author  wenzelm 
Thu, 09 Oct 1997 15:03:06 +0200  
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parent 3370  5c5fdce3a4e4 
child 3842  b55686a7b22c 
permissions  rwrr 
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(* Title: HOL/Set.thy 
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ID: $Id$ 

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Author: Tobias Nipkow 

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Copyright 1993 University of Cambridge 

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*) 

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Set = Ord + 

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(** Core syntax **) 

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types 
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'a set 

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arities 

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set :: (term) term 

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instance 

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set :: (term) {ord, minus, power} 
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syntax 
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"op :" :: ['a, 'a set] => bool ("op :") 

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consts 
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"{}" :: 'a set ("{}") 
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insert :: ['a, 'a set] => 'a set 
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Collect :: ('a => bool) => 'a set (*comprehension*) 
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Compl :: ('a set) => 'a set (*complement*) 
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Int :: ['a set, 'a set] => 'a set (infixl 70) 
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Un :: ['a set, 'a set] => 'a set (infixl 65) 
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UNION, INTER :: ['a set, 'a => 'b set] => 'b set (*general*) 
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UNION1 :: ['a => 'b set] => 'b set (binder "UN " 10) 
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INTER1 :: ['a => 'b set] => 'b set (binder "INT " 10) 
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Union, Inter :: (('a set) set) => 'a set (*of a set*) 
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Pow :: 'a set => 'a set set (*powerset*) 
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range :: ('a => 'b) => 'b set (*of function*) 
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Ball, Bex :: ['a set, 'a => bool] => bool (*bounded quantifiers*) 
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"``" :: ['a => 'b, 'a set] => ('b set) (infixr 90) 
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(*membership*) 
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"op :" :: ['a, 'a set] => bool ("(_/ : _)" [50, 51] 50) 

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(** Additional concrete syntax **) 

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syntax 
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UNIV :: 'a set 
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(* Infix syntax for nonmembership *) 

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"op ~:" :: ['a, 'a set] => bool ("op ~:") 
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"op ~:" :: ['a, 'a set] => bool ("(_/ ~: _)" [50, 51] 50) 
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"@Finset" :: args => 'a set ("{(_)}") 
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"@Coll" :: [pttrn, bool] => 'a set ("(1{_./ _})") 

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"@SetCompr" :: ['a, idts, bool] => 'a set ("(1{_ /_./ _})") 

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(* Big Intersection / Union *) 

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"@INTER" :: [pttrn, 'a set, 'b set] => 'b set ("(3INT _:_./ _)" 10) 
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"@UNION" :: [pttrn, 'a set, 'b set] => 'b set ("(3UN _:_./ _)" 10) 
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(* Bounded Quantifiers *) 

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"@Ball" :: [pttrn, 'a set, bool] => bool ("(3! _:_./ _)" [0, 0, 10] 10) 
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"@Bex" :: [pttrn, 'a set, bool] => bool ("(3? _:_./ _)" [0, 0, 10] 10) 

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"*Ball" :: [pttrn, 'a set, bool] => bool ("(3ALL _:_./ _)" [0, 0, 10] 10) 

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"*Bex" :: [pttrn, 'a set, bool] => bool ("(3EX _:_./ _)" [0, 0, 10] 10) 

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translations 

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"UNIV" == "Compl {}" 
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"range f" == "f``UNIV" 
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"x ~: y" == "~ (x : y)" 
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"{x, xs}" == "insert x {xs}" 

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"{x}" == "insert x {}" 

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"{x. P}" == "Collect (%x. P)" 

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"INT x:A. B" == "INTER A (%x. B)" 

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"UN x:A. B" == "UNION A (%x. B)" 

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"! x:A. P" == "Ball A (%x. P)" 

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"? x:A. P" == "Bex A (%x. P)" 

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"ALL x:A. P" => "Ball A (%x. P)" 

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"EX x:A. P" => "Bex A (%x. P)" 

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syntax ("" output) 
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"_setle" :: ['a set, 'a set] => bool ("op <=") 
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"_setle" :: ['a set, 'a set] => bool ("(_/ <= _)" [50, 51] 50) 
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"_setless" :: ['a set, 'a set] => bool ("op <") 
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"_setless" :: ['a set, 'a set] => bool ("(_/ < _)" [50, 51] 50) 
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syntax (symbols) 
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"_setle" :: ['a set, 'a set] => bool ("op \\<subseteq>") 
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"_setle" :: ['a set, 'a set] => bool ("(_/ \\<subseteq> _)" [50, 51] 50) 
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"_setless" :: ['a set, 'a set] => bool ("op \\<subset>") 
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"_setless" :: ['a set, 'a set] => bool ("(_/ \\<subset> _)" [50, 51] 50) 
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"op Int" :: ['a set, 'a set] => 'a set (infixl "\\<inter>" 70) 
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"op Un" :: ['a set, 'a set] => 'a set (infixl "\\<union>" 65) 

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"op :" :: ['a, 'a set] => bool ("op \\<in>") 
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"op :" :: ['a, 'a set] => bool ("(_/ \\<in> _)" [50, 51] 50) 
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"op ~:" :: ['a, 'a set] => bool ("op \\<notin>") 
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"op ~:" :: ['a, 'a set] => bool ("(_/ \\<notin> _)" [50, 51] 50) 
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"UN " :: [idts, bool] => bool ("(3\\<Union> _./ _)" 10) 

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"INT " :: [idts, bool] => bool ("(3\\<Inter> _./ _)" 10) 

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"@UNION" :: [pttrn, 'a set, 'b set] => 'b set ("(3\\<Union> _\\<in>_./ _)" 10) 

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"@INTER" :: [pttrn, 'a set, 'b set] => 'b set ("(3\\<Inter> _\\<in>_./ _)" 10) 

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Union :: (('a set) set) => 'a set ("\\<Union> _" [90] 90) 

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Inter :: (('a set) set) => 'a set ("\\<Inter> _" [90] 90) 

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"@Ball" :: [pttrn, 'a set, bool] => bool ("(3\\<forall> _\\<in>_./ _)" [0, 0, 10] 10) 
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"@Bex" :: [pttrn, 'a set, bool] => bool ("(3\\<exists> _\\<in>_./ _)" [0, 0, 10] 10) 

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syntax (symbols output) 

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"*Ball" :: [pttrn, 'a set, bool] => bool ("(3\\<forall> _\\<in>_./ _)" [0, 0, 10] 10) 
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"*Bex" :: [pttrn, 'a set, bool] => bool ("(3\\<exists> _\\<in>_./ _)" [0, 0, 10] 10) 

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translations 
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"op \\<subseteq>" => "op <= :: [_ set, _ set] => bool" 
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"op \\<subset>" => "op < :: [_ set, _ set] => bool" 
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(** Rules and definitions **) 

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rules 
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(* Isomorphisms between Predicates and Sets *) 

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mem_Collect_eq "(a : {x.P(x)}) = P(a)" 

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Collect_mem_eq "{x.x:A} = A" 

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defs 

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Ball_def "Ball A P == ! x. x:A > P(x)" 
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Bex_def "Bex A P == ? x. x:A & P(x)" 

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subset_def "A <= B == ! x:A. x:B" 

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psubset_def "A < B == (A::'a set) <= B & ~ A=B" 
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Compl_def "Compl A == {x. ~x:A}" 
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Un_def "A Un B == {x.x:A  x:B}" 
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Int_def "A Int B == {x.x:A & x:B}" 

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set_diff_def "A  B == {x. x:A & ~x:B}" 

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INTER_def "INTER A B == {y. ! x:A. y: B(x)}" 

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UNION_def "UNION A B == {y. ? x:A. y: B(x)}" 

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INTER1_def "INTER1 B == INTER {x.True} B" 
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UNION1_def "UNION1 B == UNION {x.True} B" 

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Inter_def "Inter S == (INT x:S. x)" 

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Union_def "Union S == (UN x:S. x)" 

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Pow_def "Pow A == {B. B <= A}" 

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empty_def "{} == {x. False}" 
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insert_def "insert a B == {x.x=a} Un B" 

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image_def "f``A == {y. ? x:A. y=f(x)}" 

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end 
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ML 
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local 

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(* Set inclusion *) 
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fun le_tr' (*op <=*) (Type ("fun", (Type ("set", _) :: _))) ts = 

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list_comb (Syntax.const "_setle", ts) 

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 le_tr' (*op <=*) _ _ = raise Match; 

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fun less_tr' (*op <*) (Type ("fun", (Type ("set", _) :: _))) ts = 
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list_comb (Syntax.const "_setless", ts) 

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 less_tr' (*op <*) _ _ = raise Match; 

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(* Translates between { e  x1..xn. P} and {u. ? x1..xn. u=e & P} *) 
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(* {y. ? x1..xn. y = e & P} is only translated if [0..n] subset bvs(e) *) 

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val ex_tr = snd(mk_binder_tr("? ","Ex")); 

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fun nvars(Const("_idts",_) $ _ $ idts) = nvars(idts)+1 

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 nvars(_) = 1; 

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fun setcompr_tr[e,idts,b] = 

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let val eq = Syntax.const("op =") $ Bound(nvars(idts)) $ e 

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val P = Syntax.const("op &") $ eq $ b 

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val exP = ex_tr [idts,P] 

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in Syntax.const("Collect") $ Abs("",dummyT,exP) end; 

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val ex_tr' = snd(mk_binder_tr' ("Ex","DUMMY")); 

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fun setcompr_tr'[Abs(_,_,P)] = 

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let fun ok(Const("Ex",_)$Abs(_,_,P),n) = ok(P,n+1) 

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 ok(Const("op &",_) $ (Const("op =",_) $ Bound(m) $ e) $ _, n) = 

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if n>0 andalso m=n andalso 

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((0 upto (n1)) subset add_loose_bnos(e,0,[])) 

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then () else raise Match 

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fun tr'(_ $ abs) = 

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let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr'[abs] 

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in Syntax.const("@SetCompr") $ e $ idts $ Q end 

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in ok(P,0); tr'(P) end; 

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in 

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val parse_translation = [("@SetCompr", setcompr_tr)]; 

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val print_translation = [("Collect", setcompr_tr')]; 

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val typed_print_translation = [("op <=", le_tr'), ("op <", less_tr')]; 
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val print_ast_translation = 
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map HOL.alt_ast_tr' [("@Ball", "*Ball"), ("@Bex", "*Bex")]; 

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end; 