## TPTP Problem File: SEV491^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% 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 ) ) ) )).

%------------------------------------------------------------------------------
```