%------------------------------------------------------------------------------
% File     : SET008^0 : TPTP v9.0.0. Released v3.6.0.
% Domain   : Set Theory
% Axioms   : Basic set theory definitions
% Version  : [Ben08] axioms.
% English  :

% Refs     : [BS+05] Benzmueller et al. (2005), Can a Higher-Order and a Fi
%          : [BS+08] Benzmueller et al. (2007), Combined Reasoning by Autom
%          : [Ben08] Benzmueller (2008), Email to Geoff Sutcliffe
% Source   : [Ben08]
% Names    : Typed_Set [Ben08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   28 (  14 unt;  14 typ;  14 def)
%            Number of atoms       :   35 (  18 equ;   0 cnn)
%            Maximal formula atoms :    1 (   1 avg)
%            Number of connectives :   36 (   5   ~;   3   |;   6   &;  21   @)
%                                         (   0 <=>;   1  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    1 (   1 avg;  21 nst)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :   70 (  70   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   16 (  14 usr;   1 con; 0-3 aty)
%            Number of variables   :   35 (  32   ^   1   !;   2   ?;  35   :)
% SPC      : 

% Comments :
%------------------------------------------------------------------------------
thf(in_decl,type,
    in: $i > ( $i > $o ) > $o ).

thf(in,definition,
    ( in
    = ( ^ [X: $i,M: $i > $o] : ( M @ X ) ) ) ).

thf(is_a_decl,type,
    is_a: $i > ( $i > $o ) > $o ).

thf(is_a,definition,
    ( is_a
    = ( ^ [X: $i,M: $i > $o] : ( M @ X ) ) ) ).

thf(emptyset_decl,type,
    emptyset: $i > $o ).

thf(emptyset,definition,
    ( emptyset
    = ( ^ [X: $i] : $false ) ) ).

thf(unord_pair_decl,type,
    unord_pair: $i > $i > $i > $o ).

thf(unord_pair,definition,
    ( unord_pair
    = ( ^ [X: $i,Y: $i,U: $i] :
          ( ( U = X )
          | ( U = Y ) ) ) ) ).

thf(singleton_decl,type,
    singleton: $i > $i > $o ).

thf(singleton,definition,
    ( singleton
    = ( ^ [X: $i,U: $i] : ( U = X ) ) ) ).

thf(union_decl,type,
    union: ( $i > $o ) > ( $i > $o ) > $i > $o ).

thf(union,definition,
    ( union
    = ( ^ [X: $i > $o,Y: $i > $o,U: $i] :
          ( ( X @ U )
          | ( Y @ U ) ) ) ) ).

thf(excl_union_decl,type,
    excl_union: ( $i > $o ) > ( $i > $o ) > $i > $o ).

thf(excl_union,definition,
    ( excl_union
    = ( ^ [X: $i > $o,Y: $i > $o,U: $i] :
          ( ( ( X @ U )
            & ~ ( Y @ U ) )
          | ( ~ ( X @ U )
            & ( Y @ U ) ) ) ) ) ).

thf(intersection_decl,type,
    intersection: ( $i > $o ) > ( $i > $o ) > $i > $o ).

thf(intersection,definition,
    ( intersection
    = ( ^ [X: $i > $o,Y: $i > $o,U: $i] :
          ( ( X @ U )
          & ( Y @ U ) ) ) ) ).

thf(setminus_decl,type,
    setminus: ( $i > $o ) > ( $i > $o ) > $i > $o ).

thf(setminus,definition,
    ( setminus
    = ( ^ [X: $i > $o,Y: $i > $o,U: $i] :
          ( ( X @ U )
          & ~ ( Y @ U ) ) ) ) ).

thf(complement_decl,type,
    complement: ( $i > $o ) > $i > $o ).

thf(complement,definition,
    ( complement
    = ( ^ [X: $i > $o,U: $i] :
          ~ ( X @ U ) ) ) ).

thf(disjoint_decl,type,
    disjoint: ( $i > $o ) > ( $i > $o ) > $o ).

thf(disjoint,definition,
    ( disjoint
    = ( ^ [X: $i > $o,Y: $i > $o] :
          ( ( intersection @ X @ Y )
          = emptyset ) ) ) ).

thf(subset_decl,type,
    subset: ( $i > $o ) > ( $i > $o ) > $o ).

thf(subset,definition,
    ( subset
    = ( ^ [X: $i > $o,Y: $i > $o] :
        ! [U: $i] :
          ( ( X @ U )
         => ( Y @ U ) ) ) ) ).

thf(meets_decl,type,
    meets: ( $i > $o ) > ( $i > $o ) > $o ).

thf(meets,definition,
    ( meets
    = ( ^ [X: $i > $o,Y: $i > $o] :
        ? [U: $i] :
          ( ( X @ U )
          & ( Y @ U ) ) ) ) ).

thf(misses_decl,type,
    misses: ( $i > $o ) > ( $i > $o ) > $o ).

thf(misses,definition,
    ( misses
    = ( ^ [X: $i > $o,Y: $i > $o] :
          ~ ? [U: $i] :
              ( ( X @ U )
              & ( Y @ U ) ) ) ) ).

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