TPTP Problem File: ANA103^1.p
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% File : ANA103^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Analysis
% Problem : REAL_OF_NUM_SUM
% Version : Especial.
% English :
% Refs : [Kal16] Kalisyk (2016), Email to Geoff Sutcliffe
% Source : [Kal16]
% Names : REAL_OF_NUM_SUM_.p [Kal16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 21 ( 3 unt; 14 typ; 0 def)
% Number of atoms : 20 ( 6 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 102 ( 1 ~; 0 |; 3 &; 92 @)
% ( 0 <=>; 6 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 7 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 39 ( 39 >; 0 *; 0 +; 0 <<)
% Number of symbols : 13 ( 12 usr; 1 con; 0-4 aty)
% Number of variables : 30 ( 1 ^; 22 !; 0 ?; 30 :)
% ( 7 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : Exported from core HOL Light.
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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/sets/INSERT',type,
'const/sets/INSERT':
!>[A: $tType] : ( A > ( 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/sets/EMPTY',type,
'const/sets/EMPTY':
!>[A: $tType] : ( A > $o ) ).
thf('thf_const_const/realax/real_of_num',type,
'const/realax/real_of_num': 'type/nums/num' > 'type/realax/real' ).
thf('thf_const_const/realax/real_add',type,
'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/nsum',type,
'const/iterate/nsum':
!>[A: $tType] : ( ( A > $o ) > ( A > 'type/nums/num' ) > 'type/nums/num' ) ).
thf('thf_const_const/class/COND',type,
'const/class/COND':
!>[A: $tType] : ( $o > A > A > A ) ).
thf('thf_const_const/arith/+',type,
'const/arith/+': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' ).
thf('thm/sets/FINITE_INDUCT_STRONG_',axiom,
! [A: $tType,P: ( A > $o ) > $o] :
( ( ( P @ ( 'const/sets/EMPTY' @ A ) )
& ! [A0: A,A1: A > $o] :
( ( ( P @ A1 )
& ~ ( 'const/sets/IN' @ A @ A0 @ A1 )
& ( 'const/sets/FINITE' @ A @ A1 ) )
=> ( P @ ( 'const/sets/INSERT' @ A @ A0 @ A1 ) ) ) )
=> ! [A0: A > $o] :
( ( 'const/sets/FINITE' @ A @ A0 )
=> ( P @ A0 ) ) ) ).
thf('thm/iterate/SUM_CLAUSES_1',axiom,
! [A: $tType,A0: A,A1: A > 'type/realax/real',A2: A > $o] :
( ( 'const/sets/FINITE' @ A @ A2 )
=> ( ( 'const/iterate/sum' @ A @ ( 'const/sets/INSERT' @ A @ A0 @ A2 ) @ A1 )
= ( 'const/class/COND' @ 'type/realax/real' @ ( 'const/sets/IN' @ A @ A0 @ A2 ) @ ( 'const/iterate/sum' @ A @ A2 @ A1 ) @ ( 'const/realax/real_add' @ ( A1 @ A0 ) @ ( 'const/iterate/sum' @ A @ A2 @ A1 ) ) ) ) ) ).
thf('thm/realax/REAL_OF_NUM_ADD_',axiom,
! [A: 'type/nums/num',A0: 'type/nums/num'] :
( ( 'const/realax/real_add' @ ( 'const/realax/real_of_num' @ A ) @ ( 'const/realax/real_of_num' @ A0 ) )
= ( 'const/realax/real_of_num' @ ( 'const/arith/+' @ A @ A0 ) ) ) ).
thf('thm/iterate/NSUM_CLAUSES_1',axiom,
! [A: $tType,A0: A,A1: A > 'type/nums/num',A2: A > $o] :
( ( 'const/sets/FINITE' @ A @ A2 )
=> ( ( 'const/iterate/nsum' @ A @ ( 'const/sets/INSERT' @ A @ A0 @ A2 ) @ A1 )
= ( 'const/class/COND' @ 'type/nums/num' @ ( 'const/sets/IN' @ A @ A0 @ A2 ) @ ( 'const/iterate/nsum' @ A @ A2 @ A1 ) @ ( 'const/arith/+' @ ( A1 @ A0 ) @ ( 'const/iterate/nsum' @ A @ A2 @ A1 ) ) ) ) ) ).
thf('thm/iterate/SUM_CLAUSES_0',axiom,
! [A: $tType,A0: A > 'type/realax/real'] :
( ( 'const/iterate/sum' @ A @ ( 'const/sets/EMPTY' @ A ) @ A0 )
= ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ).
thf('thm/iterate/NSUM_CLAUSES_0',axiom,
! [A: $tType,A0: A > 'type/nums/num'] :
( ( 'const/iterate/nsum' @ A @ ( 'const/sets/EMPTY' @ A ) @ A0 )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ).
thf('thm/iterate/REAL_OF_NUM_SUM_',conjecture,
! [A: $tType,A0: A > 'type/nums/num',A1: A > $o] :
( ( 'const/sets/FINITE' @ A @ A1 )
=> ( ( 'const/realax/real_of_num' @ ( 'const/iterate/nsum' @ A @ A1 @ A0 ) )
= ( 'const/iterate/sum' @ A @ A1
@ ^ [A2: A] : ( 'const/realax/real_of_num' @ ( A0 @ A2 ) ) ) ) ) ).
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