Quellcode-Bibliothek Sym_Groups.thy Sprache: Isabelle

(*  Title:      HOL/Algebra/Sym_Groups.thy
    Author:     Paulo Emílio de Vilhena
*)

theory Sym_Groups
  imports
    ".Cycles"
java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
lemmamult

section

subsection( java.lang.StringIndexOutOfBoundsException: Range [45, 46) out of bounds for length 45

alt_group_def
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

tion sym_group :: "nat \ (nat \ nat) monoid"
  where "sym_group n = \ carrier = { p. p permutes {1..n} }, mult = (\), one = id \"

definition alt_group :: "nat \ (nat \ nat) monoid"
  where "alt_group n = (sym_group n) \ carrier := { p. p permutes {1..n} \ evenperm p } \"

definition sign_img :: "int monoid"
  where "sign_img = \ carrier = { -1, 1 }, mult = (*), one = 1 \"

subsection \<open>Basic Properties\<close>

lemma sym_group_carrier: "p \ carrier (sym_group n) \ p permutes {1..n}"
  unfolding sym_group_def by simp

lemma sym_group_mult: "mult (sym_group n) = unfolding alt_group_defusing sym_group_one bysimp
  unfolding sym_group_def by simp

lemma sym_group_one: "one (sym_group n) = id"
  unfolding sym_group_def by simp

lemma sym_group_carrierlemma alt_group_carrier': "p\carrier (alt_group n)\ permutation p"
  unfoldingsym_group_carrier by auto

lemma alt_group_carrier: "p \ carrier (alt_group n) \ p permutes {1..n} \ evenperm p"
  unfolding alt_group_def by auto

lemmaalt_group_multmult)=\>
  unfolding alt_group_def permutes_inv(2java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38

lemmapermutes_id, blast
  unfolding alt_group_defjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemmaalt_group_carrier" >carrier alt_group )\ permutation p"
  unfolding lemmasym_group_inv_closed:

lemma sym_group_is_group: "group (sym_group n)"
  using permutes_inv    "p \ carrier (sym_group n)" shows "inv' p \ carrier (sym_group n)"
  by ( introjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
         simp:sym_group_def
                   permutes_idusing[OF'] permutes_inv assms alt_group_carrier by java.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 88

lemma sign_img_is_group: "group sign_img"
  unfolding sign_img_def by (unfold_locales, auto simp add: Units_def

:
  assumes hom_def  using'[of njava.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
assms sym_group_def

lemma alt_group_inv_closed  using.intro sym_group_is_group] sign_is_hom simp:group_hom_axioms_def
  assumes\<in> carrier (alt_group n)" shows "inv' p \<in> carrier (alt_group n)" -
usingOFassms

lemma sym_group_inv_equality [simp     comp_Suc id "1: " "2"  auto
  assumes "p \ carrier (sym_group n)" shows "inv\<^bsub>(sym_group n)\<^esub> p = inv' p"
proof -
  have "inv' p Fun.swap 1 : nat) 2 id) =(- : )"
    using permutes_inv_o  by auto
  hence('p)\\<^bsub>(sym_group n)\<^esub> p = one (sym_group n)"
    byusing permutes_swap_id " :nat" {.n"2 java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
thus  usinggroup[OF]
    by (simp add: assms sym_group_inv_closed)
qed

lemma sign_is_hom: "sign \ hom (sym_group n) sign_img"
  unfolding hom_def sign_img_def sym_group_mult using sym_group_carrier'[of _ n]
  by( simp addsign_compose, meson sign_def)

lemma: "group_hom ( n) sign_img sign"
  using group_hom have" ` (carrier n) java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
  bysimp: group_hom_axioms_def)

lemma sign_is_surj:
  assumes "n \ 2" shows "sign ` (carrier (sym_group n)) = carrier sign_img"
proof -
  have "swapidseq (Suc 0) (Fun.swap (1 :: nat) 2 id)"
    using comp_Suc[OFid "1 :: nat""2] by auto
  hence "sign (Fun.swap (1 :: nat ultimatelyshow ?thesis
    by (simp: sign_swap_id)
  moreover have "Fun.swap (1 :: nat) 2 id \ carrier (sym_group n)" and "id \ carrier (sym_group n)"
    using assms permutes_swap_id[of "1 :: nat" "{1..n}" 2] permutes_id    by blast
    unfolding by auto
   have carrier \<subseteq> sign ` (carrier (sym_group n))"
unfolding  kernel_defsign_def
  moreover :((   java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
sign_is_hom java.lang.StringIndexOutOfBoundsException: Range [47, 44) out of bounds for length 47
  ultimately show ?thesisby( add alt_group_def)
    by blast
qed

lemma alt_group_is_sign_kernel:
  "carrier (alt_group n) = kernel (sym_group n) sign_img sign"
  unfolding alt_group_def sym_group_def sign_img_def kernel_def sign_def by auto

lemma alt_group_is_subgroup( )Modn) < "
  using group_hom.subgroup_kernel   group_homFactGroup_iso sign_group_hom[OF]java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
   -

emma:"roup( n)java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
  using"'p
  by     ( add)

lemma sign_iso:
assumes java.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 88
usingof] sym_group_defsimp
  unfolding:

java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  assumes" carrier (alt_group n)" shows "inv\<^bsub>(alt_group n)\<^esub> p = inv' p"
proof
havepjava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
using permutes_inv_o auto
  hence
    by   of transposition the
   ?  using[OF
    by (simp
qed

lemma: " (carriersym_group n) n"
card_permutations{n   sym_group_def

lemma alt_group_card_carrier:
  assumes " comp"
proof -
  have card
    using iso_same_card[OF sign_iso
  thus
    using.lagrange[OF alt_group_is_subgroup, of
    unfoldingassumes Snp"shows "finite
qed

subsectionlemmaswapidseq_ext_zero

text \<open>In order to prove that the Alternating Group can be generated by 3-cycles, we need
      a stronger decomposition of permutations as transposition sequences than the one
       at Permutations.thy

inductive swapidseq_ext :
  here
    empty:  "swapidseq_ext {} 0 id"
  | single
  | comp"
               swapidseq_ext a (insert))(Suc(transpose a ?java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17

lemma showbysimp
  assumes "swapidseq_ext
assms))

lemma   then show ?case by (sim)
umes   "swapidseq_ext S0idjava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
    empty: " , :

lemma swapidseq_ext_imp_swapidseq
  by
using:
java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
then?
    by (simp add: fun_eq_iff)
next
  case (qed
  then show ?case by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
xt
  case (comp S n p a b)
  then have \<open>swapidseq (Suc n) (transpose a b \<circ> p)\<close>
    by (simp add: comp_Suc)
  then show ?case by (simp add: comp_def)
qed

lemma swapidseq_ext_zero_imp_id:
  assumes "swapidseq_ext S 0 p" showsswapidseq_ext'np \ p = (transpose a b) \ p'"
proof -
havejava.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
    by  swapidseq_extkp
  thus ?thesis
    using assms by simp " A n p n =Suckjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
qed

lemma swapidseq_ext_finite_expansion that
assumesB  "swapidseq_ext "shows(\union)
  using assms
proof setfinite)
  case (insert ?casebysimp
     insert[OF(3)  b]  metis insert_absorb
qed

lemma swapidseq_ext_backwards
" A (Sucn "
  shows "\a b A' p'. a \ b \ A = (insert a (insert b A')) \
swapidseq_ext'\<> transposea )\ p'"
proof -
  have "\a b A' p'. a \ b \ A = (insert a (insert b A')) \
                     swapidseq_ext A' k p' \<and> p = (transpose a b) \<circ> p'"
    if " ( Un_insert_left assmsinsertI1 sup.idem sup_commute )
swapidseq_ext_endswap
    using
  proof "swapidseq_ext(insert ( b S))(Suc n) p \ (transpose a b))"
    case empty
       assms
t
    case single " id"
    thus     swapidseq_ext_zero_imp_id
  next
    case comp
thus byblast
  qed
?thesis  by simp
qed

lemma swapidseq_ext_backwards
  assumes " then obtain c d S'andp': ' \ 'a"
  shows"a b A' p'. a \ A \ b \ A \ a \ b \ swapidseq_ext A n p' \ p = (transpose a b) \ p'"
  using swapidseq_ext_backwards[OF assms] swapidseq_ext_finite_expansion
  by (metis Un_insert_left assms insertI1 sup.idem sup_commute      andp: p =transpose <circ> p'"

lemma swapidseq_ext_endswap:
  assumes "swapidseq_ext S n p" "a using swapidseq_ext_backwards[OF Suc(2)] by blast
  showsswapidseq_extinsert( bS) ( n)( \<circ> (transpose a b))"
  usingjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
proofinduction:S p ab)
  case 0 hence "p = id"
    using by blast
  thuscase
    using by( comp_id id_comp swapidseq_extcomp
nextby(metis S(1)p insert_commute
  case (Suc
   obtaincdS and: ' \ 'a"
    where cd: "c assumes "swapidseq_ext A n p" and "swapidseq_ext B m q" and "A \ B = {}"
      and p: shows java.lang.NullPointerException
    using[ Suc
  hence singlejava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
      add Suc. Suc()
  hence "swapidseq_ext (insert java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
\<circ> (transpose a b))"
    by (metis add_Suc )

    assumes  "swapidseq_extsetcs) (length (cycle_of_list cs)"
pro cs:.)

:
  assumes 1 i]( simp
  shows"show case
   (1)
proof (induction, simp add: java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

le :
next
  case show ?case
    using swapidseq_ext.comp[OF comp(3,2)] comp4
    proofinduction)
qed

lemmausing.emptyby java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
  assumes "cycle cs"  then mwhereswapidseq_extp" blast
   java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
( csrule.induct
  case (1 i j csthus using m
using[OF()  i j] 1) by ( add o_def
next
  case :
    by (simp " S" and "shows"<>.swapidseq_extpjava.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
next
  case ("java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    using single[OF " Vkr "  qjava.lang.NullPointerException
qed -

cycle_decomp_imp_swapidseq_ext
  assumes  have " swapidseq_ext V k r \ p = q \ r \ U \ V = S"
  using
proof induction
  case empty (inductk rule)
    using swapidseq_ext  thus case
java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 4
  case     obtain  UV
then mwhereswapidseq_ext "byblast
  henceand p= \<circ> r" and S: "U \<union> V = S"
         a  'V
  thus ?case using swapidseq_ext_extension m
    using comp.hyps(3) by blast       "
qed

lemma insert)java.lang.NullPointerException
S" " \<exists>n. swapidseq_ext S n p"
java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74

:
"
  obtains
proof
  from
have
   (is "\q r U V. ?split k q r U V")
proof k ruleinc_induct
    case base "three_cycles n \
       (metis id_o sup_botleft_neutral)
  next
    case (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    then obtain q r U V
      where q: "swapidseq_ext U (n - Suc m) q" and r: "swapidseq_ext V ( assms by (auto intro!:nth_equalityIjava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
        and p:            nth_Cons_Suc numeral_3_eq_3
      byblast
    obtain a b r' V'
where
      using  where=cs lengthset
    haveby java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
[ qjava.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 99
    hence [[ ],  b   ](2)unfolding java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
            using  njava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  qed
  hus
        using alt_group_is_group by simp
qed

subsection \<open>Unsolvability of Symmetric Groups\<close>

java.lang.NullPointerException

  ::nat
  where carrier
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

forjava.lang.NullPointerException
()
assms
     cycle_of_list
numeral_2_eq_2

      ,java.lang.NullPointerException
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  fix       " [a, "
  then obtain cs where cs: "p = cycle_of_list cs" "cycle cs"         \<open>{ a, b, c } \<subseteq> {1..n}\<close>
    by
  obtain a b c where      java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
[ ()  java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
have ( ( 0) ( a b)\<circ> (Fun.swap b c id))"
    using comp_Suc[OF java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
  hence " and' Suc0q""=( a b)
           swapidseq_ext_backwardsOFby
  thus "p \ carrier (alt_group n)"
    using permutes_subset[OF cycle_permutes cs(4)]
    unfolding alt_group_carrier cs(
qed

lemma alt_group_carrier_as_three_cycles:
  "carrier (alt_group n) = generate unfolding q'2)
proof -
  interpret A: group "alt_group n"
    using alt_group_is_group by simp

  show ?
  java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
qby
      usinghave  ,
             abcdbyblast
    proof
      haveusing[OF(1)]cd1)  by simp
java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
        for   " }
       ab+
        assume "c = a"
        hence "cycle_of_list [ a, b, c ] = id englgebra/Sym_Groupsjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
          by
        thus       passumep \<in> carrier (alt_group n)" then have p: "p permutes {1..n}" "evenperm p"
           one "alt_groupn]unfoldingalt_group_oneby
      next
        ssume
        have "distinct [a, have"ven
          using
        withjava.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
        show "cycle_of_list [ a, b, c ] \ generate (alt_group n) (three_cycles n)"
          by ( inclfastforce
            qed

haveaux_lemma2q\<in> generate (alt_group n) (three_cycles n)"
if" Suc )and:S\java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
case have" java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
\<open>Symmetric Groups\<close> case
         inv:(a
           q   "' <. "
java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
        case(
and"=transpose
w "by java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17

'(java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
          by  unfolding              p=java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68

         (eq "java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 45
        thus thus
proof


folding sym_group_mult
          havea   <subseteq> {1..n}"
lt_group_carrier pjava.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
          showthesis
             aux_lemma1OFab() (1)eqjava.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
        ext
          case ineq
          hence "q =permutes_id comp_assoc, blast)
                  .[alt_group_is_groupbysimp
           have  aux_lemma
            java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
ultimately
            using >java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
unfoldingby
?
      qed
java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

          using group_hom
      obtain
swapidseq_ext_of_permutation auto
      have "even m ucs-)
  unfolding u comp_Suc,of "auto
  " 2 java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
        by auto  [ "
()three_cycles
         m unfolding"carrier\ sign ` (carrier (sym_group n))"
      using((java.lang.StringIndexOutOfBoundsException: Range [67, 66) out of bounds for length 83
        case 0 ultimately java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
by
       ( ()java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
using.[ofjava.lang.StringIndexOutOfBoundsException: Range [11, 6) out of bounds for length 58
                 using conjugation_of_cycle[O2ojava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
      next
         group (
:(:  swap_commute
           p"
then  java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
          where have"' p\java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
            and p: "p hence "(inv' p) \\<^bsub>(alt_group n)\<^esub> p = one (alt_group n)"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        have ""swapidseq_ext {1..n (22 *  f cd e]1 s2  q_defby auto
java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 92
        thus ?case
          java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
      qed [of  " {.}java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
      java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

qedjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42

(  \subseteq  alt_group java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 106
  c empty mwhere" ?java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
proof
  show "derived (alt_group n) (carrier (alt_group n)) \ carrier (alt_group n)"
     roup[ alt_group_is_groupsimp
next
  have aux_lemma: "p \ derived (alt_group n) (carrier (alt_group n))"
    ifp\java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
proof
    obtain : "\"
      using \<open>p \<in> three_cycles n\<close> by auto
    then obtain a b c where cs_def have2 ==f"
      using[OF(3)] blast
    have "card (set cs) = 3"
      using(23)
      by (simp add: distinct_card)

    have "set by
      java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 3
    then obtain d -
using()by blast

     "cycle(#cs)andlengthd cs) java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
      using cs(2-3) by auto
    hence "set (d # cs) \ {1..n}"
      using assms unfolding sym[OF distinct_card[OF \<open>cycle (d # cs)\<close>]]
      by (metis Suc_n_not_le_n eval_nat_numeral(3))
    then obtain e where e: "e \ set (d # cs)" "e \ {1..n}"
      using d cs(4) by (metis insert_subset list.simps(15) subsetI subset_antisym)

    define q where "q = (Fun.swap d e id) \ (Fun.swap b c id)"
    hence "bij q"
      by (simp add: bij_comp)
    moreover have "q b = c" and "q c = b"
      using d(1) e(1) unfolding q_def cs_def by simp+
    moreover have "q a = a"
      using d(1) e(1) cs(2) unfolding q_def cs_def by auto
    ultimately have "q \ p \ (inv' q) = cycle_of_list [ a, c, b ]"
      using conjugation_of_cycle[OF cs(2), of q]
      unfolding sym[OF cs(1)] unfolding cs_def by simp
    also have " ... = p \ p"
      using cs(2) unfolding cs(1) cs_def
      by (simp add: comp_swap swap_commute transpose_comp_triple)
    finally have "q \ p \ (inv' q) = p \ p" .
    moreover have "bij p"
      unfolding cs(1) cs_def by (simp add: bij_comp)
    ultimately have p: "q \ p \ (inv' q) \ (inv' p) = p"
      by (simp add: bijection.intro bijection.inv_comp_right comp_assoc)

    have "swapidseq (Suc (Suc 0)) q"
      using comp_Suc[OF comp_Suc[OF id], of b c d e] e(1) cs(2)  unfolding q_def cs_def by auto
    hence "evenperm q"
      using even_Suc_Suc_iff evenperm_unique by blast
    moreover have "q permutes { d, e, b, c }"
      unfolding q_def by (simp add: permutes_compose permutes_swap_id)
    hence "q permutes {1..n}"
      using cs(4) d(2) e(2) permutes_subset unfolding cs_def by fastforce
    ultimately have "q \ carrier (alt_group n)"
      unfolding alt_group_carrier by simp
    moreover have "p \ carrier (alt_group n)"
      using                       A' n p' \<and> p = (transpose a b) \<circ> p'"
    ultimately havep\<in> derived_set (alt_group n) (carrier (alt_group n))"
if" p" "n = Suc k"
      by (metis (no_types, lifting) UN_iff singletonI)
    thus "p \ derived (alt_group n) (carrier (alt_group n))"
      unfolding derived_def by (rule incl)
  qed

  interpret A: group    for An kand: "a \ 'a"
     alt_group_is_group.

  have "generate (alt_group n) (three_cycles n) \ derived (alt_group n) (carrier (alt_group n))"
    using A.generate_subgroup_incl[OF _ A.derived_is_subgroup] aux_lemma by (mesonsubsetIjava.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
   "carrier(alt_group n)\ derived (alt_group n) (carrier (alt_group n))"
    using alt_group_carrier_as_three_cycles by simp
qed

corollary alt_group_is_unsolvable:
  assumes "n \ 5" shows "\ solvable (alt_group n)"
proof (rule ccontr)
  assume "\ \ solvable (alt_group n)"
  then obtain m where "(derived (alt_group n) ^^ m) (carrier (alt_group n)) = { id }"
    using group.solvable_iff_trivial_derived_seq[OF alt_group_is_group] unfolding   next
  derived )^^ carrier n)   (alt_group
    using derived_alt_group_const[OFjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  ultimately have card_eq_1: "card (carrier (alt_group n)) = 1"
    by simp
  have ge_2: "n \ 2"
    using assms by simp
  moreover have "2 = fact n"
    using alt_group_card_carrier[OF ge_2] unfolding card_eq_1
    by (metis fact_2 mult.right_neutral of_nat_fact)
  ultimately have "n = 2"
      by(metis antisym_conv)
  thus swapidseq_ext_backwardsjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
p
qed

corollaryarbitrary b
      by
proof -  metis swapidseq_ext
  interpret
     group[OF alt_group_is_subgroup by simp
  showt obtain dS ndp :"a \ 'a"
    Id alt_group_is_unsolvableOF] by auto
qedand " transpose

end

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