TPTP Problem File: ANA132^1.p
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%------------------------------------------------------------------------------
% File : ANA132^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Analysis
% Problem : POLYNOMIAL_FUNCTION_FINITE_ROOTS
% Version : Especial.
% English :
% Refs : [Kal16] Kalisyk (2016), Email to Geoff Sutcliffe
% Source : [Kal16]
% Names : POLYNOMIAL_FUNCTION_FINITE_ROOTS_.p [Kal16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 32 ( 10 unt; 18 typ; 0 def)
% Number of atoms : 51 ( 19 equ; 0 cnn)
% Maximal formula atoms : 3 ( 3 avg)
% Number of connectives : 134 ( 3 ~; 0 |; 3 &; 124 @)
% ( 0 <=>; 4 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 45 ( 45 >; 0 *; 0 +; 0 <<)
% Number of symbols : 18 ( 16 usr; 2 con; 0-4 aty)
% Number of variables : 52 ( 9 ^; 29 !; 7 ?; 52 :)
% ( 7 !>; 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/sets/UNIV',type,
'const/sets/UNIV':
!>[A: $tType] : ( A > $o ) ).
thf('thf_const_const/sets/SETSPEC',type,
'const/sets/SETSPEC':
!>[A: $tType] : ( A > $o > A > $o ) ).
thf('thf_const_const/sets/INFINITE',type,
'const/sets/INFINITE':
!>[A: $tType] : ( ( A > $o ) > $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 ) ).
thf('thf_const_const/realax/real_sub',type,
'const/realax/real_sub': 'type/realax/real' > 'type/realax/real' > 'type/realax/real' ).
thf('thf_const_const/realax/real_pow',type,
'const/realax/real_pow': 'type/realax/real' > 'type/nums/num' > 'type/realax/real' ).
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_mul',type,
'const/realax/real_mul': '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/polynomial_function',type,
'const/iterate/polynomial_function': ( 'type/realax/real' > 'type/realax/real' ) > $o ).
thf('thf_const_const/iterate/..',type,
'const/iterate/..': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' > $o ).
thf('thf_const_const/arith/<=',type,
'const/arith/<=': 'type/nums/num' > 'type/nums/num' > $o ).
thf('thm/iterate/SUM_0_',axiom,
! [A: $tType,A0: A > $o] :
( ( 'const/iterate/sum' @ A @ A0
@ ^ [A1: A] : ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) )
= ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ).
thf('thm/calc_int/REAL_MUL_LZERO_',axiom,
! [A: 'type/realax/real'] :
( ( 'const/realax/real_mul' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ A )
= ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ).
thf('thm/iterate/SUM_EQ_',axiom,
! [A: $tType,A0: A > 'type/realax/real',A1: A > 'type/realax/real',A2: A > $o] :
( ! [A3: A] :
( ( 'const/sets/IN' @ A @ A3 @ A2 )
=> ( ( A0 @ A3 )
= ( A1 @ A3 ) ) )
=> ( ( 'const/iterate/sum' @ A @ A2 @ A0 )
= ( 'const/iterate/sum' @ A @ A2 @ A1 ) ) ) ).
thf('thm/sets/IN_ELIM_THM_1',axiom,
! [A: $tType,A0: A > $o,A1: A] :
( ( 'const/sets/IN' @ A @ A1
@ ( 'const/sets/GSPEC' @ A
@ ^ [A2: A] :
? [A3: A] : ( 'const/sets/SETSPEC' @ A @ A2 @ ( A0 @ A3 ) @ A3 ) ) )
= ( A0 @ A1 ) ) ).
thf('thm/iterate/IN_NUMSEG_',axiom,
! [A: 'type/nums/num',A0: 'type/nums/num',A1: 'type/nums/num'] :
( ( 'const/sets/IN' @ 'type/nums/num' @ A1 @ ( 'const/iterate/..' @ A @ A0 ) )
= ( ( 'const/arith/<=' @ A @ A1 )
& ( 'const/arith/<=' @ A1 @ A0 ) ) ) ).
thf('thm/iterate/REAL_POLYFUN_FINITE_ROOTS_',axiom,
! [A: 'type/nums/num',A0: 'type/nums/num' > 'type/realax/real'] :
( ( 'const/sets/FINITE' @ 'type/realax/real'
@ ( 'const/sets/GSPEC' @ 'type/realax/real'
@ ^ [A1: 'type/realax/real'] :
? [A2: 'type/realax/real'] :
( 'const/sets/SETSPEC' @ 'type/realax/real' @ A1
@ ( ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) @ A )
@ ^ [A3: 'type/nums/num'] : ( 'const/realax/real_mul' @ ( A0 @ A3 ) @ ( 'const/realax/real_pow' @ A2 @ A3 ) ) )
= ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) )
@ A2 ) ) )
= ( ? [A1: 'type/nums/num'] :
( ( 'const/sets/IN' @ 'type/nums/num' @ A1 @ ( 'const/iterate/..' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) @ A ) )
& ( ( A0 @ A1 )
!= ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ) ) ).
thf('thm/sets/real_INFINITE_',axiom,
'const/sets/INFINITE' @ 'type/realax/real' @ ( 'const/sets/UNIV' @ 'type/realax/real' ) ).
thf('thm/sets/UNIV_GSPEC_',axiom,
! [A: $tType] :
( ( 'const/sets/GSPEC' @ A
@ ^ [A0: A] :
? [A1: A] : ( 'const/sets/SETSPEC' @ A @ A0 @ $true @ A1 ) )
= ( 'const/sets/UNIV' @ A ) ) ).
thf('thm/sets/INFINITE_',axiom,
! [A: $tType,A0: A > $o] :
( ( 'const/sets/INFINITE' @ A @ A0 )
= ( ~ ( 'const/sets/FINITE' @ A @ A0 ) ) ) ).
thf('thm/iterate/polynomial_function_',axiom,
! [A: 'type/realax/real' > 'type/realax/real'] :
( ( 'const/iterate/polynomial_function' @ A )
= ( ? [A0: 'type/nums/num',A1: 'type/nums/num' > 'type/realax/real'] :
! [A2: 'type/realax/real'] :
( ( A @ A2 )
= ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) @ A0 )
@ ^ [A3: 'type/nums/num'] : ( 'const/realax/real_mul' @ ( A1 @ A3 ) @ ( 'const/realax/real_pow' @ A2 @ A3 ) ) ) ) ) ) ).
thf('thm/iterate/POLYNOMIAL_FUNCTION_SUB_',axiom,
! [A: 'type/realax/real' > 'type/realax/real',A0: 'type/realax/real' > 'type/realax/real'] :
( ( ( 'const/iterate/polynomial_function' @ A )
& ( 'const/iterate/polynomial_function' @ A0 ) )
=> ( 'const/iterate/polynomial_function'
@ ^ [A1: 'type/realax/real'] : ( 'const/realax/real_sub' @ ( A @ A1 ) @ ( A0 @ A1 ) ) ) ) ).
thf('thm/iterate/POLYNOMIAL_FUNCTION_CONST_',axiom,
! [A: 'type/realax/real'] :
( 'const/iterate/polynomial_function'
@ ^ [A0: 'type/realax/real'] : A ) ).
thf('thm/realarith/REAL_SUB_0_',axiom,
! [A: 'type/realax/real',A0: 'type/realax/real'] :
( ( ( 'const/realax/real_sub' @ A @ A0 )
= ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) )
= ( A = A0 ) ) ).
thf('thm/iterate/POLYNOMIAL_FUNCTION_FINITE_ROOTS_',conjecture,
! [A: 'type/realax/real' > 'type/realax/real',A0: 'type/realax/real'] :
( ( 'const/iterate/polynomial_function' @ A )
=> ( ( 'const/sets/FINITE' @ 'type/realax/real'
@ ( 'const/sets/GSPEC' @ 'type/realax/real'
@ ^ [A1: 'type/realax/real'] :
? [A2: 'type/realax/real'] :
( 'const/sets/SETSPEC' @ 'type/realax/real' @ A1
@ ( ( A @ A2 )
= A0 )
@ A2 ) ) )
= ( ~ ! [A1: 'type/realax/real'] :
( ( A @ A1 )
= A0 ) ) ) ) ).
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