## TPTP Problem File: SEV509^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV509^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Analysis
% Problem  : SUM_SING
% Version  : Especial.
% English  :

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

% Status   : Theorem
% Rating   : 1.00 v7.1.0
% Syntax   : Number of formulae    :   18 (   0 unit;  12 type;   0 defn)
%            Number of atoms       :   64 (   4 equality;  35 variable)
%            Maximal formula depth :   13 (   5 average)
%            Number of connectives :   51 (   1   ~;   0   |;   0   &;  49   @)
%                                         (   0 <=>;   1  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   25 (  25   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   14 (  12   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   :   19 (   0 sgn;  13   !;   0   ?;   0   ^)
%                                         (  19   :;   6  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH1_THM_EQU_NAR

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

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/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_of_num',type,(
'const/realax/real_of_num': 'type/nums/num' > 'type/realax/real' )).

'const/realax/real_add': 'type/realax/real' > 'type/realax/real' > 'type/realax/real' )).

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/iterate/sum',type,(
'const/iterate/sum':
!>[A: \$tType] :
( ( A > \$o ) > ( A > 'type/realax/real' ) > 'type/realax/real' ) )).

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

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

thf('thm/iterate/SUM_CLAUSES_0',axiom,(
! [A: \$tType,A0: A > 'type/realax/real'] :
( ( 'const/iterate/sum' @ A @ ( 'const/sets/EMPTY' @ A ) @ A0 )
= ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) )).

! [A: 'type/realax/real'] :
( ( 'const/realax/real_add' @ A @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) )
= A ) )).

thf('thm/iterate/SUM_CLAUSES_1',axiom,(
! [A: \$tType,A0: A,A1: A > 'type/realax/real',A2: A > \$o] :
( ( 'const/sets/FINITE' @ A @ A2 )
=> ( ( 'const/iterate/sum' @ A @ ( 'const/sets/INSERT' @ A @ A0 @ A2 ) @ A1 )
= ( 'const/class/COND' @ 'type/realax/real' @ ( 'const/sets/IN' @ A @ A0 @ A2 ) @ ( 'const/iterate/sum' @ A @ A2 @ A1 ) @ ( 'const/realax/real_add' @ ( A1 @ A0 ) @ ( 'const/iterate/sum' @ A @ A2 @ A1 ) ) ) ) ) )).

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

thf('thm/iterate/SUM_SING_',conjecture,(
! [A: \$tType,A0: A > 'type/realax/real',A1: A] :
( ( 'const/iterate/sum' @ A @ ( 'const/sets/INSERT' @ A @ A1 @ ( 'const/sets/EMPTY' @ A ) ) @ A0 )
= ( A0 @ A1 ) ) )).

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