TPTP Problem File: SEV491^1.p

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% File     : SEV491^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Analysis
% Problem  : BIJECTIONS_CARD_EQ
% Version  : Especial.
% English  :

% Refs     : [Kal16] Kalisyk (2016), Email to Geoff Sutcliffe
% Source   : [Kal16]
% Names    : BIJECTIONS_CARD_EQ_.p [Kal16]

% Status   : Theorem
% Rating   : 0.25 v7.5.0, 0.00 v7.2.0, 0.25 v7.1.0
% Syntax   : Number of formulae    :    8 (   0 unit;   5 type;   0 defn)
%            Number of atoms       :   91 (   8 equality;  66 variable)
%            Maximal formula depth :   16 (   8 average)
%            Number of connectives :   72 (   0   ~;   1   |;   8   &;  57   @)
%                                         (   0 <=>;   6  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   19 (  19   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    7 (   5   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   :   24 (   0 sgn;  20   !;   0   ?;   0   ^)
%                                         (  24   :;   4  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH1_THM_EQU_NAR

% Comments : Exported from core HOL Light.
%------------------------------------------------------------------------------
thf('thf_type_type/nums/num',type,(
    'type/nums/num': $tType )).

thf('thf_const_const/sets/IN',type,(
    'const/sets/IN': 
      !>[A: $tType] :
        ( A > ( A > $o ) > $o ) )).

thf('thf_const_const/sets/HAS_SIZE',type,(
    'const/sets/HAS_SIZE': 
      !>[A: $tType] :
        ( ( A > $o ) > 'type/nums/num' > $o ) )).

thf('thf_const_const/sets/FINITE',type,(
    'const/sets/FINITE': 
      !>[A: $tType] :
        ( ( A > $o ) > $o ) )).

thf('thf_const_const/sets/CARD',type,(
    'const/sets/CARD': 
      !>[A: $tType] :
        ( ( A > $o ) > 'type/nums/num' ) )).

thf('thm/sets/HAS_SIZE_',axiom,(
    ! [A: $tType,A0: A > $o,A1: 'type/nums/num'] :
      ( ( 'const/sets/HAS_SIZE' @ A @ A0 @ A1 )
      = ( ( 'const/sets/FINITE' @ A @ A0 )
        & ( ( 'const/sets/CARD' @ A @ A0 )
          = A1 ) ) ) )).

thf('thm/sets/BIJECTIONS_HAS_SIZE_EQ_',axiom,(
    ! [A: $tType,B: $tType,A0: A > $o,A1: B > $o,A2: A > B,A3: B > A] :
      ( ( ! [A4: A] :
            ( ( 'const/sets/IN' @ A @ A4 @ A0 )
           => ( ( 'const/sets/IN' @ B @ ( A2 @ A4 ) @ A1 )
              & ( ( A3 @ ( A2 @ A4 ) )
                = A4 ) ) )
        & ! [A4: B] :
            ( ( 'const/sets/IN' @ B @ A4 @ A1 )
           => ( ( 'const/sets/IN' @ A @ ( A3 @ A4 ) @ A0 )
              & ( ( A2 @ ( A3 @ A4 ) )
                = A4 ) ) ) )
     => ! [A4: 'type/nums/num'] :
          ( ( 'const/sets/HAS_SIZE' @ A @ A0 @ A4 )
          = ( 'const/sets/HAS_SIZE' @ B @ A1 @ A4 ) ) ) )).

thf('thm/sets/BIJECTIONS_CARD_EQ_',conjecture,(
    ! [A: $tType,B: $tType,A0: A > $o,A1: B > $o,A2: A > B,A3: B > A] :
      ( ( ( ( 'const/sets/FINITE' @ A @ A0 )
          | ( 'const/sets/FINITE' @ B @ A1 ) )
        & ! [A4: A] :
            ( ( 'const/sets/IN' @ A @ A4 @ A0 )
           => ( ( 'const/sets/IN' @ B @ ( A2 @ A4 ) @ A1 )
              & ( ( A3 @ ( A2 @ A4 ) )
                = A4 ) ) )
        & ! [A4: B] :
            ( ( 'const/sets/IN' @ B @ A4 @ A1 )
           => ( ( 'const/sets/IN' @ A @ ( A3 @ A4 ) @ A0 )
              & ( ( A2 @ ( A3 @ A4 ) )
                = A4 ) ) ) )
     => ( ( 'const/sets/CARD' @ A @ A0 )
        = ( 'const/sets/CARD' @ B @ A1 ) ) ) )).

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