TPTP Problem File: SEV493^1.p

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% File     : SEV493^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Analysis
% Problem  : INF_FINITE
% Version  : Especial.
% English  :

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

% Status   : Theorem
% Rating   : 0.00 v7.1.0
% Syntax   : Number of formulae    :   11 (   0 unit;   6 type;   0 defn)
%            Number of atoms       :   84 (   5 equality;  41 variable)
%            Maximal formula depth :   11 (   6 average)
%            Number of connectives :   72 (   3   ~;   1   |;   7   &;  52   @)
%                                         (   0 <=>;   9  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   13 (  13   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    8 (   6   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   :   18 (   0 sgn;  13   !;   2   ?;   0   ^)
%                                         (  18   :;   3  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH1_THM_EQU_NAR

% Comments : Exported from core HOL Light.
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thf('thf_type_type/realax/real',type,(
    'type/realax/real': $tType )).

thf('thf_const_const/sets/inf',type,(
    'const/sets/inf': ( 'type/realax/real' > $o ) > 'type/realax/real' )).

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

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

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

thf('thf_const_const/realax/real_le',type,(
    'const/realax/real_le': 'type/realax/real' > 'type/realax/real' > $o )).

thf('thm/sets/INF_',axiom,(
    ! [A: 'type/realax/real' > $o] :
      ( ( ( A
         != ( 'const/sets/EMPTY' @ 'type/realax/real' ) )
        & ? [A0: 'type/realax/real'] :
          ! [A1: 'type/realax/real'] :
            ( ( 'const/sets/IN' @ 'type/realax/real' @ A1 @ A )
           => ( 'const/realax/real_le' @ A0 @ A1 ) ) )
     => ( ! [A0: 'type/realax/real'] :
            ( ( 'const/sets/IN' @ 'type/realax/real' @ A0 @ A )
           => ( 'const/realax/real_le' @ ( 'const/sets/inf' @ A ) @ A0 ) )
        & ! [A0: 'type/realax/real'] :
            ( ! [A1: 'type/realax/real'] :
                ( ( 'const/sets/IN' @ 'type/realax/real' @ A1 @ A )
               => ( 'const/realax/real_le' @ A0 @ A1 ) )
           => ( 'const/realax/real_le' @ A0 @ ( 'const/sets/inf' @ A ) ) ) ) ) )).

thf('thm/realax/REAL_LE_TOTAL_',axiom,(
    ! [A: 'type/realax/real',A0: 'type/realax/real'] :
      ( ( 'const/realax/real_le' @ A @ A0 )
      | ( 'const/realax/real_le' @ A0 @ A ) ) )).

thf('thm/realax/REAL_LE_ANTISYM_',axiom,(
    ! [A: 'type/realax/real',A0: 'type/realax/real'] :
      ( ( ( 'const/realax/real_le' @ A @ A0 )
        & ( 'const/realax/real_le' @ A0 @ A ) )
      = ( A = A0 ) ) )).

thf('thm/sets/INF_FINITE_LEMMA_',axiom,(
    ! [A: 'type/realax/real' > $o] :
      ( ( ( 'const/sets/FINITE' @ 'type/realax/real' @ A )
        & ( A
         != ( 'const/sets/EMPTY' @ 'type/realax/real' ) ) )
     => ? [A0: 'type/realax/real'] :
          ( ( 'const/sets/IN' @ 'type/realax/real' @ A0 @ A )
          & ! [A1: 'type/realax/real'] :
              ( ( 'const/sets/IN' @ 'type/realax/real' @ A1 @ A )
             => ( 'const/realax/real_le' @ A0 @ A1 ) ) ) ) )).

thf('thm/sets/INF_FINITE_',conjecture,(
    ! [A: 'type/realax/real' > $o] :
      ( ( ( 'const/sets/FINITE' @ 'type/realax/real' @ A )
        & ( A
         != ( 'const/sets/EMPTY' @ 'type/realax/real' ) ) )
     => ( ( 'const/sets/IN' @ 'type/realax/real' @ ( 'const/sets/inf' @ A ) @ A )
        & ! [A0: 'type/realax/real'] :
            ( ( 'const/sets/IN' @ 'type/realax/real' @ A0 @ A )
           => ( 'const/realax/real_le' @ ( 'const/sets/inf' @ A ) @ A0 ) ) ) ) )).

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