TPTP Problem File: SEV505^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SEV505^1 : TPTP v9.0.0. 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 ( 3 unt; 12 typ; 1 def)
% Number of atoms : 18 ( 6 equ; 0 cnn)
% Maximal formula atoms : 7 ( 3 avg)
% Number of connectives : 79 ( 0 ~; 0 |; 4 &; 70 @)
% ( 0 <=>; 5 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 43 ( 43 >; 0 *; 0 +; 0 <<)
% Number of symbols : 12 ( 11 usr; 1 con; 0-5 aty)
% Number of variables : 22 ( 0 ^; 13 !; 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 ) ) ) ) ) ).
thf('thm/iterate/MONOIDAL_ADD_',axiom,
'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/+' ) ) ).
thf('thm/iterate/NEUTRAL_ADD_',axiom,
( ( '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 ) ) ) ) ).
%------------------------------------------------------------------------------