src/HOL/Algebra/Bij.thy
author nipkow
Sun Oct 18 12:07:25 2009 +0200 (2009-10-18)
changeset 32988 d1d4d7a08a66
parent 31754 b5260f5272a4
child 33057 764547b68538
permissions -rw-r--r--
Inv -> inv_onto, inv abbr. inv_onto UNIV.
     1 (*  Title:      HOL/Algebra/Bij.thy
     2     ID:         $Id$
     3     Author:     Florian Kammueller, with new proofs by L C Paulson
     4 *)
     5 
     6 theory Bij imports Group begin
     7 
     8 
     9 section {* Bijections of a Set, Permutation and Automorphism Groups *}
    10 
    11 constdefs
    12   Bij :: "'a set \<Rightarrow> ('a \<Rightarrow> 'a) set"
    13     --{*Only extensional functions, since otherwise we get too many.*}
    14   "Bij S \<equiv> extensional S \<inter> {f. bij_betw f S S}"
    15 
    16   BijGroup :: "'a set \<Rightarrow> ('a \<Rightarrow> 'a) monoid"
    17   "BijGroup S \<equiv>
    18     \<lparr>carrier = Bij S,
    19      mult = \<lambda>g \<in> Bij S. \<lambda>f \<in> Bij S. compose S g f,
    20      one = \<lambda>x \<in> S. x\<rparr>"
    21 
    22 
    23 declare Id_compose [simp] compose_Id [simp]
    24 
    25 lemma Bij_imp_extensional: "f \<in> Bij S \<Longrightarrow> f \<in> extensional S"
    26   by (simp add: Bij_def)
    27 
    28 lemma Bij_imp_funcset: "f \<in> Bij S \<Longrightarrow> f \<in> S \<rightarrow> S"
    29   by (auto simp add: Bij_def bij_betw_imp_funcset)
    30 
    31 
    32 subsection {*Bijections Form a Group *}
    33 
    34 lemma restrict_inv_onto_Bij: "f \<in> Bij S \<Longrightarrow> (\<lambda>x \<in> S. (inv_onto S f) x) \<in> Bij S"
    35   by (simp add: Bij_def bij_betw_inv_onto)
    36 
    37 lemma id_Bij: "(\<lambda>x\<in>S. x) \<in> Bij S "
    38   by (auto simp add: Bij_def bij_betw_def inj_on_def)
    39 
    40 lemma compose_Bij: "\<lbrakk>x \<in> Bij S; y \<in> Bij S\<rbrakk> \<Longrightarrow> compose S x y \<in> Bij S"
    41   by (auto simp add: Bij_def bij_betw_compose) 
    42 
    43 lemma Bij_compose_restrict_eq:
    44      "f \<in> Bij S \<Longrightarrow> compose S (restrict (inv_onto S f) S) f = (\<lambda>x\<in>S. x)"
    45   by (simp add: Bij_def compose_inv_onto_id)
    46 
    47 theorem group_BijGroup: "group (BijGroup S)"
    48 apply (simp add: BijGroup_def)
    49 apply (rule groupI)
    50     apply (simp add: compose_Bij)
    51    apply (simp add: id_Bij)
    52   apply (simp add: compose_Bij)
    53   apply (blast intro: compose_assoc [symmetric] dest: Bij_imp_funcset)
    54  apply (simp add: id_Bij Bij_imp_funcset Bij_imp_extensional, simp)
    55 apply (blast intro: Bij_compose_restrict_eq restrict_inv_onto_Bij)
    56 done
    57 
    58 
    59 subsection{*Automorphisms Form a Group*}
    60 
    61 lemma Bij_inv_onto_mem: "\<lbrakk> f \<in> Bij S;  x \<in> S\<rbrakk> \<Longrightarrow> inv_onto S f x \<in> S"
    62 by (simp add: Bij_def bij_betw_def inv_onto_into)
    63 
    64 lemma Bij_inv_onto_lemma:
    65  assumes eq: "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> h(g x y) = g (h x) (h y)"
    66  shows "\<lbrakk>h \<in> Bij S;  g \<in> S \<rightarrow> S \<rightarrow> S;  x \<in> S;  y \<in> S\<rbrakk>
    67         \<Longrightarrow> inv_onto S h (g x y) = g (inv_onto S h x) (inv_onto S h y)"
    68 apply (simp add: Bij_def bij_betw_def)
    69 apply (subgoal_tac "\<exists>x'\<in>S. \<exists>y'\<in>S. x = h x' & y = h y'", clarify)
    70  apply (simp add: eq [symmetric] inv_f_f funcset_mem [THEN funcset_mem], blast)
    71 done
    72 
    73 
    74 constdefs
    75   auto :: "('a, 'b) monoid_scheme \<Rightarrow> ('a \<Rightarrow> 'a) set"
    76   "auto G \<equiv> hom G G \<inter> Bij (carrier G)"
    77 
    78   AutoGroup :: "('a, 'c) monoid_scheme \<Rightarrow> ('a \<Rightarrow> 'a) monoid"
    79   "AutoGroup G \<equiv> BijGroup (carrier G) \<lparr>carrier := auto G\<rparr>"
    80 
    81 lemma (in group) id_in_auto: "(\<lambda>x \<in> carrier G. x) \<in> auto G"
    82   by (simp add: auto_def hom_def restrictI group.axioms id_Bij)
    83 
    84 lemma (in group) mult_funcset: "mult G \<in> carrier G \<rightarrow> carrier G \<rightarrow> carrier G"
    85   by (simp add:  Pi_I group.axioms)
    86 
    87 lemma (in group) restrict_inv_onto_hom:
    88       "\<lbrakk>h \<in> hom G G; h \<in> Bij (carrier G)\<rbrakk>
    89        \<Longrightarrow> restrict (inv_onto (carrier G) h) (carrier G) \<in> hom G G"
    90   by (simp add: hom_def Bij_inv_onto_mem restrictI mult_funcset
    91                 group.axioms Bij_inv_onto_lemma)
    92 
    93 lemma inv_BijGroup:
    94      "f \<in> Bij S \<Longrightarrow> m_inv (BijGroup S) f = (\<lambda>x \<in> S. (inv_onto S f) x)"
    95 apply (rule group.inv_equality)
    96 apply (rule group_BijGroup)
    97 apply (simp_all add:BijGroup_def restrict_inv_onto_Bij Bij_compose_restrict_eq)
    98 done
    99 
   100 lemma (in group) subgroup_auto:
   101       "subgroup (auto G) (BijGroup (carrier G))"
   102 proof (rule subgroup.intro)
   103   show "auto G \<subseteq> carrier (BijGroup (carrier G))"
   104     by (force simp add: auto_def BijGroup_def)
   105 next
   106   fix x y
   107   assume "x \<in> auto G" "y \<in> auto G" 
   108   thus "x \<otimes>\<^bsub>BijGroup (carrier G)\<^esub> y \<in> auto G"
   109     by (force simp add: BijGroup_def is_group auto_def Bij_imp_funcset 
   110                         group.hom_compose compose_Bij)
   111 next
   112   show "\<one>\<^bsub>BijGroup (carrier G)\<^esub> \<in> auto G" by (simp add:  BijGroup_def id_in_auto)
   113 next
   114   fix x 
   115   assume "x \<in> auto G" 
   116   thus "inv\<^bsub>BijGroup (carrier G)\<^esub> x \<in> auto G"
   117     by (simp del: restrict_apply
   118         add: inv_BijGroup auto_def restrict_inv_onto_Bij restrict_inv_onto_hom)
   119 qed
   120 
   121 theorem (in group) AutoGroup: "group (AutoGroup G)"
   122 by (simp add: AutoGroup_def subgroup.subgroup_is_group subgroup_auto 
   123               group_BijGroup)
   124 
   125 end