TPTP Problem File: SWW500_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW500_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fundamental Theorem of Algebra line 213
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : fta_213 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 213 ( 85 unt; 47 typ; 0 def)
% Number of atoms : 329 ( 90 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 188 ( 25 ~; 9 |; 17 &)
% ( 37 <=>; 100 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 12 ( 8 >; 4 *; 0 +; 0 <<)
% Number of predicates : 30 ( 29 usr; 0 prp; 1-3 aty)
% Number of functors : 14 ( 14 usr; 7 con; 0-3 aty)
% Number of variables : 278 ( 242 !; 2 ?; 278 :)
% ( 34 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:17:30
%------------------------------------------------------------------------------
%----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 (43)
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_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_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_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[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,type,
real_normed_algebra:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__normed__algebra__1,type,
real_n2089651433ebra_1:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__normed__div__algebra,type,
real_n1866405975lgebra:
!>[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_Fundamental__Theorem__Algebra__Mirabelle__jmqnahvvas_Ocsqrt,type,
fundam1563812824_csqrt: complex > complex ).
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_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_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_m____,type,
m: nat ).
tff(sy_v_n,type,
n: nat ).
tff(sy_v_na____,type,
na: nat ).
tff(sy_v_z____,type,
z: complex ).
%----Relevant facts (98)
tff(fact_0_z,axiom,
ord_less(real,norm_norm(complex,plus_plus(complex,one_one(complex),times_times(complex,b,power_power(complex,z,m)))),one_one(real)) ).
tff(fact_1_n,axiom,
na != zero_zero(nat) ).
tff(fact_2_b,axiom,
b != zero_zero(complex) ).
tff(fact_3_e,axiom,
even_odd_even(nat,na) ).
tff(fact_4__096_B_Bthesis_O_A_I_B_Bz_O_Acmod_A_I1_A_L_Ab_A_K_Az_A_094_Am_J_A_060_A1_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [Z1: complex] : ~ ord_less(real,norm_norm(complex,plus_plus(complex,one_one(complex),times_times(complex,b,power_power(complex,Z1,m)))),one_one(real)) ).
tff(fact_5_assms_I2_J,axiom,
n != zero_zero(nat) ).
tff(fact_6_norm__one,axiom,
! [A: $tType] :
( real_n2089651433ebra_1(A)
=> ( norm_norm(A,one_one(A)) = one_one(real) ) ) ).
tff(fact_7_power__one,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat] : power_power(A,one_one(A),N) = one_one(A) ) ).
tff(fact_8_add__less__cancel__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [Ba: A,C1: A,A2: A] :
( ord_less(A,plus_plus(A,A2,C1),plus_plus(A,Ba,C1))
<=> ord_less(A,A2,Ba) ) ) ).
tff(fact_9_add__less__cancel__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [Ba: A,A2: A,C1: A] :
( ord_less(A,plus_plus(A,C1,A2),plus_plus(A,C1,Ba))
<=> ord_less(A,A2,Ba) ) ) ).
tff(fact_10_power__gt1__lemma,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,A1: A] :
( ord_less(A,one_one(A),A1)
=> ord_less(A,one_one(A),times_times(A,A1,power_power(A,A1,N))) ) ) ).
tff(fact_11_power__less__power__Suc,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,A1: A] :
( ord_less(A,one_one(A),A1)
=> ord_less(A,power_power(A,A1,N),times_times(A,A1,power_power(A,A1,N))) ) ) ).
tff(fact_12_IH,axiom,
! [M1: nat] :
( ord_less(nat,M1,na)
=> ( ( M1 != zero_zero(nat) )
=> ? [Z1: complex] : ord_less(real,norm_norm(complex,plus_plus(complex,one_one(complex),times_times(complex,b,power_power(complex,Z1,M1)))),one_one(real)) ) ) ).
tff(fact_13_norm__add__less,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [S: real,Y: A,R: real,X: A] :
( ord_less(real,norm_norm(A,X),R)
=> ( ord_less(real,norm_norm(A,Y),S)
=> ord_less(real,norm_norm(A,plus_plus(A,X,Y)),plus_plus(real,R,S)) ) ) ) ).
tff(fact_14_norm__mult__less,axiom,
! [A: $tType] :
( real_normed_algebra(A)
=> ! [S: real,Y: A,R: real,X: A] :
( ord_less(real,norm_norm(A,X),R)
=> ( ord_less(real,norm_norm(A,Y),S)
=> ord_less(real,norm_norm(A,times_times(A,X,Y)),times_times(real,R,S)) ) ) ) ).
tff(fact_15_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_16_power__strict__increasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Y1: nat,X1: nat,Ba: A] :
( ord_less(A,one_one(A),Ba)
=> ( ord_less(A,power_power(A,Ba,X1),power_power(A,Ba,Y1))
<=> ord_less(nat,X1,Y1) ) ) ) ).
tff(fact_17__096m_A_126_061_A0_096,axiom,
m != zero_zero(nat) ).
tff(fact_18_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,A2: A,Ba: A] :
( ( plus_plus(A,Ba,A2) = plus_plus(A,C1,A2) )
<=> ( Ba = C1 ) ) ) ).
tff(fact_19_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,Ba: A,A2: A] :
( ( plus_plus(A,A2,Ba) = plus_plus(A,A2,C1) )
<=> ( Ba = C1 ) ) ) ).
tff(fact_20__096m_A_060_An_096,axiom,
ord_less(nat,m,na) ).
tff(fact_21_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_22_nat__zero__less__power__iff,axiom,
! [Nb: nat,X1: nat] :
( ord_less(nat,zero_zero(nat),power_power(nat,X1,Nb))
<=> ( ord_less(nat,zero_zero(nat),X1)
| ( Nb = zero_zero(nat) ) ) ) ).
tff(fact_23_norm__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ( norm_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).
tff(fact_24_norm__eq__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [X1: A] :
( ( norm_norm(A,X1) = zero_zero(real) )
<=> ( X1 = zero_zero(A) ) ) ) ).
tff(fact_25_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_26_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_27_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_28_power__0,axiom,
! [A: $tType] :
( power(A)
=> ! [A1: A] : power_power(A,A1,zero_zero(nat)) = one_one(A) ) ).
tff(fact_29_zero__less__norm__iff,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [X1: A] :
( ord_less(real,zero_zero(real),norm_norm(A,X1))
<=> ( X1 != zero_zero(A) ) ) ) ).
tff(fact_30__096_091_124_Am_A_060_An_059_Am_A_126_061_A0_A_124_093_A_061_061_062_AEX_Az_O_Acmod_A_I1_A_L_Ab_A_K_Az_A_094_Am_J_A_060_A1_096,axiom,
( ord_less(nat,m,na)
=> ( ( m != zero_zero(nat) )
=> ? [Z1: complex] : ord_less(real,norm_norm(complex,plus_plus(complex,one_one(complex),times_times(complex,b,power_power(complex,Z1,m)))),one_one(real)) ) ) ).
tff(fact_31_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X1: A] :
( ( zero_zero(A) = X1 )
<=> ( X1 = zero_zero(A) ) ) ) ).
tff(fact_32_nat__power__less__imp__less,axiom,
! [N: nat,M: nat,I: nat] :
( ord_less(nat,zero_zero(nat),I)
=> ( ord_less(nat,power_power(nat,I,M),power_power(nat,I,N))
=> ord_less(nat,M,N) ) ) ).
tff(fact_33_power__mult,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat,M: nat,A1: A] : power_power(A,A1,times_times(nat,M,N)) = power_power(A,power_power(A,A1,M),N) ) ).
tff(fact_34_add_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A1: A] : plus_plus(A,A1,zero_zero(A)) = A1 ) ).
tff(fact_35_add__0__right,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A1: A] : plus_plus(A,A1,zero_zero(A)) = A1 ) ).
tff(fact_36_add__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A1: A] : plus_plus(A,zero_zero(A),A1) = A1 ) ).
tff(fact_37_add__0__left,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A1: A] : plus_plus(A,zero_zero(A),A1) = A1 ) ).
tff(fact_38_power__one__right,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A1: A] : power_power(A,A1,one_one(nat)) = A1 ) ).
tff(fact_39_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_40_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_41_power__add,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat,M: nat,A1: A] : power_power(A,A1,plus_plus(nat,M,N)) = times_times(A,power_power(A,A1,M),power_power(A,A1,N)) ) ).
tff(fact_42_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_43_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_44_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_45_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_46_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_47_power__Suc__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,A1: A] :
( ord_less(A,zero_zero(A),A1)
=> ( ord_less(A,A1,one_one(A))
=> ord_less(A,times_times(A,A1,power_power(A,A1,N)),power_power(A,A1,N)) ) ) ) ).
tff(fact_48_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [C: A,B: A,A1: A] : times_times(A,times_times(A,A1,B),C) = times_times(A,A1,times_times(A,B,C)) ) ).
tff(fact_49_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,A1: A,B: A] :
( ( plus_plus(A,B,A1) = plus_plus(A,C,A1) )
=> ( B = C ) ) ) ).
tff(fact_50_add__imp__eq,axiom,
! [A: $tType] :
( cancel146912293up_add(A)
=> ! [C: A,B: A,A1: A] :
( ( plus_plus(A,A1,B) = plus_plus(A,A1,C) )
=> ( B = C ) ) ) ).
tff(fact_51_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,B: A,A1: A] :
( ( plus_plus(A,A1,B) = plus_plus(A,A1,C) )
=> ( B = C ) ) ) ).
tff(fact_52_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [C: A,B: A,A1: A] : plus_plus(A,plus_plus(A,A1,B),C) = plus_plus(A,A1,plus_plus(A,B,C)) ) ).
tff(fact_53_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X1: A] :
( ( one_one(A) = X1 )
<=> ( X1 = one_one(A) ) ) ) ).
tff(fact_54_add__less__imp__less__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B: A,A1: A,C: A] :
( ord_less(A,plus_plus(A,C,A1),plus_plus(A,C,B))
=> ord_less(A,A1,B) ) ) ).
tff(fact_55_add__less__imp__less__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B: A,C: A,A1: A] :
( ord_less(A,plus_plus(A,A1,C),plus_plus(A,B,C))
=> ord_less(A,A1,B) ) ) ).
tff(fact_56_add__strict__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [D: A,C: A,B: A,A1: A] :
( ord_less(A,A1,B)
=> ( ord_less(A,C,D)
=> ord_less(A,plus_plus(A,A1,C),plus_plus(A,B,D)) ) ) ) ).
tff(fact_57_add__strict__left__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [C: A,B: A,A1: A] :
( ord_less(A,A1,B)
=> ord_less(A,plus_plus(A,C,A1),plus_plus(A,C,B)) ) ) ).
tff(fact_58_add__strict__right__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [C: A,B: A,A1: A] :
( ord_less(A,A1,B)
=> ord_less(A,plus_plus(A,A1,C),plus_plus(A,B,C)) ) ) ).
tff(fact_59_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A1: A] : times_times(A,A1,one_one(A)) = A1 ) ).
tff(fact_60_mult__1__right,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A1: A] : times_times(A,A1,one_one(A)) = A1 ) ).
tff(fact_61_mult__1,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A1: A] : times_times(A,one_one(A),A1) = A1 ) ).
tff(fact_62_mult__1__left,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A1: A] : times_times(A,one_one(A),A1) = A1 ) ).
tff(fact_63_power__mult__distrib,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [N: nat,B: A,A1: A] : power_power(A,times_times(A,A1,B),N) = times_times(A,power_power(A,A1,N),power_power(A,B,N)) ) ).
tff(fact_64_power__commutes,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat,A1: A] : times_times(A,power_power(A,A1,N),A1) = times_times(A,A1,power_power(A,A1,N)) ) ).
tff(fact_65_norm__mult,axiom,
! [A: $tType] :
( real_n1866405975lgebra(A)
=> ! [Y: A,X: A] : norm_norm(A,times_times(A,X,Y)) = times_times(real,norm_norm(A,X),norm_norm(A,Y)) ) ).
tff(fact_66_norm__power,axiom,
! [A: $tType] :
( real_n1866405975lgebra(A)
=> ! [N: nat,X: A] : norm_norm(A,power_power(A,X,N)) = power_power(real,norm_norm(A,X),N) ) ).
tff(fact_67_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_68_power__less__imp__less__exp,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,M: nat,A1: A] :
( ord_less(A,one_one(A),A1)
=> ( ord_less(A,power_power(A,A1,M),power_power(A,A1,N))
=> ord_less(nat,M,N) ) ) ) ).
tff(fact_69_even__power__nat,axiom,
! [Y1: nat,X1: nat] :
( even_odd_even(nat,power_power(nat,X1,Y1))
<=> ( even_odd_even(nat,X1)
& ord_less(nat,zero_zero(nat),Y1) ) ) ).
tff(fact_70_zero__less__power__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Nb: nat,X1: A] :
( ord_less(A,zero_zero(A),power_power(A,X1,Nb))
<=> ( ( Nb = zero_zero(nat) )
| ( even_odd_even(nat,Nb)
& ( X1 != zero_zero(A) ) )
| ( ~ even_odd_even(nat,Nb)
& ord_less(A,zero_zero(A),X1) ) ) ) ) ).
tff(fact_71_neq0__conv,axiom,
! [Nb: nat] :
( ( Nb != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),Nb) ) ).
tff(fact_72_add__gr__0,axiom,
! [Nb: nat,Ma: nat] :
( ord_less(nat,zero_zero(nat),plus_plus(nat,Ma,Nb))
<=> ( ord_less(nat,zero_zero(nat),Ma)
| ord_less(nat,zero_zero(nat),Nb) ) ) ).
tff(fact_73_nat__0__less__mult__iff,axiom,
! [Nb: nat,Ma: nat] :
( ord_less(nat,zero_zero(nat),times_times(nat,Ma,Nb))
<=> ( ord_less(nat,zero_zero(nat),Ma)
& ord_less(nat,zero_zero(nat),Nb) ) ) ).
tff(fact_74_less__nat__zero__code,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_75_mult__less__cancel1,axiom,
! [Nb: nat,Ma: nat,K: nat] :
( ord_less(nat,times_times(nat,K,Ma),times_times(nat,K,Nb))
<=> ( ord_less(nat,zero_zero(nat),K)
& ord_less(nat,Ma,Nb) ) ) ).
tff(fact_76_mult__less__cancel2,axiom,
! [Nb: nat,K: nat,Ma: nat] :
( ord_less(nat,times_times(nat,Ma,K),times_times(nat,Nb,K))
<=> ( ord_less(nat,zero_zero(nat),K)
& ord_less(nat,Ma,Nb) ) ) ).
tff(fact_77_nat__add__left__cancel,axiom,
! [Nb: nat,Ma: nat,K: nat] :
( ( plus_plus(nat,K,Ma) = plus_plus(nat,K,Nb) )
<=> ( Ma = Nb ) ) ).
tff(fact_78_nat__add__right__cancel,axiom,
! [Nb: nat,K: nat,Ma: nat] :
( ( plus_plus(nat,Ma,K) = plus_plus(nat,Nb,K) )
<=> ( Ma = Nb ) ) ).
tff(fact_79_less__zeroE,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_80_add__is__0,axiom,
! [Nb: nat,Ma: nat] :
( ( plus_plus(nat,Ma,Nb) = zero_zero(nat) )
<=> ( ( Ma = zero_zero(nat) )
& ( Nb = zero_zero(nat) ) ) ) ).
tff(fact_81_mult__cancel2,axiom,
! [Nb: nat,K: nat,Ma: nat] :
( ( times_times(nat,Ma,K) = times_times(nat,Nb,K) )
<=> ( ( Ma = Nb )
| ( K = zero_zero(nat) ) ) ) ).
tff(fact_82_mult__cancel1,axiom,
! [Nb: nat,Ma: nat,K: nat] :
( ( times_times(nat,K,Ma) = times_times(nat,K,Nb) )
<=> ( ( Ma = Nb )
| ( K = zero_zero(nat) ) ) ) ).
tff(fact_83_mult__is__0,axiom,
! [Nb: nat,Ma: nat] :
( ( times_times(nat,Ma,Nb) = zero_zero(nat) )
<=> ( ( Ma = zero_zero(nat) )
| ( Nb = zero_zero(nat) ) ) ) ).
tff(fact_84_mult__0__right,axiom,
! [M: nat] : times_times(nat,M,zero_zero(nat)) = zero_zero(nat) ).
tff(fact_85_mult__0,axiom,
! [N: nat] : times_times(nat,zero_zero(nat),N) = zero_zero(nat) ).
tff(fact_86_nat__add__left__cancel__less,axiom,
! [Nb: nat,Ma: nat,K: nat] :
( ord_less(nat,plus_plus(nat,K,Ma),plus_plus(nat,K,Nb))
<=> ord_less(nat,Ma,Nb) ) ).
tff(fact_87_nat__1__eq__mult__iff,axiom,
! [Nb: nat,Ma: nat] :
( ( one_one(nat) = times_times(nat,Ma,Nb) )
<=> ( ( Ma = one_one(nat) )
& ( Nb = one_one(nat) ) ) ) ).
tff(fact_88_nat__mult__eq__1__iff,axiom,
! [Nb: nat,Ma: nat] :
( ( times_times(nat,Ma,Nb) = one_one(nat) )
<=> ( ( Ma = one_one(nat) )
& ( Nb = one_one(nat) ) ) ) ).
tff(fact_89_even__sum__nat,axiom,
! [Y1: nat,X1: nat] :
( even_odd_even(nat,plus_plus(nat,X1,Y1))
<=> ( ( even_odd_even(nat,X1)
& even_odd_even(nat,Y1) )
| ( ~ even_odd_even(nat,X1)
& ~ even_odd_even(nat,Y1) ) ) ) ).
tff(fact_90_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_91_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_92_even__product__nat,axiom,
! [Y1: nat,X1: nat] :
( even_odd_even(nat,times_times(nat,X1,Y1))
<=> ( even_odd_even(nat,X1)
| even_odd_even(nat,Y1) ) ) ).
tff(fact_93_nat__mult__1,axiom,
! [N: nat] : times_times(nat,one_one(nat),N) = N ).
tff(fact_94_nat__mult__1__right,axiom,
! [N: nat] : times_times(nat,N,one_one(nat)) = N ).
tff(fact_95_nat__add__commute,axiom,
! [N: nat,M: nat] : plus_plus(nat,M,N) = plus_plus(nat,N,M) ).
tff(fact_96_nat__mult__commute,axiom,
! [N: nat,M: nat] : times_times(nat,M,N) = times_times(nat,N,M) ).
tff(fact_97_nat__add__left__commute,axiom,
! [Z: nat,Y: nat,X: nat] : plus_plus(nat,X,plus_plus(nat,Y,Z)) = plus_plus(nat,Y,plus_plus(nat,X,Z)) ).
%----Arities (65)
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__mult,axiom,
ab_semigroup_mult(nat) ).
tff(arity_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(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___RealVector_Oreal__normed__div__algebra,axiom,
real_n1866405975lgebra(real) ).
tff(arity_RealDef_Oreal___RealVector_Oreal__normed__algebra__1,axiom,
real_n2089651433ebra_1(real) ).
tff(arity_RealDef_Oreal___RealVector_Oreal__normed__algebra,axiom,
real_normed_algebra(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___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__semidom,axiom,
linordered_semidom(real) ).
tff(arity_RealDef_Oreal___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(real) ).
tff(arity_RealDef_Oreal___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(real) ).
tff(arity_RealDef_Oreal___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(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___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___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___RealVector_Oreal__normed__algebra__1,axiom,
real_n2089651433ebra_1(complex) ).
tff(arity_Complex_Ocomplex___RealVector_Oreal__normed__algebra,axiom,
real_normed_algebra(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___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(complex) ).
tff(arity_Complex_Ocomplex___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(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___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___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,
ord_less(real,norm_norm(complex,plus_plus(complex,one_one(complex),times_times(complex,b,power_power(complex,fundam1563812824_csqrt(z),na)))),one_one(real)) ).
%------------------------------------------------------------------------------