TPTP Problem File: SWW498_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW498_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fundamental Theorem of Algebra line 194
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : fta_194 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.50 v7.1.0, 0.33 v6.4.0
% Syntax : Number of formulae : 158 ( 64 unt; 30 typ; 0 def)
% Number of atoms : 211 ( 81 equ)
% Maximal formula atoms : 5 ( 1 avg)
% Number of connectives : 95 ( 12 ~; 2 |; 9 &)
% ( 26 <=>; 46 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 3 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 14 ( 10 >; 4 *; 0 +; 0 <<)
% Number of predicates : 11 ( 10 usr; 0 prp; 1-3 aty)
% Number of functors : 15 ( 15 usr; 6 con; 0-3 aty)
% Number of variables : 176 ( 161 !; 0 ?; 176 :)
% ( 15 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:17:12
%------------------------------------------------------------------------------
%----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 (25)
tff(sy_cl_Int_Onumber,type,
number:
!>[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_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Onumber__semiring,type,
number_semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_c_Fundamental__Theorem__Algebra__Mirabelle__jmqnahvvas_Ocsqrt,type,
fundam1563812824_csqrt: complex > complex ).
tff(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( 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_Int_OBit0,type,
bit0: int > int ).
tff(sy_c_Int_OBit1,type,
bit1: int > int ).
tff(sy_c_Int_OMin,type,
min: 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_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 ).
%----Relevant facts (97)
tff(fact_0__096abs_A_I2_A_K_Ay_J_A_094_A2_A_060_061_A1_A_094_A2_096,axiom,
ord_less_eq(real,power_power(real,abs_abs(real,times_times(real,number_number_of(real,bit0(bit1(pls))),y)),number_number_of(nat,bit0(bit1(pls)))),power_power(real,one_one(real),number_number_of(nat,bit0(bit1(pls))))) ).
tff(fact_1__096abs_A_I2_A_K_Ax_J_A_094_A2_A_060_061_A1_A_094_A2_096,axiom,
ord_less_eq(real,power_power(real,abs_abs(real,times_times(real,number_number_of(real,bit0(bit1(pls))),x)),number_number_of(nat,bit0(bit1(pls)))),power_power(real,one_one(real),number_number_of(nat,bit0(bit1(pls))))) ).
tff(fact_2__0962_A_K_Ax_A_060_061_A1_096,axiom,
ord_less_eq(real,times_times(real,number_number_of(real,bit0(bit1(pls))),x),one_one(real)) ).
tff(fact_3__0962_A_K_Ay_A_060_061_A1_096,axiom,
ord_less_eq(real,times_times(real,number_number_of(real,bit0(bit1(pls))),y),one_one(real)) ).
tff(fact_4__096abs_A_I2_A_K_Ax_J_A_060_061_A1_096,axiom,
ord_less_eq(real,abs_abs(real,times_times(real,number_number_of(real,bit0(bit1(pls))),x)),one_one(real)) ).
tff(fact_5__096abs_A_I2_A_K_Ay_J_A_060_061_A1_096,axiom,
ord_less_eq(real,abs_abs(real,times_times(real,number_number_of(real,bit0(bit1(pls))),y)),one_one(real)) ).
tff(fact_6_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_7_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_8_le__special_I2_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Ya: int] :
( ord_less_eq(A,one_one(A),number_number_of(A,Ya))
<=> ord_less_eq(int,bit1(pls),Ya) ) ) ).
tff(fact_9_le__special_I4_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Xa: int] :
( ord_less_eq(A,number_number_of(A,Xa),one_one(A))
<=> ord_less_eq(int,Xa,bit1(pls)) ) ) ).
tff(fact_10_numeral__1__eq__1,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,bit1(pls)) = one_one(A) ) ) ).
tff(fact_11_four__x__squared,axiom,
! [X: real] : times_times(real,number_number_of(real,bit0(bit0(bit1(pls)))),power_power(real,X,number_number_of(nat,bit0(bit1(pls))))) = power_power(real,times_times(real,number_number_of(real,bit0(bit1(pls))),X),number_number_of(nat,bit0(bit1(pls)))) ).
tff(fact_12_two__realpow__ge__one,axiom,
! [N: nat] : ord_less_eq(real,one_one(real),power_power(real,number_number_of(real,bit0(bit1(pls))),N)) ).
tff(fact_13_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [X: A] : times_times(A,X,X) = power_power(A,X,number_number_of(nat,bit0(bit1(pls)))) ) ).
tff(fact_14_power2__eq__square,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A1: A] : power_power(A,A1,number_number_of(nat,bit0(bit1(pls)))) = times_times(A,A1,A1) ) ).
tff(fact_15_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N: nat,X: A] : power_power(A,X,times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)) = times_times(A,power_power(A,X,N),power_power(A,X,N)) ) ).
tff(fact_16__096_N1_A_060_061_A2_A_K_Ax_096,axiom,
ord_less_eq(real,number_number_of(real,min),times_times(real,number_number_of(real,bit0(bit1(pls))),x)) ).
tff(fact_17_Bit0__Pls,axiom,
bit0(pls) = pls ).
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,K: int] :
( ( bit1(K) = bit1(L) )
<=> ( K = L ) ) ).
tff(fact_20_mult__Pls,axiom,
! [W: int] : times_times(int,pls,W) = pls ).
tff(fact_21_mult__Bit0,axiom,
! [L1: int,K3: int] : times_times(int,bit0(K3),L1) = bit0(times_times(int,K3,L1)) ).
tff(fact_22_rel__simps_I48_J,axiom,
! [L: int,K: int] :
( ( bit0(K) = bit0(L) )
<=> ( K = L ) ) ).
tff(fact_23_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_24_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_25_rel__simps_I46_J,axiom,
! [K3: int] : bit1(K3) != pls ).
tff(fact_26_rel__simps_I39_J,axiom,
! [L1: int] : pls != bit1(L1) ).
tff(fact_27_rel__simps_I50_J,axiom,
! [L1: int,K3: int] : bit1(K3) != bit0(L1) ).
tff(fact_28_rel__simps_I49_J,axiom,
! [L1: int,K3: int] : bit0(K3) != bit1(L1) ).
tff(fact_29_rel__simps_I44_J,axiom,
! [K: int] :
( ( bit0(K) = pls )
<=> ( K = pls ) ) ).
tff(fact_30_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls = bit0(L) )
<=> ( pls = L ) ) ).
tff(fact_31_rel__simps_I34_J,axiom,
! [L: int,K: int] :
( ord_less_eq(int,bit1(K),bit1(L))
<=> ord_less_eq(int,K,L) ) ).
tff(fact_32_rel__simps_I19_J,axiom,
ord_less_eq(int,pls,pls) ).
tff(fact_33_rel__simps_I31_J,axiom,
! [L: int,K: int] :
( ord_less_eq(int,bit0(K),bit0(L))
<=> ord_less_eq(int,K,L) ) ).
tff(fact_34_rel__simps_I47_J,axiom,
! [K: int] :
( ( bit1(K) = min )
<=> ( K = min ) ) ).
tff(fact_35_rel__simps_I43_J,axiom,
! [L: int] :
( ( min = bit1(L) )
<=> ( min = L ) ) ).
tff(fact_36_Bit1__Min,axiom,
bit1(min) = min ).
tff(fact_37_rel__simps_I37_J,axiom,
pls != min ).
tff(fact_38_rel__simps_I40_J,axiom,
min != pls ).
tff(fact_39_rel__simps_I45_J,axiom,
! [K3: int] : bit0(K3) != min ).
tff(fact_40_rel__simps_I42_J,axiom,
! [L1: int] : min != bit0(L1) ).
tff(fact_41_rel__simps_I24_J,axiom,
ord_less_eq(int,min,min) ).
tff(fact_42__096_N1_A_060_061_A2_A_K_Ay_096,axiom,
ord_less_eq(real,number_number_of(real,min),times_times(real,number_number_of(real,bit0(bit1(pls))),y)) ).
tff(fact_43_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_44_rel__simps_I22_J,axiom,
! [K: int] :
( ord_less_eq(int,pls,bit1(K))
<=> ord_less_eq(int,pls,K) ) ).
tff(fact_45_nat__numeral__1__eq__1,axiom,
number_number_of(nat,bit1(pls)) = one_one(nat) ).
tff(fact_46_rel__simps_I32_J,axiom,
! [L: int,K: int] :
( ord_less_eq(int,bit0(K),bit1(L))
<=> ord_less_eq(int,K,L) ) ).
tff(fact_47_rel__simps_I27_J,axiom,
! [K: int] :
( ord_less_eq(int,bit0(K),pls)
<=> ord_less_eq(int,K,pls) ) ).
tff(fact_48_rel__simps_I21_J,axiom,
! [K: int] :
( ord_less_eq(int,pls,bit0(K))
<=> ord_less_eq(int,pls,K) ) ).
tff(fact_49_le__nat__number__of,axiom,
! [V3: int,V2: int] :
( ord_less_eq(nat,number_number_of(nat,V2),number_number_of(nat,V3))
<=> ( ~ ord_less_eq(int,V2,V3)
=> ord_less_eq(int,V2,pls) ) ) ).
tff(fact_50_rel__simps_I26_J,axiom,
! [K: int] :
( ord_less_eq(int,min,bit1(K))
<=> ord_less_eq(int,min,K) ) ).
tff(fact_51_rel__simps_I30_J,axiom,
! [K: int] :
( ord_less_eq(int,bit1(K),min)
<=> ord_less_eq(int,K,min) ) ).
tff(fact_52_rel__simps_I23_J,axiom,
ord_less_eq(int,min,pls) ).
tff(fact_53_rel__simps_I20_J,axiom,
~ ord_less_eq(int,pls,min) ).
tff(fact_54_rel__simps_I28_J,axiom,
! [K: int] :
( ord_less_eq(int,bit0(K),min)
<=> ord_less_eq(int,K,min) ) ).
tff(fact_55_abs__minus__one,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ( abs_abs(A,number_number_of(A,min)) = one_one(A) ) ) ).
tff(fact_56_abs__power__minus__one,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [N: nat] : abs_abs(A,power_power(A,number_number_of(A,min),N)) = one_one(A) ) ).
tff(fact_57_power2__abs,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] : power_power(A,abs_abs(A,A1),number_number_of(nat,bit0(bit1(pls)))) = power_power(A,A1,number_number_of(nat,bit0(bit1(pls)))) ) ).
tff(fact_58_abs__power2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] : abs_abs(A,power_power(A,A1,number_number_of(nat,bit0(bit1(pls))))) = power_power(A,A1,number_number_of(nat,bit0(bit1(pls)))) ) ).
tff(fact_59_eq__number__of__Pls__Min,axiom,
number_number_of(int,pls) != number_number_of(int,min) ).
tff(fact_60_less__eq__number__of__int__code,axiom,
! [L: int,K: int] :
( ord_less_eq(int,number_number_of(int,K),number_number_of(int,L))
<=> ord_less_eq(int,K,L) ) ).
tff(fact_61_zmult__eq__1__iff,axiom,
! [N1: int,M1: int] :
( ( times_times(int,M1,N1) = one_one(int) )
<=> ( ( ( M1 = one_one(int) )
& ( N1 = one_one(int) ) )
| ( ( M1 = number_number_of(int,min) )
& ( N1 = number_number_of(int,min) ) ) ) ) ).
tff(fact_62_zpower__zpower,axiom,
! [Z: nat,Y: nat,X: int] : power_power(int,power_power(int,X,Y),Z) = power_power(int,X,times_times(nat,Y,Z)) ).
tff(fact_63_pos__zmult__eq__1__iff__lemma,axiom,
! [N: int,M: int] :
( ( times_times(int,M,N) = one_one(int) )
=> ( ( M = one_one(int) )
| ( M = number_number_of(int,min) ) ) ) ).
tff(fact_64_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Q: nat,P: nat,X: A] : power_power(A,power_power(A,X,P),Q) = power_power(A,X,times_times(nat,P,Q)) ) ).
tff(fact_65_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_66_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_67_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [X: A] : power_power(A,X,one_one(nat)) = X ) ).
tff(fact_68_one__is__num__one,axiom,
one_one(int) = number_number_of(int,bit1(pls)) ).
tff(fact_69_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_70_semiring__mult__number__of,axiom,
! [A: $tType] :
( number_semiring(A)
=> ! [V1: int,V: int] :
( ord_less_eq(int,pls,V)
=> ( ord_less_eq(int,pls,V1)
=> ( times_times(A,number_number_of(A,V),number_number_of(A,V1)) = number_number_of(A,times_times(int,V,V1)) ) ) ) ) ).
tff(fact_71_Numeral1__eq1__nat,axiom,
one_one(nat) = number_number_of(nat,bit1(pls)) ).
tff(fact_72_power__m1__even,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [N: nat] : power_power(A,number_number_of(A,min),times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)) = one_one(A) ) ).
tff(fact_73_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),times_times(A,Rx,Ry)) = times_times(A,times_times(A,Lx,Rx),times_times(A,Ly,Ry)) ) ).
tff(fact_74_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),times_times(A,Rx,Ry)) = times_times(A,Rx,times_times(A,times_times(A,Lx,Ly),Ry)) ) ).
tff(fact_75_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),times_times(A,Rx,Ry)) = times_times(A,Lx,times_times(A,Ly,times_times(A,Rx,Ry))) ) ).
tff(fact_76_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),Rx) = times_times(A,times_times(A,Lx,Rx),Ly) ) ).
tff(fact_77_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),Rx) = times_times(A,Lx,times_times(A,Ly,Rx)) ) ).
tff(fact_78_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Lx: A] : times_times(A,Lx,times_times(A,Rx,Ry)) = times_times(A,times_times(A,Lx,Rx),Ry) ) ).
tff(fact_79_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Lx: A] : times_times(A,Lx,times_times(A,Rx,Ry)) = times_times(A,Rx,times_times(A,Lx,Ry)) ) ).
tff(fact_80_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [B: A,A1: A] : times_times(A,A1,B) = times_times(A,B,A1) ) ).
tff(fact_81_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_82_power__even__eq,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat,A1: A] : power_power(A,A1,times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)) = power_power(A,power_power(A,A1,N),number_number_of(nat,bit0(bit1(pls)))) ) ).
tff(fact_83_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_84_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A] : times_times(A,A1,one_one(A)) = A1 ) ).
tff(fact_85_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A] : times_times(A,one_one(A),A1) = A1 ) ).
tff(fact_86_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Q: nat,Y: A,X: A] : power_power(A,times_times(A,X,Y),Q) = times_times(A,power_power(A,X,Q),power_power(A,Y,Q)) ) ).
tff(fact_87_mult__numeral__1,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : times_times(A,number_number_of(A,bit1(pls)),A1) = A1 ) ).
tff(fact_88_mult__numeral__1__right,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : times_times(A,A1,number_number_of(A,bit1(pls))) = A1 ) ).
tff(fact_89_semiring__numeral__1__eq__1,axiom,
! [A: $tType] :
( number_semiring(A)
=> ( number_number_of(A,bit1(pls)) = one_one(A) ) ) ).
tff(fact_90_semiring__norm_I110_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ( one_one(A) = number_number_of(A,bit1(pls)) ) ) ).
tff(fact_91_power3__eq__cube,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A1: A] : power_power(A,A1,number_number_of(nat,bit1(bit1(pls)))) = times_times(A,times_times(A,A1,A1),A1) ) ).
tff(fact_92_xy,axiom,
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) ).
tff(fact_93_abs__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ( abs_abs(A,one_one(A)) = one_one(A) ) ) ).
tff(fact_94_abs__mult__self,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] : times_times(A,abs_abs(A,A1),abs_abs(A,A1)) = times_times(A,A1,A1) ) ).
tff(fact_95_power__one,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat] : power_power(A,one_one(A),N) = one_one(A) ) ).
tff(fact_96_csqrt,axiom,
! [Z: complex] : power_power(complex,fundam1563812824_csqrt(Z),number_number_of(nat,bit0(bit1(pls)))) = Z ).
%----Arities (28)
tff(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(arity_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(int) ).
tff(arity_Int_Oint___Int_Onumber__semiring,axiom,
number_semiring(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___Int_Oring__char__0,axiom,
ring_char_0(int) ).
tff(arity_Int_Oint___Int_Onumber__ring,axiom,
number_ring(int) ).
tff(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
tff(arity_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(nat) ).
tff(arity_Nat_Onat___Int_Onumber__semiring,axiom,
number_semiring(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___Int_Onumber,axiom,
number(nat) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom(real) ).
tff(arity_RealDef_Oreal___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(real) ).
tff(arity_RealDef_Oreal___Int_Onumber__semiring,axiom,
number_semiring(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___Int_Oring__char__0,axiom,
ring_char_0(real) ).
tff(arity_RealDef_Oreal___Int_Onumber__ring,axiom,
number_ring(real) ).
tff(arity_RealDef_Oreal___Int_Onumber,axiom,
number(real) ).
tff(arity_Complex_Ocomplex___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(complex) ).
tff(arity_Complex_Ocomplex___Int_Onumber__semiring,axiom,
number_semiring(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___Int_Oring__char__0,axiom,
ring_char_0(complex) ).
tff(arity_Complex_Ocomplex___Int_Onumber__ring,axiom,
number_ring(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,
ord_less_eq(real,times_times(real,number_number_of(real,bit0(bit0(bit1(pls)))),power_power(real,x,number_number_of(nat,bit0(bit1(pls))))),one_one(real)) ).
%------------------------------------------------------------------------------