src/HOL/Algebra/Bij.thy
 author wenzelm Sun Nov 26 21:08:32 2017 +0100 (17 months ago) changeset 67091 1393c2340eec parent 63167 0909deb8059b child 67443 3abf6a722518 permissions -rw-r--r--
more symbols;
```     1 (*  Title:      HOL/Algebra/Bij.thy
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```     2     Author:     Florian Kammueller, with new proofs by L C Paulson
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```     3 *)
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```     4
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```     5 theory Bij
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```     6 imports Group
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```     7 begin
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```     8
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```     9 section \<open>Bijections of a Set, Permutation and Automorphism Groups\<close>
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```    10
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```    11 definition
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```    12   Bij :: "'a set \<Rightarrow> ('a \<Rightarrow> 'a) set"
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```    13     \<comment>\<open>Only extensional functions, since otherwise we get too many.\<close>
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```    14    where "Bij S = extensional S \<inter> {f. bij_betw f S S}"
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```    15
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```    16 definition
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```    17   BijGroup :: "'a set \<Rightarrow> ('a \<Rightarrow> 'a) monoid"
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```    18   where "BijGroup S =
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```    19     \<lparr>carrier = Bij S,
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```    20      mult = \<lambda>g \<in> Bij S. \<lambda>f \<in> Bij S. compose S g f,
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```    21      one = \<lambda>x \<in> S. x\<rparr>"
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```    22
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```    23
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```    24 declare Id_compose [simp] compose_Id [simp]
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```    25
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```    26 lemma Bij_imp_extensional: "f \<in> Bij S \<Longrightarrow> f \<in> extensional S"
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```    27   by (simp add: Bij_def)
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```    28
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```    29 lemma Bij_imp_funcset: "f \<in> Bij S \<Longrightarrow> f \<in> S \<rightarrow> S"
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```    30   by (auto simp add: Bij_def bij_betw_imp_funcset)
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```    31
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```    32
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```    33 subsection \<open>Bijections Form a Group\<close>
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```    34
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```    35 lemma restrict_inv_into_Bij: "f \<in> Bij S \<Longrightarrow> (\<lambda>x \<in> S. (inv_into S f) x) \<in> Bij S"
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```    36   by (simp add: Bij_def bij_betw_inv_into)
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```    37
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```    38 lemma id_Bij: "(\<lambda>x\<in>S. x) \<in> Bij S "
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```    39   by (auto simp add: Bij_def bij_betw_def inj_on_def)
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```    40
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```    41 lemma compose_Bij: "\<lbrakk>x \<in> Bij S; y \<in> Bij S\<rbrakk> \<Longrightarrow> compose S x y \<in> Bij S"
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```    42   by (auto simp add: Bij_def bij_betw_compose)
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```    43
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```    44 lemma Bij_compose_restrict_eq:
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```    45      "f \<in> Bij S \<Longrightarrow> compose S (restrict (inv_into S f) S) f = (\<lambda>x\<in>S. x)"
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```    46   by (simp add: Bij_def compose_inv_into_id)
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```    47
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```    48 theorem group_BijGroup: "group (BijGroup S)"
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```    49 apply (simp add: BijGroup_def)
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```    50 apply (rule groupI)
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```    51     apply (simp add: compose_Bij)
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```    52    apply (simp add: id_Bij)
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```    53   apply (simp add: compose_Bij)
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```    54   apply (blast intro: compose_assoc [symmetric] dest: Bij_imp_funcset)
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```    55  apply (simp add: id_Bij Bij_imp_funcset Bij_imp_extensional, simp)
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```    56 apply (blast intro: Bij_compose_restrict_eq restrict_inv_into_Bij)
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```    57 done
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```    58
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```    59
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```    60 subsection\<open>Automorphisms Form a Group\<close>
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```    61
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```    62 lemma Bij_inv_into_mem: "\<lbrakk> f \<in> Bij S;  x \<in> S\<rbrakk> \<Longrightarrow> inv_into S f x \<in> S"
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```    63 by (simp add: Bij_def bij_betw_def inv_into_into)
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```    64
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```    65 lemma Bij_inv_into_lemma:
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```    66  assumes eq: "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> h(g x y) = g (h x) (h y)"
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```    67  shows "\<lbrakk>h \<in> Bij S;  g \<in> S \<rightarrow> S \<rightarrow> S;  x \<in> S;  y \<in> S\<rbrakk>
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```    68         \<Longrightarrow> inv_into S h (g x y) = g (inv_into S h x) (inv_into S h y)"
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```    69 apply (simp add: Bij_def bij_betw_def)
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```    70 apply (subgoal_tac "\<exists>x'\<in>S. \<exists>y'\<in>S. x = h x' \<and> y = h y'", clarify)
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```    71  apply (simp add: eq [symmetric] inv_f_f funcset_mem [THEN funcset_mem], blast)
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```    72 done
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```    73
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```    74
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```    75 definition
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```    76   auto :: "('a, 'b) monoid_scheme \<Rightarrow> ('a \<Rightarrow> 'a) set"
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```    77   where "auto G = hom G G \<inter> Bij (carrier G)"
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```    78
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```    79 definition
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```    80   AutoGroup :: "('a, 'c) monoid_scheme \<Rightarrow> ('a \<Rightarrow> 'a) monoid"
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```    81   where "AutoGroup G = BijGroup (carrier G) \<lparr>carrier := auto G\<rparr>"
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```    82
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```    83 lemma (in group) id_in_auto: "(\<lambda>x \<in> carrier G. x) \<in> auto G"
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```    84   by (simp add: auto_def hom_def restrictI group.axioms id_Bij)
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```    85
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```    86 lemma (in group) mult_funcset: "mult G \<in> carrier G \<rightarrow> carrier G \<rightarrow> carrier G"
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```    87   by (simp add:  Pi_I group.axioms)
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```    88
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```    89 lemma (in group) restrict_inv_into_hom:
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```    90       "\<lbrakk>h \<in> hom G G; h \<in> Bij (carrier G)\<rbrakk>
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```    91        \<Longrightarrow> restrict (inv_into (carrier G) h) (carrier G) \<in> hom G G"
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```    92   by (simp add: hom_def Bij_inv_into_mem restrictI mult_funcset
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```    93                 group.axioms Bij_inv_into_lemma)
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```    94
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```    95 lemma inv_BijGroup:
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```    96      "f \<in> Bij S \<Longrightarrow> m_inv (BijGroup S) f = (\<lambda>x \<in> S. (inv_into S f) x)"
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```    97 apply (rule group.inv_equality)
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```    98 apply (rule group_BijGroup)
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```    99 apply (simp_all add:BijGroup_def restrict_inv_into_Bij Bij_compose_restrict_eq)
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```   100 done
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```   101
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```   102 lemma (in group) subgroup_auto:
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```   103       "subgroup (auto G) (BijGroup (carrier G))"
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```   104 proof (rule subgroup.intro)
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```   105   show "auto G \<subseteq> carrier (BijGroup (carrier G))"
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```   106     by (force simp add: auto_def BijGroup_def)
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```   107 next
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```   108   fix x y
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```   109   assume "x \<in> auto G" "y \<in> auto G"
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```   110   thus "x \<otimes>\<^bsub>BijGroup (carrier G)\<^esub> y \<in> auto G"
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```   111     by (force simp add: BijGroup_def is_group auto_def Bij_imp_funcset
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```   112                         group.hom_compose compose_Bij)
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```   113 next
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```   114   show "\<one>\<^bsub>BijGroup (carrier G)\<^esub> \<in> auto G" by (simp add:  BijGroup_def id_in_auto)
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```   115 next
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```   116   fix x
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```   117   assume "x \<in> auto G"
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```   118   thus "inv\<^bsub>BijGroup (carrier G)\<^esub> x \<in> auto G"
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```   119     by (simp del: restrict_apply
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```   120         add: inv_BijGroup auto_def restrict_inv_into_Bij restrict_inv_into_hom)
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```   121 qed
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```   122
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```   123 theorem (in group) AutoGroup: "group (AutoGroup G)"
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```   124 by (simp add: AutoGroup_def subgroup.subgroup_is_group subgroup_auto
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```   125               group_BijGroup)
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```   126
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```   127 end
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