## TPTP Problem File: SEV512^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV512^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Analysis
% Problem  : SUM_IMAGE_NONZERO
% Version  : Especial.
% English  :

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

% Status   : Theorem
% Rating   : 0.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax   : Number of formulae    :   19 (   0 unit;  14 type;   1 defn)
%            Number of atoms       :  112 (  10 equality;  69 variable)
%            Maximal formula depth :   19 (   6 average)
%            Number of connectives :   89 (   2   ~;   0   |;   8   &;  74   @)
%                                         (   0 <=>;   5  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   44 (  44   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   16 (  14   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   :   29 (   0 sgn;  17   !;   0   ?;   0   ^)
%                                         (  29   :;  12  !>;   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/trivia/o',type,(
'const/trivia/o':
!>[B: \$tType,A: \$tType,C: \$tType] :
( ( B > C ) > ( A > B ) > A > C ) )).

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

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

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

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('thm/iterate/ITERATE_IMAGE_NONZERO_',axiom,(
! [C: \$tType,A: \$tType,B: \$tType,A0: C > C > C] :
( ( 'const/iterate/monoidal' @ C @ A0 )
=> ! [A1: B > C,A2: A > B,A3: A > \$o] :
( ( ( 'const/sets/FINITE' @ A @ A3 )
& ! [A4: A,A5: A] :
( ( ( 'const/sets/IN' @ A @ A4 @ A3 )
& ( 'const/sets/IN' @ A @ A5 @ A3 )
& ( A4 != A5 )
& ( ( A2 @ A4 )
= ( A2 @ A5 ) ) )
=> ( ( A1 @ ( A2 @ A4 ) )
= ( 'const/iterate/neutral' @ C @ A0 ) ) ) )
=> ( ( 'const/iterate/iterate' @ B @ C @ A0 @ ( 'const/sets/IMAGE' @ A @ B @ A2 @ A3 ) @ A1 )
= ( 'const/iterate/iterate' @ A @ C @ A0 @ A3 @ ( 'const/trivia/o' @ B @ A @ C @ A1 @ A2 ) ) ) ) ) )).

( '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' ) ) )).

( ( 'const/iterate/neutral' @ 'type/realax/real' @ 'const/realax/real_add' )
= ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) )).

thf('thm/iterate/SUM_IMAGE_NONZERO_',conjecture,(
! [A: \$tType,B: \$tType,A0: B > 'type/realax/real',A1: A > B,A2: A > \$o] :
( ( ( 'const/sets/FINITE' @ A @ A2 )
& ! [A3: A,A4: A] :
( ( ( 'const/sets/IN' @ A @ A3 @ A2 )
& ( 'const/sets/IN' @ A @ A4 @ A2 )
& ( A3 != A4 )
& ( ( A1 @ A3 )
= ( A1 @ A4 ) ) )
=> ( ( A0 @ ( A1 @ A3 ) )
= ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) )
=> ( ( 'const/iterate/sum' @ B @ ( 'const/sets/IMAGE' @ A @ B @ A1 @ A2 ) @ A0 )
= ( 'const/iterate/sum' @ A @ A2 @ ( 'const/trivia/o' @ B @ A @ 'type/realax/real' @ A0 @ A1 ) ) ) ) )).

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