src/HOL/Algebra/Bij.thy
author haftmann
Mon Mar 01 13:40:23 2010 +0100 (2010-03-01)
changeset 35416 d8d7d1b785af
parent 33057 764547b68538
child 35848 5443079512ea
permissions -rw-r--r--
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
wenzelm@14706
     1
(*  Title:      HOL/Algebra/Bij.thy
paulson@13945
     2
    Author:     Florian Kammueller, with new proofs by L C Paulson
paulson@13945
     3
*)
paulson@13945
     4
ballarin@20318
     5
theory Bij imports Group begin
paulson@13945
     6
ballarin@20318
     7
ballarin@27717
     8
section {* Bijections of a Set, Permutation and Automorphism Groups *}
paulson@13945
     9
haftmann@35416
    10
definition Bij :: "'a set \<Rightarrow> ('a \<Rightarrow> 'a) set" where
paulson@13945
    11
    --{*Only extensional functions, since otherwise we get too many.*}
paulson@14963
    12
  "Bij S \<equiv> extensional S \<inter> {f. bij_betw f S S}"
paulson@13945
    13
haftmann@35416
    14
definition BijGroup :: "'a set \<Rightarrow> ('a \<Rightarrow> 'a) monoid" where
paulson@14963
    15
  "BijGroup S \<equiv>
paulson@14963
    16
    \<lparr>carrier = Bij S,
paulson@14963
    17
     mult = \<lambda>g \<in> Bij S. \<lambda>f \<in> Bij S. compose S g f,
paulson@14963
    18
     one = \<lambda>x \<in> S. x\<rparr>"
paulson@13945
    19
paulson@13945
    20
paulson@13945
    21
declare Id_compose [simp] compose_Id [simp]
paulson@13945
    22
paulson@14963
    23
lemma Bij_imp_extensional: "f \<in> Bij S \<Longrightarrow> f \<in> extensional S"
wenzelm@14666
    24
  by (simp add: Bij_def)
paulson@13945
    25
paulson@14963
    26
lemma Bij_imp_funcset: "f \<in> Bij S \<Longrightarrow> f \<in> S \<rightarrow> S"
paulson@14853
    27
  by (auto simp add: Bij_def bij_betw_imp_funcset)
paulson@13945
    28
paulson@13945
    29
wenzelm@14666
    30
subsection {*Bijections Form a Group *}
paulson@13945
    31
nipkow@33057
    32
lemma restrict_inv_into_Bij: "f \<in> Bij S \<Longrightarrow> (\<lambda>x \<in> S. (inv_into S f) x) \<in> Bij S"
nipkow@33057
    33
  by (simp add: Bij_def bij_betw_inv_into)
paulson@13945
    34
paulson@13945
    35
lemma id_Bij: "(\<lambda>x\<in>S. x) \<in> Bij S "
paulson@14853
    36
  by (auto simp add: Bij_def bij_betw_def inj_on_def)
paulson@13945
    37
paulson@14963
    38
lemma compose_Bij: "\<lbrakk>x \<in> Bij S; y \<in> Bij S\<rbrakk> \<Longrightarrow> compose S x y \<in> Bij S"
paulson@14853
    39
  by (auto simp add: Bij_def bij_betw_compose) 
paulson@13945
    40
paulson@13945
    41
lemma Bij_compose_restrict_eq:
nipkow@33057
    42
     "f \<in> Bij S \<Longrightarrow> compose S (restrict (inv_into S f) S) f = (\<lambda>x\<in>S. x)"
nipkow@33057
    43
  by (simp add: Bij_def compose_inv_into_id)
paulson@13945
    44
paulson@13945
    45
theorem group_BijGroup: "group (BijGroup S)"
wenzelm@14666
    46
apply (simp add: BijGroup_def)
paulson@13945
    47
apply (rule groupI)
paulson@13945
    48
    apply (simp add: compose_Bij)
paulson@13945
    49
   apply (simp add: id_Bij)
paulson@13945
    50
  apply (simp add: compose_Bij)
nipkow@31754
    51
  apply (blast intro: compose_assoc [symmetric] dest: Bij_imp_funcset)
paulson@13945
    52
 apply (simp add: id_Bij Bij_imp_funcset Bij_imp_extensional, simp)
nipkow@33057
    53
apply (blast intro: Bij_compose_restrict_eq restrict_inv_into_Bij)
paulson@13945
    54
done
paulson@13945
    55
paulson@13945
    56
paulson@13945
    57
subsection{*Automorphisms Form a Group*}
paulson@13945
    58
nipkow@33057
    59
lemma Bij_inv_into_mem: "\<lbrakk> f \<in> Bij S;  x \<in> S\<rbrakk> \<Longrightarrow> inv_into S f x \<in> S"
nipkow@33057
    60
by (simp add: Bij_def bij_betw_def inv_into_into)
paulson@13945
    61
nipkow@33057
    62
lemma Bij_inv_into_lemma:
paulson@14963
    63
 assumes eq: "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> h(g x y) = g (h x) (h y)"
paulson@14963
    64
 shows "\<lbrakk>h \<in> Bij S;  g \<in> S \<rightarrow> S \<rightarrow> S;  x \<in> S;  y \<in> S\<rbrakk>
nipkow@33057
    65
        \<Longrightarrow> inv_into S h (g x y) = g (inv_into S h x) (inv_into S h y)"
paulson@14853
    66
apply (simp add: Bij_def bij_betw_def)
paulson@14853
    67
apply (subgoal_tac "\<exists>x'\<in>S. \<exists>y'\<in>S. x = h x' & y = h y'", clarify)
nipkow@32988
    68
 apply (simp add: eq [symmetric] inv_f_f funcset_mem [THEN funcset_mem], blast)
paulson@13945
    69
done
paulson@13945
    70
paulson@14963
    71
haftmann@35416
    72
definition auto :: "('a, 'b) monoid_scheme \<Rightarrow> ('a \<Rightarrow> 'a) set" where
paulson@14963
    73
  "auto G \<equiv> hom G G \<inter> Bij (carrier G)"
paulson@13945
    74
haftmann@35416
    75
definition AutoGroup :: "('a, 'c) monoid_scheme \<Rightarrow> ('a \<Rightarrow> 'a) monoid" where
paulson@14963
    76
  "AutoGroup G \<equiv> BijGroup (carrier G) \<lparr>carrier := auto G\<rparr>"
paulson@13945
    77
paulson@14963
    78
lemma (in group) id_in_auto: "(\<lambda>x \<in> carrier G. x) \<in> auto G"
wenzelm@14666
    79
  by (simp add: auto_def hom_def restrictI group.axioms id_Bij)
paulson@13945
    80
paulson@14963
    81
lemma (in group) mult_funcset: "mult G \<in> carrier G \<rightarrow> carrier G \<rightarrow> carrier G"
paulson@13945
    82
  by (simp add:  Pi_I group.axioms)
paulson@13945
    83
nipkow@33057
    84
lemma (in group) restrict_inv_into_hom:
paulson@14963
    85
      "\<lbrakk>h \<in> hom G G; h \<in> Bij (carrier G)\<rbrakk>
nipkow@33057
    86
       \<Longrightarrow> restrict (inv_into (carrier G) h) (carrier G) \<in> hom G G"
nipkow@33057
    87
  by (simp add: hom_def Bij_inv_into_mem restrictI mult_funcset
nipkow@33057
    88
                group.axioms Bij_inv_into_lemma)
paulson@13945
    89
paulson@13945
    90
lemma inv_BijGroup:
nipkow@33057
    91
     "f \<in> Bij S \<Longrightarrow> m_inv (BijGroup S) f = (\<lambda>x \<in> S. (inv_into S f) x)"
paulson@13945
    92
apply (rule group.inv_equality)
paulson@13945
    93
apply (rule group_BijGroup)
nipkow@33057
    94
apply (simp_all add:BijGroup_def restrict_inv_into_Bij Bij_compose_restrict_eq)
paulson@13945
    95
done
paulson@13945
    96
paulson@14963
    97
lemma (in group) subgroup_auto:
paulson@14963
    98
      "subgroup (auto G) (BijGroup (carrier G))"
paulson@14963
    99
proof (rule subgroup.intro)
paulson@14963
   100
  show "auto G \<subseteq> carrier (BijGroup (carrier G))"
paulson@14963
   101
    by (force simp add: auto_def BijGroup_def)
paulson@14963
   102
next
paulson@14963
   103
  fix x y
paulson@14963
   104
  assume "x \<in> auto G" "y \<in> auto G" 
paulson@14963
   105
  thus "x \<otimes>\<^bsub>BijGroup (carrier G)\<^esub> y \<in> auto G"
paulson@14963
   106
    by (force simp add: BijGroup_def is_group auto_def Bij_imp_funcset 
paulson@14963
   107
                        group.hom_compose compose_Bij)
paulson@14963
   108
next
paulson@14963
   109
  show "\<one>\<^bsub>BijGroup (carrier G)\<^esub> \<in> auto G" by (simp add:  BijGroup_def id_in_auto)
paulson@14963
   110
next
paulson@14963
   111
  fix x 
paulson@14963
   112
  assume "x \<in> auto G" 
paulson@14963
   113
  thus "inv\<^bsub>BijGroup (carrier G)\<^esub> x \<in> auto G"
paulson@14963
   114
    by (simp del: restrict_apply
nipkow@33057
   115
        add: inv_BijGroup auto_def restrict_inv_into_Bij restrict_inv_into_hom)
paulson@14963
   116
qed
paulson@13945
   117
paulson@14963
   118
theorem (in group) AutoGroup: "group (AutoGroup G)"
paulson@14963
   119
by (simp add: AutoGroup_def subgroup.subgroup_is_group subgroup_auto 
paulson@14963
   120
              group_BijGroup)
paulson@13945
   121
paulson@13945
   122
end