TPTP Problem File: ANA130^1.p
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% File : ANA130^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Analysis
% Problem : POLYNOMIAL_FUNCTION_INDUCT
% Version : Especial.
% English :
% Refs : [Kal16] Kalisyk (2016), Email to Geoff Sutcliffe
% Source : [Kal16]
% Names : POLYNOMIAL_FUNCTION_INDUCT_.p [Kal16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 28 ( 11 unt; 15 typ; 0 def)
% Number of atoms : 21 ( 11 equ; 0 cnn)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 139 ( 0 ~; 0 |; 6 &; 126 @)
% ( 0 <=>; 7 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 38 ( 38 >; 0 *; 0 +; 0 <<)
% Number of symbols : 14 ( 13 usr; 1 con; 0-3 aty)
% Number of variables : 45 ( 7 ^; 35 !; 2 ?; 45 :)
% ( 1 !>; 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/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/realax/real_add',type,
'const/realax/real_add': 'type/realax/real' > 'type/realax/real' > 'type/realax/real' ).
thf('thf_const_const/nums/SUC',type,
'const/nums/SUC': 'type/nums/num' > 'type/nums/num' ).
thf('thf_const_const/nums/NUMERAL',type,
'const/nums/NUMERAL': 'type/nums/num' > 'type/nums/num' ).
thf('thf_const_const/nums/BIT1',type,
'const/nums/BIT1': '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' > 'type/nums/num' ).
thf('thf_const_const/arith/<=',type,
'const/arith/<=': 'type/nums/num' > 'type/nums/num' > $o ).
thf('thm/nums/num_INDUCTION_',axiom,
! [P: 'type/nums/num' > $o] :
( ( ( P @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
& ! [A: 'type/nums/num'] :
( ( P @ A )
=> ( P @ ( 'const/nums/SUC' @ A ) ) ) )
=> ! [A: 'type/nums/num'] : ( P @ A ) ) ).
thf('thm/iterate/SUM_RMUL_',axiom,
! [A: $tType,A0: A > 'type/realax/real',A1: 'type/realax/real',A2: A > $o] :
( ( 'const/iterate/sum' @ A @ A2
@ ^ [A3: A] : ( 'const/realax/real_mul' @ ( A0 @ A3 ) @ A1 ) )
= ( 'const/realax/real_mul' @ ( 'const/iterate/sum' @ A @ A2 @ A0 ) @ A1 ) ) ).
thf('thm/realax/REAL_MUL_ASSOC_',axiom,
! [A: 'type/realax/real',A0: 'type/realax/real',A1: 'type/realax/real'] :
( ( 'const/realax/real_mul' @ A @ ( 'const/realax/real_mul' @ A0 @ A1 ) )
= ( 'const/realax/real_mul' @ ( 'const/realax/real_mul' @ A @ A0 ) @ A1 ) ) ).
thf('thm/real/REAL_POW_1_',axiom,
! [A: 'type/realax/real'] :
( ( 'const/realax/real_pow' @ A @ ( 'const/nums/NUMERAL' @ ( 'const/nums/BIT1' @ 'const/nums/_0' ) ) )
= A ) ).
thf('thm/real/REAL_POW_ADD_',axiom,
! [A: 'type/realax/real',A0: 'type/nums/num',A1: 'type/nums/num'] :
( ( 'const/realax/real_pow' @ A @ ( 'const/arith/+' @ A0 @ A1 ) )
= ( 'const/realax/real_mul' @ ( 'const/realax/real_pow' @ A @ A0 ) @ ( 'const/realax/real_pow' @ A @ A1 ) ) ) ).
thf('thm/iterate/SUM_OFFSET_',axiom,
! [A: 'type/nums/num',A0: 'type/nums/num' > 'type/realax/real',A1: 'type/nums/num',A2: 'type/nums/num'] :
( ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/arith/+' @ A1 @ A ) @ ( 'const/arith/+' @ A2 @ A ) ) @ A0 )
= ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ A1 @ A2 )
@ ^ [A3: 'type/nums/num'] : ( A0 @ ( 'const/arith/+' @ A3 @ A ) ) ) ) ).
thf('thm/realax/real_pow_0',axiom,
! [A: 'type/realax/real'] :
( ( 'const/realax/real_pow' @ A @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ ( 'const/nums/BIT1' @ 'const/nums/_0' ) ) ) ) ).
thf('thm/arith/ADD1_',axiom,
! [A: 'type/nums/num'] :
( ( 'const/nums/SUC' @ A )
= ( 'const/arith/+' @ A @ ( 'const/nums/NUMERAL' @ ( 'const/nums/BIT1' @ 'const/nums/_0' ) ) ) ) ).
thf('thm/iterate/SUM_CLAUSES_LEFT_',axiom,
! [A: 'type/nums/num' > 'type/realax/real',A0: 'type/nums/num',A1: 'type/nums/num'] :
( ( 'const/arith/<=' @ A0 @ A1 )
=> ( ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ A0 @ A1 ) @ A )
= ( 'const/realax/real_add' @ ( A @ A0 ) @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/arith/+' @ A0 @ ( 'const/nums/NUMERAL' @ ( 'const/nums/BIT1' @ 'const/nums/_0' ) ) ) @ A1 ) @ A ) ) ) ) ).
thf('thm/arith/LE_0_',axiom,
! [A: 'type/nums/num'] : ( 'const/arith/<=' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) @ A ) ).
thf('thm/iterate/SUM_SING_NUMSEG_',axiom,
! [A: 'type/nums/num' > 'type/realax/real',A0: 'type/nums/num'] :
( ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ A0 @ A0 ) @ A )
= ( 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_INDUCT_',conjecture,
! [P: ( 'type/realax/real' > 'type/realax/real' ) > $o] :
( ( ( P
@ ^ [A: 'type/realax/real'] : A )
& ! [A: 'type/realax/real'] :
( P
@ ^ [A0: 'type/realax/real'] : A )
& ! [A: 'type/realax/real' > 'type/realax/real',A0: 'type/realax/real' > 'type/realax/real'] :
( ( ( P @ A )
& ( P @ A0 ) )
=> ( P
@ ^ [A1: 'type/realax/real'] : ( 'const/realax/real_add' @ ( A @ A1 ) @ ( A0 @ A1 ) ) ) )
& ! [A: 'type/realax/real' > 'type/realax/real',A0: 'type/realax/real' > 'type/realax/real'] :
( ( ( P @ A )
& ( P @ A0 ) )
=> ( P
@ ^ [A1: 'type/realax/real'] : ( 'const/realax/real_mul' @ ( A @ A1 ) @ ( A0 @ A1 ) ) ) ) )
=> ! [A: 'type/realax/real' > 'type/realax/real'] :
( ( 'const/iterate/polynomial_function' @ A )
=> ( P @ A ) ) ) ).
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