TPTP Problem File: SEV475^1.p

View Solutions - Solve Problem

%------------------------------------------------------------------------------
% File     : SEV475^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Analysis
% Problem  : EXISTS_SUBSET_IMAGE_INJ
% Version  : Especial.
% English  :

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

% Status   : Theorem
% Rating   : 0.75 v7.5.0, 0.00 v7.2.0, 0.25 v7.1.0
% Syntax   : Number of formulae    :    7 (   0 unit;   4 type;   0 defn)
%            Number of atoms       :   80 (  10 equality;  57 variable)
%            Maximal formula depth :   17 (   9 average)
%            Number of connectives :   57 (   0   ~;   0   |;   7   &;  48   @)
%                                         (   0 <=>;   2  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   23 (  23   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    6 (   4   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   :   24 (   0 sgn;  16   !;   3   ?;   0   ^)
%                                         (  24   :;   5  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH1_THM_EQU_NAR

% Comments : Exported from core HOL Light.
%------------------------------------------------------------------------------
thf('thf_const_const/trivia/I',type,(
    'const/trivia/I': 
      !>[A: $tType] :
        ( A > A ) )).

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

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

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

thf('thm/trivia/I_THM_',axiom,(
    ! [A: $tType,A0: A] :
      ( ( 'const/trivia/I' @ A @ A0 )
      = A0 ) )).

thf('thm/sets/SUBSET_IMAGE_INJ_',axiom,(
    ! [B: $tType,A: $tType,A0: A > B,A1: B > $o,A2: A > $o] :
      ( ( 'const/sets/SUBSET' @ B @ A1 @ ( 'const/sets/IMAGE' @ A @ B @ A0 @ A2 ) )
      = ( ? [A3: A > $o] :
            ( ( 'const/sets/SUBSET' @ A @ A3 @ A2 )
            & ! [A4: A,A5: A] :
                ( ( ( 'const/sets/IN' @ A @ A4 @ A3 )
                  & ( 'const/sets/IN' @ A @ A5 @ A3 ) )
               => ( ( ( A0 @ A4 )
                    = ( A0 @ A5 ) )
                  = ( A4 = A5 ) ) )
            & ( A1
              = ( 'const/sets/IMAGE' @ A @ B @ A0 @ A3 ) ) ) ) ) )).

thf('thm/sets/EXISTS_SUBSET_IMAGE_INJ_',conjecture,(
    ! [A: $tType,A0: $tType,P: ( A > $o ) > $o,A1: A0 > A,A2: A0 > $o] :
      ( ( ? [A3: A > $o] :
            ( ( 'const/sets/SUBSET' @ A @ A3 @ ( 'const/sets/IMAGE' @ A0 @ A @ A1 @ A2 ) )
            & ( P @ A3 ) ) )
      = ( ? [A3: A0 > $o] :
            ( ( 'const/sets/SUBSET' @ A0 @ A3 @ A2 )
            & ! [A4: A0,A5: A0] :
                ( ( ( 'const/sets/IN' @ A0 @ A4 @ A3 )
                  & ( 'const/sets/IN' @ A0 @ A5 @ A3 ) )
               => ( ( ( A1 @ A4 )
                    = ( A1 @ A5 ) )
                  = ( A4 = A5 ) ) )
            & ( P @ ( 'const/sets/IMAGE' @ A0 @ A @ A1 @ A3 ) ) ) ) ) )).

%------------------------------------------------------------------------------