## TPTP Problem File: SEV479^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV479^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Analysis
% Problem  : HAS_SIZE_SUC
% Version  : Especial.
% English  :

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

% Status   : Theorem
% Rating   : 1.00 v7.2.0, 0.75 v7.1.0
% Syntax   : Number of formulae    :   26 (   0 unit;  12 type;   0 defn)
%            Number of atoms       :  156 (  21 equality;  90 variable)
%            Maximal formula depth :   13 (   6 average)
%            Number of connectives :  105 (   5   ~;   1   |;   3   &;  94   @)
%                                         (   0 <=>;   2  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   34 (  34   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   14 (  12   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   :   46 (   0 sgn;  37   !;   1   ?;   0   ^)
%                                         (  46   :;   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/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/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/EMPTY',type,(
'const/sets/EMPTY':
!>[A: \$tType] :
( A > \$o ) )).

thf('thf_const_const/sets/DELETE',type,(
'const/sets/DELETE':
!>[A: \$tType] :
( ( A > \$o ) > A > 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/MEMBER_NOT_EMPTY_',axiom,(
! [A: \$tType,A0: A > \$o] :
( ( ? [A1: A] :
( 'const/sets/IN' @ A @ A1 @ A0 ) )
= ( A0
!= ( 'const/sets/EMPTY' @ A ) ) ) )).

thf('thm/sets/IN_',axiom,(
! [A: \$tType,P: A > \$o,A0: A] :
( ( 'const/sets/IN' @ A @ A0 @ P )
= ( P @ A0 ) ) )).

thf('thm/sets/IN_DELETE_',axiom,(
! [A: \$tType,A0: A > \$o,A1: A,A2: A] :
( ( 'const/sets/IN' @ A @ A1 @ ( 'const/sets/DELETE' @ A @ A0 @ A2 ) )
= ( ( 'const/sets/IN' @ A @ A1 @ A0 )
& ( A1 != A2 ) ) ) )).

thf('thm/sets/IN_INSERT_',axiom,(
! [A: \$tType,A0: A,A1: A,A2: A > \$o] :
( ( 'const/sets/IN' @ A @ A0 @ ( 'const/sets/INSERT' @ A @ A1 @ A2 ) )
= ( ( A0 = A1 )
| ( 'const/sets/IN' @ A @ A0 @ A2 ) ) ) )).

thf('thm/sets/EXTENSION_',axiom,(
! [A: \$tType,A0: A > \$o,A1: A > \$o] :
( ( A0 = A1 )
= ( ! [A2: A] :
( ( 'const/sets/IN' @ A @ A2 @ A0 )
= ( 'const/sets/IN' @ A @ A2 @ A1 ) ) ) ) )).

thf('thm/sets/FINITE_DELETE_',axiom,(
! [A: \$tType,A0: A > \$o,A1: A] :
( ( 'const/sets/FINITE' @ A @ ( 'const/sets/DELETE' @ A @ A0 @ A1 ) )
= ( 'const/sets/FINITE' @ A @ A0 ) ) )).

thf('thm/sets/CARD_CLAUSES_1',axiom,(
! [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 ) ) ) ) ) )).

thf('thm/nums/SUC_INJ_',axiom,(
! [A: 'type/nums/num',A0: 'type/nums/num'] :
( ( ( 'const/nums/SUC' @ A )
= ( 'const/nums/SUC' @ A0 ) )
= ( A = A0 ) ) )).

thf('thm/sets/NOT_IN_EMPTY_',axiom,(
! [A: \$tType,A0: A] :
~ ( 'const/sets/IN' @ A @ A0 @ ( 'const/sets/EMPTY' @ A ) ) )).

thf('thm/nums/NOT_SUC_',axiom,(
! [A: 'type/nums/num'] :
( ( 'const/nums/SUC' @ A )
!= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) )).

thf('thm/sets/CARD_CLAUSES_0',axiom,(
! [A: \$tType] :
( ( 'const/sets/CARD' @ A @ ( 'const/sets/EMPTY' @ A ) )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) )).

thf('thm/sets/FINITE_RULES_0',axiom,(
! [A: \$tType] :
( 'const/sets/FINITE' @ A @ ( 'const/sets/EMPTY' @ A ) ) )).

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/HAS_SIZE_SUC_',conjecture,(
! [A: \$tType,A0: A > \$o,A1: 'type/nums/num'] :
( ( 'const/sets/HAS_SIZE' @ A @ A0 @ ( 'const/nums/SUC' @ A1 ) )
= ( ( A0
!= ( 'const/sets/EMPTY' @ A ) )
& ! [A2: A] :
( ( 'const/sets/IN' @ A @ A2 @ A0 )
=> ( 'const/sets/HAS_SIZE' @ A @ ( 'const/sets/DELETE' @ A @ A0 @ A2 ) @ A1 ) ) ) ) )).

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