--- a/src/HOL/Algebra/Coset.thy Thu May 01 08:39:37 2003 +0200
+++ b/src/HOL/Algebra/Coset.thy Thu May 01 10:29:44 2003 +0200
@@ -491,23 +491,4 @@
apply (auto dest: coset.setrcos_inv_mult_group_eq simp add: setinv_closed)
done
-(*????????????????
-theorem factorgroup_is_group: "H <| G ==> group (G Mod H)"
-apply (frule normal_imp_coset)
-apply (simp add: FactGroup_def)
-apply (rule group.intro)
-apply (rule magma.intro)
-apply (simp add: );
- apply (simp add: restrictI coset.setmult_closed)
- apply (rule semigroup.intro)
- apply (simp add: restrictI coset.setmult_closed)
- apply (simp add: coset.setmult_closed coset.setrcos_assoc)
-apply (rule group_axioms.intro)
- apply (simp add: restrictI setinv_closed)
- apply (simp add: normal_imp_subgroup subgroup_in_rcosets)
- apply (simp add: setinv_closed coset.setrcos_inv_mult_group_eq)
-apply (simp add: normal_imp_subgroup subgroup_in_rcosets coset.setrcos_mult_eq)
-done
-*)
-
end
--- a/src/HOL/Algebra/Group.thy Thu May 01 08:39:37 2003 +0200
+++ b/src/HOL/Algebra/Group.thy Thu May 01 10:29:44 2003 +0200
@@ -99,29 +99,23 @@
lemma (in monoid) Units_inv_closed [intro, simp]:
"x \<in> Units G ==> inv x \<in> carrier G"
- apply (unfold Units_def m_inv_def)
- apply auto
+ apply (unfold Units_def m_inv_def, auto)
apply (rule theI2, fast)
- apply (fast intro: inv_unique)
- apply fast
+ apply (fast intro: inv_unique, fast)
done
lemma (in monoid) Units_l_inv:
"x \<in> Units G ==> inv x \<otimes> x = \<one>"
- apply (unfold Units_def m_inv_def)
- apply auto
+ apply (unfold Units_def m_inv_def, auto)
apply (rule theI2, fast)
- apply (fast intro: inv_unique)
- apply fast
+ apply (fast intro: inv_unique, fast)
done
lemma (in monoid) Units_r_inv:
"x \<in> Units G ==> x \<otimes> inv x = \<one>"
- apply (unfold Units_def m_inv_def)
- apply auto
+ apply (unfold Units_def m_inv_def, auto)
apply (rule theI2, fast)
- apply (fast intro: inv_unique)
- apply fast
+ apply (fast intro: inv_unique, fast)
done
lemma (in monoid) Units_inv_Units [intro, simp]:
@@ -354,6 +348,14 @@
"[| x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>"
by (rule Units_inv_comm) auto
+lemma (in group) m_inv_equality:
+ "[|y \<otimes> x = \<one>; x \<in> carrier G; y \<in> carrier G|] ==> inv x = y"
+apply (simp add: m_inv_def)
+apply (rule the_equality)
+ apply (simp add: inv_comm [of y x])
+apply (rule r_cancel [THEN iffD1], auto)
+done
+
text {* Power *}
lemma (in group) int_pow_def2:
@@ -594,6 +596,15 @@
"[| h \<in> hom G H; x \<in> carrier G |] ==> h x \<in> carrier H"
by (auto simp add: hom_def funcset_mem)
+lemma compose_hom:
+ "[|group G; h \<in> hom G G; h' \<in> hom G G; h' \<in> carrier G -> carrier G|]
+ ==> compose (carrier G) h h' \<in> hom G G"
+apply (simp (no_asm_simp) add: hom_def)
+apply (intro conjI)
+ apply (force simp add: funcset_compose hom_def)
+apply (simp add: compose_def group.axioms hom_mult funcset_mem)
+done
+
locale group_hom = group G + group H + var h +
assumes homh: "h \<in> hom G H"
notes hom_mult [simp] = hom_mult [OF homh]
--- a/src/HOL/Algebra/ROOT.ML Thu May 01 08:39:37 2003 +0200
+++ b/src/HOL/Algebra/ROOT.ML Thu May 01 10:29:44 2003 +0200
@@ -8,6 +8,7 @@
no_document use_thy "FuncSet";
use_thy "Sylow"; (* Groups *)
+use_thy "Bij"; (* Automorphism Groups *)
use_thy "UnivPoly"; (* Rings and polynomials *)
(* Old development, based on axiomatic type classes.
--- a/src/HOL/GroupTheory/Bij.thy Thu May 01 08:39:37 2003 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,131 +0,0 @@
-(* Title: HOL/GroupTheory/Bij
- ID: $Id$
- Author: Florian Kammueller, with new proofs by L C Paulson
-*)
-
-
-header{*Bijections of a Set, Permutation Groups, Automorphism Groups*}
-
-theory Bij = Group:
-
-constdefs
- Bij :: "'a set => (('a => 'a)set)"
- --{*Only extensional functions, since otherwise we get too many.*}
- "Bij S == extensional S \<inter> {f. f`S = S & inj_on f S}"
-
- BijGroup :: "'a set => (('a => 'a) group)"
- "BijGroup S == (| carrier = Bij S,
- sum = %g: Bij S. %f: Bij S. compose S g f,
- gminus = %f: Bij S. %x: S. (Inv S f) x,
- zero = %x: S. x |)"
-
-
-declare Id_compose [simp] compose_Id [simp]
-
-lemma Bij_imp_extensional: "f \<in> Bij S ==> f \<in> extensional S"
-by (simp add: Bij_def)
-
-lemma Bij_imp_funcset: "f \<in> Bij S ==> f \<in> S -> S"
-by (auto simp add: Bij_def Pi_def)
-
-lemma Bij_imp_apply: "f \<in> Bij S ==> f ` S = S"
-by (simp add: Bij_def)
-
-lemma Bij_imp_inj_on: "f \<in> Bij S ==> inj_on f S"
-by (simp add: Bij_def)
-
-lemma BijI: "[| f \<in> extensional(S); f`S = S; inj_on f S |] ==> f \<in> Bij S"
-by (simp add: Bij_def)
-
-
-subsection{*Bijections Form a Group*}
-
-lemma restrict_Inv_Bij: "f \<in> Bij S ==> (%x:S. (Inv S f) x) \<in> Bij S"
-apply (simp add: Bij_def)
-apply (intro conjI)
-txt{*Proving @{term "restrict (Inv S f) S ` S = S"}*}
- apply (rule equalityI)
- apply (force simp add: Inv_mem) --{*first inclusion*}
- apply (rule subsetI) --{*second inclusion*}
- apply (rule_tac x = "f x" in image_eqI)
- apply (force intro: simp add: Inv_f_f)
- apply blast
-txt{*Remaining goal: @{term "inj_on (restrict (Inv S f) S) S"}*}
-apply (rule inj_onI)
-apply (auto elim: Inv_injective)
-done
-
-lemma id_Bij: "(\<lambda>x\<in>S. x) \<in> Bij S "
-apply (rule BijI)
-apply (auto simp add: inj_on_def)
-done
-
-lemma compose_Bij: "[| x \<in> Bij S; y \<in> Bij S|] ==> compose S x y \<in> Bij S"
-apply (rule BijI)
- apply (simp add: compose_extensional)
- apply (blast del: equalityI
- intro: surj_compose dest: Bij_imp_apply Bij_imp_inj_on)
-apply (blast intro: inj_on_compose dest: Bij_imp_apply Bij_imp_inj_on)
-done
-
-theorem group_BijGroup: "group (BijGroup S)"
-apply (simp add: group_def semigroup_def group_axioms_def
- BijGroup_def restrictI compose_Bij restrict_Inv_Bij id_Bij)
-apply (auto intro!: compose_Bij)
- apply (blast intro: compose_assoc [symmetric] Bij_imp_funcset)
- apply (simp add: Bij_def compose_Inv_id)
-apply (simp add: Id_compose Bij_imp_funcset Bij_imp_extensional)
-done
-
-
-subsection{*Automorphisms Form a Group*}
-
-lemma Bij_Inv_mem: "[| f \<in> Bij S; x : S |] ==> Inv S f x : S"
-by (simp add: Bij_def Inv_mem)
-
-lemma Bij_Inv_lemma:
- assumes eq: "!!x y. [|x \<in> S; y \<in> S|] ==> h(g x y) = g (h x) (h y)"
- shows "[| h \<in> Bij S; g \<in> S \<rightarrow> S \<rightarrow> S; x \<in> S; y \<in> S |]
- ==> Inv S h (g x y) = g (Inv S h x) (Inv S h y)"
-apply (simp add: Bij_def)
-apply (subgoal_tac "EX x':S. EX y':S. x = h x' & y = h y'")
- apply clarify
- apply (simp add: eq [symmetric] Inv_f_f funcset_mem [THEN funcset_mem], blast)
-done
-
-constdefs
- auto :: "('a,'b)group_scheme => ('a => 'a)set"
- "auto G == hom G G Int Bij (carrier G)"
-
- AutoGroup :: "[('a,'c) group_scheme] => ('a=>'a) group"
- "AutoGroup G == BijGroup (carrier G) (|carrier := auto G |)"
-
-lemma id_in_auto: "group G ==> (%x: carrier G. x) \<in> auto G"
-by (simp add: auto_def hom_def restrictI semigroup.sum_closed
- group.axioms id_Bij)
-
-lemma restrict_Inv_hom:
- "[|group G; h \<in> hom G G; h \<in> Bij (carrier G)|]
- ==> restrict (Inv (carrier G) h) (carrier G) \<in> hom G G"
-by (simp add: hom_def Bij_Inv_mem restrictI semigroup.sum_closed
- semigroup.sum_funcset group.axioms Bij_Inv_lemma)
-
-lemma subgroup_auto:
- "group G ==> subgroup (auto G) (BijGroup (carrier G))"
-apply (rule group.subgroupI)
- apply (rule group_BijGroup)
- apply (force simp add: auto_def BijGroup_def)
- apply (blast intro: dest: id_in_auto)
- apply (simp add: auto_def BijGroup_def restrict_Inv_Bij
- restrict_Inv_hom)
-apply (simp add: auto_def BijGroup_def compose_Bij)
-apply (simp add: hom_def compose_def Pi_def group.axioms)
-done
-
-theorem AutoGroup: "group G ==> group (AutoGroup G)"
-apply (drule subgroup_auto)
-apply (simp add: subgroup_def AutoGroup_def)
-done
-
-end
-
--- a/src/HOL/GroupTheory/ROOT.ML Thu May 01 08:39:37 2003 +0200
+++ b/src/HOL/GroupTheory/ROOT.ML Thu May 01 10:29:44 2003 +0200
@@ -1,3 +1,3 @@
no_document use_thy "FuncSet";
-use_thy "Bij";
+use_thy "Group";
--- a/src/HOL/IsaMakefile Thu May 01 08:39:37 2003 +0200
+++ b/src/HOL/IsaMakefile Thu May 01 10:29:44 2003 +0200
@@ -284,7 +284,6 @@
$(LOG)/HOL-GroupTheory.gz: $(OUT)/HOL \
Library/Primes.thy Library/FuncSet.thy \
- GroupTheory/Bij.thy \
GroupTheory/Group.thy \
GroupTheory/ROOT.ML \
GroupTheory/document/root.tex
@@ -341,6 +340,7 @@
HOL-Algebra: HOL $(LOG)/HOL-Algebra.gz
$(LOG)/HOL-Algebra.gz: $(OUT)/HOL Algebra/ROOT.ML \
+ Algebra/Bij.thy \
Algebra/CRing.thy \
Algebra/Coset.thy \
Algebra/Exponent.thy \