Quelle Sym_Groups.thy Sprache: Isabelle

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

theoryHOL-Combinatorics
    Solvable_Groups
    "HOL-Combinatorics.Cycles"
    Solvable_Groups
begin

section \<open>Symmetric Groups\<close>

subsection \<open>Definitions\<close>

abbreviation inv' :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)"
  where "inv' f \ Hilbert_Choice.inv f"

definition 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

sym_group_mult: "mult (sym_group n) = (\)"
  unfolding sym_group_def by simp

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

lemma sym_group_carrier': "p \ carrier (sym_group n) \ permutation p"
  unfolding sym_group_carrier permutation_permutes by auto

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

lemma alt_group_mult: "mult (alt_group n) = (\)"
  unfolding alt_group_def using

lemma java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
    java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53

alt_group_carrier  carrier)
  unfolding alt_group_carrier permutation_permutes   unfolding  permutation_permutesjava.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58

lemma : " (alt_group n) (
  using permutes_inv_o)
  by (auto intro!: groupI

                    comp_assoc)

lemma sign_img_is_group: "group sign_img"
': p\ permutation p"

sym_group_inv_closedjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
assumes
  using assms permutes_invauto!: groupI

lemma alt_group_inv_closed:
  assumes add:  permutes_compose
   evenperm_inv alt_group_carrier assmsauto

lemma sym_group_inv_equality [simp]:
  assumes "p \ carrier (sym_group n)" shows "inv\<^bsub>(sym_group n)\<^esub> p = inv' p"
proof -
  have "inv' p \ p = id"
    using assms permutes_inv_o(2) sym_group_def by auto
  hence "(inv' p) \\<^bsub>(sym_group n)\<^esub> p = one (sym_group n)"
    by (simp add: sym_group_def)
  thus ?thesis  using group.inv_equality[OF sym_group_is_group]
    by (simp)
qed

lemma sign_is_homlemmasym_group_inv_closed
  unfolding sign_img_def sym_group_mult sym_group_carrier_]
  by (auto simp add: sign_compose, meson sign_def)

lemma sign_group_hom  using permutes_inv by auto
   group_hom[OF sign_img_is_group
  by( add )

lemma sign_is_surj:
  assumes "p
proof
  have "swapidseq (Suc 0) (Fun using evenperm_inv[ alt_group_carrier'] permutes_inv alt_group_carrier by auto
    using[OF, of :nat]byjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
Fun(:nat  -:int
    by (simp assms(2)sym_group_def
  moreover have "inv )
     assms[of": ""1.} 2]permutes_id
    unfolding sym_group_carrier by auto
  ultimately    ?thesis .inv_equality sym_group_is_group
    using auto: sign_compose meson sign_def sign_group_homsym_group
  moreoverhave sign (sym_group)\<subseteq> carrier sign_img"
    using sign_is_hom unfolding hom_def ( addgroup_hom_axioms_def , of 2
  thesisby addjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
by
qed

lemma alt_group_is_sign_kernel sym_group_carrier
    ultimately" sign_img\<subseteq> sign ` (carrier (sym_group n))"
  unfolding alt_group_def sym_group_defsign_img_defkernel_def  by auto

lemmaalt_group_is_subgroup "subgroup carrier alt_group n))(sym_groupn)"
  using group_hom.subgroup_kernel[OF sign_group_hom]
  unfolding alt_group_is_sign_kernel by blast

lemma alt_group_is_group: "group (alt_group n)"    usingsign_is_hom unfolding hom_defby auto
  using group.subgroup_imp_group[OF sym_group_is_group alt_group_is_subgroup]
   simp:java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31

lemma
"sym_groupn (carrier (alt_group )\cong>sign_imgjava.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 88
using.[OF sign_is_surj assms]
  unfolding alt_group_is_sign_kernel .

lemma alt_group_inv_equality:
  assumes "p \ carrier (alt_group n)" shows "inv\<^bsub>(alt_group n)\<^esub> p = inv' p"
oof
l alt_group_is_group " alt_groupn)
    using assms permutes_inv_o(2) alt_group_def by auto
  hence "inv ) \\<^bsub>(alt_group n)\<^esub> p = one (alt_group n)"
    by simp: alt_group_def sym_group_def
  thusjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
    by  assumes "n \ 2" shows "(sym_group n) Mod (carrier (alt_group n)) \ sign_img"
qed

lemma sym_group_card_carrier: "card (carrier (sym_group n)) = fact n"
   card_permutations[ "{1..n}" n] unfolding by simp

lemma alt_group_card_carrier
  assumes
proof-
  have "card (rcosets\<^bsub>sym_group n\<^esub> (carrier (alt_group n))) = 2"
    using iso_same_card[OF sign_iso "\
  thus -
       "inv' \ p = id"
    unfolding order_def sym_group_card_carrier by simp
qed

subsection     assms(2) alt_group_def by

text\<open>In order to prove that the Alternating Group can be generated by 3-cycles, we need
      astronger decomposition permutations as sequences than one
      proposed at Permutationsthusthesis group.inv_equality alt_group_is_group]

inductive
  where
    empty sym_group_card_carriercard (sym_group)=fact
  |  using[of "{..}"n]unfolding by simp
| :   \<lbrakk> swapidseq_ext S n p; a \<noteq> b \<rbrakk> \<Longrightarrow>
                 have" (rcosets\<^bsub>sym_group n\<^esub> (carrier (alt_group n))) = 2"

lemma    ?thesis group sym_group_is_group n]
   "swapidseq_extS "shows S"


:
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  using assmsproposedPermutations.. \<close>

java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  \<open>swapidseq n p\<close> if \<open>swapidseq_ext S n p\<close>:   \<lbrakk> swapidseq_ext S n p; a \<noteq> b \<rbrakk> \<Longrightarrow> (insert b S)( n) (transpose b) \<circ> p)"
using that proof induction
  case empty
  then show ?ase
    by (simp add: fun_eq_iff)
next
  case (single S n p
  then ?case by java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
next
  case (comp S n p a b)
  thenhave\<open>swapidseq (Suc n) (transpose a b \<circ> p)\<close>
    by  usingassms by (induction (auto
p add: comp_def
qed

lemma "finiteS"showsswapidseq_ext  "
  assumes "swapidseq_ext S 0 p" shows "p = id"
proof -
  have "\ swapidseq_ext S n p; n = 0 \ \ p = id" for n
    by (induction ruleusingassms by (induct set "finite, fastforce,simp add single)
  thus ?thesis
    usingassms simp
qed

lemma swapidseq_ext_finite_expansion:
  assumes "finite case empty
  using assms
proof (induct set: "finite", simp)
  case (insert b B) show ?case  then show ?ase
    using insert single[OF insert
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma swapidseq_ext_backwards:
  assumes "swapidseq_ext A (Suc n) p"
nejava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
                      A' '\<and> p = (transpose a b) \<circ> p'"
   " swapidseq_ext S n p; n = 0 \ \ p = id" for n
  have "\a b A' p'. a \ b \ A = (insert a (insert b A')) \
                     A' '\<and> p = (transpose a b) \<circ> p'"
    ifswapidseq_ext""   "
    for A n k and p :: "'a \ 'a"
    using
  proof (induction   "finite "andAnp  "swapidseq_ext ( <> B)np"
    case (induct: "", simp
    thus  simp
  next
    case singleusing single insert,ofby( Un_insert_right)
    thus
  next
    case  swapidseq_ext )p
    thus ?case by blast
  qed
  thus ?thesis using assms by simp
qed

lemma swapidseq_ext_backwards':
  assumes "swapidseq_ext A (Suc n) A' n p' p = (transpose a b) \ p'"and p=( b \ p'"
  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
bymetis supswapidseq_ext_finite

lemma:
  assumes  that
  shows a (insert) Suc( \<circ> (transpose a b))"
using
proof
  case 0 hencep="
    using by blast
  thuscomp
     ?case  blast
next  thus usingassms
  case lemma':
' :"a
    where java.lang.StringIndexOutOfBoundsException: Index 142 out of bounds for length 142
       "= c d\java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
swapidseq_ext_backwards
  hence " ( a insert )Sucn p
assms
  hence "swapidseq_ext (induction n arbitrary )
     swapidseq_ext_zero_imp_id
    by   ?
  thus 0 metisid_comp.)
     metis  )
qed

lemma then   'and p' :"a\ 'a"
java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
   "swapidseq_ext(A \ B) (n + m) (p \ q)"
  using assms(1,3)
proof (induction swapidseq_ext_backwardsOF(2)] by blast
  case show ?case
    using swapidseq_ext.single[OF single(3)] single(2,4) by autoby(simp:SucIH.prems2)
next
  case comp show ?case
    using swapidseq_ext.comp[OF comp(3,2)]                 (transpose c d \<circ> p'
    by (metisUn_insert_left add_Suc insert_disjoint(1) o_assoc
qed

lemma swapidseq_ext_of_cycles:
   "cycle cs" shows ( cs cs- 1)(cycle_of_list
  using assms
proofof (induct rule cycle_of_listinduct
  case (lemmaswapidseq_ext_extension
comp[OF1(1), of j 1(2) by( add: o_def)
next
  case "_1" show?
    by (simp, metis eq_id_iffusingassms13java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
next
  case ("2_2" v) show ?case
    using single[OF empty, of v] by (simp, metis eq_id_iff)
qed

lemmacycle_decomp_imp_swapidseq_ext
  assumes
  usingcase comp (4
(induction
  case
     swapidseq_ext byblast
next
  case (comp I p cs)
obtainm  m: " I m p"by
  hence "swapidseq_ext (set cs) usingassms
    using comp.hyps(proofinduct : cycle_of_list)
   ?case swapidseq_ext_extension
         comp 11,of]12 by (imp: )
qed

lemmaswapidseq_ext_of_permutation
  assumesppermutes "finiteS \n S n "
  using cycle_decomp_imp_swapidseq_ext[OF cycle_decomposition[OF assms]] .

lemma split_swapidseq_ext:
  assumes "k \ n" and "swapidseq_ext S n p"
  obtains q r U V where "swapidseq_ext U (n - k) q" andswapidseq_ext  "and p=q r" and "U \ V = S"
proof
  from lemma:
  have\>U   n-)
   ( assms()
  proof  : inc_induct
    casebase ?
      by (metis diff_self_eq_0
  next
    case (step m)
then qrU
      where   obtain  m: " I mp by blastlast
         p:" q\ r" and S: "U \ V = S"
      by blast
obtainbr 'java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
wherea\<noteq> b" and r': "V = (insert a (insert b V'))" "swapidseq_ext V' m r'" "r = (transpose a b) \<circ> r'"
      using
ave"swapidseq_ext ( a (insert b U)) (n - m) (q (transpose a b))"
      using   assumes "p permutesS and"finite S shows"\<exists>n. swapidseq_ext S n p"
    hence "?split m (q using cycle_decomp_imp_swapidseq_ext[OF cycle_decomposition[OF assms]] .
      using r' S unfolding p by fastforce
    thus ?case by blast
  qed
  thus ?thesislemmasplit_swapidseq_ext
      assumesk \<le> n" and "swapidseq_ext S n p"
qed

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

text   "\q r U V. swapidseq_ext U (n - k) q \ swapidseq_ext V k r \ p = q \ r \ U \ V = S"

abbreviation three_cycles    (induct: )
  where
           by diff_self_eq_0. empty

lemma stupid_lemma:
  assumes "length cs = 3" java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 23
using :)
    (metis Suc_lessI less_2_cases not_less_eq nth_Cons_0
numeral_2_eq_2)

lemma three_cycles_incl: "three_cycles n \ carrier (alt_group n)"       blast
proof
  fix       "a \ b" and r': "V = (insert a (insert b V'))" "swapidseq_ext V' m r'" "r = (transpose a b) \ r'"
  then obtaincs cs: "p = cycle_of_list " "cycle cs"" cs = 3" "set cs \ {1..n}"
    by auto
  obtain a b c where cs_def: "cs = [ a, b, c ]"
    using stupid_lemma[OF cs      usingswapidseq_ext_endswapOF \<open>a \<noteq> b\<close>] step(2) by (metis Suc_diff_Suc)
  have "swapidseq (Suc (Suc 0)) ((transpose a b) \ (Fun.swap b c id))"
    usingcomp_Suc[F comp_SucOFidofb cab cs) cs_def by simp
  hence "evenperm p"
    using cs(1) unfolding cs_def by (simp add: evenperm_unique)
<in> carrier (alt_groupn)"
    using permutes_subset[OF cycle_permutes cs(4)]
    unfolding alt_group_carrier cs(1) by simp
qed

lemma alt_group_carrier_as_three_cycles:
  "carrier (alt_group n) = generate (alt_group n) (three_cycles n)"
proof -
  interprett ?thesis
java.lang.StringIndexOutOfBoundsException: Range [9, 4) out of bounds for length 36

  show ?thesis
  proof
    show "generate (alt_group n) (text \We show that symmetric groups (<^term>\sym_group n\) are unsolvable for \<^term>\n \ 5\.\
      using abbreviation three_cycles " \ (nat \ nat) set"
    show" (alt_group n) \ generate (alt_group n) (three_cycles n)"
    proof
      have aux_lemma1: "cycle_of_list [a, b, c] \ generate (alt_group n) (three_cycles n)"
        if "a \ b" "b \ c" "{ a, b, c } \ {1..n}"
         q :: "nat nat" and a b c
      proof(ases
        assume "c using assms by (auto intro!: nth_equalityI)
        hence"cycle_of_list [ a, b, c ] = id"
          by (           nth_Cons_Sucnumeral_2_eq_2 numeral_3_eq_3)
   thus "cycle_of_list [a, b c ] \ generate (alt_group n) (three_cycles n)"
proof
      next
        assume "c \ a"
  havedistinct b,c]
          using \<open>a \<noteq> b\<close> and \<open>b \<noteq> c\<close> and \<open>c \<noteq> a\<close> by auto
        with
        show "cycle_of_list [ a, b, c ] \ generate (alt_group n) (three_cycles n)"
          by (intro incl) fastforce auto
      qed

      have aux_lemma2: "q \ generate (alt_group n) (three_cycles n)"
        if seq: "swapidseq_ext S (Suc usingstupid_lemmaOFcs3)]byauto
        for S :: "nat set" and q :: "nat "swapidseqSucSuc)(transpose \<circ> (Fun.swap b c id))"
      proof-
        obtain a b q' where ab: "a \ b" "a \ S" "b \ S"
q:"swapidseq_extS(Suc 0) ' "  (ranspose)\<>q'"
usingswapidseq_ext_backwards'[ seq] auto
        obtain c d where cd(1) by simp
          and q: "q = (transpose a b) \ (Fun.swap c d id)"
          using swapidseq_ext_backwards'[OF q'(1)]
            swapidseq_ext_zero_imp_id
'java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
          by fastforce

proof
        thus ?thesis
        proof cases
          case eq
          then have "q = cycle_of_list [ a, b, d ]"
            unfolding  simp
          moreover have "{ a,b,d } \ {1..n}"
            using  S  blast
          ultimately show
             aux_lemma1 ab () eq
        next
          caseineq
          hence "q = cycle_of_list [ a, b, c ] \ cycle_of_list [ b, c, d ]"
            unfolding q by (simp add: swap_nilpotent o_assoc)
          moreoverhave{ a,b, c }\<subseteq> {1..n}" and "{ b, c, d } \<subseteq> {1..n}"
            using cd S by blast
          ultimately show ?thesis
usingeng/.thy
            unfolding alt_group_mult by simp
        qed
      qed

fix  " \ carrier (alt_group n)" then have p: "p permutes {1..n}" "evenperm p"
        unfolding alt_group_carrier by auto
                using[of " simp
        using a "c \ a"
e m"
        using            \<open>a \<noteq> b\<close> and \<open>b \<noteq> c\<close> and \<open>c \<noteq> a\<close> by auto \<open>{ a, b, c } \<subseteq> {1..n}\<close>
      thenintro)
y auto
            haveaux_lemma2: " \ generate (alt_group n) (three_cycles n)"
        using        if: swapidseq_ext Suc   " \ {1..n}"
      proof (induct k arbitrary: p)
         0 then "p= id"
          using by
        showjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
          using generate.oneand   "nv Hilbert_Choice.f"
java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
      java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
        case Sucm)
           q: q  ( ab)\<circ> (Fun.swap c d id)"
           auto
        then obtainq r U V
          where q: "swapidseq_ext U (2 * m) q" and r: "swapidseq_ext where "sign_img = \<lparr> carrier = { -1, 1 }, mult = (*), one = 1 \<rparr>"subsection \<open>Basic Properties\<close>           unfolding q)
             :" \ r" and UV: "U \ V = {1..n}"
          lemma:" (sym_groupn)= idjava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
        "swapidseq_ext{)q
lemmaalt_group_carrier:p\<in carrier alt_group\<longleftrightarrow> p permutes {1..n} \<and> evenperm p"
         case
          using        proof cases
qed
    folding using by
  edmoreover"a , d}\<
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

theorem:
          ltimately thesis
using[ ab1]cd  by simp auto!: groupI
           java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
using   simp
next
have:p
using alt_group_carrier  auto
-
    obtain-
      using \<open>p \<in> three_cycles n\<close> by auto
then ab cwhere ?thesis
      usinghence "(inv' p) \<otimes
    have "card (set cs) = 3"
      usingcs-)
      by             alt_group_mult by simp

        by  add)
      using   by (auto simp a, sign_def
where:d
using4

    " (d #cs"  using[OF]by
      sing2-)byauto -
\<noteq> {1..n}"
      usingassms sym[OF    singcomp_Suc id,  "1 : " " java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
bymetis ( n) ((three_cycles)
    thenusing     havecarrier
      usingcs ( insert_subset.15  subset_antisym

    q where(swapjava.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
    ence " q"
      by (simp add: bij_comp)
have   " q c = b usingswapidseq_ext_zero_imp_id bysimp
      usingd(1) e1qed
      " alt_group ) kernel (ym_group )sign_imgsignjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
using1)e1 (2)  q_def by auto
    ultimately have "q \ p \ (inv' q) = cycle_of_list [ a, c, b ]"
Fcs, q]
      unfolding sym[OF cs(1)] unfolding    group_hom[OF]
    also have " .java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      cs2 unfolding
       simpcomp_swaptranspose_comp_triple)
    finally
    erhavebij
      unfolding (1)cs_defsimp:        then q  U java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
imately invp \<circ> p = id"
      by( add bijection bijection. comp_assocjava.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72

    have"swapidseq (Suc (Suc 0)) q"
obcd](1)(    byauto
    hence "evenperm q"
using UV swapidseq_ext_finite_expansion[OF[OFr ]by
using
      unfolding by ( add permutes_compose)
henceq permutes{1n}java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
      using -
java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
      unfolding by simp
    moreover have "p \ carrier (alt_group n)"
      using
    ultimately have "p[OF sym_group_is_group , of n]
      using     unfolding  by simp
java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
    thustext \<open>In order to prove that the Alternating Group can be generated by 3-cycles, we need
      unfoldingderived_def incl
  qed

  interpret A: group "alt_group n"
     alt_group_is_group

   alt_group)subseteq ( n) (arrier n)java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 106
    usingusing by () (auto
  thus" (alt_group n)
    using alt_group_carrier_as_three_cycles by simp
qed

corollary alt_group_is_unsolvable
  assumes
proofjava.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
  assume "\ \ solvable (alt_group n)"
  then obtainm where (thencase
    using group
moreover "derived(lt_group ) ^^java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    using derived_alt_group_const[  hen ?casesing.derived_in_carrier] java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
  ultimately ""<in> three_cycles n" for p
    bysimp
  havege_2n < 2java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
    using assms by simp
  moreover "2 fact njava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
    using stupid_lemma cs byjava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
    by (metis       cs2-)
  ultimately have "n = 2"
      java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 42
  thus False
    using assms   have \<open>swapidseq (Suc n) (transpose a b \<circ> p)\<close>    by( add comp_Sucthen show? bysimp: lemma:

:
  assumes thesis
proof- assmsjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
  hence (  " and length (#) 4 and " {1..n} = n"
     groupcanonical_inj_is_hom sym_group_is_group] alt_group_def
   ?thesis
    usingId. alt_group_is_unsolvable assmsauto
qed

end

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