author paulson Fri May 02 10:25:42 2003 +0200 (2003-05-02) changeset 13945 5433b2755e98 parent 13944 9b34607cd83e child 13946 b75562218711
moved Bij.thy from HOL/GroupTheory
 src/HOL/Algebra/Bij.thy file | annotate | diff | revisions
```     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/Algebra/Bij.thy	Fri May 02 10:25:42 2003 +0200
1.3 @@ -0,0 +1,148 @@
1.4 +(*  Title:      HOL/Algebra/Bij
1.5 +    ID:         \$Id\$
1.6 +    Author:     Florian Kammueller, with new proofs by L C Paulson
1.7 +*)
1.8 +
1.9 +
1.10 +header{*Bijections of a Set, Permutation Groups, Automorphism Groups*}
1.11 +
1.12 +theory Bij = Group:
1.13 +
1.14 +constdefs
1.15 +  Bij :: "'a set => (('a => 'a)set)"
1.16 +    --{*Only extensional functions, since otherwise we get too many.*}
1.17 +    "Bij S == extensional S \<inter> {f. f`S = S & inj_on f S}"
1.18 +
1.19 +   BijGroup ::  "'a set => (('a => 'a) monoid)"
1.20 +   "BijGroup S == (| carrier = Bij S,
1.21 +		     mult  = %g: Bij S. %f: Bij S. compose S g f,
1.22 +		     one = %x: S. x |)"
1.23 +
1.24 +
1.25 +declare Id_compose [simp] compose_Id [simp]
1.26 +
1.27 +lemma Bij_imp_extensional: "f \<in> Bij S ==> f \<in> extensional S"
1.29 +
1.30 +lemma Bij_imp_funcset: "f \<in> Bij S ==> f \<in> S -> S"
1.31 +by (auto simp add: Bij_def Pi_def)
1.32 +
1.33 +lemma Bij_imp_apply: "f \<in> Bij S ==> f ` S = S"
1.35 +
1.36 +lemma Bij_imp_inj_on: "f \<in> Bij S ==> inj_on f S"
1.38 +
1.39 +lemma BijI: "[| f \<in> extensional(S); f`S = S; inj_on f S |] ==> f \<in> Bij S"
1.41 +
1.42 +
1.43 +subsection{*Bijections Form a Group*}
1.44 +
1.45 +lemma restrict_Inv_Bij: "f \<in> Bij S ==> (%x:S. (Inv S f) x) \<in> Bij S"
1.47 +apply (intro conjI)
1.48 +txt{*Proving @{term "restrict (Inv S f) S ` S = S"}*}
1.49 + apply (rule equalityI)
1.50 +  apply (force simp add: Inv_mem) --{*first inclusion*}
1.51 + apply (rule subsetI)   --{*second inclusion*}
1.52 + apply (rule_tac x = "f x" in image_eqI)
1.53 +  apply (force intro:  simp add: Inv_f_f, blast)
1.54 +txt{*Remaining goal: @{term "inj_on (restrict (Inv S f) S) S"}*}
1.55 +apply (rule inj_onI)
1.56 +apply (auto elim: Inv_injective)
1.57 +done
1.58 +
1.59 +lemma id_Bij: "(\<lambda>x\<in>S. x) \<in> Bij S "
1.60 +apply (rule BijI)
1.61 +apply (auto simp add: inj_on_def)
1.62 +done
1.63 +
1.64 +lemma compose_Bij: "[| x \<in> Bij S; y \<in> Bij S|] ==> compose S x y \<in> Bij S"
1.65 +apply (rule BijI)
1.66 +  apply (simp add: compose_extensional)
1.67 + apply (blast del: equalityI
1.68 +              intro: surj_compose dest: Bij_imp_apply Bij_imp_inj_on)
1.69 +apply (blast intro: inj_on_compose dest: Bij_imp_apply Bij_imp_inj_on)
1.70 +done
1.71 +
1.72 +lemma Bij_compose_restrict_eq:
1.73 +     "f \<in> Bij S ==> compose S (restrict (Inv S f) S) f = (\<lambda>x\<in>S. x)"
1.74 +apply (rule compose_Inv_id)
1.75 + apply (simp add: Bij_imp_inj_on)
1.77 +done
1.78 +
1.79 +theorem group_BijGroup: "group (BijGroup S)"
1.81 +apply (rule groupI)
1.82 +    apply (simp add: compose_Bij)
1.83 +   apply (simp add: id_Bij)
1.84 +  apply (simp add: compose_Bij)
1.85 +  apply (blast intro: compose_assoc [symmetric] Bij_imp_funcset)
1.86 + apply (simp add: id_Bij Bij_imp_funcset Bij_imp_extensional, simp)
1.87 +apply (blast intro: Bij_compose_restrict_eq restrict_Inv_Bij)
1.88 +done
1.89 +
1.90 +
1.91 +subsection{*Automorphisms Form a Group*}
1.92 +
1.93 +lemma Bij_Inv_mem: "[|  f \<in> Bij S;  x : S |] ==> Inv S f x : S"
1.94 +by (simp add: Bij_def Inv_mem)
1.95 +
1.96 +lemma Bij_Inv_lemma:
1.97 + assumes eq: "!!x y. [|x \<in> S; y \<in> S|] ==> h(g x y) = g (h x) (h y)"
1.98 + shows "[| h \<in> Bij S;  g \<in> S \<rightarrow> S \<rightarrow> S;  x \<in> S;  y \<in> S |]
1.99 +        ==> Inv S h (g x y) = g (Inv S h x) (Inv S h y)"
1.101 +apply (subgoal_tac "EX x':S. EX y':S. x = h x' & y = h y'", clarify)
1.102 + apply (simp add: eq [symmetric] Inv_f_f funcset_mem [THEN funcset_mem], blast)
1.103 +done
1.104 +
1.105 +constdefs
1.106 + auto :: "('a,'b) monoid_scheme => ('a => 'a)set"
1.107 +  "auto G == hom G G \<inter> Bij (carrier G)"
1.108 +
1.109 +  AutoGroup :: "[('a,'c) monoid_scheme] => ('a=>'a) monoid"
1.110 +  "AutoGroup G == BijGroup (carrier G) (|carrier := auto G |)"
1.111 +
1.112 +lemma id_in_auto: "group G ==> (%x: carrier G. x) \<in> auto G"
1.113 +  by (simp add: auto_def hom_def restrictI group.axioms id_Bij)
1.114 +
1.115 +lemma mult_funcset: "group G ==> mult G \<in> carrier G -> carrier G -> carrier G"
1.116 +  by (simp add:  Pi_I group.axioms)
1.117 +
1.118 +lemma restrict_Inv_hom:
1.119 +      "[|group G; h \<in> hom G G; h \<in> Bij (carrier G)|]
1.120 +       ==> restrict (Inv (carrier G) h) (carrier G) \<in> hom G G"
1.121 +  by (simp add: hom_def Bij_Inv_mem restrictI mult_funcset
1.122 +                group.axioms Bij_Inv_lemma)
1.123 +
1.124 +lemma inv_BijGroup:
1.125 +     "f \<in> Bij S ==> m_inv (BijGroup S) f = (%x: S. (Inv S f) x)"
1.126 +apply (rule group.inv_equality)
1.127 +apply (rule group_BijGroup)
1.128 +apply (simp_all add: BijGroup_def restrict_Inv_Bij Bij_compose_restrict_eq)
1.129 +done
1.130 +
1.131 +lemma subgroup_auto:
1.132 +      "group G ==> subgroup (auto G) (BijGroup (carrier G))"
1.133 +apply (rule group.subgroupI)
1.134 +    apply (rule group_BijGroup)
1.135 +   apply (force simp add: auto_def BijGroup_def)
1.136 +  apply (blast intro: dest: id_in_auto)
1.137 + apply (simp del: restrict_apply
1.138 +	     add: inv_BijGroup auto_def restrict_Inv_Bij restrict_Inv_hom)
1.139 +apply (simp add: BijGroup_def auto_def Bij_imp_funcset compose_hom compose_Bij)
1.140 +done
1.141 +
1.142 +theorem AutoGroup: "group G ==> group (AutoGroup G)"