## TPTP Problem File: SEV478^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV478^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Analysis
% Problem  : CARD_CLAUSES_
% Version  : Especial.
% English  :

% Refs     : [Kal16] Kalisyk (2016), Email to Geoff Sutcliffe
% Source   : [Kal16]
% Names    : CARD_CLAUSES_1.p [Kal16]

% Status   : Theorem
% Rating   : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax   : Number of formulae    :   14 (   0 unit;  11 type;   0 defn)
%            Number of atoms       :   91 (   6 equality;  61 variable)
%            Maximal formula depth :   18 (   6 average)
%            Number of connectives :   77 (   1   ~;   0   |;   1   &;  71   @)
%                                         (   0 <=>;   4  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   28 (  28   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   13 (  11   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   :   24 (   1 sgn;  14   !;   0   ?;   2   ^)
%                                         (  24   :;   8  !>;   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/ITSET',type,(
'const/sets/ITSET':
!>[A: \$tType,A0: \$tType] :
( ( A > A0 > A0 ) > ( A > \$o ) > A0 > A0 ) )).

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

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

thf('thf_const_const/nums/SUC',type,(
'const/nums/SUC': 'type/nums/num' > 'type/nums/num' )).

thf('thf_const_const/nums/NUMERAL',type,(
'const/nums/NUMERAL': 'type/nums/num' > 'type/nums/num' )).

thf('thf_const_const/nums/_0',type,(
'const/nums/_0': 'type/nums/num' )).

thf('thf_const_const/class/COND',type,(
'const/class/COND':
!>[A: \$tType] :
( \$o > A > A > A ) )).

thf('thm/sets/CARD_',axiom,(
! [A: \$tType,A0: A > \$o] :
( ( 'const/sets/CARD' @ A @ A0 )
= ( 'const/sets/ITSET' @ A @ 'type/nums/num'
@ ^ [A1: A,A2: 'type/nums/num'] :
( 'const/nums/SUC' @ A2 )
@ A0
@ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) )).

thf('thm/sets/FINITE_RECURSION_',axiom,(
! [A: \$tType,B: \$tType,A0: A > B > B,A1: B] :
( ! [A2: A,A3: A,A4: B] :
( ( A2 != A3 )
=> ( ( A0 @ A2 @ ( A0 @ A3 @ A4 ) )
= ( A0 @ A3 @ ( A0 @ A2 @ A4 ) ) ) )
=> ( ( ( 'const/sets/ITSET' @ A @ B @ A0 @ ( 'const/sets/EMPTY' @ A ) @ A1 )
= A1 )
& ! [A2: A,A3: A > \$o] :
( ( 'const/sets/FINITE' @ A @ A3 )
=> ( ( 'const/sets/ITSET' @ A @ B @ A0 @ ( 'const/sets/INSERT' @ A @ A2 @ A3 ) @ A1 )
= ( 'const/class/COND' @ B @ ( 'const/sets/IN' @ A @ A2 @ A3 ) @ ( 'const/sets/ITSET' @ A @ B @ A0 @ A3 @ A1 ) @ ( A0 @ A2 @ ( 'const/sets/ITSET' @ A @ B @ A0 @ A3 @ A1 ) ) ) ) ) ) ) )).

thf('thm/sets/CARD_CLAUSES_1',conjecture,(
! [A: \$tType,A0: A,A1: A > \$o] :
( ( 'const/sets/FINITE' @ A @ A1 )
=> ( ( 'const/sets/CARD' @ A @ ( 'const/sets/INSERT' @ A @ A0 @ A1 ) )
= ( 'const/class/COND' @ 'type/nums/num' @ ( 'const/sets/IN' @ A @ A0 @ A1 ) @ ( 'const/sets/CARD' @ A @ A1 ) @ ( 'const/nums/SUC' @ ( 'const/sets/CARD' @ A @ A1 ) ) ) ) ) )).

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