--- a/src/HOL/Algebra/Bij.thy Fri Apr 23 20:52:04 2004 +0200
+++ b/src/HOL/Algebra/Bij.thy Fri Apr 23 21:46:04 2004 +0200
@@ -3,41 +3,41 @@
Author: Florian Kammueller, with new proofs by L C Paulson
*)
-
-header{*Bijections of a Set, Permutation Groups, Automorphism Groups*}
+header {* Bijections of a Set, Permutation Groups, Automorphism Groups *}
theory Bij = Group:
constdefs
- Bij :: "'a set => (('a => 'a)set)"
+ 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}"
+ "Bij S == extensional S \<inter> {f. f`S = S & inj_on f S}"
- BijGroup :: "'a set => (('a => 'a) monoid)"
- "BijGroup S == (| carrier = Bij S,
- mult = %g: Bij S. %f: Bij S. compose S g f,
- one = %x: S. x |)"
+ BijGroup :: "'a set => ('a => 'a) monoid"
+ "BijGroup S ==
+ (| carrier = Bij S,
+ mult = %g: Bij S. %f: Bij S. compose S g f,
+ one = %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)
+ 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)
+ by (auto simp add: Bij_def Pi_def)
lemma Bij_imp_apply: "f \<in> Bij S ==> f ` S = S"
-by (simp add: Bij_def)
+ by (simp add: Bij_def)
lemma Bij_imp_inj_on: "f \<in> Bij S ==> inj_on f S"
-by (simp add: Bij_def)
+ 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)
+ by (simp add: Bij_def)
-subsection{*Bijections Form a Group*}
+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)
@@ -60,7 +60,7 @@
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 (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)
@@ -70,44 +70,44 @@
"f \<in> Bij S ==> compose S (restrict (Inv S f) S) f = (\<lambda>x\<in>S. x)"
apply (rule compose_Inv_id)
apply (simp add: Bij_imp_inj_on)
-apply (simp add: Bij_imp_apply)
+apply (simp add: Bij_imp_apply)
done
theorem group_BijGroup: "group (BijGroup S)"
-apply (simp add: BijGroup_def)
+apply (simp add: BijGroup_def)
apply (rule groupI)
apply (simp add: compose_Bij)
apply (simp add: id_Bij)
apply (simp add: compose_Bij)
apply (blast intro: compose_assoc [symmetric] Bij_imp_funcset)
apply (simp add: id_Bij Bij_imp_funcset Bij_imp_extensional, simp)
-apply (blast intro: Bij_compose_restrict_eq restrict_Inv_Bij)
+apply (blast intro: Bij_compose_restrict_eq restrict_Inv_Bij)
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)
+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 |]
+ 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 (simp add: Bij_def)
apply (subgoal_tac "EX x':S. EX y':S. x = h x' & y = h y'", clarify)
apply (simp add: eq [symmetric] Inv_f_f funcset_mem [THEN funcset_mem], blast)
done
constdefs
- auto :: "('a,'b) monoid_scheme => ('a => 'a)set"
+ auto :: "('a, 'b) monoid_scheme => ('a => 'a) set"
"auto G == hom G G \<inter> Bij (carrier G)"
- AutoGroup :: "[('a,'c) monoid_scheme] => ('a=>'a) monoid"
+ AutoGroup :: "('a, 'c) monoid_scheme => ('a => 'a) monoid"
"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 group.axioms id_Bij)
+ by (simp add: auto_def hom_def restrictI group.axioms id_Bij)
lemma mult_funcset: "group G ==> mult G \<in> carrier G -> carrier G -> carrier G"
by (simp add: Pi_I group.axioms)
@@ -122,27 +122,26 @@
"f \<in> Bij S ==> m_inv (BijGroup S) f = (%x: S. (Inv S f) x)"
apply (rule group.inv_equality)
apply (rule group_BijGroup)
-apply (simp_all add: BijGroup_def restrict_Inv_Bij Bij_compose_restrict_eq)
+apply (simp_all add: BijGroup_def restrict_Inv_Bij Bij_compose_restrict_eq)
done
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 (rule group.subgroupI)
+ apply (rule group_BijGroup)
+ apply (force simp add: auto_def BijGroup_def)
+ apply (blast intro: dest: id_in_auto)
apply (simp del: restrict_apply
- add: inv_BijGroup auto_def restrict_Inv_Bij restrict_Inv_hom)
+ add: inv_BijGroup auto_def restrict_Inv_Bij restrict_Inv_hom)
apply (simp add: BijGroup_def auto_def Bij_imp_funcset compose_hom compose_Bij)
done
theorem AutoGroup: "group G ==> group (AutoGroup G)"
-apply (simp add: AutoGroup_def)
+apply (simp add: AutoGroup_def)
apply (rule Group.subgroup.groupI)
-apply (erule subgroup_auto)
-apply (insert Bij.group_BijGroup [of "carrier G"])
-apply (simp_all add: group_def)
+apply (erule subgroup_auto)
+apply (insert Bij.group_BijGroup [of "carrier G"])
+apply (simp_all add: group_def)
done
end
-