TPTP Problem File: SWW503_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW503_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fundamental Theorem of Algebra line 235
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : fta_235 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.25 v7.1.0, 0.33 v6.4.0
% Syntax : Number of formulae : 228 ( 88 unt; 54 typ; 0 def)
% Number of atoms : 367 ( 85 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 225 ( 32 ~; 7 |; 21 &)
% ( 30 <=>; 135 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 16 ( 10 >; 6 *; 0 +; 0 <<)
% Number of predicates : 36 ( 35 usr; 0 prp; 1-3 aty)
% Number of functors : 15 ( 15 usr; 6 con; 0-3 aty)
% Number of variables : 292 ( 248 !; 2 ?; 292 :)
% ( 42 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:17:55
%------------------------------------------------------------------------------
%----Should-be-implicit typings (4)
tff(ty_tc_Complex_Ocomplex,type,
complex: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_RealDef_Oreal,type,
real: $tType ).
%----Explicit typings (50)
tff(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Power_Opower,type,
power:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__field,type,
real_field:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ono__zero__divisors,type,
no_zero_divisors:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Ofield__inverse__zero,type,
field_inverse_zero:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__algebra__1,type,
real_algebra_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__normed__field,type,
real_normed_field:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__normed__vector,type,
real_normed_vector:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_11004092258visors:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere216010020id_add:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__normed__algebra__1,type,
real_n2089651433ebra_1:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Odivision__ring__inverse__zero,type,
divisi14063676e_zero:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__normed__div__algebra,type,
real_n1866405975lgebra:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Olinordered__field__inverse__zero,type,
linord1117847801e_zero:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A: $tType] : $o ).
tff(sy_c_Fields_Oinverse__class_Odivide,type,
inverse_divide:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_NthRoot_Oroot,type,
root: ( nat * real ) > real ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Parity_Oeven__odd__class_Oeven,type,
even_odd_even:
!>[A: $tType] : ( A > $o ) ).
tff(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( ( A * nat ) > A ) ).
tff(sy_c_RealVector_Onorm__class_Onorm,type,
norm_norm:
!>[A: $tType] : ( A > real ) ).
tff(sy_c_RealVector_Oof__real,type,
of_real:
!>[A: $tType] : ( real > A ) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_b,type,
b: complex ).
tff(sy_v_n,type,
n: nat ).
tff(sy_v_na____,type,
na: nat ).
tff(sy_v_v____,type,
v: complex ).
%----Relevant facts (98)
tff(fact_0__096root_An_A_Icmod_Ab_J_A_094_An_A_061_Acmod_Ab_096,axiom,
power_power(real,root(na,norm_norm(complex,b)),na) = norm_norm(complex,b) ).
tff(fact_1_n,axiom,
na != zero_zero(nat) ).
tff(fact_2_b,axiom,
b != zero_zero(complex) ).
tff(fact_3_o,axiom,
~ even_odd_even(nat,na) ).
tff(fact_4_assms_I2_J,axiom,
n != zero_zero(nat) ).
tff(fact_5_of__real__divide,axiom,
! [A: $tType] :
( ( field_inverse_zero(A)
& real_field(A) )
=> ! [Y: real,X1: real] : of_real(A,inverse_divide(real,X1,Y)) = inverse_divide(A,of_real(A,X1),of_real(A,Y)) ) ).
tff(fact_6_of__real__power,axiom,
! [A: $tType] :
( real_algebra_1(A)
=> ! [N: nat,X1: real] : of_real(A,power_power(real,X1,N)) = power_power(A,of_real(A,X1),N) ) ).
tff(fact_7_th0,axiom,
norm_norm(complex,inverse_divide(complex,of_real(complex,norm_norm(complex,b)),b)) = one_one(real) ).
tff(fact_8_complex__of__real__power,axiom,
! [N: nat,X1: real] : power_power(complex,of_real(complex,X1),N) = of_real(complex,power_power(real,X1,N)) ).
tff(fact_9_of__real__eq__iff,axiom,
! [A: $tType] :
( real_algebra_1(A)
=> ! [Y1: real,X: real] :
( ( of_real(A,X) = of_real(A,Y1) )
<=> ( X = Y1 ) ) ) ).
tff(fact_10_v,axiom,
ord_less(real,norm_norm(complex,plus_plus(complex,inverse_divide(complex,of_real(complex,norm_norm(complex,b)),b),power_power(complex,v,na))),one_one(real)) ).
tff(fact_11_norm__divide,axiom,
! [A: $tType] :
( ( field_inverse_zero(A)
& real_normed_field(A) )
=> ! [B: A,A1: A] : norm_norm(A,inverse_divide(A,A1,B)) = inverse_divide(real,norm_norm(A,A1),norm_norm(A,B)) ) ).
tff(fact_12_norm__power,axiom,
! [A: $tType] :
( real_n1866405975lgebra(A)
=> ! [N: nat,X1: A] : norm_norm(A,power_power(A,X1,N)) = power_power(real,norm_norm(A,X1),N) ) ).
tff(fact_13_power__divide,axiom,
! [A: $tType] :
( field_inverse_zero(A)
=> ! [N: nat,B: A,A1: A] : power_power(A,inverse_divide(A,A1,B),N) = inverse_divide(A,power_power(A,A1,N),power_power(A,B,N)) ) ).
tff(fact_14__096EX_Av_O_Acmod_A_Icomplex__of__real_A_Icmod_Ab_J_A_P_Ab_A_L_Av_A_094_An_J_A_060_A1_096,axiom,
? [V: complex] : ord_less(real,norm_norm(complex,plus_plus(complex,inverse_divide(complex,of_real(complex,norm_norm(complex,b)),b),power_power(complex,V,na))),one_one(real)) ).
tff(fact_15__096_B_Bthesis_O_A_I_B_Bv_O_Acmod_A_Icomplex__of__real_A_Icmod_Ab_J_A_P_Ab_A_L_Av_A_094_An_J_A_060_A1_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [V: complex] : ~ ord_less(real,norm_norm(complex,plus_plus(complex,inverse_divide(complex,of_real(complex,norm_norm(complex,b)),b),power_power(complex,V,na))),one_one(real)) ).
tff(fact_16_power__one,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat] : power_power(A,one_one(A),N) = one_one(A) ) ).
tff(fact_17_norm__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ( norm_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).
tff(fact_18_norm__eq__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [X: A] :
( ( norm_norm(A,X) = zero_zero(real) )
<=> ( X = zero_zero(A) ) ) ) ).
tff(fact_19_of__real__0,axiom,
! [A: $tType] :
( real_algebra_1(A)
=> ( of_real(A,zero_zero(real)) = zero_zero(A) ) ) ).
tff(fact_20_of__real__eq__0__iff,axiom,
! [A: $tType] :
( real_algebra_1(A)
=> ! [X: real] :
( ( of_real(A,X) = zero_zero(A) )
<=> ( X = zero_zero(real) ) ) ) ).
tff(fact_21_of__real__add,axiom,
! [A: $tType] :
( real_algebra_1(A)
=> ! [Y: real,X1: real] : of_real(A,plus_plus(real,X1,Y)) = plus_plus(A,of_real(A,X1),of_real(A,Y)) ) ).
tff(fact_22_power__eq__0__iff,axiom,
! [A: $tType] :
( ( power(A)
& mult_zero(A)
& no_zero_divisors(A)
& zero_neq_one(A) )
=> ! [Nb: nat,A2: A] :
( ( power_power(A,A2,Nb) = zero_zero(A) )
<=> ( ( A2 = zero_zero(A) )
& ( Nb != zero_zero(nat) ) ) ) ) ).
tff(fact_23_power__0,axiom,
! [A: $tType] :
( power(A)
=> ! [A1: A] : power_power(A,A1,zero_zero(nat)) = one_one(A) ) ).
tff(fact_24_zero__less__norm__iff,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [X: A] :
( ord_less(real,zero_zero(real),norm_norm(A,X))
<=> ( X != zero_zero(A) ) ) ) ).
tff(fact_25_norm__one,axiom,
! [A: $tType] :
( real_n2089651433ebra_1(A)
=> ( norm_norm(A,one_one(A)) = one_one(real) ) ) ).
tff(fact_26_of__real__1,axiom,
! [A: $tType] :
( real_algebra_1(A)
=> ( of_real(A,one_one(real)) = one_one(A) ) ) ).
tff(fact_27_power__0__left,axiom,
! [A: $tType] :
( ( power(A)
& semiring_0(A) )
=> ! [N: nat] :
( ( ( N = zero_zero(nat) )
=> ( power_power(A,zero_zero(A),N) = one_one(A) ) )
& ( ( N != zero_zero(nat) )
=> ( power_power(A,zero_zero(A),N) = zero_zero(A) ) ) ) ) ).
tff(fact_28_zero__less__power,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,A1: A] :
( ord_less(A,zero_zero(A),A1)
=> ord_less(A,zero_zero(A),power_power(A,A1,N)) ) ) ).
tff(fact_29_power__inject__exp,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Nb: nat,Ma: nat,A2: A] :
( ord_less(A,one_one(A),A2)
=> ( ( power_power(A,A2,Ma) = power_power(A,A2,Nb) )
<=> ( Ma = Nb ) ) ) ) ).
tff(fact_30_power__strict__increasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Y1: nat,X: nat,Ba: A] :
( ord_less(A,one_one(A),Ba)
=> ( ord_less(A,power_power(A,Ba,X),power_power(A,Ba,Y1))
<=> ord_less(nat,X,Y1) ) ) ) ).
tff(fact_31_one__less__power,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,A1: A] :
( ord_less(A,one_one(A),A1)
=> ( ord_less(nat,zero_zero(nat),N)
=> ord_less(A,one_one(A),power_power(A,A1,N)) ) ) ) ).
tff(fact_32_power__less__imp__less__exp,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,M1: nat,A1: A] :
( ord_less(A,one_one(A),A1)
=> ( ord_less(A,power_power(A,A1,M1),power_power(A,A1,N))
=> ord_less(nat,M1,N) ) ) ) ).
tff(fact_33_power__strict__increasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A1: A,N1: nat,N: nat] :
( ord_less(nat,N,N1)
=> ( ord_less(A,one_one(A),A1)
=> ord_less(A,power_power(A,A1,N),power_power(A,A1,N1)) ) ) ) ).
tff(fact_34_power__strict__decreasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A1: A,N1: nat,N: nat] :
( ord_less(nat,N,N1)
=> ( ord_less(A,zero_zero(A),A1)
=> ( ord_less(A,A1,one_one(A))
=> ord_less(A,power_power(A,A1,N1),power_power(A,A1,N)) ) ) ) ) ).
tff(fact_35_norm__add__less,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [S: real,Y: A,R: real,X1: A] :
( ord_less(real,norm_norm(A,X1),R)
=> ( ord_less(real,norm_norm(A,Y),S)
=> ord_less(real,norm_norm(A,plus_plus(A,X1,Y)),plus_plus(real,R,S)) ) ) ) ).
tff(fact_36_field__power__not__zero,axiom,
! [A: $tType] :
( ring_11004092258visors(A)
=> ! [N: nat,A1: A] :
( ( A1 != zero_zero(A) )
=> ( power_power(A,A1,N) != zero_zero(A) ) ) ) ).
tff(fact_37_power__one__over,axiom,
! [A: $tType] :
( field_inverse_zero(A)
=> ! [N: nat,A1: A] : inverse_divide(A,one_one(A),power_power(A,A1,N)) = power_power(A,inverse_divide(A,one_one(A),A1),N) ) ).
tff(fact_38_nonzero__power__divide,axiom,
! [A: $tType] :
( field(A)
=> ! [N: nat,A1: A,B: A] :
( ( B != zero_zero(A) )
=> ( power_power(A,inverse_divide(A,A1,B),N) = inverse_divide(A,power_power(A,A1,N),power_power(A,B,N)) ) ) ) ).
tff(fact_39_nonzero__norm__divide,axiom,
! [A: $tType] :
( real_normed_field(A)
=> ! [A1: A,B: A] :
( ( B != zero_zero(A) )
=> ( norm_norm(A,inverse_divide(A,A1,B)) = inverse_divide(real,norm_norm(A,A1),norm_norm(A,B)) ) ) ) ).
tff(fact_40_divide__self__if,axiom,
! [A: $tType] :
( divisi14063676e_zero(A)
=> ! [A1: A] :
( ( ( A1 = zero_zero(A) )
=> ( inverse_divide(A,A1,A1) = zero_zero(A) ) )
& ( ( A1 != zero_zero(A) )
=> ( inverse_divide(A,A1,A1) = one_one(A) ) ) ) ) ).
tff(fact_41_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ord_less(A,zero_zero(A),plus_plus(A,A2,A2))
<=> ord_less(A,zero_zero(A),A2) ) ) ).
tff(fact_42_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ord_less(A,plus_plus(A,A2,A2),zero_zero(A))
<=> ord_less(A,A2,zero_zero(A)) ) ) ).
tff(fact_43_zero__less__power__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Nb: nat,X: A] :
( ord_less(A,zero_zero(A),power_power(A,X,Nb))
<=> ( ( Nb = zero_zero(nat) )
| ( even_odd_even(nat,Nb)
& ( X != zero_zero(A) ) )
| ( ~ even_odd_even(nat,Nb)
& ord_less(A,zero_zero(A),X) ) ) ) ) ).
tff(fact_44_divide__zero,axiom,
! [A: $tType] :
( divisi14063676e_zero(A)
=> ! [A1: A] : inverse_divide(A,A1,zero_zero(A)) = zero_zero(A) ) ).
tff(fact_45_divide__zero__left,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A1: A] : inverse_divide(A,zero_zero(A),A1) = zero_zero(A) ) ).
tff(fact_46_add__less__cancel__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [Ba: A,C: A,A2: A] :
( ord_less(A,plus_plus(A,A2,C),plus_plus(A,Ba,C))
<=> ord_less(A,A2,Ba) ) ) ).
tff(fact_47_add__less__cancel__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [Ba: A,A2: A,C: A] :
( ord_less(A,plus_plus(A,C,A2),plus_plus(A,C,Ba))
<=> ord_less(A,A2,Ba) ) ) ).
tff(fact_48_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,A2: A,Ba: A] :
( ( plus_plus(A,Ba,A2) = plus_plus(A,C,A2) )
<=> ( Ba = C ) ) ) ).
tff(fact_49_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,Ba: A,A2: A] :
( ( plus_plus(A,A2,Ba) = plus_plus(A,A2,C) )
<=> ( Ba = C ) ) ) ).
tff(fact_50_even__sum__nat,axiom,
! [Y1: nat,X: nat] :
( even_odd_even(nat,plus_plus(nat,X,Y1))
<=> ( ( even_odd_even(nat,X)
& even_odd_even(nat,Y1) )
| ( ~ even_odd_even(nat,X)
& ~ even_odd_even(nat,Y1) ) ) ) ).
tff(fact_51_even__add,axiom,
! [Nb: nat,Ma: nat] :
( even_odd_even(nat,plus_plus(nat,Ma,Nb))
<=> ( even_odd_even(nat,Ma)
<=> even_odd_even(nat,Nb) ) ) ).
tff(fact_52_odd__add,axiom,
! [Nb: nat,Ma: nat] :
( ~ even_odd_even(nat,plus_plus(nat,Ma,Nb))
<=> ~ ( ~ even_odd_even(nat,Ma)
<=> ~ even_odd_even(nat,Nb) ) ) ).
tff(fact_53_double__zero__sym,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ( zero_zero(A) = plus_plus(A,A2,A2) )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_54_nat__zero__less__power__iff,axiom,
! [Nb: nat,X: nat] :
( ord_less(nat,zero_zero(nat),power_power(nat,X,Nb))
<=> ( ord_less(nat,zero_zero(nat),X)
| ( Nb = zero_zero(nat) ) ) ) ).
tff(fact_55_even__power__nat,axiom,
! [Y1: nat,X: nat] :
( even_odd_even(nat,power_power(nat,X,Y1))
<=> ( even_odd_even(nat,X)
& ord_less(nat,zero_zero(nat),Y1) ) ) ).
tff(fact_56_zero__less__power__nat__eq,axiom,
! [Nb: nat,X: nat] :
( ord_less(nat,zero_zero(nat),power_power(nat,X,Nb))
<=> ( ( Nb = zero_zero(nat) )
| ord_less(nat,zero_zero(nat),X) ) ) ).
tff(fact_57_odd__1__nat,axiom,
~ even_odd_even(nat,one_one(nat)) ).
tff(fact_58_nat__power__less__imp__less,axiom,
! [N: nat,M1: nat,I: nat] :
( ord_less(nat,zero_zero(nat),I)
=> ( ord_less(nat,power_power(nat,I,M1),power_power(nat,I,N))
=> ord_less(nat,M1,N) ) ) ).
tff(fact_59_power__one__right,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A1: A] : power_power(A,A1,one_one(nat)) = A1 ) ).
tff(fact_60_odd__pos,axiom,
! [N: nat] :
( ~ even_odd_even(nat,N)
=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_61_norm__not__less__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [X1: A] : ~ ord_less(real,norm_norm(A,X1),zero_zero(real)) ) ).
tff(fact_62_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X: A] :
( ( zero_zero(A) = X )
<=> ( X = zero_zero(A) ) ) ) ).
tff(fact_63_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,A1: A,B: A] :
( ( plus_plus(A,B,A1) = plus_plus(A,C1,A1) )
=> ( B = C1 ) ) ) ).
tff(fact_64_add__imp__eq,axiom,
! [A: $tType] :
( cancel146912293up_add(A)
=> ! [C1: A,B: A,A1: A] :
( ( plus_plus(A,A1,B) = plus_plus(A,A1,C1) )
=> ( B = C1 ) ) ) ).
tff(fact_65_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,B: A,A1: A] :
( ( plus_plus(A,A1,B) = plus_plus(A,A1,C1) )
=> ( B = C1 ) ) ) ).
tff(fact_66_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [C1: A,B: A,A1: A] : plus_plus(A,plus_plus(A,A1,B),C1) = plus_plus(A,A1,plus_plus(A,B,C1)) ) ).
tff(fact_67_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X: A] :
( ( one_one(A) = X )
<=> ( X = one_one(A) ) ) ) ).
tff(fact_68_nonzero__of__real__divide,axiom,
! [A: $tType] :
( real_field(A)
=> ! [X1: real,Y: real] :
( ( Y != zero_zero(real) )
=> ( of_real(A,inverse_divide(real,X1,Y)) = inverse_divide(A,of_real(A,X1),of_real(A,Y)) ) ) ) ).
tff(fact_69_add_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A1: A] : plus_plus(A,A1,zero_zero(A)) = A1 ) ).
tff(fact_70_add__0__right,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A1: A] : plus_plus(A,A1,zero_zero(A)) = A1 ) ).
tff(fact_71_add__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A1: A] : plus_plus(A,zero_zero(A),A1) = A1 ) ).
tff(fact_72_add__0__left,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A1: A] : plus_plus(A,zero_zero(A),A1) = A1 ) ).
tff(fact_73_add__less__imp__less__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B: A,A1: A,C1: A] :
( ord_less(A,plus_plus(A,C1,A1),plus_plus(A,C1,B))
=> ord_less(A,A1,B) ) ) ).
tff(fact_74_add__less__imp__less__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B: A,C1: A,A1: A] :
( ord_less(A,plus_plus(A,A1,C1),plus_plus(A,B,C1))
=> ord_less(A,A1,B) ) ) ).
tff(fact_75_add__strict__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [D: A,C1: A,B: A,A1: A] :
( ord_less(A,A1,B)
=> ( ord_less(A,C1,D)
=> ord_less(A,plus_plus(A,A1,C1),plus_plus(A,B,D)) ) ) ) ).
tff(fact_76_add__strict__left__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [C1: A,B: A,A1: A] :
( ord_less(A,A1,B)
=> ord_less(A,plus_plus(A,C1,A1),plus_plus(A,C1,B)) ) ) ).
tff(fact_77_add__strict__right__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [C1: A,B: A,A1: A] :
( ord_less(A,A1,B)
=> ord_less(A,plus_plus(A,A1,C1),plus_plus(A,B,C1)) ) ) ).
tff(fact_78_add__divide__distrib,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C1: A,B: A,A1: A] : inverse_divide(A,plus_plus(A,A1,B),C1) = plus_plus(A,inverse_divide(A,A1,C1),inverse_divide(A,B,C1)) ) ).
tff(fact_79_divide__1,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A1: A] : inverse_divide(A,A1,one_one(A)) = A1 ) ).
tff(fact_80_even__zero__nat,axiom,
even_odd_even(nat,zero_zero(nat)) ).
tff(fact_81_add__neg__neg,axiom,
! [A: $tType] :
( ordere216010020id_add(A)
=> ! [B: A,A1: A] :
( ord_less(A,A1,zero_zero(A))
=> ( ord_less(A,B,zero_zero(A))
=> ord_less(A,plus_plus(A,A1,B),zero_zero(A)) ) ) ) ).
tff(fact_82_add__pos__pos,axiom,
! [A: $tType] :
( ordere216010020id_add(A)
=> ! [B: A,A1: A] :
( ord_less(A,zero_zero(A),A1)
=> ( ord_less(A,zero_zero(A),B)
=> ord_less(A,zero_zero(A),plus_plus(A,A1,B)) ) ) ) ).
tff(fact_83_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C1: A,A1: A,B: A] :
( ord_less(A,B,A1)
=> ( ord_less(A,C1,zero_zero(A))
=> ord_less(A,inverse_divide(A,A1,C1),inverse_divide(A,B,C1)) ) ) ) ).
tff(fact_84_divide__strict__right__mono,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C1: A,B: A,A1: A] :
( ord_less(A,A1,B)
=> ( ord_less(A,zero_zero(A),C1)
=> ord_less(A,inverse_divide(A,A1,C1),inverse_divide(A,B,C1)) ) ) ) ).
tff(fact_85_divide__neg__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,X1: A] :
( ord_less(A,X1,zero_zero(A))
=> ( ord_less(A,Y,zero_zero(A))
=> ord_less(A,zero_zero(A),inverse_divide(A,X1,Y)) ) ) ) ).
tff(fact_86_divide__neg__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,X1: A] :
( ord_less(A,X1,zero_zero(A))
=> ( ord_less(A,zero_zero(A),Y)
=> ord_less(A,inverse_divide(A,X1,Y),zero_zero(A)) ) ) ) ).
tff(fact_87_divide__pos__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,X1: A] :
( ord_less(A,zero_zero(A),X1)
=> ( ord_less(A,Y,zero_zero(A))
=> ord_less(A,inverse_divide(A,X1,Y),zero_zero(A)) ) ) ) ).
tff(fact_88_divide__pos__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,X1: A] :
( ord_less(A,zero_zero(A),X1)
=> ( ord_less(A,zero_zero(A),Y)
=> ord_less(A,zero_zero(A),inverse_divide(A,X1,Y)) ) ) ) ).
tff(fact_89_divide__less__cancel,axiom,
! [A: $tType] :
( linord1117847801e_zero(A)
=> ! [Ba: A,C: A,A2: A] :
( ord_less(A,inverse_divide(A,A2,C),inverse_divide(A,Ba,C))
<=> ( ( ord_less(A,zero_zero(A),C)
=> ord_less(A,A2,Ba) )
& ( ord_less(A,C,zero_zero(A))
=> ord_less(A,Ba,A2) )
& ( C != zero_zero(A) ) ) ) ) ).
tff(fact_90_divide__less__0__iff,axiom,
! [A: $tType] :
( linord1117847801e_zero(A)
=> ! [Ba: A,A2: A] :
( ord_less(A,inverse_divide(A,A2,Ba),zero_zero(A))
<=> ( ( ord_less(A,zero_zero(A),A2)
& ord_less(A,Ba,zero_zero(A)) )
| ( ord_less(A,A2,zero_zero(A))
& ord_less(A,zero_zero(A),Ba) ) ) ) ) ).
tff(fact_91_zero__less__divide__iff,axiom,
! [A: $tType] :
( linord1117847801e_zero(A)
=> ! [Ba: A,A2: A] :
( ord_less(A,zero_zero(A),inverse_divide(A,A2,Ba))
<=> ( ( ord_less(A,zero_zero(A),A2)
& ord_less(A,zero_zero(A),Ba) )
| ( ord_less(A,A2,zero_zero(A))
& ord_less(A,Ba,zero_zero(A)) ) ) ) ) ).
tff(fact_92_right__inverse__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A2: A,Ba: A] :
( ( Ba != zero_zero(A) )
=> ( ( inverse_divide(A,A2,Ba) = one_one(A) )
<=> ( A2 = Ba ) ) ) ) ).
tff(fact_93_divide__self,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A1: A] :
( ( A1 != zero_zero(A) )
=> ( inverse_divide(A,A1,A1) = one_one(A) ) ) ) ).
tff(fact_94_gt__half__sum,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B: A,A1: A] :
( ord_less(A,A1,B)
=> ord_less(A,inverse_divide(A,plus_plus(A,A1,B),plus_plus(A,one_one(A),one_one(A))),B) ) ) ).
tff(fact_95_less__half__sum,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B: A,A1: A] :
( ord_less(A,A1,B)
=> ord_less(A,A1,inverse_divide(A,plus_plus(A,A1,B),plus_plus(A,one_one(A),one_one(A)))) ) ) ).
tff(fact_96_power__less__zero__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Nb: nat,X: A] :
( ord_less(A,power_power(A,X,Nb),zero_zero(A))
<=> ( ~ even_odd_even(nat,Nb)
& ord_less(A,X,zero_zero(A)) ) ) ) ).
tff(fact_97_IH,axiom,
! [M: nat] :
( ord_less(nat,M,na)
=> ( ( M != zero_zero(nat) )
=> ? [Z: complex] : ord_less(real,norm_norm(complex,plus_plus(complex,one_one(complex),times_times(complex,b,power_power(complex,Z,M)))),one_one(real)) ) ) ).
%----Arities (73)
tff(arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(nat) ).
tff(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(nat) ).
tff(arity_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add(nat) ).
tff(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(nat) ).
tff(arity_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(nat) ).
tff(arity_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom(nat) ).
tff(arity_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(nat) ).
tff(arity_Nat_Onat___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(nat) ).
tff(arity_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(nat) ).
tff(arity_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one(nat) ).
tff(arity_Nat_Onat___Groups_Omonoid__mult,axiom,
monoid_mult(nat) ).
tff(arity_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring__0,axiom,
semiring_0(nat) ).
tff(arity_Nat_Onat___Rings_Omult__zero,axiom,
mult_zero(nat) ).
tff(arity_Nat_Onat___Power_Opower,axiom,
power(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_Nat_Onat___Groups_Oone,axiom,
one(nat) ).
tff(arity_RealDef_Oreal___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(real) ).
tff(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(real) ).
tff(arity_RealDef_Oreal___Fields_Olinordered__field__inverse__zero,axiom,
linord1117847801e_zero(real) ).
tff(arity_RealDef_Oreal___RealVector_Oreal__normed__div__algebra,axiom,
real_n1866405975lgebra(real) ).
tff(arity_RealDef_Oreal___Fields_Odivision__ring__inverse__zero,axiom,
divisi14063676e_zero(real) ).
tff(arity_RealDef_Oreal___RealVector_Oreal__normed__algebra__1,axiom,
real_n2089651433ebra_1(real) ).
tff(arity_RealDef_Oreal___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add(real) ).
tff(arity_RealDef_Oreal___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(real) ).
tff(arity_RealDef_Oreal___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(real) ).
tff(arity_RealDef_Oreal___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(real) ).
tff(arity_RealDef_Oreal___RealVector_Oreal__normed__vector,axiom,
real_normed_vector(real) ).
tff(arity_RealDef_Oreal___RealVector_Oreal__normed__field,axiom,
real_normed_field(real) ).
tff(arity_RealDef_Oreal___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(real) ).
tff(arity_RealDef_Oreal___RealVector_Oreal__algebra__1,axiom,
real_algebra_1(real) ).
tff(arity_RealDef_Oreal___Fields_Ofield__inverse__zero,axiom,
field_inverse_zero(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__semidom,axiom,
linordered_semidom(real) ).
tff(arity_RealDef_Oreal___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(real) ).
tff(arity_RealDef_Oreal___Fields_Olinordered__field,axiom,
linordered_field(real) ).
tff(arity_RealDef_Oreal___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(real) ).
tff(arity_RealDef_Oreal___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom(real) ).
tff(arity_RealDef_Oreal___RealVector_Oreal__field,axiom,
real_field(real) ).
tff(arity_RealDef_Oreal___Fields_Odivision__ring,axiom,
division_ring(real) ).
tff(arity_RealDef_Oreal___Rings_Ozero__neq__one,axiom,
zero_neq_one(real) ).
tff(arity_RealDef_Oreal___Groups_Omonoid__mult,axiom,
monoid_mult(real) ).
tff(arity_RealDef_Oreal___Groups_Omonoid__add,axiom,
monoid_add(real) ).
tff(arity_RealDef_Oreal___Rings_Osemiring__0,axiom,
semiring_0(real) ).
tff(arity_RealDef_Oreal___Rings_Omult__zero,axiom,
mult_zero(real) ).
tff(arity_RealDef_Oreal___Fields_Ofield,axiom,
field(real) ).
tff(arity_RealDef_Oreal___Power_Opower,axiom,
power(real) ).
tff(arity_RealDef_Oreal___Groups_Ozero,axiom,
zero(real) ).
tff(arity_RealDef_Oreal___Groups_Oone,axiom,
one(real) ).
tff(arity_Complex_Ocomplex___RealVector_Oreal__normed__div__algebra,axiom,
real_n1866405975lgebra(complex) ).
tff(arity_Complex_Ocomplex___Fields_Odivision__ring__inverse__zero,axiom,
divisi14063676e_zero(complex) ).
tff(arity_Complex_Ocomplex___RealVector_Oreal__normed__algebra__1,axiom,
real_n2089651433ebra_1(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(complex) ).
tff(arity_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(complex) ).
tff(arity_Complex_Ocomplex___RealVector_Oreal__normed__vector,axiom,
real_normed_vector(complex) ).
tff(arity_Complex_Ocomplex___RealVector_Oreal__normed__field,axiom,
real_normed_field(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(complex) ).
tff(arity_Complex_Ocomplex___RealVector_Oreal__algebra__1,axiom,
real_algebra_1(complex) ).
tff(arity_Complex_Ocomplex___Fields_Ofield__inverse__zero,axiom,
field_inverse_zero(complex) ).
tff(arity_Complex_Ocomplex___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(complex) ).
tff(arity_Complex_Ocomplex___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(complex) ).
tff(arity_Complex_Ocomplex___RealVector_Oreal__field,axiom,
real_field(complex) ).
tff(arity_Complex_Ocomplex___Fields_Odivision__ring,axiom,
division_ring(complex) ).
tff(arity_Complex_Ocomplex___Rings_Ozero__neq__one,axiom,
zero_neq_one(complex) ).
tff(arity_Complex_Ocomplex___Groups_Omonoid__mult,axiom,
monoid_mult(complex) ).
tff(arity_Complex_Ocomplex___Groups_Omonoid__add,axiom,
monoid_add(complex) ).
tff(arity_Complex_Ocomplex___Rings_Osemiring__0,axiom,
semiring_0(complex) ).
tff(arity_Complex_Ocomplex___Rings_Omult__zero,axiom,
mult_zero(complex) ).
tff(arity_Complex_Ocomplex___Fields_Ofield,axiom,
field(complex) ).
tff(arity_Complex_Ocomplex___Power_Opower,axiom,
power(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ozero,axiom,
zero(complex) ).
tff(arity_Complex_Ocomplex___Groups_Oone,axiom,
one(complex) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (1)
tff(conj_0,conjecture,
power_power(complex,inverse_divide(complex,v,of_real(complex,root(na,norm_norm(complex,b)))),na) = inverse_divide(complex,power_power(complex,v,na),of_real(complex,norm_norm(complex,b))) ).
%------------------------------------------------------------------------------