## TPTP Problem File: SEV513^1.p

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```%------------------------------------------------------------------------------
% File     : SEV513^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Analysis
% Problem  : SUM_CASES
% Version  : Especial.
% English  :

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

% Status   : Theorem
% Rating   : 0.00 v7.5.0, 0.33 v7.2.0, 0.25 v7.1.0
% Syntax   : Number of formulae    :   14 (   0 unit;  10 type;   1 defn)
%            Number of atoms       :  111 (   3 equality;  76 variable)
%            Maximal formula depth :   24 (   8 average)
%            Number of connectives :  103 (   2   ~;   0   |;   4   &;  94   @)
%                                         (   0 <=>;   3  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   40 (  40   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   12 (  10   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   :   32 (   0 sgn;  13   !;   4   ?;   6   ^)
%                                         (  32   :;   9  !>;   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/sets/SETSPEC',type,(
'const/sets/SETSPEC':
!>[A: \$tType] :
( 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/GSPEC',type,(
'const/sets/GSPEC':
!>[A: \$tType] :
( ( A > \$o ) > A > \$o ) )).

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

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

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

thf('thf_const_const/iterate/iterate',type,(
'const/iterate/iterate':
!>[A: \$tType,A0: \$tType] :
( ( A0 > A0 > A0 ) > ( A > \$o ) > ( A > A0 ) > A0 ) )).

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

thf('thm/iterate/ITERATE_CASES_',axiom,(
! [A: \$tType,B: \$tType,A0: B > B > B] :
( ( 'const/iterate/monoidal' @ B @ A0 )
=> ! [A1: A > \$o,P: A > \$o,A2: A > B,A3: A > B] :
( ( 'const/sets/FINITE' @ A @ A1 )
=> ( ( 'const/iterate/iterate' @ A @ B @ A0 @ A1
@ ^ [A4: A] :
( 'const/class/COND' @ B @ ( P @ A4 ) @ ( A2 @ A4 ) @ ( A3 @ A4 ) ) )
= ( A0
@ ( 'const/iterate/iterate' @ A @ B @ A0
@ ( 'const/sets/GSPEC' @ A
@ ^ [A4: A] :
? [A5: A] :
( 'const/sets/SETSPEC' @ A @ A4
@ ( ( 'const/sets/IN' @ A @ A5 @ A1 )
& ( P @ A5 ) )
@ A5 ) )
@ A2 )
@ ( 'const/iterate/iterate' @ A @ B @ A0
@ ( 'const/sets/GSPEC' @ A
@ ^ [A4: A] :
? [A5: A] :
( 'const/sets/SETSPEC' @ A @ A4
@ ( ( 'const/sets/IN' @ A @ A5 @ A1 )
& ~ ( P @ A5 ) )
@ A5 ) )
@ A3 ) ) ) ) ) )).

( 'const/iterate/monoidal' @ 'type/realax/real' @ 'const/realax/real_add' )).

thf('thm/iterate/sum_',definition,(
! [A: \$tType] :
( ( 'const/iterate/sum' @ A )
= ( 'const/iterate/iterate' @ A @ 'type/realax/real' @ 'const/realax/real_add' ) ) )).

thf('thm/iterate/SUM_CASES_',conjecture,(
! [A: \$tType,A0: A > \$o,P: A > \$o,A1: A > 'type/realax/real',A2: A > 'type/realax/real'] :
( ( 'const/sets/FINITE' @ A @ A0 )
=> ( ( 'const/iterate/sum' @ A @ A0
@ ^ [A3: A] :
( 'const/class/COND' @ 'type/realax/real' @ ( P @ A3 ) @ ( A1 @ A3 ) @ ( A2 @ A3 ) ) )
@ ( 'const/iterate/sum' @ A
@ ( 'const/sets/GSPEC' @ A
@ ^ [A3: A] :
? [A4: A] :
( 'const/sets/SETSPEC' @ A @ A3
@ ( ( 'const/sets/IN' @ A @ A4 @ A0 )
& ( P @ A4 ) )
@ A4 ) )
@ A1 )
@ ( 'const/iterate/sum' @ A
@ ( 'const/sets/GSPEC' @ A
@ ^ [A3: A] :
? [A4: A] :
( 'const/sets/SETSPEC' @ A @ A3
@ ( ( 'const/sets/IN' @ A @ A4 @ A0 )
& ~ ( P @ A4 ) )
@ A4 ) )
@ A2 ) ) ) ) )).

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