TPTP Problem File: SWW497_5.p
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%------------------------------------------------------------------------------
% File : SWW497_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fundamental Theorem of Algebra line 186
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : fta_186 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.25 v7.1.0, 0.67 v6.4.0
% Syntax : Number of formulae : 229 ( 116 unt; 51 typ; 0 def)
% Number of atoms : 266 ( 129 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 96 ( 8 ~; 0 |; 12 &)
% ( 23 <=>; 53 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 23 ( 15 >; 8 *; 0 +; 0 <<)
% Number of predicates : 25 ( 24 usr; 0 prp; 1-2 aty)
% Number of functors : 22 ( 22 usr; 6 con; 0-4 aty)
% Number of variables : 234 ( 200 !; 0 ?; 234 :)
% ( 34 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:17:02
%------------------------------------------------------------------------------
%----Should-be-implicit typings (6)
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 ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (45)
tff(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Onumber,type,
number:
!>[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_Osemiring,type,
semiring:
!>[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_Rings_Osemiring__1,type,
semiring_1:
!>[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_Int_Onumber__semiring,type,
number_semiring:
!>[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_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__ring__strict,type,
linord581940658strict:
!>[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_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_c_Complex_Ocomplex_OComplex,type,
complex1: ( real * real ) > complex ).
tff(sy_c_Complex_Ocomplex_Ocomplex__case,type,
complex_case:
!>[T: $tType] : ( ( fun(real,fun(real,T)) * complex ) > T ) ).
tff(sy_c_Complex_Ocomplex_Ocomplex__rec,type,
complex_rec:
!>[T: $tType] : ( ( fun(real,fun(real,T)) * complex ) > T ) ).
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_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_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_aa,type,
aa:
!>[A: $tType,B1: $tType] : ( ( fun(A,B1) * A ) > B1 ) ).
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 (96)
tff(fact_0_assms,axiom,
norm_norm(complex,z) = one_one(real) ).
tff(fact_1_z,axiom,
z = complex1(x,y) ).
tff(fact_2_one__power2,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( power_power(A,one_one(A),number_number_of(nat,bit0(bit1(pls)))) = one_one(A) ) ) ).
tff(fact_3_one__add__one__is__two,axiom,
! [A: $tType] :
( number_ring(A)
=> ( plus_plus(A,one_one(A),one_one(A)) = number_number_of(A,bit0(bit1(pls))) ) ) ).
tff(fact_4_add__special_I2_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int] : plus_plus(A,one_one(A),number_number_of(A,W)) = number_number_of(A,plus_plus(int,bit1(pls),W)) ) ).
tff(fact_5_add__special_I3_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [V: int] : plus_plus(A,number_number_of(A,V),one_one(A)) = number_number_of(A,plus_plus(int,V,bit1(pls))) ) ).
tff(fact_6_numeral__1__eq__1,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,bit1(pls)) = one_one(A) ) ) ).
tff(fact_7_semiring__one__add__one__is__two,axiom,
! [A: $tType] :
( number_semiring(A)
=> ( plus_plus(A,one_one(A),one_one(A)) = number_number_of(A,bit0(bit1(pls))) ) ) ).
tff(fact_8_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_9_rel__simps_I38_J,axiom,
! [L1: int] :
( ( pls = bit0(L1) )
<=> ( pls = L1 ) ) ).
tff(fact_10_rel__simps_I44_J,axiom,
! [K1: int] :
( ( bit0(K1) = pls )
<=> ( K1 = pls ) ) ).
tff(fact_11_rel__simps_I49_J,axiom,
! [L: int,K: int] : bit0(K) != bit1(L) ).
tff(fact_12_rel__simps_I50_J,axiom,
! [L: int,K: int] : bit1(K) != bit0(L) ).
tff(fact_13_rel__simps_I39_J,axiom,
! [L: int] : pls != bit1(L) ).
tff(fact_14_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_15_rel__simps_I51_J,axiom,
! [L1: int,K1: int] :
( ( bit1(K1) = bit1(L1) )
<=> ( K1 = L1 ) ) ).
tff(fact_16_rel__simps_I48_J,axiom,
! [L1: int,K1: int] :
( ( bit0(K1) = bit0(L1) )
<=> ( K1 = L1 ) ) ).
tff(fact_17__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_18_rel__simps_I46_J,axiom,
! [K: int] : bit1(K) != pls ).
tff(fact_19_add__Bit0__Bit0,axiom,
! [L: int,K: int] : plus_plus(int,bit0(K),bit0(L)) = bit0(plus_plus(int,K,L)) ).
tff(fact_20_add__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A,W: int,V: int] : plus_plus(A,number_number_of(A,V),plus_plus(A,number_number_of(A,W),Z)) = plus_plus(A,number_number_of(A,plus_plus(int,V,W)),Z) ) ).
tff(fact_21_add__number__of__eq,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : plus_plus(A,number_number_of(A,V),number_number_of(A,W)) = number_number_of(A,plus_plus(int,V,W)) ) ).
tff(fact_22_nat__numeral__1__eq__1,axiom,
number_number_of(nat,bit1(pls)) = one_one(nat) ).
tff(fact_23_add__Bit1__Bit0,axiom,
! [L: int,K: int] : plus_plus(int,bit1(K),bit0(L)) = bit1(plus_plus(int,K,L)) ).
tff(fact_24_add__Bit0__Bit1,axiom,
! [L: int,K: int] : plus_plus(int,bit0(K),bit1(L)) = bit1(plus_plus(int,K,L)) ).
tff(fact_25_nat__1__add__1,axiom,
plus_plus(nat,one_one(nat),one_one(nat)) = number_number_of(nat,bit0(bit1(pls))) ).
tff(fact_26_Bit1__def,axiom,
! [K: int] : bit1(K) = plus_plus(int,plus_plus(int,one_one(int),K),K) ).
tff(fact_27_plus__numeral__code_I9_J,axiom,
! [W: int,V: int] : plus_plus(int,number_number_of(int,V),number_number_of(int,W)) = number_number_of(int,plus_plus(int,V,W)) ).
tff(fact_28_add__Pls__right,axiom,
! [K: int] : plus_plus(int,K,pls) = K ).
tff(fact_29_add__Pls,axiom,
! [K: int] : plus_plus(int,pls,K) = K ).
tff(fact_30_Bit0__def,axiom,
! [K: int] : bit0(K) = plus_plus(int,K,K) ).
tff(fact_31_one__is__num__one,axiom,
one_one(int) = number_number_of(int,bit1(pls)) ).
tff(fact_32_Numeral1__eq1__nat,axiom,
one_one(nat) = number_number_of(nat,bit1(pls)) ).
tff(fact_33_number__of__reorient,axiom,
! [A: $tType] :
( number(A)
=> ! [Xa: A,W1: int] :
( ( number_number_of(A,W1) = Xa )
<=> ( Xa = number_number_of(A,W1) ) ) ) ).
tff(fact_34_number__of__add,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : number_number_of(A,plus_plus(int,V,W)) = plus_plus(A,number_number_of(A,V),number_number_of(A,W)) ) ).
tff(fact_35_add__numeral__0,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : plus_plus(A,number_number_of(A,pls),A1) = A1 ) ).
tff(fact_36_add__numeral__0__right,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : plus_plus(A,A1,number_number_of(A,pls)) = A1 ) ).
tff(fact_37_number__of__Bit1,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int] : number_number_of(A,bit1(W)) = plus_plus(A,plus_plus(A,one_one(A),number_number_of(A,W)),number_number_of(A,W)) ) ).
tff(fact_38_semiring__numeral__1__eq__1,axiom,
! [A: $tType] :
( number_semiring(A)
=> ( number_number_of(A,bit1(pls)) = one_one(A) ) ) ).
tff(fact_39_semiring__norm_I110_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ( one_one(A) = number_number_of(A,bit1(pls)) ) ) ).
tff(fact_40_norm__one,axiom,
! [A: $tType] :
( real_n2089651433ebra_1(A)
=> ( norm_norm(A,one_one(A)) = one_one(real) ) ) ).
tff(fact_41_complex__add,axiom,
! [D: real,C: real,B: real,A1: real] : plus_plus(complex,complex1(A1,B),complex1(C,D)) = complex1(plus_plus(real,A1,C),plus_plus(real,B,D)) ).
tff(fact_42_power__one,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat] : power_power(A,one_one(A),N) = one_one(A) ) ).
tff(fact_43_csqrt,axiom,
! [Z: complex] : power_power(complex,fundam1563812824_csqrt(Z),number_number_of(nat,bit0(bit1(pls)))) = Z ).
tff(fact_44_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_45_complex_Oinject,axiom,
! [Real21: real,Real11: real,Real2: real,Real1: real] :
( ( complex1(Real1,Real2) = complex1(Real11,Real21) )
<=> ( ( Real1 = Real11 )
& ( Real2 = Real21 ) ) ) ).
tff(fact_46_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,A2: A,B2: A] :
( ( plus_plus(A,B2,A2) = plus_plus(A,C1,A2) )
<=> ( B2 = C1 ) ) ) ).
tff(fact_47_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,B2: A,A2: A] :
( ( plus_plus(A,A2,B2) = plus_plus(A,A2,C1) )
<=> ( B2 = C1 ) ) ) ).
tff(fact_48_number__of__is__id,axiom,
! [K: int] : number_number_of(int,K) = K ).
tff(fact_49_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_50_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_51_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_52_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_53_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [Xa: A] :
( ( one_one(A) = Xa )
<=> ( Xa = one_one(A) ) ) ) ).
tff(fact_54_power__one__right,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A1: A] : power_power(A,A1,one_one(nat)) = A1 ) ).
tff(fact_55_complex__norm,axiom,
! [Y: real,X: real] : norm_norm(complex,complex1(X,Y)) = sqrt(plus_plus(real,power_power(real,X,number_number_of(nat,bit0(bit1(pls)))),power_power(real,Y,number_number_of(nat,bit0(bit1(pls)))))) ).
tff(fact_56_nat__add__left__cancel,axiom,
! [N1: nat,M1: nat,K1: nat] :
( ( plus_plus(nat,K1,M1) = plus_plus(nat,K1,N1) )
<=> ( M1 = N1 ) ) ).
tff(fact_57_nat__add__right__cancel,axiom,
! [N1: nat,K1: nat,M1: nat] :
( ( plus_plus(nat,M1,K1) = plus_plus(nat,N1,K1) )
<=> ( M1 = N1 ) ) ).
tff(fact_58_complex_Osimps_I2_J,axiom,
! [A: $tType,Real2: real,Real1: real,F1: fun(real,fun(real,A))] : complex_case(A,F1,complex1(Real1,Real2)) = aa(real,A,aa(real,fun(real,A),F1,Real1),Real2) ).
tff(fact_59_nat__add__assoc,axiom,
! [K: nat,N: nat,M: nat] : plus_plus(nat,plus_plus(nat,M,N),K) = plus_plus(nat,M,plus_plus(nat,N,K)) ).
tff(fact_60_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)) ).
tff(fact_61_nat__add__commute,axiom,
! [N: nat,M: nat] : plus_plus(nat,M,N) = plus_plus(nat,N,M) ).
tff(fact_62_real__sqrt__one,axiom,
sqrt(one_one(real)) = one_one(real) ).
tff(fact_63_real__sqrt__eq__1__iff,axiom,
! [Xa: real] :
( ( sqrt(Xa) = one_one(real) )
<=> ( Xa = one_one(real) ) ) ).
tff(fact_64_real__sqrt__eq__iff,axiom,
! [Ya: real,Xa: real] :
( ( sqrt(Xa) = sqrt(Ya) )
<=> ( Xa = Ya ) ) ).
tff(fact_65_real__sqrt__power,axiom,
! [K: nat,X: real] : sqrt(power_power(real,X,K)) = power_power(real,sqrt(X),K) ).
tff(fact_66_real__sqrt__sum__squares__mult__squared__eq,axiom,
! [Ya1: real,Xa1: real,Y: real,X: real] : power_power(real,sqrt(times_times(real,plus_plus(real,power_power(real,X,number_number_of(nat,bit0(bit1(pls)))),power_power(real,Y,number_number_of(nat,bit0(bit1(pls))))),plus_plus(real,power_power(real,Xa1,number_number_of(nat,bit0(bit1(pls)))),power_power(real,Ya1,number_number_of(nat,bit0(bit1(pls))))))),number_number_of(nat,bit0(bit1(pls)))) = times_times(real,plus_plus(real,power_power(real,X,number_number_of(nat,bit0(bit1(pls)))),power_power(real,Y,number_number_of(nat,bit0(bit1(pls))))),plus_plus(real,power_power(real,Xa1,number_number_of(nat,bit0(bit1(pls)))),power_power(real,Ya1,number_number_of(nat,bit0(bit1(pls)))))) ).
tff(fact_67_complex_Orecs,axiom,
! [A: $tType,Real2: real,Real1: real,F1: fun(real,fun(real,A))] : complex_rec(A,F1,complex1(Real1,Real2)) = aa(real,A,aa(real,fun(real,A),F1,Real1),Real2) ).
tff(fact_68_real__sqrt__sum__squares__eq__cancel,axiom,
! [Y: real,X: real] :
( ( sqrt(plus_plus(real,power_power(real,X,number_number_of(nat,bit0(bit1(pls)))),power_power(real,Y,number_number_of(nat,bit0(bit1(pls)))))) = X )
=> ( Y = zero_zero(real) ) ) ).
tff(fact_69_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_70_double__eq__0__iff,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ( plus_plus(A,A2,A2) = zero_zero(A) )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_71_arith__simps_I32_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : times_times(A,number_number_of(A,V),number_number_of(A,W)) = number_number_of(A,times_times(int,V,W)) ) ).
tff(fact_72_mult__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A,W: int,V: int] : times_times(A,number_number_of(A,V),times_times(A,number_number_of(A,W),Z)) = times_times(A,number_number_of(A,times_times(int,V,W)),Z) ) ).
tff(fact_73_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_74_real__sqrt__zero,axiom,
sqrt(zero_zero(real)) = zero_zero(real) ).
tff(fact_75_real__sqrt__eq__0__iff,axiom,
! [Xa: real] :
( ( sqrt(Xa) = zero_zero(real) )
<=> ( Xa = zero_zero(real) ) ) ).
tff(fact_76_Complex__eq__number__of,axiom,
! [W1: int,B2: real,A2: real] :
( ( complex1(A2,B2) = number_number_of(complex,W1) )
<=> ( ( A2 = number_number_of(real,W1) )
& ( B2 = zero_zero(real) ) ) ) ).
tff(fact_77_left__distrib__number__of,axiom,
! [B1: $tType] :
( ( number(B1)
& semiring(B1) )
=> ! [V: int,B: B1,A1: B1] : times_times(B1,plus_plus(B1,A1,B),number_number_of(B1,V)) = plus_plus(B1,times_times(B1,A1,number_number_of(B1,V)),times_times(B1,B,number_number_of(B1,V))) ) ).
tff(fact_78_right__distrib__number__of,axiom,
! [B1: $tType] :
( ( number(B1)
& semiring(B1) )
=> ! [C: B1,B: B1,V: int] : times_times(B1,number_number_of(B1,V),plus_plus(B1,B,C)) = plus_plus(B1,times_times(B1,number_number_of(B1,V),B),times_times(B1,number_number_of(B1,V),C)) ) ).
tff(fact_79_number__of__Pls,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_80_norm__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ( norm_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).
tff(fact_81_norm__eq__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [Xa: A] :
( ( norm_norm(A,Xa) = zero_zero(real) )
<=> ( Xa = zero_zero(A) ) ) ) ).
tff(fact_82_Complex__eq__1,axiom,
! [B2: real,A2: real] :
( ( complex1(A2,B2) = one_one(complex) )
<=> ( ( A2 = one_one(real) )
& ( B2 = zero_zero(real) ) ) ) ).
tff(fact_83_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_84_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_85_power2__eq__square__number__of,axiom,
! [B1: $tType] :
( ( monoid_mult(B1)
& number(B1) )
=> ! [W: int] : power_power(B1,number_number_of(B1,W),number_number_of(nat,bit0(bit1(pls)))) = times_times(B1,number_number_of(B1,W),number_number_of(B1,W)) ) ).
tff(fact_86_real__sqrt__mult,axiom,
! [Y: real,X: real] : sqrt(times_times(real,X,Y)) = times_times(real,sqrt(X),sqrt(Y)) ).
tff(fact_87_number__of__mult,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : number_number_of(A,times_times(int,V,W)) = times_times(A,number_number_of(A,V),number_number_of(A,W)) ) ).
tff(fact_88_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_89_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [Ya: A,Xa: A] :
( ( plus_plus(A,times_times(A,Xa,Xa),times_times(A,Ya,Ya)) = zero_zero(A) )
<=> ( ( Xa = zero_zero(A) )
& ( Ya = zero_zero(A) ) ) ) ) ).
tff(fact_90_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A1: A] : times_times(A,A1,one_one(A)) = A1 ) ).
tff(fact_91_mult__1__right,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A1: A] : times_times(A,A1,one_one(A)) = A1 ) ).
tff(fact_92_mult__1,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A1: A] : times_times(A,one_one(A),A1) = A1 ) ).
tff(fact_93_mult__1__left,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A1: A] : times_times(A,one_one(A),A1) = A1 ) ).
tff(fact_94_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_95_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)) ) ).
%----Arities (79)
tff(arity_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(int) ).
tff(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(int) ).
tff(arity_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(int) ).
tff(arity_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord581940658strict(int) ).
tff(arity_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
tff(arity_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(int) ).
tff(arity_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(int) ).
tff(arity_Int_Oint___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(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___Groups_Omonoid__mult,axiom,
monoid_mult(int) ).
tff(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(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_Osemiring,axiom,
semiring(int) ).
tff(arity_Int_Oint___Power_Opower,axiom,
power(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
tff(arity_Int_Oint___Groups_Oone,axiom,
one(int) ).
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___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___Int_Onumber__semiring,axiom,
number_semiring(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___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
tff(arity_Nat_Onat___Rings_Omult__zero,axiom,
mult_zero(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring,axiom,
semiring(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_Nat_Onat___Groups_Oone,axiom,
one(nat) ).
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___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___Rings_Olinordered__ring__strict,axiom,
linord581940658strict(real) ).
tff(arity_RealDef_Oreal___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(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___Int_Onumber__semiring,axiom,
number_semiring(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___Rings_Osemiring__1,axiom,
semiring_1(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_Osemiring,axiom,
semiring(real) ).
tff(arity_RealDef_Oreal___Power_Opower,axiom,
power(real) ).
tff(arity_RealDef_Oreal___Groups_Ozero,axiom,
zero(real) ).
tff(arity_RealDef_Oreal___Int_Onumber,axiom,
number(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___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_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___Int_Onumber__semiring,axiom,
number_semiring(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___Rings_Osemiring__1,axiom,
semiring_1(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_Osemiring,axiom,
semiring(complex) ).
tff(arity_Complex_Ocomplex___Power_Opower,axiom,
power(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ozero,axiom,
zero(complex) ).
tff(arity_Complex_Ocomplex___Int_Onumber,axiom,
number(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,
plus_plus(real,power_power(real,x,number_number_of(nat,bit0(bit1(pls)))),power_power(real,y,number_number_of(nat,bit0(bit1(pls))))) = one_one(real) ).
%------------------------------------------------------------------------------