## TPTP Problem File: SEV505^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV505^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Analysis
% Problem  : NSUM_UNION_NONZERO
% Version  : Especial.
% English  :

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

% Status   : Theorem
% Rating   : 0.00 v7.5.0, 0.33 v7.2.0, 0.25 v7.1.0
% Syntax   : Number of formulae    :   17 (   0 unit;  12 type;   1 defn)
%            Number of atoms       :   96 (   6 equality;  57 variable)
%            Maximal formula depth :   17 (   6 average)
%            Number of connectives :   79 (   0   ~;   0   |;   4   &;  70   @)
%                                         (   0 <=>;   5  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   43 (  43   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   14 (  12   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   :   22 (   0 sgn;  13   !;   0   ?;   0   ^)
%                                         (  22   :;   9  !>;   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/UNION',type,(
'const/sets/UNION':
!>[A: \$tType] :
( ( A > \$o ) > ( A > \$o ) > A > \$o ) )).

thf('thf_const_const/sets/INTER',type,(
'const/sets/INTER':
!>[A: \$tType] :
( ( A > \$o ) > ( 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/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/nsum',type,(
'const/iterate/nsum':
!>[A: \$tType] :
( ( A > \$o ) > ( A > 'type/nums/num' ) > 'type/nums/num' ) )).

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('thf_const_const/arith/+',type,(
'const/arith/+': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' )).

thf('thm/iterate/ITERATE_UNION_NONZERO_',axiom,(
! [A: \$tType,B: \$tType,A0: B > B > B] :
( ( 'const/iterate/monoidal' @ B @ A0 )
=> ! [A1: A > B,A2: A > \$o,A3: A > \$o] :
( ( ( 'const/sets/FINITE' @ A @ A2 )
& ( 'const/sets/FINITE' @ A @ A3 )
& ! [A4: A] :
( ( 'const/sets/IN' @ A @ A4 @ ( 'const/sets/INTER' @ A @ A2 @ A3 ) )
=> ( ( A1 @ A4 )
= ( 'const/iterate/neutral' @ B @ A0 ) ) ) )
=> ( ( 'const/iterate/iterate' @ A @ B @ A0 @ ( 'const/sets/UNION' @ A @ A2 @ A3 ) @ A1 )
= ( A0 @ ( 'const/iterate/iterate' @ A @ B @ A0 @ A2 @ A1 ) @ ( 'const/iterate/iterate' @ A @ B @ A0 @ A3 @ A1 ) ) ) ) ) )).

( 'const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+' )).

thf('thm/iterate/nsum_',definition,(
! [A: \$tType] :
( ( 'const/iterate/nsum' @ A )
= ( 'const/iterate/iterate' @ A @ 'type/nums/num' @ 'const/arith/+' ) ) )).

( ( 'const/iterate/neutral' @ 'type/nums/num' @ 'const/arith/+' )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )).

thf('thm/iterate/NSUM_UNION_NONZERO_',conjecture,(
! [A: \$tType,A0: A > 'type/nums/num',A1: A > \$o,A2: A > \$o] :
( ( ( 'const/sets/FINITE' @ A @ A1 )
& ( 'const/sets/FINITE' @ A @ A2 )
& ! [A3: A] :
( ( 'const/sets/IN' @ A @ A3 @ ( 'const/sets/INTER' @ A @ A1 @ A2 ) )
=> ( ( A0 @ A3 )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) )
=> ( ( 'const/iterate/nsum' @ A @ ( 'const/sets/UNION' @ A @ A1 @ A2 ) @ A0 )
= ( 'const/arith/+' @ ( 'const/iterate/nsum' @ A @ A1 @ A0 ) @ ( 'const/iterate/nsum' @ A @ A2 @ A0 ) ) ) ) )).

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