TPTP Problem File: SWW480_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW480_5 : TPTP v9.0.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fundamental Theorem of Algebra line 27
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : fta_27 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 199 ( 88 unt; 40 typ; 0 def)
% Number of atoms : 282 ( 97 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 136 ( 13 ~; 1 |; 15 &)
% ( 50 <=>; 57 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 13 ( 10 >; 3 *; 0 +; 0 <<)
% Number of predicates : 21 ( 20 usr; 0 prp; 1-3 aty)
% Number of functors : 15 ( 15 usr; 6 con; 0-3 aty)
% Number of variables : 183 ( 160 !; 0 ?; 183 :)
% ( 23 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:13:55
%------------------------------------------------------------------------------
%----Should-be-implicit typings (5)
tff(ty_tc_Complex_Ocomplex,type,
complex: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_RealDef_Oreal,type,
real: $tType ).
%----Explicit typings (35)
tff(sy_cl_Int_Onumber,type,
number:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oidom,type,
idom:
!>[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_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Onumber__ring,type,
number_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Onumber__semiring,type,
number_semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ono__zero__divisors,type,
no_zero_divisors:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_11004092258visors:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
tff(sy_c_Complex_Ocomplex_OComplex,type,
complex1: ( real * real ) > complex ).
tff(sy_c_Fundamental__Theorem__Algebra__Mirabelle__jmqnahvvas_Ocsqrt,type,
fundam1563812824_csqrt: complex > complex ).
tff(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Int_OBit0,type,
bit0: int > int ).
tff(sy_c_Int_OBit1,type,
bit1: int > int ).
tff(sy_c_Int_OPls,type,
pls: int ).
tff(sy_c_Int_Onumber__class_Onumber__of,type,
number_number_of:
!>[A: $tType] : ( int > A ) ).
tff(sy_c_NthRoot_Osqrt,type,
sqrt: real > real ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( ( A * nat ) > 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_x____,type,
x: real ).
tff(sy_v_y____,type,
y: real ).
tff(sy_v_z,type,
z: complex ).
%----Relevant facts (97)
tff(fact_0__096_126_A0_A_060_061_Ax_096,axiom,
~ ord_less_eq(real,zero_zero(real),x) ).
tff(fact_1_xy,axiom,
z = complex1(x,y) ).
tff(fact_2__096sqrt_A_I_N_Ax_J_A_094_A2_A_061_A_N_Ax_096,axiom,
power_power(real,sqrt(uminus_uminus(real,x)),number_number_of(nat,bit0(bit1(pls)))) = uminus_uminus(real,x) ).
tff(fact_3__096_B_Bthesis_O_A_I_B_Bx_Ay_O_Az_A_061_AComplex_Ax_Ay_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [X1: real,Y1: real] : ( z != complex1(X1,Y1) ) ).
tff(fact_4_calculation,axiom,
( ord_less_eq(real,zero_zero(real),x)
=> ( power_power(complex,fundam1563812824_csqrt(z),number_number_of(nat,bit0(bit1(pls)))) = z ) ) ).
tff(fact_5_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_6_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls = bit0(L) )
<=> ( pls = L ) ) ).
tff(fact_7_rel__simps_I44_J,axiom,
! [K3: int] :
( ( bit0(K3) = pls )
<=> ( K3 = pls ) ) ).
tff(fact_8_rel__simps_I49_J,axiom,
! [L1: int,K: int] : ( bit0(K) != bit1(L1) ) ).
tff(fact_9_rel__simps_I50_J,axiom,
! [L1: int,K: int] : ( bit1(K) != bit0(L1) ) ).
tff(fact_10_rel__simps_I39_J,axiom,
! [L1: int] : ( pls != bit1(L1) ) ).
tff(fact_11_rel__simps_I46_J,axiom,
! [K: int] : ( bit1(K) != pls ) ).
tff(fact_12_zero__power2,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( power_power(A,zero_zero(A),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(A) ) ) ).
tff(fact_13_zero__eq__power2,axiom,
! [A: $tType] :
( ring_11004092258visors(A)
=> ! [A2: A] :
( ( power_power(A,A2,number_number_of(nat,bit0(bit1(pls)))) = zero_zero(A) )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_14_power2__minus,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [A1: A] : ( power_power(A,uminus_uminus(A,A1),number_number_of(nat,bit0(bit1(pls)))) = power_power(A,A1,number_number_of(nat,bit0(bit1(pls)))) ) ) ).
tff(fact_15_power2__eq__iff,axiom,
! [A: $tType] :
( idom(A)
=> ! [Ya: A,Xa: A] :
( ( power_power(A,Xa,number_number_of(nat,bit0(bit1(pls)))) = power_power(A,Ya,number_number_of(nat,bit0(bit1(pls)))) )
<=> ( ( Xa = Ya )
| ( Xa = uminus_uminus(A,Ya) ) ) ) ) ).
tff(fact_16_y0,axiom,
y = zero_zero(real) ).
tff(fact_17_x0,axiom,
ord_less_eq(real,zero_zero(real),uminus_uminus(real,x)) ).
tff(fact_18_eq__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& ring_char_0(A) )
=> ! [Ya: int,Xa: int] :
( ( number_number_of(A,Xa) = number_number_of(A,Ya) )
<=> ( Xa = Ya ) ) ) ).
tff(fact_19_rel__simps_I51_J,axiom,
! [L: int,K3: int] :
( ( bit1(K3) = bit1(L) )
<=> ( K3 = L ) ) ).
tff(fact_20_minus__Pls,axiom,
uminus_uminus(int,pls) = pls ).
tff(fact_21_minus__Bit0,axiom,
! [K: int] : ( uminus_uminus(int,bit0(K)) = bit0(uminus_uminus(int,K)) ) ).
tff(fact_22_rel__simps_I48_J,axiom,
! [L: int,K3: int] :
( ( bit0(K3) = bit0(L) )
<=> ( K3 = L ) ) ).
tff(fact_23_le__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Ya: int,Xa: int] :
( ord_less_eq(A,number_number_of(A,Xa),number_number_of(A,Ya))
<=> ord_less_eq(int,Xa,Ya) ) ) ).
tff(fact_24_arith__simps_I30_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int] : ( uminus_uminus(A,number_number_of(A,W1)) = number_number_of(A,uminus_uminus(int,W1)) ) ) ).
tff(fact_25_nat__number__of__Pls,axiom,
number_number_of(nat,pls) = zero_zero(nat) ).
tff(fact_26_number__of__Pls,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_27_le__special_I3_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Xa: int] :
( ord_less_eq(A,number_number_of(A,Xa),zero_zero(A))
<=> ord_less_eq(int,Xa,pls) ) ) ).
tff(fact_28_le__special_I1_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Ya: int] :
( ord_less_eq(A,zero_zero(A),number_number_of(A,Ya))
<=> ord_less_eq(int,pls,Ya) ) ) ).
tff(fact_29_number__of__minus,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int] : ( number_number_of(A,uminus_uminus(int,W1)) = uminus_uminus(A,number_number_of(A,W1)) ) ) ).
tff(fact_30_Pls__def,axiom,
pls = zero_zero(int) ).
tff(fact_31_semiring__norm_I113_J,axiom,
zero_zero(nat) = number_number_of(nat,pls) ).
tff(fact_32_number__of__reorient,axiom,
! [A: $tType] :
( number(A)
=> ! [Xa: A,W: int] :
( ( number_number_of(A,W) = Xa )
<=> ( Xa = number_number_of(A,W) ) ) ) ).
tff(fact_33_semiring__numeral__0__eq__0,axiom,
! [A: $tType] :
( number_semiring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_34_semiring__norm_I112_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ( zero_zero(A) = number_number_of(A,pls) ) ) ).
tff(fact_35_power2__eq__imp__eq,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Y: A,X: A] :
( ( power_power(A,X,number_number_of(nat,bit0(bit1(pls)))) = power_power(A,Y,number_number_of(nat,bit0(bit1(pls)))) )
=> ( ord_less_eq(A,zero_zero(A),X)
=> ( ord_less_eq(A,zero_zero(A),Y)
=> ( X = Y ) ) ) ) ) ).
tff(fact_36_power2__le__imp__le,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,power_power(A,X,number_number_of(nat,bit0(bit1(pls)))),power_power(A,Y,number_number_of(nat,bit0(bit1(pls)))))
=> ( ord_less_eq(A,zero_zero(A),Y)
=> ord_less_eq(A,X,Y) ) ) ) ).
tff(fact_37_zero__le__power2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] : ord_less_eq(A,zero_zero(A),power_power(A,A1,number_number_of(nat,bit0(bit1(pls))))) ) ).
tff(fact_38_real__sqrt__pow2__iff,axiom,
! [Xa: real] :
( ( power_power(real,sqrt(Xa),number_number_of(nat,bit0(bit1(pls)))) = Xa )
<=> ord_less_eq(real,zero_zero(real),Xa) ) ).
tff(fact_39_real__sqrt__unique,axiom,
! [X: real,Y: real] :
( ( power_power(real,Y,number_number_of(nat,bit0(bit1(pls)))) = X )
=> ( ord_less_eq(real,zero_zero(real),Y)
=> ( sqrt(X) = Y ) ) ) ).
tff(fact_40_real__sqrt__pow2,axiom,
! [X: real] :
( ord_less_eq(real,zero_zero(real),X)
=> ( power_power(real,sqrt(X),number_number_of(nat,bit0(bit1(pls)))) = X ) ) ).
tff(fact_41_real__sqrt__ge__0__iff,axiom,
! [Ya: real] :
( ord_less_eq(real,zero_zero(real),sqrt(Ya))
<=> ord_less_eq(real,zero_zero(real),Ya) ) ).
tff(fact_42_real__sqrt__le__0__iff,axiom,
! [Xa: real] :
( ord_less_eq(real,sqrt(Xa),zero_zero(real))
<=> ord_less_eq(real,Xa,zero_zero(real)) ) ).
tff(fact_43_power__eq__0__iff__number__of,axiom,
! [A: $tType] :
( ( power(A)
& mult_zero(A)
& no_zero_divisors(A)
& zero_neq_one(A) )
=> ! [W: int,A2: A] :
( ( power_power(A,A2,number_number_of(nat,W)) = zero_zero(A) )
<=> ( ( A2 = zero_zero(A) )
& ( number_number_of(nat,W) != zero_zero(nat) ) ) ) ) ).
tff(fact_44_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A2: A] :
( ord_less_eq(A,zero_zero(A),uminus_uminus(A,A2))
<=> ord_less_eq(A,A2,zero_zero(A)) ) ) ).
tff(fact_45_le__minus__self__iff,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ord_less_eq(A,A2,uminus_uminus(A,A2))
<=> ord_less_eq(A,A2,zero_zero(A)) ) ) ).
tff(fact_46_neg__equal__iff__equal,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B1: A,A2: A] :
( ( uminus_uminus(A,A2) = uminus_uminus(A,B1) )
<=> ( A2 = B1 ) ) ) ).
tff(fact_47_real__sqrt__eq__iff,axiom,
! [Ya: real,Xa: real] :
( ( sqrt(Xa) = sqrt(Ya) )
<=> ( Xa = Ya ) ) ).
tff(fact_48_neg__equal__zero,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ( uminus_uminus(A,A2) = A2 )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_49_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A2: A] :
( ( uminus_uminus(A,A2) = zero_zero(A) )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_50_equal__neg__zero,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ( A2 = uminus_uminus(A,A2) )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_51_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A2: A] :
( ( zero_zero(A) = uminus_uminus(A,A2) )
<=> ( zero_zero(A) = A2 ) ) ) ).
tff(fact_52_minus__zero,axiom,
! [A: $tType] :
( group_add(A)
=> ( uminus_uminus(A,zero_zero(A)) = zero_zero(A) ) ) ).
tff(fact_53_neg__le__iff__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A2: A,B1: A] :
( ord_less_eq(A,uminus_uminus(A,B1),uminus_uminus(A,A2))
<=> ord_less_eq(A,A2,B1) ) ) ).
tff(fact_54_rel__simps_I34_J,axiom,
! [L: int,K3: int] :
( ord_less_eq(int,bit1(K3),bit1(L))
<=> ord_less_eq(int,K3,L) ) ).
tff(fact_55_rel__simps_I19_J,axiom,
ord_less_eq(int,pls,pls) ).
tff(fact_56_rel__simps_I31_J,axiom,
! [L: int,K3: int] :
( ord_less_eq(int,bit0(K3),bit0(L))
<=> ord_less_eq(int,K3,L) ) ).
tff(fact_57_real__sqrt__le__iff,axiom,
! [Ya: real,Xa: real] :
( ord_less_eq(real,sqrt(Xa),sqrt(Ya))
<=> ord_less_eq(real,Xa,Ya) ) ).
tff(fact_58_real__sqrt__eq__0__iff,axiom,
! [Xa: real] :
( ( sqrt(Xa) = zero_zero(real) )
<=> ( Xa = zero_zero(real) ) ) ).
tff(fact_59_real__sqrt__zero,axiom,
sqrt(zero_zero(real)) = zero_zero(real) ).
tff(fact_60_minus__le__self__iff,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ord_less_eq(A,uminus_uminus(A,A2),A2)
<=> ord_less_eq(A,zero_zero(A),A2) ) ) ).
tff(fact_61_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A2: A] :
( ord_less_eq(A,uminus_uminus(A,A2),zero_zero(A))
<=> ord_less_eq(A,zero_zero(A),A2) ) ) ).
tff(fact_62_rel__simps_I22_J,axiom,
! [K3: int] :
( ord_less_eq(int,pls,bit1(K3))
<=> ord_less_eq(int,pls,K3) ) ).
tff(fact_63_rel__simps_I32_J,axiom,
! [L: int,K3: int] :
( ord_less_eq(int,bit0(K3),bit1(L))
<=> ord_less_eq(int,K3,L) ) ).
tff(fact_64_rel__simps_I21_J,axiom,
! [K3: int] :
( ord_less_eq(int,pls,bit0(K3))
<=> ord_less_eq(int,pls,K3) ) ).
tff(fact_65_rel__simps_I27_J,axiom,
! [K3: int] :
( ord_less_eq(int,bit0(K3),pls)
<=> ord_less_eq(int,K3,pls) ) ).
tff(fact_66_le__nat__number__of,axiom,
! [V1: int,V: int] :
( ord_less_eq(nat,number_number_of(nat,V),number_number_of(nat,V1))
<=> ( ~ ord_less_eq(int,V,V1)
=> ord_less_eq(int,V,pls) ) ) ).
tff(fact_67_eq__0__number__of,axiom,
! [V: int] :
( ( zero_zero(nat) = number_number_of(nat,V) )
<=> ord_less_eq(int,V,pls) ) ).
tff(fact_68_eq__number__of__0,axiom,
! [V: int] :
( ( number_number_of(nat,V) = zero_zero(nat) )
<=> ord_less_eq(int,V,pls) ) ).
tff(fact_69_minus__numeral__code_I5_J,axiom,
! [W1: int] : ( uminus_uminus(int,number_number_of(int,W1)) = number_number_of(int,uminus_uminus(int,W1)) ) ).
tff(fact_70_less__eq__number__of__int__code,axiom,
! [L: int,K3: int] :
( ord_less_eq(int,number_number_of(int,K3),number_number_of(int,L))
<=> ord_less_eq(int,K3,L) ) ).
tff(fact_71_zero__is__num__zero,axiom,
zero_zero(int) = number_number_of(int,pls) ).
tff(fact_72_less__eq__int__code_I16_J,axiom,
! [K2: int,K1: int] :
( ord_less_eq(int,bit1(K1),bit1(K2))
<=> ord_less_eq(int,K1,K2) ) ).
tff(fact_73_less__eq__int__code_I13_J,axiom,
! [K2: int,K1: int] :
( ord_less_eq(int,bit0(K1),bit0(K2))
<=> ord_less_eq(int,K1,K2) ) ).
tff(fact_74_less__eq__int__code_I14_J,axiom,
! [K2: int,K1: int] :
( ord_less_eq(int,bit0(K1),bit1(K2))
<=> ord_less_eq(int,K1,K2) ) ).
tff(fact_75_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [Xa: A] :
( ( zero_zero(A) = Xa )
<=> ( Xa = zero_zero(A) ) ) ) ).
tff(fact_76_minus__equation__iff,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B1: A,A2: A] :
( ( uminus_uminus(A,A2) = B1 )
<=> ( uminus_uminus(A,B1) = A2 ) ) ) ).
tff(fact_77_equation__minus__iff,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B1: A,A2: A] :
( ( A2 = uminus_uminus(A,B1) )
<=> ( B1 = uminus_uminus(A,A2) ) ) ) ).
tff(fact_78_minus__minus,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A1: A] : ( uminus_uminus(A,uminus_uminus(A,A1)) = A1 ) ) ).
tff(fact_79_le__imp__neg__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B: A,A1: A] :
( ord_less_eq(A,A1,B)
=> ord_less_eq(A,uminus_uminus(A,B),uminus_uminus(A,A1)) ) ) ).
tff(fact_80_minus__le__iff,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B1: A,A2: A] :
( ord_less_eq(A,uminus_uminus(A,A2),B1)
<=> ord_less_eq(A,uminus_uminus(A,B1),A2) ) ) ).
tff(fact_81_le__minus__iff,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B1: A,A2: A] :
( ord_less_eq(A,A2,uminus_uminus(A,B1))
<=> ord_less_eq(A,B1,uminus_uminus(A,A2)) ) ) ).
tff(fact_82_real__sqrt__le__mono,axiom,
! [Y: real,X: real] :
( ord_less_eq(real,X,Y)
=> ord_less_eq(real,sqrt(X),sqrt(Y)) ) ).
tff(fact_83_real__sqrt__power,axiom,
! [K: nat,X: real] : ( sqrt(power_power(real,X,K)) = power_power(real,sqrt(X),K) ) ).
tff(fact_84_real__sqrt__minus,axiom,
! [X: real] : ( sqrt(uminus_uminus(real,X)) = uminus_uminus(real,sqrt(X)) ) ).
tff(fact_85_real__sqrt__eq__zero__cancel,axiom,
! [X: real] :
( ord_less_eq(real,zero_zero(real),X)
=> ( ( sqrt(X) = zero_zero(real) )
=> ( X = zero_zero(real) ) ) ) ).
tff(fact_86_real__sqrt__ge__zero,axiom,
! [X: real] :
( ord_less_eq(real,zero_zero(real),X)
=> ord_less_eq(real,zero_zero(real),sqrt(X)) ) ).
tff(fact_87_power__eq__0__iff,axiom,
! [A: $tType] :
( ( power(A)
& mult_zero(A)
& no_zero_divisors(A)
& zero_neq_one(A) )
=> ! [N1: nat,A2: A] :
( ( power_power(A,A2,N1) = zero_zero(A) )
<=> ( ( A2 = zero_zero(A) )
& ( N1 != zero_zero(nat) ) ) ) ) ).
tff(fact_88_realpow__square__minus__le,axiom,
! [X: real,U: real] : ord_less_eq(real,uminus_uminus(real,power_power(real,U,number_number_of(nat,bit0(bit1(pls))))),power_power(real,X,number_number_of(nat,bit0(bit1(pls))))) ).
tff(fact_89_complex__minus,axiom,
! [B: real,A1: real] : ( uminus_uminus(complex,complex1(A1,B)) = complex1(uminus_uminus(real,A1),uminus_uminus(real,B)) ) ).
tff(fact_90_Complex__eq__number__of,axiom,
! [W: int,B1: real,A2: real] :
( ( complex1(A2,B1) = number_number_of(complex,W) )
<=> ( ( A2 = number_number_of(real,W) )
& ( B1 = zero_zero(real) ) ) ) ).
tff(fact_91_complex_Oinject,axiom,
! [Real21: real,Real11: real,Real2: real,Real1: real] :
( ( complex1(Real1,Real2) = complex1(Real11,Real21) )
<=> ( ( Real1 = Real11 )
& ( Real2 = Real21 ) ) ) ).
tff(fact_92_Complex__eq__0,axiom,
! [B1: real,A2: real] :
( ( complex1(A2,B1) = zero_zero(complex) )
<=> ( ( A2 = zero_zero(real) )
& ( B1 = zero_zero(real) ) ) ) ).
tff(fact_93_number__of__is__id,axiom,
! [K: int] : ( number_number_of(int,K) = K ) ).
tff(fact_94_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_95_complex__zero__def,axiom,
zero_zero(complex) = complex1(zero_zero(real),zero_zero(real)) ).
tff(fact_96_power__mono,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,B: A,A1: A] :
( ord_less_eq(A,A1,B)
=> ( ord_less_eq(A,zero_zero(A),A1)
=> ord_less_eq(A,power_power(A,A1,N),power_power(A,B,N)) ) ) ) ).
%----Arities (59)
tff(arity_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(int) ).
tff(arity_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(int) ).
tff(arity_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add(int) ).
tff(arity_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
tff(arity_Int_Oint___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(int) ).
tff(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(arity_Int_Oint___Int_Onumber__semiring,axiom,
number_semiring(int) ).
tff(arity_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one(int) ).
tff(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
tff(arity_Int_Oint___Groups_Ogroup__add,axiom,
group_add(int) ).
tff(arity_Int_Oint___Rings_Omult__zero,axiom,
mult_zero(int) ).
tff(arity_Int_Oint___Int_Oring__char__0,axiom,
ring_char_0(int) ).
tff(arity_Int_Oint___Int_Onumber__ring,axiom,
number_ring(int) ).
tff(arity_Int_Oint___Rings_Oring__1,axiom,
ring_1(int) ).
tff(arity_Int_Oint___Power_Opower,axiom,
power(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Int_Oint___Rings_Oidom,axiom,
idom(int) ).
tff(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
tff(arity_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom(nat) ).
tff(arity_Nat_Onat___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(nat) ).
tff(arity_Nat_Onat___Int_Onumber__semiring,axiom,
number_semiring(nat) ).
tff(arity_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(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___Int_Onumber,axiom,
number(nat) ).
tff(arity_RealDef_Oreal___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(real) ).
tff(arity_RealDef_Oreal___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(real) ).
tff(arity_RealDef_Oreal___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__semidom,axiom,
linordered_semidom(real) ).
tff(arity_RealDef_Oreal___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom(real) ).
tff(arity_RealDef_Oreal___Int_Onumber__semiring,axiom,
number_semiring(real) ).
tff(arity_RealDef_Oreal___Rings_Ozero__neq__one,axiom,
zero_neq_one(real) ).
tff(arity_RealDef_Oreal___Rings_Osemiring__1,axiom,
semiring_1(real) ).
tff(arity_RealDef_Oreal___Groups_Ogroup__add,axiom,
group_add(real) ).
tff(arity_RealDef_Oreal___Rings_Omult__zero,axiom,
mult_zero(real) ).
tff(arity_RealDef_Oreal___Int_Oring__char__0,axiom,
ring_char_0(real) ).
tff(arity_RealDef_Oreal___Int_Onumber__ring,axiom,
number_ring(real) ).
tff(arity_RealDef_Oreal___Rings_Oring__1,axiom,
ring_1(real) ).
tff(arity_RealDef_Oreal___Power_Opower,axiom,
power(real) ).
tff(arity_RealDef_Oreal___Groups_Ozero,axiom,
zero(real) ).
tff(arity_RealDef_Oreal___Rings_Oidom,axiom,
idom(real) ).
tff(arity_RealDef_Oreal___Int_Onumber,axiom,
number(real) ).
tff(arity_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(complex) ).
tff(arity_Complex_Ocomplex___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(complex) ).
tff(arity_Complex_Ocomplex___Int_Onumber__semiring,axiom,
number_semiring(complex) ).
tff(arity_Complex_Ocomplex___Rings_Ozero__neq__one,axiom,
zero_neq_one(complex) ).
tff(arity_Complex_Ocomplex___Rings_Osemiring__1,axiom,
semiring_1(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ogroup__add,axiom,
group_add(complex) ).
tff(arity_Complex_Ocomplex___Rings_Omult__zero,axiom,
mult_zero(complex) ).
tff(arity_Complex_Ocomplex___Int_Oring__char__0,axiom,
ring_char_0(complex) ).
tff(arity_Complex_Ocomplex___Int_Onumber__ring,axiom,
number_ring(complex) ).
tff(arity_Complex_Ocomplex___Rings_Oring__1,axiom,
ring_1(complex) ).
tff(arity_Complex_Ocomplex___Power_Opower,axiom,
power(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ozero,axiom,
zero(complex) ).
tff(arity_Complex_Ocomplex___Rings_Oidom,axiom,
idom(complex) ).
tff(arity_Complex_Ocomplex___Int_Onumber,axiom,
number(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,fundam1563812824_csqrt(z),number_number_of(nat,bit0(bit1(pls)))) = z ).
%------------------------------------------------------------------------------