## TPTP Problem File: SEV508^1.p

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```%------------------------------------------------------------------------------
% File     : SEV508^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Analysis
% Problem  : SUM_LT_ALL
% Version  : Especial.
% English  :

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

% Status   : Theorem
% Rating   : 0.00 v7.1.0
% Syntax   : Number of formulae    :   13 (   0 unit;   8 type;   0 defn)
%            Number of atoms       :   75 (   4 equality;  51 variable)
%            Maximal formula depth :   13 (   6 average)
%            Number of connectives :   64 (   2   ~;   0   |;   5   &;  52   @)
%                                         (   0 <=>;   5  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   22 (  22   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   10 (   8   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   :   23 (   0 sgn;  16   !;   2   ?;   0   ^)
%                                         (  23   :;   5  !>;   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_const_const/trivia/I',type,(
'const/trivia/I':
!>[A: \$tType] :
( A > A ) )).

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_lt',type,(
'const/realax/real_lt': 'type/realax/real' > 'type/realax/real' > \$o )).

thf('thf_const_const/realax/real_le',type,(
'const/realax/real_le': 'type/realax/real' > 'type/realax/real' > \$o )).

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

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

thf('thm/iterate/SUM_LT_',axiom,(
! [A: \$tType,A0: A > 'type/realax/real',A1: A > 'type/realax/real',A2: A > \$o] :
( ( ( 'const/sets/FINITE' @ A @ A2 )
& ! [A3: A] :
( ( 'const/sets/IN' @ A @ A3 @ A2 )
=> ( 'const/realax/real_le' @ ( A0 @ A3 ) @ ( A1 @ A3 ) ) )
& ? [A3: A] :
( ( 'const/sets/IN' @ A @ A3 @ A2 )
& ( 'const/realax/real_lt' @ ( A0 @ A3 ) @ ( A1 @ A3 ) ) ) )
=> ( 'const/realax/real_lt' @ ( 'const/iterate/sum' @ A @ A2 @ A0 ) @ ( 'const/iterate/sum' @ A @ A2 @ A1 ) ) ) )).

thf('thm/realarith/REAL_LT_IMP_LE_',axiom,(
! [A: 'type/realax/real',A0: 'type/realax/real'] :
( ( 'const/realax/real_lt' @ A @ A0 )
=> ( 'const/realax/real_le' @ A @ A0 ) ) )).

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/iterate/SUM_LT_ALL_',conjecture,(
! [A: \$tType,A0: A > 'type/realax/real',A1: A > 'type/realax/real',A2: A > \$o] :
( ( ( 'const/sets/FINITE' @ A @ A2 )
& ( A2
!= ( 'const/sets/EMPTY' @ A ) )
& ! [A3: A] :
( ( 'const/sets/IN' @ A @ A3 @ A2 )
=> ( 'const/realax/real_lt' @ ( A0 @ A3 ) @ ( A1 @ A3 ) ) ) )
=> ( 'const/realax/real_lt' @ ( 'const/iterate/sum' @ A @ A2 @ A0 ) @ ( 'const/iterate/sum' @ A @ A2 @ A1 ) ) ) )).

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