author paulson Fri, 14 May 2004 16:50:33 +0200 changeset 14747 2eaff69d3d10 parent 14746 9f7b31cf74d8 child 14748 001323d6d75b
removal of locale coset
 src/HOL/Algebra/Coset.thy file | annotate | diff | comparison | revisions src/HOL/Algebra/Sylow.thy file | annotate | diff | comparison | revisions
```--- a/src/HOL/Algebra/Coset.thy	Fri May 14 16:50:13 2004 +0200
+++ b/src/HOL/Algebra/Coset.thy	Fri May 14 16:50:33 2004 +0200
@@ -3,24 +3,24 @@
Author:     Florian Kammueller, with new proofs by L C Paulson
*)

theory Coset = Group + Exponent:

declare (in group) l_inv [simp] and r_inv [simp]

constdefs (structure G)
-  r_coset    :: "[_,'a set, 'a] => 'a set"
-  "r_coset G H a == (% x. x \<otimes> a) ` H"
+  r_coset    :: "[_, 'a set, 'a] => 'a set"    (infixl "#>\<index>" 60)
+  "H #> a == (% x. x \<otimes> a) ` H"

-  l_coset    :: "[_, 'a, 'a set] => 'a set"
-  "l_coset G a H == (% x. a \<otimes> x) ` H"
+  l_coset    :: "[_, 'a, 'a set] => 'a set"    (infixl "<#\<index>" 60)
+  "a <# H == (% x. a \<otimes> x) ` H"

rcosets  :: "[_, 'a set] => ('a set)set"
"rcosets G H == r_coset G H ` (carrier G)"

-  set_mult  :: "[_, 'a set ,'a set] => 'a set"
-  "set_mult G H K == (%(x,y). x \<otimes> y) ` (H \<times> K)"
+  set_mult  :: "[_, 'a set ,'a set] => 'a set" (infixl "<#>\<index>" 60)
+  "H <#> K == (%(x,y). x \<otimes> y) ` (H \<times> K)"

set_inv   :: "[_,'a set] => 'a set"
"set_inv G H == m_inv G ` H"
@@ -30,120 +30,89 @@
(\<forall>x \<in> carrier G. r_coset G H x = l_coset G x H)"

syntax (xsymbols)
-  normal  :: "['a set, ('a,'b) monoid_scheme] => bool" (infixl "\<lhd>" 60)
-
-locale coset = group G +
-  fixes rcos      :: "['a set, 'a] => 'a set"     (infixl "#>" 60)
-    and lcos      :: "['a, 'a set] => 'a set"     (infixl "<#" 60)
-    and setmult   :: "['a set, 'a set] => 'a set" (infixl "<#>" 60)
-  defines rcos_def: "H #> x == r_coset G H x"
-      and lcos_def: "x <# H == l_coset G x H"
-      and setmult_def: "H <#> K == set_mult G H K"
-
-text {*
-  Locale @{term coset} provides only syntax.
-  Logically, groups are cosets.
-*}
-lemma (in group) is_coset:
-  "coset G"
-  by (rule coset.intro)
-
-text{*Lemmas useful for Sylow's Theorem*}
-
-lemma card_inj:
-     "[|f \<in> A\<rightarrow>B; inj_on f A; finite A; finite B|] ==> card(A) <= card(B)"
-apply (rule card_inj_on_le)
-done
-
-lemma card_bij:
-     "[|f \<in> A\<rightarrow>B; inj_on f A;
-        g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B|] ==> card(A) = card(B)"
-by (blast intro: card_inj order_antisym)
+  normal :: "['a set, ('a,'b) monoid_scheme] => bool"  (infixl "\<lhd>" 60)

-subsection {*Lemmas Using *}
+subsection {*Lemmas Using Locale Constants*}

-lemma (in coset) r_coset_eq: "H #> a = {b . \<exists>h\<in>H. h \<otimes> a = b}"
-by (auto simp add: rcos_def r_coset_def)
+lemma (in group) r_coset_eq: "H #> a = {b . \<exists>h\<in>H. h \<otimes> a = b}"

-lemma (in coset) l_coset_eq: "a <# H = {b . \<exists>h\<in>H. a \<otimes> h = b}"
-by (auto simp add: lcos_def l_coset_def)
+lemma (in group) l_coset_eq: "a <# H = {b . \<exists>h\<in>H. a \<otimes> h = b}"

-lemma (in coset) setrcos_eq: "rcosets G H = {C . \<exists>a\<in>carrier G. C = H #> a}"
-by (auto simp add: rcosets_def rcos_def)
+lemma (in group) setrcos_eq: "rcosets G H = {C . \<exists>a\<in>carrier G. C = H #> a}"

-lemma (in coset) set_mult_eq: "H <#> K = {c . \<exists>h\<in>H. \<exists>k\<in>K. c = h \<otimes> k}"
-by (simp add: setmult_def set_mult_def image_def)
+lemma (in group) set_mult_eq: "H <#> K = {c . \<exists>h\<in>H. \<exists>k\<in>K. c = h \<otimes> k}"

-lemma (in coset) coset_mult_assoc:
-     "[| M <= carrier G; g \<in> carrier G; h \<in> carrier G |]
+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_eq m_assoc)
+by (force simp add: r_coset_def m_assoc)

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

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

-lemma (in coset) coset_mult_inv2:
-     "[| M #> x = M #> y;  x \<in> carrier G;  y \<in> carrier G;  M <= carrier G |]
+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 coset) coset_join1:
-     "[| H #> x = H;  x \<in> carrier G;  subgroup H G |] ==> x\<in>H"
+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)
done

-text{*Locales don't currently work with @{text rule_tac}, so we
-must prove this lemma separately.*}
-lemma (in coset) solve_equation:
+lemma (in group) solve_equation:
"\<lbrakk>subgroup H G; x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> \<exists>h\<in>H. h \<otimes> x = y"
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 coset) coset_join2:
+lemma (in group) coset_join2:
"[| x \<in> carrier G;  subgroup H G;  x\<in>H |] ==> H #> x = H"
by (force simp add: subgroup.m_closed r_coset_eq solve_equation)

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

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

-lemma (in coset) setrcosI:
-     "[| H <= carrier G; x \<in> carrier G |] ==> H #> x \<in> rcosets G H"
+lemma (in group) setrcosI:
+     "[| H \<subseteq> carrier G; x \<in> carrier G |] ==> H #> x \<in> rcosets G H"

text{*Really needed?*}
-lemma (in coset) transpose_inv:
+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 coset) repr_independence:
+lemma (in group) repr_independence:
"[| y \<in> H #> x;  x \<in> carrier G; subgroup H G |] ==> H #> x = H #> y"
by (auto simp add: r_coset_eq m_assoc [symmetric]
subgroup.subset [THEN subsetD]
subgroup.m_closed solve_equation)

-lemma (in coset) rcos_self: "[| x \<in> carrier G; subgroup H G |] ==> x \<in> H #> x"
+lemma (in group) rcos_self: "[| x \<in> carrier G; subgroup H G |] ==> x \<in> H #> x"
apply (blast intro: l_one subgroup.subset [THEN subsetD]
subgroup.one_closed)
@@ -152,60 +121,85 @@

subsection {* Normal subgroups *}

-(*????????????????
-text "Allows use of theorems proved in the locale coset"
-lemma subgroup_imp_coset: "subgroup H G ==> coset G"
-apply (drule subgroup_imp_group)
-done
-*)
-
lemma normal_imp_subgroup: "H <| G ==> subgroup H G"

-
-(*????????????????
-lemmas normal_imp_group = normal_imp_subgroup [THEN subgroup_imp_group]
-lemmas normal_imp_coset = normal_imp_subgroup [THEN subgroup_imp_coset]
-*)
-
-lemma (in coset) normal_iff:
-     "(H <| G) = (subgroup H G & (\<forall>x \<in> carrier G. H #> x = x <# H))"
-by (simp add: lcos_def rcos_def normal_def)
-
-lemma (in coset) normal_imp_rcos_eq_lcos:
-     "[| H <| G; x \<in> carrier G |] ==> H #> x = x <# H"
-by (simp add: lcos_def rcos_def normal_def)
-
-lemma (in coset) normal_inv_op_closed1:
+lemma (in group) normal_inv_op_closed1:
"\<lbrakk>H \<lhd> G; x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> (inv x) \<otimes> h \<otimes> x \<in> H"
-apply (auto simp add: l_coset_eq r_coset_eq normal_iff)
+apply (auto simp add: l_coset_def r_coset_def normal_def)
apply (drule bspec, assumption)
apply (drule equalityD1 [THEN subsetD], blast, clarify)
apply (simp add: m_assoc subgroup.subset [THEN subsetD])
-apply (erule subst)
apply (simp add: m_assoc [symmetric] subgroup.subset [THEN subsetD])
done

-lemma (in coset) normal_inv_op_closed2:
+lemma (in group) normal_inv_op_closed2:
"\<lbrakk>H \<lhd> G; x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> h \<otimes> (inv x) \<in> H"
apply (drule normal_inv_op_closed1 [of H "(inv x)"])
apply auto
done

-lemma (in coset) lcos_m_assoc:
-     "[| M <= carrier G; g \<in> carrier G; h \<in> carrier G |]
-      ==> g <# (h <# M) = (g \<otimes> h) <# M"
-by (force simp add: l_coset_eq m_assoc)
+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_imp_subgroup normal_inv_op_closed2)
+next
+  assume ?rhs
+  hence sg: "subgroup N G"
+    and closed: "!!x. x\<in>carrier G ==> \<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 (simp add: sg normal_def l_coset_def r_coset_def, clarify)
+    fix x
+    assume x: "x \<in> carrier G"
+    show "(\<lambda>n. n \<otimes> x) ` N = op \<otimes> x ` N"
+    proof
+      show "(\<lambda>n. n \<otimes> x) ` N \<subseteq> op \<otimes> x ` N"
+      proof clarify
+        fix n
+        assume n: "n \<in> N"
+        show "n \<otimes> x \<in> op \<otimes> x ` N"
+        proof
+          show "n \<otimes> x = x \<otimes> (inv x \<otimes> n \<otimes> x)"
+            by (simp add: x n m_assoc [symmetric] sb [THEN subsetD])
+          with closed [of "inv x"]
+          show "inv x \<otimes> n \<otimes> x \<in> N" by (simp add: x n)
+        qed
+      qed
+    next
+      show "op \<otimes> x ` N \<subseteq> (\<lambda>n. n \<otimes> x) ` N"
+      proof clarify
+        fix n
+        assume n: "n \<in> N"
+        show "x \<otimes> n \<in> (\<lambda>n. n \<otimes> x) ` N"
+        proof
+          show "x \<otimes> n = (x \<otimes> n \<otimes> inv x) \<otimes> x"
+            by (simp add: x n m_assoc sb [THEN subsetD])
+          show "x \<otimes> n \<otimes> inv x \<in> N" by (simp add: x n closed)
+        qed
+      qed
+    qed
+  qed
+qed

-lemma (in coset) lcos_mult_one: "M <= carrier G ==> \<one> <# M = M"
+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 coset) l_coset_subset_G:
-     "[| H <= carrier G; x \<in> carrier G |] ==> x <# H <= carrier G"
-by (auto simp add: l_coset_eq subsetD)
+lemma (in group) lcos_mult_one: "M \<subseteq> carrier G ==> \<one> <# M = M"

-lemma (in coset) l_coset_swap:
+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:
"[| y \<in> x <# H;  x \<in> carrier G;  subgroup H G |] ==> x \<in> y <# H"
assume "\<exists>h\<in>H. x \<otimes> h = y"
@@ -221,23 +215,23 @@
qed
qed

-lemma (in coset) l_coset_carrier:
+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_eq m_assoc
+by (auto simp add: l_coset_def m_assoc
subgroup.subset [THEN subsetD] subgroup.m_closed)

-lemma (in coset) l_repr_imp_subset:
+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_eq)
+  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_eq m_assoc
+    by (auto simp add: l_coset_def m_assoc
subgroup.subset [THEN subsetD] subgroup.m_closed)
qed

-lemma (in coset) l_repr_independence:
+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
@@ -247,11 +241,11 @@
show "y <# H \<subseteq> x <# H" by (rule l_repr_imp_subset [OF y x sb])
qed

-lemma (in coset) setmult_subset_G:
-     "[| H <= carrier G; K <= carrier G |] ==> H <#> K <= carrier G"
+lemma (in group) setmult_subset_G:
+     "[| H \<subseteq> carrier G; K \<subseteq> carrier G |] ==> H <#> K \<subseteq> carrier G"
by (auto simp add: set_mult_eq subsetD)

-lemma (in coset) subgroup_mult_id: "subgroup H G ==> H <#> H = H"
+lemma (in group) subgroup_mult_id: "subgroup H G ==> H <#> H = H"
apply (auto simp add: subgroup.m_closed set_mult_eq Sigma_def image_def)
apply (rule_tac x = x in bexI)
apply (rule bexI [of _ "\<one>"])
@@ -260,21 +254,21 @@
done

-text {* Set of inverses of an @{text r_coset}. *}
+subsubsection {* Set of inverses of an @{text r_coset}. *}

-lemma (in coset) rcos_inv:
+lemma (in group) rcos_inv:
assumes normalHG: "H <| G"
-      and xinG:     "x \<in> carrier G"
+      and x:     "x \<in> carrier G"
shows "set_inv G (H #> x) = H #> (inv x)"
proof -
-  have H_subset: "H <= carrier G"
+  have H_subset: "H \<subseteq> carrier G"
by (rule subgroup.subset [OF normal_imp_subgroup, OF normalHG])
show ?thesis
proof (auto simp add: r_coset_eq image_def set_inv_def)
fix h
assume "h \<in> H"
hence "((inv x) \<otimes> (inv h) \<otimes> x) \<otimes> inv x = inv (h \<otimes> x)"
-        by (simp add: xinG m_assoc inv_mult_group H_subset [THEN subsetD])
+        by (simp add: x m_assoc inv_mult_group H_subset [THEN subsetD])
thus "\<exists>j\<in>H. j \<otimes> inv x = inv (h \<otimes> x)"
using prems
by (blast intro: normal_inv_op_closed1 normal_imp_subgroup
@@ -283,7 +277,7 @@
fix h
assume "h \<in> H"
hence eq: "(x \<otimes> (inv h) \<otimes> (inv x)) \<otimes> x = x \<otimes> inv h"
-        by (simp add: xinG m_assoc H_subset [THEN subsetD])
+        by (simp add: x m_assoc H_subset [THEN subsetD])
hence "(\<exists>j\<in>H. j \<otimes> x = inv  (h \<otimes> (inv x))) \<and> h \<otimes> inv x = inv (inv (h \<otimes> (inv x)))"
using prems
by (simp add: m_assoc inv_mult_group H_subset [THEN subsetD],
@@ -293,24 +287,7 @@
qed
qed

-(*The old proof is something like this, but the rule_tac calls make
-illegal references to implicit structures.
-lemma (in coset) rcos_inv:
-     "[| H <| G; x \<in> carrier G |] ==> set_inv G (H #> x) = H #> (inv x)"
-apply (frule normal_imp_subgroup)
-apply (auto simp add: r_coset_eq image_def set_inv_def)
-apply (rule_tac x = "(inv x) \<otimes> (inv h) \<otimes> x" in bexI)
-apply (simp add: m_assoc inv_mult_group subgroup.subset [THEN subsetD])
-apply (rule_tac x = "inv  (h \<otimes> (inv x)) " in exI)
-apply (simp add: inv_mult_group subgroup.subset [THEN subsetD])
-apply (rule_tac x = "x \<otimes> (inv h) \<otimes> (inv x)" in bexI)
-apply (simp add: m_assoc subgroup.subset [THEN subsetD])
-done
-*)
-
-lemma (in coset) rcos_inv2:
+lemma (in group) rcos_inv2:
"[| H <| G; K \<in> rcosets G H; x \<in> K |] ==> set_inv G K = H #> (inv x)"
apply (subgoal_tac "x : carrier G")
@@ -323,47 +300,46 @@
done

-text {* Some rules for @{text "<#>"} with @{text "#>"} or @{text "<#"}. *}
+subsubsection {* Some rules for @{text "<#>"} with @{text "#>"} or @{text "<#"}. *}

-lemma (in coset) setmult_rcos_assoc:
-     "[| H <= carrier G; K <= carrier G; x \<in> carrier G |]
+lemma (in group) setmult_rcos_assoc:
+     "[| H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G |]
==> H <#> (K #> x) = (H <#> K) #> x"
-apply (auto simp add: rcos_def r_coset_def setmult_def set_mult_def)
+apply (auto simp add: r_coset_def set_mult_def)
done

-lemma (in coset) rcos_assoc_lcos:
-     "[| H <= carrier G; K <= carrier G; x \<in> carrier G |]
+lemma (in group) rcos_assoc_lcos:
+     "[| H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G |]
==> (H #> x) <#> K = H <#> (x <# K)"
-apply (auto simp add: rcos_def r_coset_def lcos_def l_coset_def
-                      setmult_def set_mult_def Sigma_def image_def)
+apply (auto simp add: r_coset_def l_coset_def set_mult_def Sigma_def image_def)
apply (force intro!: exI bexI simp add: m_assoc)+
done

-lemma (in coset) rcos_mult_step1:
+lemma (in group) rcos_mult_step1:
"[| H <| G; x \<in> carrier G; y \<in> carrier G |]
==> (H #> x) <#> (H #> y) = (H <#> (x <# H)) #> y"
by (simp add: setmult_rcos_assoc normal_imp_subgroup [THEN subgroup.subset]
r_coset_subset_G l_coset_subset_G rcos_assoc_lcos)

-lemma (in coset) rcos_mult_step2:
+lemma (in group) rcos_mult_step2:
"[| H <| G; x \<in> carrier G; y \<in> carrier G |]
==> (H <#> (x <# H)) #> y = (H <#> (H #> x)) #> y"

-lemma (in coset) rcos_mult_step3:
+lemma (in group) rcos_mult_step3:
"[| H <| G; x \<in> carrier G; y \<in> carrier G |]
==> (H <#> (H #> x)) #> y = H #> (x \<otimes> y)"
by (simp add: setmult_rcos_assoc r_coset_subset_G coset_mult_assoc
setmult_subset_G  subgroup_mult_id
subgroup.subset normal_imp_subgroup)

-lemma (in coset) rcos_sum:
+lemma (in group) rcos_sum:
"[| H <| G; x \<in> carrier G; y \<in> carrier G |]
==> (H #> x) <#> (H #> y) = H #> (x \<otimes> y)"
by (simp add: rcos_mult_step1 rcos_mult_step2 rcos_mult_step3)

-lemma (in coset) setrcos_mult_eq: "[|H <| G; M \<in> rcosets G H|] ==> H <#> M = M"
+lemma (in group) setrcos_mult_eq: "[|H <| G; M \<in> rcosets G H|] ==> H <#> M = M"
-- {* generalizes @{text subgroup_mult_id} *}
by (auto simp add: setrcos_eq normal_imp_subgroup subgroup.subset
setmult_rcos_assoc subgroup_mult_id)
@@ -371,23 +347,21 @@

subsection {*Lemmas Leading to Lagrange's Theorem *}

-lemma (in coset) setrcos_part_G: "subgroup H G ==> \<Union>rcosets G H = carrier G"
+lemma (in group) setrcos_part_G: "subgroup H G ==> \<Union>rcosets G H = carrier G"
apply (rule equalityI)
apply (force simp add: subgroup.subset [THEN subsetD]
-                       setrcos_eq r_coset_eq)
+                       setrcos_eq r_coset_def)
apply (auto simp add: setrcos_eq subgroup.subset rcos_self)
done

-lemma (in coset) cosets_finite:
-     "[| c \<in> rcosets G H;  H <= carrier G;  finite (carrier G) |] ==> finite c"
+lemma (in group) cosets_finite:
+     "[| c \<in> rcosets G H;  H \<subseteq> carrier G;  finite (carrier G) |] ==> finite c"
apply (simp (no_asm_simp) add: r_coset_subset_G [THEN finite_subset])
done

-text{*The next two lemmas support the proof of @{text card_cosets_equal},
-since we can't use @{text rule_tac} with explicit terms for @{term f} and
-@{term g}.*}
-lemma (in coset) inj_on_f:
+text{*The next two lemmas support the proof of @{text card_cosets_equal}.*}
+lemma (in group) inj_on_f:
"[|H \<subseteq> carrier G;  a \<in> carrier G|] ==> 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")
@@ -395,27 +369,28 @@
done

-lemma (in coset) inj_on_g:
+lemma (in group) inj_on_g:
"[|H \<subseteq> carrier G;  a \<in> carrier G|] ==> inj_on (\<lambda>y. y \<otimes> a) H"
by (force simp add: inj_on_def subsetD)

-lemma (in coset) card_cosets_equal:
-     "[| c \<in> rcosets G H;  H <= carrier G; finite(carrier G) |]
+lemma (in group) card_cosets_equal:
+     "[| c \<in> rcosets G H;  H \<subseteq> carrier G; finite(carrier G) |]
==> card c = card H"
apply (rule card_bij_eq)
apply (rule inj_on_f, assumption+)
-    apply (force simp add: m_assoc subsetD r_coset_eq)
+    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_eq)
+  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

+
subsection{*Two distinct right cosets are disjoint*}

-lemma (in coset) rcos_equation:
+lemma (in group) rcos_equation:
"[|subgroup H G;  a \<in> carrier G;  b \<in> carrier G;  ha \<otimes> a = h \<otimes> b;
h \<in> H;  ha \<in> H;  hb \<in> H|]
==> \<exists>h\<in>H. h \<otimes> b = hb \<otimes> a"
@@ -424,30 +399,31 @@
done

-lemma (in coset) rcos_disjoint:
+lemma (in group) rcos_disjoint:
"[|subgroup H G; a \<in> rcosets G H; b \<in> rcosets G H; a\<noteq>b|] ==> a \<inter> b = {}"
apply (blast intro: rcos_equation sym)
done

-lemma (in coset) setrcos_subset_PowG:
-     "subgroup H G  ==> rcosets G H <= Pow(carrier G)"
+lemma (in group) setrcos_subset_PowG:
+     "subgroup H G  ==> rcosets G H \<subseteq> Pow(carrier G)"
apply (blast dest: r_coset_subset_G subgroup.subset)
done

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

constdefs
FactGroup :: "[('a,'b) monoid_scheme, 'a set] => ('a set) monoid"
(infixl "Mod" 60)
+    --{*Actually defined for groups rather than monoids*}
"FactGroup G H ==
(| carrier = rcosets G H,
mult = (%X: rcosets G H. %Y: rcosets G H. set_mult G X Y),
-       one = H (*,
-       m_inv = (%X: rcosets G H. set_inv G X) *) |)"
+       one = H |)"

-lemma (in coset) setmult_closed:
+
+lemma (in group) setmult_closed:
"[| H <| G; K1 \<in> rcosets G H; K2 \<in> rcosets G H |]
==> K1 <#> K2 \<in> rcosets G H"
by (auto simp add: normal_imp_subgroup [THEN subgroup.subset]
@@ -455,9 +431,9 @@

lemma (in group) setinv_closed:
"[| H <| G; K \<in> rcosets G H |] ==> set_inv G K \<in> rcosets G H"
-                 subgroup.subset coset.rcos_inv [OF is_coset]
-                 coset.setrcos_eq [OF is_coset])
+                   subgroup.subset rcos_inv
+                   setrcos_eq)

(*
The old version is no longer valid: "group G" has to be an explicit premise.
@@ -468,7 +444,7 @@
subgroup.subset coset.rcos_inv coset.setrcos_eq)
*)

-lemma (in coset) setrcos_assoc:
+lemma (in group) setrcos_assoc:
"[|H <| G; M1 \<in> rcosets G H; M2 \<in> rcosets G H; M3 \<in> rcosets G H|]
==> M1 <#> M2 <#> M3 = M1 <#> (M2 <#> M3)"
by (auto simp add: setrcos_eq rcos_sum normal_imp_subgroup
@@ -479,10 +455,10 @@
proof -
assume sub: "subgroup H G"
have "r_coset G H \<one> = H"
-    by (rule coset.coset_join2)
-      (auto intro: sub subgroup.one_closed is_coset)
+    by (rule coset_join2)
+       (auto intro: sub subgroup.one_closed)
then show ?thesis
-    by (auto simp add: coset.setrcos_eq [OF is_coset])
+    by (auto simp add: setrcos_eq)
qed

(*
@@ -497,7 +473,7 @@
done
*)

-lemma (in coset) setrcos_inv_mult_group_eq:
+lemma (in group) setrcos_inv_mult_group_eq:
"[|H <| G; M \<in> rcosets G H|] ==> set_inv G M <#> M = H"
by (auto simp add: setrcos_eq rcos_inv rcos_sum normal_imp_subgroup
subgroup.subset)
@@ -513,15 +489,29 @@
*)
theorem (in group) factorgroup_is_group:
"H <| G ==> group (G Mod H)"
-apply (insert is_coset)
apply (rule groupI)
-  apply (simp add: restrictI coset.setmult_closed coset.setrcos_assoc)
+  apply (simp add: restrictI setmult_closed setrcos_assoc)
-   subgroup_in_rcosets coset.setrcos_mult_eq)
-apply (auto dest: coset.setrcos_inv_mult_group_eq simp add: setinv_closed)
+                  subgroup_in_rcosets setrcos_mult_eq)
+apply (auto dest: setrcos_inv_mult_group_eq simp add: setinv_closed)
done

+lemma (in group) inv_FactGroup:
+     "N <| G ==> X \<in> carrier (G Mod N) ==> inv\<^bsub>G Mod N\<^esub> X = set_inv G X"
+apply (rule group.inv_equality [OF factorgroup_is_group])
+    setrcos_inv_mult_group_eq group.intro [OF prems])
+done
+
+text{*The coset map is a homomorphism from @{term G} to the quotient group
+  @{term "G Mod N"}*}
+lemma (in group) r_coset_hom_Mod:
+  assumes N: "N <| G"
+  shows "(r_coset G N) \<in> hom G (G Mod N)"
+by (simp add: FactGroup_def rcosets_def Pi_def hom_def
+           rcos_sum group.intro prems)
+
end```
```--- a/src/HOL/Algebra/Sylow.thy	Fri May 14 16:50:13 2004 +0200
+++ b/src/HOL/Algebra/Sylow.thy	Fri May 14 16:50:33 2004 +0200
@@ -17,7 +17,7 @@
order :: "('a, 'b) semigroup_scheme => nat"
"order S == card (carrier S)"

-theorem (in coset) lagrange:
+theorem (in group) lagrange:
"[| finite(carrier G); subgroup H G |]
==> card(rcosets G H) * card(H) = order(G)"
apply (simp (no_asm_simp) add: order_def setrcos_part_G [symmetric])
@@ -32,12 +32,12 @@

text{*The combinatorial argument is in theory Exponent*}

-locale sylow = coset +
+locale sylow = group +
fixes p and a and m and calM and RelM
assumes prime_p:   "p \<in> prime"
and order_G:   "order(G) = (p^a) * m"
and finite_G [iff]:  "finite (carrier G)"
-  defines "calM == {s. s <= carrier(G) & card(s) = p^a}"
+  defines "calM == {s. s \<subseteq> carrier(G) & card(s) = p^a}"
and "RelM == {(N1,N2). N1 \<in> calM & N2 \<in> calM &
(\<exists>g \<in> carrier(G). N1 = (N2 #> g) )}"

@@ -64,7 +64,7 @@
apply (blast intro: RelM_refl RelM_sym RelM_trans)
done

-lemma (in sylow) M_subset_calM_prep: "M' \<in> calM // RelM  ==> M' <= calM"
+lemma (in sylow) M_subset_calM_prep: "M' \<in> calM // RelM  ==> M' \<subseteq> calM"
apply (unfold RelM_def)
apply (blast elim!: quotientE)
done
@@ -79,7 +79,7 @@
and M1_in_M:    "M1 \<in> M"
defines "H == {g. g\<in>carrier G & M1 #> g = M1}"

-lemma (in sylow_central) M_subset_calM: "M <= calM"
+lemma (in sylow_central) M_subset_calM: "M \<subseteq> calM"
by (rule M_in_quot [THEN M_subset_calM_prep])

lemma (in sylow_central) card_M1: "card(M1) = p^a"
@@ -97,7 +97,7 @@
done

-lemma (in sylow_central) M1_subset_G [simp]: "M1 <= carrier G"
+lemma (in sylow_central) M1_subset_G [simp]: "M1 \<subseteq> carrier G"
apply (rule subsetD [THEN PowD])
apply (rule_tac [2] M1_in_M)
apply (rule M_subset_calM [THEN subset_trans])
@@ -332,7 +332,7 @@
apply (simp add: H_I coset_mult_inv2 M1_subset_G H_elem_map_carrier)
done

-lemma (in sylow_central) calM_subset_PowG: "calM <= Pow(carrier G)"
+lemma (in sylow_central) calM_subset_PowG: "calM \<subseteq> Pow(carrier G)"

@@ -352,7 +352,7 @@
lemma (in sylow_central) index_lem: "card(M) * card(H) = order(G)"
by (simp add: cardMeqIndexH lagrange H_is_subgroup)

-lemma (in sylow_central) lemma_leq1: "p^a <= card(H)"
+lemma (in sylow_central) lemma_leq1: "p^a \<le> card(H)"
apply (rule dvd_imp_le)
apply (rule div_combine [OF prime_p not_dvd_M])
prefer 2 apply (blast intro: subgroup.finite_imp_card_positive H_is_subgroup)
@@ -360,7 +360,7 @@
zero_less_m)
done

-lemma (in sylow_central) lemma_leq2: "card(H) <= p^a"
+lemma (in sylow_central) lemma_leq2: "card(H) \<le> p^a"
apply (subst card_M1 [symmetric])
apply (cut_tac M1_inj_H)
apply (blast intro!: M1_subset_G intro:```