src/HOL/Algebra/Bij.thy
 author paulson Fri May 02 10:25:42 2003 +0200 (2003-05-02) changeset 13945 5433b2755e98 child 14666 65f8680c3f16 permissions -rw-r--r--
moved Bij.thy from HOL/GroupTheory
```     1 (*  Title:      HOL/Algebra/Bij
```
```     2     ID:         \$Id\$
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```     3     Author:     Florian Kammueller, with new proofs by L C Paulson
```
```     4 *)
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```     5
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```     6
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```     7 header{*Bijections of a Set, Permutation Groups, Automorphism Groups*}
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```     8
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```     9 theory Bij = Group:
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```    10
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```    11 constdefs
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```    12   Bij :: "'a set => (('a => 'a)set)"
```
```    13     --{*Only extensional functions, since otherwise we get too many.*}
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```    14     "Bij S == extensional S \<inter> {f. f`S = S & inj_on f S}"
```
```    15
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```    16    BijGroup ::  "'a set => (('a => 'a) monoid)"
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```    17    "BijGroup S == (| carrier = Bij S,
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```    18 		     mult  = %g: Bij S. %f: Bij S. compose S g f,
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```    19 		     one = %x: S. x |)"
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```    20
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```    21
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```    22 declare Id_compose [simp] compose_Id [simp]
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```    23
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```    24 lemma Bij_imp_extensional: "f \<in> Bij S ==> f \<in> extensional S"
```
```    25 by (simp add: Bij_def)
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```    26
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```    27 lemma Bij_imp_funcset: "f \<in> Bij S ==> f \<in> S -> S"
```
```    28 by (auto simp add: Bij_def Pi_def)
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```    29
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```    30 lemma Bij_imp_apply: "f \<in> Bij S ==> f ` S = S"
```
```    31 by (simp add: Bij_def)
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```    32
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```    33 lemma Bij_imp_inj_on: "f \<in> Bij S ==> inj_on f S"
```
```    34 by (simp add: Bij_def)
```
```    35
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```    36 lemma BijI: "[| f \<in> extensional(S); f`S = S; inj_on f S |] ==> f \<in> Bij S"
```
```    37 by (simp add: Bij_def)
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```    38
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```    39
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```    40 subsection{*Bijections Form a Group*}
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```    41
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```    42 lemma restrict_Inv_Bij: "f \<in> Bij S ==> (%x:S. (Inv S f) x) \<in> Bij S"
```
```    43 apply (simp add: Bij_def)
```
```    44 apply (intro conjI)
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```    45 txt{*Proving @{term "restrict (Inv S f) S ` S = S"}*}
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```    46  apply (rule equalityI)
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```    47   apply (force simp add: Inv_mem) --{*first inclusion*}
```
```    48  apply (rule subsetI)   --{*second inclusion*}
```
```    49  apply (rule_tac x = "f x" in image_eqI)
```
```    50   apply (force intro:  simp add: Inv_f_f, blast)
```
```    51 txt{*Remaining goal: @{term "inj_on (restrict (Inv S f) S) S"}*}
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```    52 apply (rule inj_onI)
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```    53 apply (auto elim: Inv_injective)
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```    54 done
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```    55
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```    56 lemma id_Bij: "(\<lambda>x\<in>S. x) \<in> Bij S "
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```    57 apply (rule BijI)
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```    58 apply (auto simp add: inj_on_def)
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```    59 done
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```    60
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```    61 lemma compose_Bij: "[| x \<in> Bij S; y \<in> Bij S|] ==> compose S x y \<in> Bij S"
```
```    62 apply (rule BijI)
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```    63   apply (simp add: compose_extensional)
```
```    64  apply (blast del: equalityI
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```    65               intro: surj_compose dest: Bij_imp_apply Bij_imp_inj_on)
```
```    66 apply (blast intro: inj_on_compose dest: Bij_imp_apply Bij_imp_inj_on)
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```    67 done
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```    68
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```    69 lemma Bij_compose_restrict_eq:
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```    70      "f \<in> Bij S ==> compose S (restrict (Inv S f) S) f = (\<lambda>x\<in>S. x)"
```
```    71 apply (rule compose_Inv_id)
```
```    72  apply (simp add: Bij_imp_inj_on)
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```    73 apply (simp add: Bij_imp_apply)
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```    74 done
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```    75
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```    76 theorem group_BijGroup: "group (BijGroup S)"
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```    77 apply (simp add: BijGroup_def)
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```    78 apply (rule groupI)
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```    79     apply (simp add: compose_Bij)
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```    80    apply (simp add: id_Bij)
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```    81   apply (simp add: compose_Bij)
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```    82   apply (blast intro: compose_assoc [symmetric] Bij_imp_funcset)
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```    83  apply (simp add: id_Bij Bij_imp_funcset Bij_imp_extensional, simp)
```
```    84 apply (blast intro: Bij_compose_restrict_eq restrict_Inv_Bij)
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```    85 done
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```    86
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```    87
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```    88 subsection{*Automorphisms Form a Group*}
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```    89
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```    90 lemma Bij_Inv_mem: "[|  f \<in> Bij S;  x : S |] ==> Inv S f x : S"
```
```    91 by (simp add: Bij_def Inv_mem)
```
```    92
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```    93 lemma Bij_Inv_lemma:
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```    94  assumes eq: "!!x y. [|x \<in> S; y \<in> S|] ==> h(g x y) = g (h x) (h y)"
```
```    95  shows "[| h \<in> Bij S;  g \<in> S \<rightarrow> S \<rightarrow> S;  x \<in> S;  y \<in> S |]
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```    96         ==> Inv S h (g x y) = g (Inv S h x) (Inv S h y)"
```
```    97 apply (simp add: Bij_def)
```
```    98 apply (subgoal_tac "EX x':S. EX y':S. x = h x' & y = h y'", clarify)
```
```    99  apply (simp add: eq [symmetric] Inv_f_f funcset_mem [THEN funcset_mem], blast)
```
```   100 done
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```   101
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```   102 constdefs
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```   103  auto :: "('a,'b) monoid_scheme => ('a => 'a)set"
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```   104   "auto G == hom G G \<inter> Bij (carrier G)"
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```   105
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```   106   AutoGroup :: "[('a,'c) monoid_scheme] => ('a=>'a) monoid"
```
```   107   "AutoGroup G == BijGroup (carrier G) (|carrier := auto G |)"
```
```   108
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```   109 lemma id_in_auto: "group G ==> (%x: carrier G. x) \<in> auto G"
```
```   110   by (simp add: auto_def hom_def restrictI group.axioms id_Bij)
```
```   111
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```   112 lemma mult_funcset: "group G ==> mult G \<in> carrier G -> carrier G -> carrier G"
```
```   113   by (simp add:  Pi_I group.axioms)
```
```   114
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```   115 lemma restrict_Inv_hom:
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```   116       "[|group G; h \<in> hom G G; h \<in> Bij (carrier G)|]
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```   117        ==> restrict (Inv (carrier G) h) (carrier G) \<in> hom G G"
```
```   118   by (simp add: hom_def Bij_Inv_mem restrictI mult_funcset
```
```   119                 group.axioms Bij_Inv_lemma)
```
```   120
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```   121 lemma inv_BijGroup:
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```   122      "f \<in> Bij S ==> m_inv (BijGroup S) f = (%x: S. (Inv S f) x)"
```
```   123 apply (rule group.inv_equality)
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```   124 apply (rule group_BijGroup)
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```   125 apply (simp_all add: BijGroup_def restrict_Inv_Bij Bij_compose_restrict_eq)
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```   126 done
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```   127
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```   128 lemma subgroup_auto:
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```   129       "group G ==> subgroup (auto G) (BijGroup (carrier G))"
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```   130 apply (rule group.subgroupI)
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```   131     apply (rule group_BijGroup)
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```   132    apply (force simp add: auto_def BijGroup_def)
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```   133   apply (blast intro: dest: id_in_auto)
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```   134  apply (simp del: restrict_apply
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```   135 	     add: inv_BijGroup auto_def restrict_Inv_Bij restrict_Inv_hom)
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```   136 apply (simp add: BijGroup_def auto_def Bij_imp_funcset compose_hom compose_Bij)
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```   137 done
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```   138
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```   139 theorem AutoGroup: "group G ==> group (AutoGroup G)"
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```   140 apply (simp add: AutoGroup_def)
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```   141 apply (rule Group.subgroup.groupI)
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```   142 apply (erule subgroup_auto)
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```   143 apply (insert Bij.group_BijGroup [of "carrier G"])
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```   144 apply (simp_all add: group_def)
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```   145 done
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```   146
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```   147 end
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```   148
```