Theory Bij

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

theory Bij
imports Group

section ‹Bijections of a Set, Permutation and Automorphism Groups›

  Bij :: "'a set  ('a  'a) set"
    ― ‹Only extensional functions, since otherwise we get too many.›
   where "Bij S = extensional S  {f. bij_betw f S S}"

  BijGroup :: "'a set  ('a  'a) monoid"
  where "BijGroup S =
    carrier = Bij S,
     mult = λg  Bij S. λf  Bij S. compose S g f,
     one = λx  S. x"

declare Id_compose [simp] compose_Id [simp]

lemma Bij_imp_extensional: "f  Bij S  f  extensional S"
  by (simp add: Bij_def)

lemma Bij_imp_funcset: "f  Bij S  f  S  S"
  by (auto simp add: Bij_def bij_betw_imp_funcset)

subsection ‹Bijections Form a Group›

lemma restrict_inv_into_Bij: "f  Bij S  (λx  S. (inv_into S f) x)  Bij S"
  by (simp add: Bij_def bij_betw_inv_into)

lemma id_Bij: "(λxS. x)  Bij S "
  by (auto simp add: Bij_def bij_betw_def inj_on_def)

lemma compose_Bij: "x  Bij S; y  Bij S  compose S x y  Bij S"
  by (auto simp add: Bij_def bij_betw_compose) 

lemma Bij_compose_restrict_eq:
     "f  Bij S  compose S (restrict (inv_into S f) S) f = (λxS. x)"
  by (simp add: Bij_def compose_inv_into_id)

theorem group_BijGroup: "group (BijGroup S)"
  apply (simp add: BijGroup_def)
  apply (rule groupI)
      apply (auto simp: compose_Bij id_Bij Bij_imp_funcset Bij_imp_extensional compose_assoc [symmetric])
  apply (blast intro: Bij_compose_restrict_eq restrict_inv_into_Bij)

subsection‹Automorphisms Form a Group›

lemma Bij_inv_into_mem: " f  Bij S;  x  S  inv_into S f x  S"
by (simp add: Bij_def bij_betw_def inv_into_into)

lemma Bij_inv_into_lemma:
  assumes eq: "x y. x  S; y  S  h(g x y) = g (h x) (h y)"
      and hg: "h  Bij S" "g  S  S  S" and "x  S" "y  S"
  shows "inv_into S h (g x y) = g (inv_into S h x) (inv_into S h y)"
proof -
  have "h ` S = S"
    by (metis (no_types) Bij_def Int_iff assms(2) bij_betw_def mem_Collect_eq)
  with x  S y  S have "x'S. y'S. x = h x'  y = h y'"
    by auto
  then show ?thesis
    using assms
    by (auto simp add: Bij_def bij_betw_def eq [symmetric] inv_f_f funcset_mem [THEN funcset_mem])

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

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

lemma (in group) id_in_auto: "(λx  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)
        restrict (inv_into (carrier G) h) (carrier G)  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 = (λx  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)

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)
  fix x y
  assume "x  auto G" "y  auto G" 
  thus "x BijGroup (carrier G)y  auto G"
    by (force simp add: BijGroup_def is_group auto_def Bij_imp_funcset 
                        group.hom_compose compose_Bij)
  show "𝟭BijGroup (carrier G) auto G" by (simp add:  BijGroup_def id_in_auto)
  fix x 
  assume "x  auto G" 
  thus "invBijGroup (carrier G)x  auto G"
    by (simp del: restrict_apply
        add: inv_BijGroup auto_def restrict_inv_into_Bij restrict_inv_into_hom)

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