)"
by (auto intro!: bexI simp add: FactGroup_def RCOSETS_def image_eq_UN)
qed
qed

text{*If @{term h} is a homomorphism from @{term G} onto @{term H}, then the
quotient group @{term "G Mod (kernel G H h)"} is isomorphic to @{term H}.*}
theorem (in group_hom) FactGroup_iso:
"h ` carrier G = carrier H
\<Longrightarrow> (\<lambda>X. contents (h`X)) \<in> (G Mod (kernel G H h)) \<cong> H"
by (simp add: iso_def FactGroup_hom FactGroup_inj_on bij_betw_def
FactGroup_onto)

end
```