Quellcode-Bibliothek Bij.thy Sprache: Isabelle

(*  Title:      HOL/Algebra/Bij.thy
    Author:     Florian Kammueller, with new proofs by L C Paulson
*)

theory Bij
imports
begin

section \<open>Bijections of a Set, Permutation and Automorphism Groups\<close>

definition
  Bij :: "'a set \ ('a \ 'a) set"
    \<comment> \<open>Only extensional functions, since otherwise we get too many.\<close>
   where "Bij S = extensional S \ {f. bij_betw f S S}"

definition
  BijGroup :: "'a set \ ('a \ 'a) monoid"
  where "BijGroup S =
    \<lparr>carrier = Bij S,
     mult = \<lambda>g \<in> Bij S. \<lambda>f \<in> Bij S. compose S g f,
     one = \<lambda>x \<in> S. x\<rparr>"

declare Id_compose [simp] compose_Id [simp]

lemma Bij_imp_extensional: "f \ Bij S \ f \ extensional S"
section

lemma: "f\<> Bij S \ f \ S \ S"
  by (auto simp add: Bij_def bij_betw_imp_funcset)

subsection

lemma restrict_inv_into_Bij
  by (simp add: Bij_def bij_betw_inv_into "BijS=extensional S \ {f. bij_betw f S S}"

lemma id_Bij: "(\x\S. x) \ Bij S "
by(auto simp:Bij_def inj_on_def

lemmamult\<lambda>g \<in> Bij S. \<lambda>f \<in> Bij S. compose S g f,
  by     one = \<lambda>x \<in> S. x\<rparr>"

lemma Bij_compose_restrict_eq:

by( add: Bij_def compose_inv_into_id

theoremgroup_BijGroup: "group(BijGroup S)"
  apply (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  apply java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
      apply (auto simp: compose_Bij id_Bij Bij_imp_funcset compose_assoc[])
  apply( intro Bij_compose_restrict_eq restrict_inv_into_Bij
    by ( simpadd BijGroup_def)

subsection\<open>Automorphisms Form a Group\<close>

assumex\<in> auto G" "y \<in> auto G"
(imp add bij_betw_definv_into_into

lemma Bij_inv_into_lemmaby( simpjava.lang.StringIndexOutOfBoundsException: Range [22, 20) out of bounds for length 71
  assumes
      and x
x\<in> auto G"
proof    "\<^bsub>BijGroup(carrier G)<^esub> x \ auto G"
  have "h ` S (simp del: restrict_apply
by ( (no_types Bij_def assms)bij_betw_def)
  with \<open>x \<in> S\<close> \<open>y \<in> S\<close> have "\<exists>x'\<in>S. \<exists>y'\<in>S. x = h x' \<and> y = h y'"
    by auto
  then show ( add AutoGroup_def.subgroup_is_group
    using assms
    by (auto simp add:              )
qed

definition
  auto :: "('a, 'b) monoid_scheme \ ('a \ 'a) set"
  where "auto G = hom G G \ Bij (carrier G)"

definition
  AutoGroup :: "('a, 'c) monoid_scheme \ ('a \ 'a) monoid"
  where "AutoGroup G = BijGroup (carrier G) \carrier := auto G\"

lemma (in group) id_in_auto: "(\x \ carrier G. x) \ auto G"
  by (simp add: auto_def hom_def restrictI group.axioms id_Bij)

lemma (in group) mult_funcset: "mult G \ carrier G \ carrier G \ carrier G"
  by (simp add:  Pi_I group.axioms)

lemma (in group) restrict_inv_into_hom:
      "\h \ hom G G; h \ Bij (carrier G)\
       \<Longrightarrow> restrict (inv_into (carrier G) h) (carrier G) \<in> hom G G"
  by (simp add: hom_def Bij_inv_into_mem restrictI mult_funcset
                group.axioms Bij_inv_into_lemma)

lemma inv_BijGroup:
     "f \ Bij S \ m_inv (BijGroup S) f = (\x \ S. (inv_into S f) x)"
apply (rule group.inv_equality [OF group_BijGroup])
apply (simp_all add:BijGroup_def restrict_inv_into_Bij Bij_compose_restrict_eq)
done

lemma (in group) subgroup_auto:
      "subgroup (auto G) (BijGroup (carrier G))"
proof (rule subgroup.intro)
  show "auto G \ carrier (BijGroup (carrier G))"
    by (force simp add: auto_def BijGroup_def)
next
  fix x y
  assume "x \ auto G" "y \ auto G"
  thus "x \\<^bsub>BijGroup (carrier G)\<^esub> y \ auto G"
    by (force simp add: BijGroup_def is_group auto_def Bij_imp_funcset
                        group.hom_compose compose_Bij)
next
  show "\\<^bsub>BijGroup (carrier G)\<^esub> \ auto G" by (simp add: BijGroup_def id_in_auto)
next
  fix x
  assume "x \ auto G"
  thus "inv\<^bsub>BijGroup (carrier G)\<^esub> x \ auto G"
    by (simp del: restrict_apply
        add: inv_BijGroup auto_def restrict_inv_into_Bij restrict_inv_into_hom)
qed

theorem (in group) AutoGroup: "group (AutoGroup G)"
by (simp add: AutoGroup_def subgroup.subgroup_is_group subgroup_auto
              group_BijGroup)

end

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