TPTP Problem File: NUM924_2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : NUM924_2 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 102, 500 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : s2s_500_tff_l102 [Bla11]
% Status : ContradictoryAxioms
% Rating : 0.00 v8.1.0, 0.09 v7.5.0, 0.12 v7.4.0, 0.00 v5.4.0, 0.33 v5.3.0
% Syntax : Number of formulae : 745 ( 358 unt; 46 typ; 0 def)
% Number of atoms : 1328 ( 588 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 751 ( 122 ~; 33 |; 62 &)
% ( 157 <=>; 377 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 4 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 56 ( 31 >; 25 *; 0 +; 0 <<)
% Number of predicates : 14 ( 13 usr; 0 prp; 1-3 aty)
% Number of functors : 30 ( 30 usr; 12 con; 0-2 aty)
% Number of variables : 1346 (1342 !; 4 ?;1346 :)
% SPC : TF0_CAX_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:20:52
%------------------------------------------------------------------------------
%----Should-be-implicit typings (3)
tff(ty_ty_tc__Int__Oint,type,
int: $tType ).
tff(ty_ty_tc__Nat__Onat,type,
nat: $tType ).
tff(ty_ty_tc__RealDef__Oreal,type,
real: $tType ).
%----Explicit typings (43)
tff(sy_c_Groups_Ominus__class_Ominus_000tc__Int__Oint,type,
minus_minus_int: ( int * int ) > int ).
tff(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
minus_minus_nat: ( nat * nat ) > nat ).
tff(sy_c_Groups_Ominus__class_Ominus_000tc__RealDef__Oreal,type,
minus_minus_real: ( real * real ) > real ).
tff(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,
one_one_int: int ).
tff(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
one_one_nat: nat ).
tff(sy_c_Groups_Oone__class_Oone_000tc__RealDef__Oreal,type,
one_one_real: real ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
plus_plus_int: ( int * int ) > int ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
plus_plus_nat: ( nat * nat ) > nat ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__RealDef__Oreal,type,
plus_plus_real: ( real * real ) > real ).
tff(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
times_times_int: ( int * int ) > int ).
tff(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
times_times_nat: ( nat * nat ) > nat ).
tff(sy_c_Groups_Otimes__class_Otimes_000tc__RealDef__Oreal,type,
times_times_real: ( real * real ) > real ).
tff(sy_c_Groups_Ozero__class_Ozero_000tc__Int__Oint,type,
zero_zero_int: int ).
tff(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_nat: nat ).
tff(sy_c_Groups_Ozero__class_Ozero_000tc__RealDef__Oreal,type,
zero_zero_real: real ).
tff(sy_c_IntPrimes_Ozcong,type,
zcong: ( int * int * int ) > $o ).
tff(sy_c_IntPrimes_Ozprime,type,
zprime: int > $o ).
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_000tc__Int__Oint,type,
number_number_of_int: int > int ).
tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Nat__Onat,type,
number_number_of_nat: int > nat ).
tff(sy_c_Int_Onumber__class_Onumber__of_000tc__RealDef__Oreal,type,
number267125858f_real: int > real ).
tff(sy_c_Orderings_Oord__class_Oless_000tc__Int__Oint,type,
ord_less_int: ( int * int ) > $o ).
tff(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
ord_less_nat: ( nat * nat ) > $o ).
tff(sy_c_Orderings_Oord__class_Oless_000tc__RealDef__Oreal,type,
ord_less_real: ( real * real ) > $o ).
tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Int__Oint,type,
ord_less_eq_int: ( int * int ) > $o ).
tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
ord_less_eq_nat: ( nat * nat ) > $o ).
tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__RealDef__Oreal,type,
ord_less_eq_real: ( real * real ) > $o ).
tff(sy_c_Power_Opower__class_Opower_000tc__Int__Oint,type,
power_power_int: ( int * nat ) > int ).
tff(sy_c_Power_Opower__class_Opower_000tc__Nat__Onat,type,
power_power_nat: ( nat * nat ) > nat ).
tff(sy_c_Power_Opower__class_Opower_000tc__RealDef__Oreal,type,
power_power_real: ( real * nat ) > real ).
tff(sy_c_Residues_OLegendre,type,
legendre: ( int * int ) > int ).
tff(sy_c_Residues_OQuadRes,type,
quadRes: ( int * int ) > $o ).
tff(sy_c_Rings_Odvd__class_Odvd_000tc__Int__Oint,type,
dvd_dvd_int: ( int * int ) > $o ).
tff(sy_c_Rings_Odvd__class_Odvd_000tc__Nat__Onat,type,
dvd_dvd_nat: ( nat * nat ) > $o ).
tff(sy_c_Rings_Odvd__class_Odvd_000tc__RealDef__Oreal,type,
dvd_dvd_real: ( real * real ) > $o ).
tff(sy_c_TwoSquares__Mirabelle__xuacyfoabs_Ois__sum2sq,type,
twoSqu512355103sum2sq: int > $o ).
tff(sy_v_m,type,
m: int ).
tff(sy_v_s1____,type,
s1: int ).
tff(sy_v_s____,type,
s: int ).
tff(sy_v_t____,type,
t: int ).
%----Relevant facts (698)
tff(fact_0__096t_A_060_A0_096,axiom,
ord_less_int(t,zero_zero_int) ).
tff(fact_1_calculation_I1_J,axiom,
ord_less_int(t,one_one_int) ).
tff(fact_2__096_I4_A_K_Am_A_L_A1_J_A_K_At_A_060_A_I4_A_K_Am_A_L_A1_J_A_K_A0_096,axiom,
ord_less_int(times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),t),times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),zero_zero_int)) ).
tff(fact_3_t,axiom,
plus_plus_int(power_power_int(s,number_number_of_nat(bit0(bit1(pls)))),one_one_int) = times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),t) ).
tff(fact_4_calculation_I2_J,axiom,
( ( t = zero_zero_int )
=> ( plus_plus_int(power_power_int(s,number_number_of_nat(bit0(bit1(pls)))),one_one_int) = zero_zero_int ) ) ).
tff(fact_5__096_126_A1_A_060_061_At_096,axiom,
~ ord_less_eq_int(one_one_int,t) ).
tff(fact_6_p0,axiom,
ord_less_int(zero_zero_int,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) ).
tff(fact_7_not__sum__power2__lt__zero,axiom,
! [X: real,Y: real] : ~ ord_less_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))))),zero_zero_real) ).
tff(fact_8_not__sum__power2__lt__zero,axiom,
! [X: int,Y: int] : ~ ord_less_int(plus_plus_int(power_power_int(X,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y,number_number_of_nat(bit0(bit1(pls))))),zero_zero_int) ).
tff(fact_9_sum__power2__gt__zero__iff,axiom,
! [X_2: real,Y_2: real] :
( ord_less_real(zero_zero_real,plus_plus_real(power_power_real(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_2,number_number_of_nat(bit0(bit1(pls))))))
<=> ( ( X_2 != zero_zero_real )
| ( Y_2 != zero_zero_real ) ) ) ).
tff(fact_10_sum__power2__gt__zero__iff,axiom,
! [X_2: int,Y_2: int] :
( ord_less_int(zero_zero_int,plus_plus_int(power_power_int(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_2,number_number_of_nat(bit0(bit1(pls))))))
<=> ( ( X_2 != zero_zero_int )
| ( Y_2 != zero_zero_int ) ) ) ).
tff(fact_11_sum__power2__eq__zero__iff,axiom,
! [X_2: real,Y_2: real] :
( ( plus_plus_real(power_power_real(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_2,number_number_of_nat(bit0(bit1(pls))))) = zero_zero_real )
<=> ( ( X_2 = zero_zero_real )
& ( Y_2 = zero_zero_real ) ) ) ).
tff(fact_12_sum__power2__eq__zero__iff,axiom,
! [X_2: int,Y_2: int] :
( ( plus_plus_int(power_power_int(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_2,number_number_of_nat(bit0(bit1(pls))))) = zero_zero_int )
<=> ( ( X_2 = zero_zero_int )
& ( Y_2 = zero_zero_int ) ) ) ).
tff(fact_13_power2__less__0,axiom,
! [A_56: real] : ~ ord_less_real(power_power_real(A_56,number_number_of_nat(bit0(bit1(pls)))),zero_zero_real) ).
tff(fact_14_power2__less__0,axiom,
! [A_56: int] : ~ ord_less_int(power_power_int(A_56,number_number_of_nat(bit0(bit1(pls)))),zero_zero_int) ).
tff(fact_15_zero__less__power2,axiom,
! [A_13: real] :
( ord_less_real(zero_zero_real,power_power_real(A_13,number_number_of_nat(bit0(bit1(pls)))))
<=> ( A_13 != zero_zero_real ) ) ).
tff(fact_16_zero__less__power2,axiom,
! [A_13: int] :
( ord_less_int(zero_zero_int,power_power_int(A_13,number_number_of_nat(bit0(bit1(pls)))))
<=> ( A_13 != zero_zero_int ) ) ).
tff(fact_17_one__power2,axiom,
power_power_real(one_one_real,number_number_of_nat(bit0(bit1(pls)))) = one_one_real ).
tff(fact_18_one__power2,axiom,
power_power_nat(one_one_nat,number_number_of_nat(bit0(bit1(pls)))) = one_one_nat ).
tff(fact_19_one__power2,axiom,
power_power_int(one_one_int,number_number_of_nat(bit0(bit1(pls)))) = one_one_int ).
tff(fact_20_zero__power2,axiom,
power_power_real(zero_zero_real,number_number_of_nat(bit0(bit1(pls)))) = zero_zero_real ).
tff(fact_21_zero__power2,axiom,
power_power_nat(zero_zero_nat,number_number_of_nat(bit0(bit1(pls)))) = zero_zero_nat ).
tff(fact_22_zero__power2,axiom,
power_power_int(zero_zero_int,number_number_of_nat(bit0(bit1(pls)))) = zero_zero_int ).
tff(fact_23_zero__eq__power2,axiom,
! [A_13: real] :
( ( power_power_real(A_13,number_number_of_nat(bit0(bit1(pls)))) = zero_zero_real )
<=> ( A_13 = zero_zero_real ) ) ).
tff(fact_24_zero__eq__power2,axiom,
! [A_13: int] :
( ( power_power_int(A_13,number_number_of_nat(bit0(bit1(pls)))) = zero_zero_int )
<=> ( A_13 = zero_zero_int ) ) ).
tff(fact_25_add__special_I2_J,axiom,
! [W_16: int] : plus_plus_real(one_one_real,number267125858f_real(W_16)) = number267125858f_real(plus_plus_int(bit1(pls),W_16)) ).
tff(fact_26_add__special_I2_J,axiom,
! [W_16: int] : plus_plus_int(one_one_int,number_number_of_int(W_16)) = number_number_of_int(plus_plus_int(bit1(pls),W_16)) ).
tff(fact_27_add__special_I3_J,axiom,
! [V_20: int] : plus_plus_real(number267125858f_real(V_20),one_one_real) = number267125858f_real(plus_plus_int(V_20,bit1(pls))) ).
tff(fact_28_add__special_I3_J,axiom,
! [V_20: int] : plus_plus_int(number_number_of_int(V_20),one_one_int) = number_number_of_int(plus_plus_int(V_20,bit1(pls))) ).
tff(fact_29_t__l__p,axiom,
ord_less_int(t,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) ).
tff(fact_30__096_B_Bthesis_O_A_I_B_Bt_O_As_A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_L_A1_,axiom,
~ ! [T: int] : plus_plus_int(power_power_int(s,number_number_of_nat(bit0(bit1(pls)))),one_one_int) != times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),T) ).
tff(fact_31_p,axiom,
zprime(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) ).
tff(fact_32_qf1pt,axiom,
twoSqu512355103sum2sq(times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),t)) ).
tff(fact_33_zle__refl,axiom,
! [W: int] : ord_less_eq_int(W,W) ).
tff(fact_34_number__of__is__id,axiom,
! [K_1: int] : number_number_of_int(K_1) = K_1 ).
tff(fact_35_zmult__commute,axiom,
! [Z: int,W: int] : times_times_int(Z,W) = times_times_int(W,Z) ).
tff(fact_36_zle__linear,axiom,
! [Z: int,W: int] :
( ord_less_eq_int(Z,W)
| ord_less_eq_int(W,Z) ) ).
tff(fact_37_times__numeral__code_I5_J,axiom,
! [V_4: int,W: int] : times_times_int(number_number_of_int(V_4),number_number_of_int(W)) = number_number_of_int(times_times_int(V_4,W)) ).
tff(fact_38_less__eq__number__of__int__code,axiom,
! [K: int,L_1: int] :
( ord_less_eq_int(number_number_of_int(K),number_number_of_int(L_1))
<=> ord_less_eq_int(K,L_1) ) ).
tff(fact_39_le__number__of,axiom,
! [X_2: int,Y_2: int] :
( ord_less_eq_real(number267125858f_real(X_2),number267125858f_real(Y_2))
<=> ord_less_eq_int(X_2,Y_2) ) ).
tff(fact_40_le__number__of,axiom,
! [X_2: int,Y_2: int] :
( ord_less_eq_int(number_number_of_int(X_2),number_number_of_int(Y_2))
<=> ord_less_eq_int(X_2,Y_2) ) ).
tff(fact_41_zmult__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] : times_times_int(times_times_int(Z1,Z2),Z3) = times_times_int(Z1,times_times_int(Z2,Z3)) ).
tff(fact_42_zle__trans,axiom,
! [K_1: int,I: int,J: int] :
( ord_less_eq_int(I,J)
=> ( ord_less_eq_int(J,K_1)
=> ord_less_eq_int(I,K_1) ) ) ).
tff(fact_43_zle__antisym,axiom,
! [Z: int,W: int] :
( ord_less_eq_int(Z,W)
=> ( ord_less_eq_int(W,Z)
=> ( Z = W ) ) ) ).
tff(fact_44_zpower__zadd__distrib,axiom,
! [X: int,Y: nat,Z: nat] : power_power_int(X,plus_plus_nat(Y,Z)) = times_times_int(power_power_int(X,Y),power_power_int(X,Z)) ).
tff(fact_45_less__eq__int__code_I16_J,axiom,
! [K1: int,K2: int] :
( ord_less_eq_int(bit1(K1),bit1(K2))
<=> ord_less_eq_int(K1,K2) ) ).
tff(fact_46_rel__simps_I34_J,axiom,
! [K: int,L_1: int] :
( ord_less_eq_int(bit1(K),bit1(L_1))
<=> ord_less_eq_int(K,L_1) ) ).
tff(fact_47_rel__simps_I19_J,axiom,
ord_less_eq_int(pls,pls) ).
tff(fact_48_less__eq__int__code_I13_J,axiom,
! [K1: int,K2: int] :
( ord_less_eq_int(bit0(K1),bit0(K2))
<=> ord_less_eq_int(K1,K2) ) ).
tff(fact_49_rel__simps_I31_J,axiom,
! [K: int,L_1: int] :
( ord_less_eq_int(bit0(K),bit0(L_1))
<=> ord_less_eq_int(K,L_1) ) ).
tff(fact_50_zless__le,axiom,
! [Z_1: int,W_1: int] :
( ord_less_int(Z_1,W_1)
<=> ( ord_less_eq_int(Z_1,W_1)
& ( Z_1 != W_1 ) ) ) ).
tff(fact_51_zadd__left__mono,axiom,
! [K_1: int,I: int,J: int] :
( ord_less_eq_int(I,J)
=> ord_less_eq_int(plus_plus_int(K_1,I),plus_plus_int(K_1,J)) ) ).
tff(fact_52_eq__number__of__0,axiom,
! [V_7: int] :
( ( number_number_of_nat(V_7) = zero_zero_nat )
<=> ord_less_eq_int(V_7,pls) ) ).
tff(fact_53_eq__0__number__of,axiom,
! [V_7: int] :
( ( zero_zero_nat = number_number_of_nat(V_7) )
<=> ord_less_eq_int(V_7,pls) ) ).
tff(fact_54_semiring__mult__number__of,axiom,
! [V_19: int,V_18: int] :
( ord_less_eq_int(pls,V_18)
=> ( ord_less_eq_int(pls,V_19)
=> ( times_times_real(number267125858f_real(V_18),number267125858f_real(V_19)) = number267125858f_real(times_times_int(V_18,V_19)) ) ) ) ).
tff(fact_55_semiring__mult__number__of,axiom,
! [V_19: int,V_18: int] :
( ord_less_eq_int(pls,V_18)
=> ( ord_less_eq_int(pls,V_19)
=> ( times_times_nat(number_number_of_nat(V_18),number_number_of_nat(V_19)) = number_number_of_nat(times_times_int(V_18,V_19)) ) ) ) ).
tff(fact_56_semiring__mult__number__of,axiom,
! [V_19: int,V_18: int] :
( ord_less_eq_int(pls,V_18)
=> ( ord_less_eq_int(pls,V_19)
=> ( times_times_int(number_number_of_int(V_18),number_number_of_int(V_19)) = number_number_of_int(times_times_int(V_18,V_19)) ) ) ) ).
tff(fact_57_mult__number__of__left,axiom,
! [V_17: int,W_15: int,Z_9: real] : times_times_real(number267125858f_real(V_17),times_times_real(number267125858f_real(W_15),Z_9)) = times_times_real(number267125858f_real(times_times_int(V_17,W_15)),Z_9) ).
tff(fact_58_mult__number__of__left,axiom,
! [V_17: int,W_15: int,Z_9: int] : times_times_int(number_number_of_int(V_17),times_times_int(number_number_of_int(W_15),Z_9)) = times_times_int(number_number_of_int(times_times_int(V_17,W_15)),Z_9) ).
tff(fact_59_arith__simps_I32_J,axiom,
! [V_16: int,W_14: int] : times_times_real(number267125858f_real(V_16),number267125858f_real(W_14)) = number267125858f_real(times_times_int(V_16,W_14)) ).
tff(fact_60_arith__simps_I32_J,axiom,
! [V_16: int,W_14: int] : times_times_int(number_number_of_int(V_16),number_number_of_int(W_14)) = number_number_of_int(times_times_int(V_16,W_14)) ).
tff(fact_61_number__of__mult,axiom,
! [V_15: int,W_13: int] : number267125858f_real(times_times_int(V_15,W_13)) = times_times_real(number267125858f_real(V_15),number267125858f_real(W_13)) ).
tff(fact_62_number__of__mult,axiom,
! [V_15: int,W_13: int] : number_number_of_int(times_times_int(V_15,W_13)) = times_times_int(number_number_of_int(V_15),number_number_of_int(W_13)) ).
tff(fact_63_sum__squares__le__zero__iff,axiom,
! [X_2: real,Y_2: real] :
( ord_less_eq_real(plus_plus_real(times_times_real(X_2,X_2),times_times_real(Y_2,Y_2)),zero_zero_real)
<=> ( ( X_2 = zero_zero_real )
& ( Y_2 = zero_zero_real ) ) ) ).
tff(fact_64_sum__squares__le__zero__iff,axiom,
! [X_2: int,Y_2: int] :
( ord_less_eq_int(plus_plus_int(times_times_int(X_2,X_2),times_times_int(Y_2,Y_2)),zero_zero_int)
<=> ( ( X_2 = zero_zero_int )
& ( Y_2 = zero_zero_int ) ) ) ).
tff(fact_65_sum__squares__ge__zero,axiom,
! [X_24: real,Y_16: real] : ord_less_eq_real(zero_zero_real,plus_plus_real(times_times_real(X_24,X_24),times_times_real(Y_16,Y_16))) ).
tff(fact_66_sum__squares__ge__zero,axiom,
! [X_24: int,Y_16: int] : ord_less_eq_int(zero_zero_int,plus_plus_int(times_times_int(X_24,X_24),times_times_int(Y_16,Y_16))) ).
tff(fact_67_le__special_I3_J,axiom,
! [X_2: int] :
( ord_less_eq_real(number267125858f_real(X_2),zero_zero_real)
<=> ord_less_eq_int(X_2,pls) ) ).
tff(fact_68_le__special_I3_J,axiom,
! [X_2: int] :
( ord_less_eq_int(number_number_of_int(X_2),zero_zero_int)
<=> ord_less_eq_int(X_2,pls) ) ).
tff(fact_69_le__special_I1_J,axiom,
! [Y_2: int] :
( ord_less_eq_real(zero_zero_real,number267125858f_real(Y_2))
<=> ord_less_eq_int(pls,Y_2) ) ).
tff(fact_70_le__special_I1_J,axiom,
! [Y_2: int] :
( ord_less_eq_int(zero_zero_int,number_number_of_int(Y_2))
<=> ord_less_eq_int(pls,Y_2) ) ).
tff(fact_71_less__0__number__of,axiom,
! [V_7: int] :
( ord_less_nat(zero_zero_nat,number_number_of_nat(V_7))
<=> ord_less_int(pls,V_7) ) ).
tff(fact_72_le__number__of__eq__not__less,axiom,
! [V_7: int,W_1: int] :
( ord_less_eq_real(number267125858f_real(V_7),number267125858f_real(W_1))
<=> ~ ord_less_real(number267125858f_real(W_1),number267125858f_real(V_7)) ) ).
tff(fact_73_le__number__of__eq__not__less,axiom,
! [V_7: int,W_1: int] :
( ord_less_eq_nat(number_number_of_nat(V_7),number_number_of_nat(W_1))
<=> ~ ord_less_nat(number_number_of_nat(W_1),number_number_of_nat(V_7)) ) ).
tff(fact_74_le__number__of__eq__not__less,axiom,
! [V_7: int,W_1: int] :
( ord_less_eq_int(number_number_of_int(V_7),number_number_of_int(W_1))
<=> ~ ord_less_int(number_number_of_int(W_1),number_number_of_int(V_7)) ) ).
tff(fact_75_rel__simps_I22_J,axiom,
! [K: int] :
( ord_less_eq_int(pls,bit1(K))
<=> ord_less_eq_int(pls,K) ) ).
tff(fact_76_less__eq__int__code_I14_J,axiom,
! [K1: int,K2: int] :
( ord_less_eq_int(bit0(K1),bit1(K2))
<=> ord_less_eq_int(K1,K2) ) ).
tff(fact_77_rel__simps_I32_J,axiom,
! [K: int,L_1: int] :
( ord_less_eq_int(bit0(K),bit1(L_1))
<=> ord_less_eq_int(K,L_1) ) ).
tff(fact_78_rel__simps_I27_J,axiom,
! [K: int] :
( ord_less_eq_int(bit0(K),pls)
<=> ord_less_eq_int(K,pls) ) ).
tff(fact_79_rel__simps_I21_J,axiom,
! [K: int] :
( ord_less_eq_int(pls,bit0(K))
<=> ord_less_eq_int(pls,K) ) ).
tff(fact_80_zadd__zless__mono,axiom,
! [Z_8: int,Z: int,W_12: int,W: int] :
( ord_less_int(W_12,W)
=> ( ord_less_eq_int(Z_8,Z)
=> ord_less_int(plus_plus_int(W_12,Z_8),plus_plus_int(W,Z)) ) ) ).
tff(fact_81_nat__number__of__Pls,axiom,
number_number_of_nat(pls) = zero_zero_nat ).
tff(fact_82_semiring__norm_I113_J,axiom,
zero_zero_nat = number_number_of_nat(pls) ).
tff(fact_83_le__special_I4_J,axiom,
! [X_2: int] :
( ord_less_eq_real(number267125858f_real(X_2),one_one_real)
<=> ord_less_eq_int(X_2,bit1(pls)) ) ).
tff(fact_84_le__special_I4_J,axiom,
! [X_2: int] :
( ord_less_eq_int(number_number_of_int(X_2),one_one_int)
<=> ord_less_eq_int(X_2,bit1(pls)) ) ).
tff(fact_85_le__special_I2_J,axiom,
! [Y_2: int] :
( ord_less_eq_real(one_one_real,number267125858f_real(Y_2))
<=> ord_less_eq_int(bit1(pls),Y_2) ) ).
tff(fact_86_le__special_I2_J,axiom,
! [Y_2: int] :
( ord_less_eq_int(one_one_int,number_number_of_int(Y_2))
<=> ord_less_eq_int(bit1(pls),Y_2) ) ).
tff(fact_87_nat__1__add__1,axiom,
plus_plus_nat(one_one_nat,one_one_nat) = number_number_of_nat(bit0(bit1(pls))) ).
tff(fact_88_mult__Pls,axiom,
! [W: int] : times_times_int(pls,W) = pls ).
tff(fact_89_mult__Bit0,axiom,
! [K_1: int,L: int] : times_times_int(bit0(K_1),L) = bit0(times_times_int(K_1,L)) ).
tff(fact_90_less__number__of__int__code,axiom,
! [K: int,L_1: int] :
( ord_less_int(number_number_of_int(K),number_number_of_int(L_1))
<=> ord_less_int(K,L_1) ) ).
tff(fact_91_zmult__1__right,axiom,
! [Z: int] : times_times_int(Z,one_one_int) = Z ).
tff(fact_92_zmult__1,axiom,
! [Z: int] : times_times_int(one_one_int,Z) = Z ).
tff(fact_93_plus__numeral__code_I9_J,axiom,
! [V_4: int,W: int] : plus_plus_int(number_number_of_int(V_4),number_number_of_int(W)) = number_number_of_int(plus_plus_int(V_4,W)) ).
tff(fact_94_zadd__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] : times_times_int(plus_plus_int(Z1,Z2),W) = plus_plus_int(times_times_int(Z1,W),times_times_int(Z2,W)) ).
tff(fact_95_zadd__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] : times_times_int(W,plus_plus_int(Z1,Z2)) = plus_plus_int(times_times_int(W,Z1),times_times_int(W,Z2)) ).
tff(fact_96_rel__simps_I29_J,axiom,
! [K: int] :
( ord_less_eq_int(bit1(K),pls)
<=> ord_less_int(K,pls) ) ).
tff(fact_97_rel__simps_I5_J,axiom,
! [K: int] :
( ord_less_int(pls,bit1(K))
<=> ord_less_eq_int(pls,K) ) ).
tff(fact_98_less__eq__int__code_I15_J,axiom,
! [K1: int,K2: int] :
( ord_less_eq_int(bit1(K1),bit0(K2))
<=> ord_less_int(K1,K2) ) ).
tff(fact_99_rel__simps_I33_J,axiom,
! [K: int,L_1: int] :
( ord_less_eq_int(bit1(K),bit0(L_1))
<=> ord_less_int(K,L_1) ) ).
tff(fact_100_less__int__code_I14_J,axiom,
! [K1: int,K2: int] :
( ord_less_int(bit0(K1),bit1(K2))
<=> ord_less_eq_int(K1,K2) ) ).
tff(fact_101_rel__simps_I15_J,axiom,
! [K: int,L_1: int] :
( ord_less_int(bit0(K),bit1(L_1))
<=> ord_less_eq_int(K,L_1) ) ).
tff(fact_102_less__nat__number__of,axiom,
! [V_7: int,V_6: int] :
( ord_less_nat(number_number_of_nat(V_7),number_number_of_nat(V_6))
<=> ( ( ord_less_int(V_7,V_6)
=> ord_less_int(pls,V_6) )
& ord_less_int(V_7,V_6) ) ) ).
tff(fact_103_int__one__le__iff__zero__less,axiom,
! [Z_1: int] :
( ord_less_eq_int(one_one_int,Z_1)
<=> ord_less_int(zero_zero_int,Z_1) ) ).
tff(fact_104_nat__numeral__1__eq__1,axiom,
number_number_of_nat(bit1(pls)) = one_one_nat ).
tff(fact_105_Numeral1__eq1__nat,axiom,
one_one_nat = number_number_of_nat(bit1(pls)) ).
tff(fact_106_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ord_less_int(W,Z)
=> ord_less_eq_int(plus_plus_int(W,one_one_int),Z) ) ).
tff(fact_107_add1__zle__eq,axiom,
! [W_1: int,Z_1: int] :
( ord_less_eq_int(plus_plus_int(W_1,one_one_int),Z_1)
<=> ord_less_int(W_1,Z_1) ) ).
tff(fact_108_zle__add1__eq__le,axiom,
! [W_1: int,Z_1: int] :
( ord_less_int(W_1,plus_plus_int(Z_1,one_one_int))
<=> ord_less_eq_int(W_1,Z_1) ) ).
tff(fact_109_semiring__add__number__of,axiom,
! [V_14: int,V_13: int] :
( ord_less_eq_int(pls,V_13)
=> ( ord_less_eq_int(pls,V_14)
=> ( plus_plus_real(number267125858f_real(V_13),number267125858f_real(V_14)) = number267125858f_real(plus_plus_int(V_13,V_14)) ) ) ) ).
tff(fact_110_semiring__add__number__of,axiom,
! [V_14: int,V_13: int] :
( ord_less_eq_int(pls,V_13)
=> ( ord_less_eq_int(pls,V_14)
=> ( plus_plus_nat(number_number_of_nat(V_13),number_number_of_nat(V_14)) = number_number_of_nat(plus_plus_int(V_13,V_14)) ) ) ) ).
tff(fact_111_semiring__add__number__of,axiom,
! [V_14: int,V_13: int] :
( ord_less_eq_int(pls,V_13)
=> ( ord_less_eq_int(pls,V_14)
=> ( plus_plus_int(number_number_of_int(V_13),number_number_of_int(V_14)) = number_number_of_int(plus_plus_int(V_13,V_14)) ) ) ) ).
tff(fact_112_add__nat__number__of,axiom,
! [V_5: int,V_4: int] :
( ( ord_less_int(V_4,pls)
=> ( plus_plus_nat(number_number_of_nat(V_4),number_number_of_nat(V_5)) = number_number_of_nat(V_5) ) )
& ( ~ ord_less_int(V_4,pls)
=> ( ( ord_less_int(V_5,pls)
=> ( plus_plus_nat(number_number_of_nat(V_4),number_number_of_nat(V_5)) = number_number_of_nat(V_4) ) )
& ( ~ ord_less_int(V_5,pls)
=> ( plus_plus_nat(number_number_of_nat(V_4),number_number_of_nat(V_5)) = number_number_of_nat(plus_plus_int(V_4,V_5)) ) ) ) ) ) ).
tff(fact_113_le__imp__0__less,axiom,
! [Z: int] :
( ord_less_eq_int(zero_zero_int,Z)
=> ord_less_int(zero_zero_int,plus_plus_int(one_one_int,Z)) ) ).
tff(fact_114_eq__number__of,axiom,
! [X_2: int,Y_2: int] :
( ( number267125858f_real(X_2) = number267125858f_real(Y_2) )
<=> ( X_2 = Y_2 ) ) ).
tff(fact_115_eq__number__of,axiom,
! [X_2: int,Y_2: int] :
( ( number_number_of_int(X_2) = number_number_of_int(Y_2) )
<=> ( X_2 = Y_2 ) ) ).
tff(fact_116_number__of__reorient,axiom,
! [W_1: int,X_2: real] :
( ( number267125858f_real(W_1) = X_2 )
<=> ( X_2 = number267125858f_real(W_1) ) ) ).
tff(fact_117_number__of__reorient,axiom,
! [W_1: int,X_2: int] :
( ( number_number_of_int(W_1) = X_2 )
<=> ( X_2 = number_number_of_int(W_1) ) ) ).
tff(fact_118_number__of__reorient,axiom,
! [W_1: int,X_2: nat] :
( ( number_number_of_nat(W_1) = X_2 )
<=> ( X_2 = number_number_of_nat(W_1) ) ) ).
tff(fact_119_rel__simps_I51_J,axiom,
! [K: int,L_1: int] :
( ( bit1(K) = bit1(L_1) )
<=> ( K = L_1 ) ) ).
tff(fact_120_rel__simps_I48_J,axiom,
! [K: int,L_1: int] :
( ( bit0(K) = bit0(L_1) )
<=> ( K = L_1 ) ) ).
tff(fact_121_zless__linear,axiom,
! [X: int,Y: int] :
( ord_less_int(X,Y)
| ( X = Y )
| ord_less_int(Y,X) ) ).
tff(fact_122_sum__squares__eq__zero__iff,axiom,
! [X_2: real,Y_2: real] :
( ( plus_plus_real(times_times_real(X_2,X_2),times_times_real(Y_2,Y_2)) = zero_zero_real )
<=> ( ( X_2 = zero_zero_real )
& ( Y_2 = zero_zero_real ) ) ) ).
tff(fact_123_sum__squares__eq__zero__iff,axiom,
! [X_2: int,Y_2: int] :
( ( plus_plus_int(times_times_int(X_2,X_2),times_times_int(Y_2,Y_2)) = zero_zero_int )
<=> ( ( X_2 = zero_zero_int )
& ( Y_2 = zero_zero_int ) ) ) ).
tff(fact_124_left__distrib__number__of,axiom,
! [A_55: real,B_20: real,V_12: int] : times_times_real(plus_plus_real(A_55,B_20),number267125858f_real(V_12)) = plus_plus_real(times_times_real(A_55,number267125858f_real(V_12)),times_times_real(B_20,number267125858f_real(V_12))) ).
tff(fact_125_left__distrib__number__of,axiom,
! [A_55: nat,B_20: nat,V_12: int] : times_times_nat(plus_plus_nat(A_55,B_20),number_number_of_nat(V_12)) = plus_plus_nat(times_times_nat(A_55,number_number_of_nat(V_12)),times_times_nat(B_20,number_number_of_nat(V_12))) ).
tff(fact_126_left__distrib__number__of,axiom,
! [A_55: int,B_20: int,V_12: int] : times_times_int(plus_plus_int(A_55,B_20),number_number_of_int(V_12)) = plus_plus_int(times_times_int(A_55,number_number_of_int(V_12)),times_times_int(B_20,number_number_of_int(V_12))) ).
tff(fact_127_right__distrib__number__of,axiom,
! [V_11: int,B_19: real,C_13: real] : times_times_real(number267125858f_real(V_11),plus_plus_real(B_19,C_13)) = plus_plus_real(times_times_real(number267125858f_real(V_11),B_19),times_times_real(number267125858f_real(V_11),C_13)) ).
tff(fact_128_right__distrib__number__of,axiom,
! [V_11: int,B_19: nat,C_13: nat] : times_times_nat(number_number_of_nat(V_11),plus_plus_nat(B_19,C_13)) = plus_plus_nat(times_times_nat(number_number_of_nat(V_11),B_19),times_times_nat(number_number_of_nat(V_11),C_13)) ).
tff(fact_129_right__distrib__number__of,axiom,
! [V_11: int,B_19: int,C_13: int] : times_times_int(number_number_of_int(V_11),plus_plus_int(B_19,C_13)) = plus_plus_int(times_times_int(number_number_of_int(V_11),B_19),times_times_int(number_number_of_int(V_11),C_13)) ).
tff(fact_130_zadd__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] : plus_plus_int(plus_plus_int(Z1,Z2),Z3) = plus_plus_int(Z1,plus_plus_int(Z2,Z3)) ).
tff(fact_131_zadd__left__commute,axiom,
! [X: int,Y: int,Z: int] : plus_plus_int(X,plus_plus_int(Y,Z)) = plus_plus_int(Y,plus_plus_int(X,Z)) ).
tff(fact_132_zadd__commute,axiom,
! [Z: int,W: int] : plus_plus_int(Z,W) = plus_plus_int(W,Z) ).
tff(fact_133_zero__is__num__zero,axiom,
zero_zero_int = number_number_of_int(pls) ).
tff(fact_134_zmult__zless__mono2,axiom,
! [K_1: int,I: int,J: int] :
( ord_less_int(I,J)
=> ( ord_less_int(zero_zero_int,K_1)
=> ord_less_int(times_times_int(K_1,I),times_times_int(K_1,J)) ) ) ).
tff(fact_135_power2__eq__imp__eq,axiom,
! [X_23: real,Y_15: real] :
( ( power_power_real(X_23,number_number_of_nat(bit0(bit1(pls)))) = power_power_real(Y_15,number_number_of_nat(bit0(bit1(pls)))) )
=> ( ord_less_eq_real(zero_zero_real,X_23)
=> ( ord_less_eq_real(zero_zero_real,Y_15)
=> ( X_23 = Y_15 ) ) ) ) ).
tff(fact_136_power2__eq__imp__eq,axiom,
! [X_23: nat,Y_15: nat] :
( ( power_power_nat(X_23,number_number_of_nat(bit0(bit1(pls)))) = power_power_nat(Y_15,number_number_of_nat(bit0(bit1(pls)))) )
=> ( ord_less_eq_nat(zero_zero_nat,X_23)
=> ( ord_less_eq_nat(zero_zero_nat,Y_15)
=> ( X_23 = Y_15 ) ) ) ) ).
tff(fact_137_power2__eq__imp__eq,axiom,
! [X_23: int,Y_15: int] :
( ( power_power_int(X_23,number_number_of_nat(bit0(bit1(pls)))) = power_power_int(Y_15,number_number_of_nat(bit0(bit1(pls)))) )
=> ( ord_less_eq_int(zero_zero_int,X_23)
=> ( ord_less_eq_int(zero_zero_int,Y_15)
=> ( X_23 = Y_15 ) ) ) ) ).
tff(fact_138_power2__le__imp__le,axiom,
! [X_22: real,Y_14: real] :
( ord_less_eq_real(power_power_real(X_22,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_14,number_number_of_nat(bit0(bit1(pls)))))
=> ( ord_less_eq_real(zero_zero_real,Y_14)
=> ord_less_eq_real(X_22,Y_14) ) ) ).
tff(fact_139_power2__le__imp__le,axiom,
! [X_22: nat,Y_14: nat] :
( ord_less_eq_nat(power_power_nat(X_22,number_number_of_nat(bit0(bit1(pls)))),power_power_nat(Y_14,number_number_of_nat(bit0(bit1(pls)))))
=> ( ord_less_eq_nat(zero_zero_nat,Y_14)
=> ord_less_eq_nat(X_22,Y_14) ) ) ).
tff(fact_140_power2__le__imp__le,axiom,
! [X_22: int,Y_14: int] :
( ord_less_eq_int(power_power_int(X_22,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_14,number_number_of_nat(bit0(bit1(pls)))))
=> ( ord_less_eq_int(zero_zero_int,Y_14)
=> ord_less_eq_int(X_22,Y_14) ) ) ).
tff(fact_141_zero__le__power2,axiom,
! [A_54: real] : ord_less_eq_real(zero_zero_real,power_power_real(A_54,number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_142_zero__le__power2,axiom,
! [A_54: int] : ord_less_eq_int(zero_zero_int,power_power_int(A_54,number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_143_power2__less__imp__less,axiom,
! [X_21: real,Y_13: real] :
( ord_less_real(power_power_real(X_21,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_13,number_number_of_nat(bit0(bit1(pls)))))
=> ( ord_less_eq_real(zero_zero_real,Y_13)
=> ord_less_real(X_21,Y_13) ) ) ).
tff(fact_144_power2__less__imp__less,axiom,
! [X_21: nat,Y_13: nat] :
( ord_less_nat(power_power_nat(X_21,number_number_of_nat(bit0(bit1(pls)))),power_power_nat(Y_13,number_number_of_nat(bit0(bit1(pls)))))
=> ( ord_less_eq_nat(zero_zero_nat,Y_13)
=> ord_less_nat(X_21,Y_13) ) ) ).
tff(fact_145_power2__less__imp__less,axiom,
! [X_21: int,Y_13: int] :
( ord_less_int(power_power_int(X_21,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_13,number_number_of_nat(bit0(bit1(pls)))))
=> ( ord_less_eq_int(zero_zero_int,Y_13)
=> ord_less_int(X_21,Y_13) ) ) ).
tff(fact_146_sum__power2__le__zero__iff,axiom,
! [X_2: real,Y_2: real] :
( ord_less_eq_real(plus_plus_real(power_power_real(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_2,number_number_of_nat(bit0(bit1(pls))))),zero_zero_real)
<=> ( ( X_2 = zero_zero_real )
& ( Y_2 = zero_zero_real ) ) ) ).
tff(fact_147_sum__power2__le__zero__iff,axiom,
! [X_2: int,Y_2: int] :
( ord_less_eq_int(plus_plus_int(power_power_int(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_2,number_number_of_nat(bit0(bit1(pls))))),zero_zero_int)
<=> ( ( X_2 = zero_zero_int )
& ( Y_2 = zero_zero_int ) ) ) ).
tff(fact_148_sum__power2__ge__zero,axiom,
! [X_20: real,Y_12: real] : ord_less_eq_real(zero_zero_real,plus_plus_real(power_power_real(X_20,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_12,number_number_of_nat(bit0(bit1(pls)))))) ).
tff(fact_149_sum__power2__ge__zero,axiom,
! [X_20: int,Y_12: int] : ord_less_eq_int(zero_zero_int,plus_plus_int(power_power_int(X_20,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_12,number_number_of_nat(bit0(bit1(pls)))))) ).
tff(fact_150_sum__squares__gt__zero__iff,axiom,
! [X_2: real,Y_2: real] :
( ord_less_real(zero_zero_real,plus_plus_real(times_times_real(X_2,X_2),times_times_real(Y_2,Y_2)))
<=> ( ( X_2 != zero_zero_real )
| ( Y_2 != zero_zero_real ) ) ) ).
tff(fact_151_sum__squares__gt__zero__iff,axiom,
! [X_2: int,Y_2: int] :
( ord_less_int(zero_zero_int,plus_plus_int(times_times_int(X_2,X_2),times_times_int(Y_2,Y_2)))
<=> ( ( X_2 != zero_zero_int )
| ( Y_2 != zero_zero_int ) ) ) ).
tff(fact_152_not__sum__squares__lt__zero,axiom,
! [X_19: real,Y_11: real] : ~ ord_less_real(plus_plus_real(times_times_real(X_19,X_19),times_times_real(Y_11,Y_11)),zero_zero_real) ).
tff(fact_153_not__sum__squares__lt__zero,axiom,
! [X_19: int,Y_11: int] : ~ ord_less_int(plus_plus_int(times_times_int(X_19,X_19),times_times_int(Y_11,Y_11)),zero_zero_int) ).
tff(fact_154_mult__numeral__1,axiom,
! [A_53: real] : times_times_real(number267125858f_real(bit1(pls)),A_53) = A_53 ).
tff(fact_155_mult__numeral__1,axiom,
! [A_53: int] : times_times_int(number_number_of_int(bit1(pls)),A_53) = A_53 ).
tff(fact_156_mult__numeral__1__right,axiom,
! [A_52: real] : times_times_real(A_52,number267125858f_real(bit1(pls))) = A_52 ).
tff(fact_157_mult__numeral__1__right,axiom,
! [A_52: int] : times_times_int(A_52,number_number_of_int(bit1(pls))) = A_52 ).
tff(fact_158_one__is__num__one,axiom,
one_one_int = number_number_of_int(bit1(pls)) ).
tff(fact_159_mult__Bit1,axiom,
! [K_1: int,L: int] : times_times_int(bit1(K_1),L) = plus_plus_int(bit0(times_times_int(K_1,L)),L) ).
tff(fact_160_pos__zmult__eq__1__iff,axiom,
! [N_1: int,Ma: int] :
( ord_less_int(zero_zero_int,Ma)
=> ( ( times_times_int(Ma,N_1) = one_one_int )
<=> ( ( Ma = one_one_int )
& ( N_1 = one_one_int ) ) ) ) ).
tff(fact_161_double__eq__0__iff,axiom,
! [A_13: real] :
( ( plus_plus_real(A_13,A_13) = zero_zero_real )
<=> ( A_13 = zero_zero_real ) ) ).
tff(fact_162_double__eq__0__iff,axiom,
! [A_13: int] :
( ( plus_plus_int(A_13,A_13) = zero_zero_int )
<=> ( A_13 = zero_zero_int ) ) ).
tff(fact_163_rel__simps_I46_J,axiom,
! [K_1: int] : bit1(K_1) != pls ).
tff(fact_164_rel__simps_I39_J,axiom,
! [L: int] : pls != bit1(L) ).
tff(fact_165_rel__simps_I50_J,axiom,
! [K_1: int,L: int] : bit1(K_1) != bit0(L) ).
tff(fact_166_rel__simps_I49_J,axiom,
! [K_1: int,L: int] : bit0(K_1) != bit1(L) ).
tff(fact_167_rel__simps_I44_J,axiom,
! [K: int] :
( ( bit0(K) = pls )
<=> ( K = pls ) ) ).
tff(fact_168_rel__simps_I38_J,axiom,
! [L_1: int] :
( ( pls = bit0(L_1) )
<=> ( pls = L_1 ) ) ).
tff(fact_169_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_170_Pls__def,axiom,
pls = zero_zero_int ).
tff(fact_171_less__int__code_I16_J,axiom,
! [K1: int,K2: int] :
( ord_less_int(bit1(K1),bit1(K2))
<=> ord_less_int(K1,K2) ) ).
tff(fact_172_rel__simps_I17_J,axiom,
! [K: int,L_1: int] :
( ord_less_int(bit1(K),bit1(L_1))
<=> ord_less_int(K,L_1) ) ).
tff(fact_173_rel__simps_I2_J,axiom,
~ ord_less_int(pls,pls) ).
tff(fact_174_less__int__code_I13_J,axiom,
! [K1: int,K2: int] :
( ord_less_int(bit0(K1),bit0(K2))
<=> ord_less_int(K1,K2) ) ).
tff(fact_175_rel__simps_I14_J,axiom,
! [K: int,L_1: int] :
( ord_less_int(bit0(K),bit0(L_1))
<=> ord_less_int(K,L_1) ) ).
tff(fact_176_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
tff(fact_177_add__Pls__right,axiom,
! [K_1: int] : plus_plus_int(K_1,pls) = K_1 ).
tff(fact_178_add__Pls,axiom,
! [K_1: int] : plus_plus_int(pls,K_1) = K_1 ).
tff(fact_179_add__Bit0__Bit0,axiom,
! [K_1: int,L: int] : plus_plus_int(bit0(K_1),bit0(L)) = bit0(plus_plus_int(K_1,L)) ).
tff(fact_180_Bit0__def,axiom,
! [K_1: int] : bit0(K_1) = plus_plus_int(K_1,K_1) ).
tff(fact_181_zadd__0__right,axiom,
! [Z: int] : plus_plus_int(Z,zero_zero_int) = Z ).
tff(fact_182_zadd__0,axiom,
! [Z: int] : plus_plus_int(zero_zero_int,Z) = Z ).
tff(fact_183_zadd__strict__right__mono,axiom,
! [K_1: int,I: int,J: int] :
( ord_less_int(I,J)
=> ord_less_int(plus_plus_int(I,K_1),plus_plus_int(J,K_1)) ) ).
tff(fact_184_double__number__of__Bit0,axiom,
! [W_11: int] : times_times_real(plus_plus_real(one_one_real,one_one_real),number267125858f_real(W_11)) = number267125858f_real(bit0(W_11)) ).
tff(fact_185_double__number__of__Bit0,axiom,
! [W_11: int] : times_times_int(plus_plus_int(one_one_int,one_one_int),number_number_of_int(W_11)) = number_number_of_int(bit0(W_11)) ).
tff(fact_186_power3__eq__cube,axiom,
! [A_51: real] : power_power_real(A_51,number_number_of_nat(bit1(bit1(pls)))) = times_times_real(times_times_real(A_51,A_51),A_51) ).
tff(fact_187_power3__eq__cube,axiom,
! [A_51: nat] : power_power_nat(A_51,number_number_of_nat(bit1(bit1(pls)))) = times_times_nat(times_times_nat(A_51,A_51),A_51) ).
tff(fact_188_power3__eq__cube,axiom,
! [A_51: int] : power_power_int(A_51,number_number_of_nat(bit1(bit1(pls)))) = times_times_int(times_times_int(A_51,A_51),A_51) ).
tff(fact_189_semiring__mult__2,axiom,
! [Z_7: real] : times_times_real(number267125858f_real(bit0(bit1(pls))),Z_7) = plus_plus_real(Z_7,Z_7) ).
tff(fact_190_semiring__mult__2,axiom,
! [Z_7: nat] : times_times_nat(number_number_of_nat(bit0(bit1(pls))),Z_7) = plus_plus_nat(Z_7,Z_7) ).
tff(fact_191_semiring__mult__2,axiom,
! [Z_7: int] : times_times_int(number_number_of_int(bit0(bit1(pls))),Z_7) = plus_plus_int(Z_7,Z_7) ).
tff(fact_192_mult__2,axiom,
! [Z_6: real] : times_times_real(number267125858f_real(bit0(bit1(pls))),Z_6) = plus_plus_real(Z_6,Z_6) ).
tff(fact_193_mult__2,axiom,
! [Z_6: int] : times_times_int(number_number_of_int(bit0(bit1(pls))),Z_6) = plus_plus_int(Z_6,Z_6) ).
tff(fact_194_semiring__mult__2__right,axiom,
! [Z_5: real] : times_times_real(Z_5,number267125858f_real(bit0(bit1(pls)))) = plus_plus_real(Z_5,Z_5) ).
tff(fact_195_semiring__mult__2__right,axiom,
! [Z_5: nat] : times_times_nat(Z_5,number_number_of_nat(bit0(bit1(pls)))) = plus_plus_nat(Z_5,Z_5) ).
tff(fact_196_semiring__mult__2__right,axiom,
! [Z_5: int] : times_times_int(Z_5,number_number_of_int(bit0(bit1(pls)))) = plus_plus_int(Z_5,Z_5) ).
tff(fact_197_mult__2__right,axiom,
! [Z_4: real] : times_times_real(Z_4,number267125858f_real(bit0(bit1(pls)))) = plus_plus_real(Z_4,Z_4) ).
tff(fact_198_mult__2__right,axiom,
! [Z_4: int] : times_times_int(Z_4,number_number_of_int(bit0(bit1(pls)))) = plus_plus_int(Z_4,Z_4) ).
tff(fact_199_power2__eq__square,axiom,
! [A_50: real] : power_power_real(A_50,number_number_of_nat(bit0(bit1(pls)))) = times_times_real(A_50,A_50) ).
tff(fact_200_power2__eq__square,axiom,
! [A_50: nat] : power_power_nat(A_50,number_number_of_nat(bit0(bit1(pls)))) = times_times_nat(A_50,A_50) ).
tff(fact_201_power2__eq__square,axiom,
! [A_50: int] : power_power_int(A_50,number_number_of_nat(bit0(bit1(pls)))) = times_times_int(A_50,A_50) ).
tff(fact_202_even__less__0__iff,axiom,
! [A_13: real] :
( ord_less_real(plus_plus_real(A_13,A_13),zero_zero_real)
<=> ord_less_real(A_13,zero_zero_real) ) ).
tff(fact_203_even__less__0__iff,axiom,
! [A_13: int] :
( ord_less_int(plus_plus_int(A_13,A_13),zero_zero_int)
<=> ord_less_int(A_13,zero_zero_int) ) ).
tff(fact_204_semiring__numeral__0__eq__0,axiom,
number267125858f_real(pls) = zero_zero_real ).
tff(fact_205_semiring__numeral__0__eq__0,axiom,
number_number_of_nat(pls) = zero_zero_nat ).
tff(fact_206_semiring__numeral__0__eq__0,axiom,
number_number_of_int(pls) = zero_zero_int ).
tff(fact_207_number__of__Pls,axiom,
number267125858f_real(pls) = zero_zero_real ).
tff(fact_208_number__of__Pls,axiom,
number_number_of_int(pls) = zero_zero_int ).
tff(fact_209_semiring__norm_I112_J,axiom,
zero_zero_real = number267125858f_real(pls) ).
tff(fact_210_semiring__norm_I112_J,axiom,
zero_zero_int = number_number_of_int(pls) ).
tff(fact_211_add__numeral__0,axiom,
! [A_49: real] : plus_plus_real(number267125858f_real(pls),A_49) = A_49 ).
tff(fact_212_add__numeral__0,axiom,
! [A_49: int] : plus_plus_int(number_number_of_int(pls),A_49) = A_49 ).
tff(fact_213_add__numeral__0__right,axiom,
! [A_48: real] : plus_plus_real(A_48,number267125858f_real(pls)) = A_48 ).
tff(fact_214_add__numeral__0__right,axiom,
! [A_48: int] : plus_plus_int(A_48,number_number_of_int(pls)) = A_48 ).
tff(fact_215_less__number__of,axiom,
! [X_2: int,Y_2: int] :
( ord_less_real(number267125858f_real(X_2),number267125858f_real(Y_2))
<=> ord_less_int(X_2,Y_2) ) ).
tff(fact_216_less__number__of,axiom,
! [X_2: int,Y_2: int] :
( ord_less_int(number_number_of_int(X_2),number_number_of_int(Y_2))
<=> ord_less_int(X_2,Y_2) ) ).
tff(fact_217_add__number__of__left,axiom,
! [V_10: int,W_10: int,Z_3: real] : plus_plus_real(number267125858f_real(V_10),plus_plus_real(number267125858f_real(W_10),Z_3)) = plus_plus_real(number267125858f_real(plus_plus_int(V_10,W_10)),Z_3) ).
tff(fact_218_add__number__of__left,axiom,
! [V_10: int,W_10: int,Z_3: int] : plus_plus_int(number_number_of_int(V_10),plus_plus_int(number_number_of_int(W_10),Z_3)) = plus_plus_int(number_number_of_int(plus_plus_int(V_10,W_10)),Z_3) ).
tff(fact_219_add__number__of__eq,axiom,
! [V_9: int,W_9: int] : plus_plus_real(number267125858f_real(V_9),number267125858f_real(W_9)) = number267125858f_real(plus_plus_int(V_9,W_9)) ).
tff(fact_220_add__number__of__eq,axiom,
! [V_9: int,W_9: int] : plus_plus_int(number_number_of_int(V_9),number_number_of_int(W_9)) = number_number_of_int(plus_plus_int(V_9,W_9)) ).
tff(fact_221_number__of__add,axiom,
! [V_8: int,W_8: int] : number267125858f_real(plus_plus_int(V_8,W_8)) = plus_plus_real(number267125858f_real(V_8),number267125858f_real(W_8)) ).
tff(fact_222_number__of__add,axiom,
! [V_8: int,W_8: int] : number_number_of_int(plus_plus_int(V_8,W_8)) = plus_plus_int(number_number_of_int(V_8),number_number_of_int(W_8)) ).
tff(fact_223_rel__simps_I12_J,axiom,
! [K: int] :
( ord_less_int(bit1(K),pls)
<=> ord_less_int(K,pls) ) ).
tff(fact_224_less__int__code_I15_J,axiom,
! [K1: int,K2: int] :
( ord_less_int(bit1(K1),bit0(K2))
<=> ord_less_int(K1,K2) ) ).
tff(fact_225_rel__simps_I16_J,axiom,
! [K: int,L_1: int] :
( ord_less_int(bit1(K),bit0(L_1))
<=> ord_less_int(K,L_1) ) ).
tff(fact_226_bin__less__0__simps_I4_J,axiom,
! [W_1: int] :
( ord_less_int(bit1(W_1),zero_zero_int)
<=> ord_less_int(W_1,zero_zero_int) ) ).
tff(fact_227_rel__simps_I10_J,axiom,
! [K: int] :
( ord_less_int(bit0(K),pls)
<=> ord_less_int(K,pls) ) ).
tff(fact_228_rel__simps_I4_J,axiom,
! [K: int] :
( ord_less_int(pls,bit0(K))
<=> ord_less_int(pls,K) ) ).
tff(fact_229_bin__less__0__simps_I1_J,axiom,
~ ord_less_int(pls,zero_zero_int) ).
tff(fact_230_bin__less__0__simps_I3_J,axiom,
! [W_1: int] :
( ord_less_int(bit0(W_1),zero_zero_int)
<=> ord_less_int(W_1,zero_zero_int) ) ).
tff(fact_231_add__Bit1__Bit0,axiom,
! [K_1: int,L: int] : plus_plus_int(bit1(K_1),bit0(L)) = bit1(plus_plus_int(K_1,L)) ).
tff(fact_232_add__Bit0__Bit1,axiom,
! [K_1: int,L: int] : plus_plus_int(bit0(K_1),bit1(L)) = bit1(plus_plus_int(K_1,L)) ).
tff(fact_233_int__0__less__1,axiom,
ord_less_int(zero_zero_int,one_one_int) ).
tff(fact_234_Bit1__def,axiom,
! [K_1: int] : bit1(K_1) = plus_plus_int(plus_plus_int(one_one_int,K_1),K_1) ).
tff(fact_235_odd__nonzero,axiom,
! [Z: int] : plus_plus_int(plus_plus_int(one_one_int,Z),Z) != zero_zero_int ).
tff(fact_236_zless__add1__eq,axiom,
! [W_1: int,Z_1: int] :
( ord_less_int(W_1,plus_plus_int(Z_1,one_one_int))
<=> ( ord_less_int(W_1,Z_1)
| ( W_1 = Z_1 ) ) ) ).
tff(fact_237_power2__eq__square__number__of,axiom,
! [W_7: int] : power_power_real(number267125858f_real(W_7),number_number_of_nat(bit0(bit1(pls)))) = times_times_real(number267125858f_real(W_7),number267125858f_real(W_7)) ).
tff(fact_238_power2__eq__square__number__of,axiom,
! [W_7: int] : power_power_nat(number_number_of_nat(W_7),number_number_of_nat(bit0(bit1(pls)))) = times_times_nat(number_number_of_nat(W_7),number_number_of_nat(W_7)) ).
tff(fact_239_power2__eq__square__number__of,axiom,
! [W_7: int] : power_power_int(number_number_of_int(W_7),number_number_of_nat(bit0(bit1(pls)))) = times_times_int(number_number_of_int(W_7),number_number_of_int(W_7)) ).
tff(fact_240_power2__sum,axiom,
! [X_18: real,Y_10: real] : power_power_real(plus_plus_real(X_18,Y_10),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_real(plus_plus_real(power_power_real(X_18,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_10,number_number_of_nat(bit0(bit1(pls))))),times_times_real(times_times_real(number267125858f_real(bit0(bit1(pls))),X_18),Y_10)) ).
tff(fact_241_power2__sum,axiom,
! [X_18: nat,Y_10: nat] : power_power_nat(plus_plus_nat(X_18,Y_10),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_nat(plus_plus_nat(power_power_nat(X_18,number_number_of_nat(bit0(bit1(pls)))),power_power_nat(Y_10,number_number_of_nat(bit0(bit1(pls))))),times_times_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls))),X_18),Y_10)) ).
tff(fact_242_power2__sum,axiom,
! [X_18: int,Y_10: int] : power_power_int(plus_plus_int(X_18,Y_10),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_int(plus_plus_int(power_power_int(X_18,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_10,number_number_of_nat(bit0(bit1(pls))))),times_times_int(times_times_int(number_number_of_int(bit0(bit1(pls))),X_18),Y_10)) ).
tff(fact_243_number__of__Bit0,axiom,
! [W_6: int] : number267125858f_real(bit0(W_6)) = plus_plus_real(plus_plus_real(zero_zero_real,number267125858f_real(W_6)),number267125858f_real(W_6)) ).
tff(fact_244_number__of__Bit0,axiom,
! [W_6: int] : number_number_of_int(bit0(W_6)) = plus_plus_int(plus_plus_int(zero_zero_int,number_number_of_int(W_6)),number_number_of_int(W_6)) ).
tff(fact_245_number__of__Bit1,axiom,
! [W_5: int] : number267125858f_real(bit1(W_5)) = plus_plus_real(plus_plus_real(one_one_real,number267125858f_real(W_5)),number267125858f_real(W_5)) ).
tff(fact_246_number__of__Bit1,axiom,
! [W_5: int] : number_number_of_int(bit1(W_5)) = plus_plus_int(plus_plus_int(one_one_int,number_number_of_int(W_5)),number_number_of_int(W_5)) ).
tff(fact_247_semiring__numeral__1__eq__1,axiom,
number267125858f_real(bit1(pls)) = one_one_real ).
tff(fact_248_semiring__numeral__1__eq__1,axiom,
number_number_of_nat(bit1(pls)) = one_one_nat ).
tff(fact_249_semiring__numeral__1__eq__1,axiom,
number_number_of_int(bit1(pls)) = one_one_int ).
tff(fact_250_numeral__1__eq__1,axiom,
number267125858f_real(bit1(pls)) = one_one_real ).
tff(fact_251_numeral__1__eq__1,axiom,
number_number_of_int(bit1(pls)) = one_one_int ).
tff(fact_252_semiring__norm_I110_J,axiom,
one_one_real = number267125858f_real(bit1(pls)) ).
tff(fact_253_semiring__norm_I110_J,axiom,
one_one_int = number_number_of_int(bit1(pls)) ).
tff(fact_254_odd__less__0,axiom,
! [Z_1: int] :
( ord_less_int(plus_plus_int(plus_plus_int(one_one_int,Z_1),Z_1),zero_zero_int)
<=> ord_less_int(Z_1,zero_zero_int) ) ).
tff(fact_255_less__special_I3_J,axiom,
! [X_2: int] :
( ord_less_real(number267125858f_real(X_2),zero_zero_real)
<=> ord_less_int(X_2,pls) ) ).
tff(fact_256_less__special_I3_J,axiom,
! [X_2: int] :
( ord_less_int(number_number_of_int(X_2),zero_zero_int)
<=> ord_less_int(X_2,pls) ) ).
tff(fact_257_less__special_I1_J,axiom,
! [Y_2: int] :
( ord_less_real(zero_zero_real,number267125858f_real(Y_2))
<=> ord_less_int(pls,Y_2) ) ).
tff(fact_258_less__special_I1_J,axiom,
! [Y_2: int] :
( ord_less_int(zero_zero_int,number_number_of_int(Y_2))
<=> ord_less_int(pls,Y_2) ) ).
tff(fact_259_semiring__one__add__one__is__two,axiom,
plus_plus_real(one_one_real,one_one_real) = number267125858f_real(bit0(bit1(pls))) ).
tff(fact_260_semiring__one__add__one__is__two,axiom,
plus_plus_nat(one_one_nat,one_one_nat) = number_number_of_nat(bit0(bit1(pls))) ).
tff(fact_261_semiring__one__add__one__is__two,axiom,
plus_plus_int(one_one_int,one_one_int) = number_number_of_int(bit0(bit1(pls))) ).
tff(fact_262_one__add__one__is__two,axiom,
plus_plus_real(one_one_real,one_one_real) = number267125858f_real(bit0(bit1(pls))) ).
tff(fact_263_one__add__one__is__two,axiom,
plus_plus_int(one_one_int,one_one_int) = number_number_of_int(bit0(bit1(pls))) ).
tff(fact_264_less__special_I4_J,axiom,
! [X_2: int] :
( ord_less_real(number267125858f_real(X_2),one_one_real)
<=> ord_less_int(X_2,bit1(pls)) ) ).
tff(fact_265_less__special_I4_J,axiom,
! [X_2: int] :
( ord_less_int(number_number_of_int(X_2),one_one_int)
<=> ord_less_int(X_2,bit1(pls)) ) ).
tff(fact_266_less__special_I2_J,axiom,
! [Y_2: int] :
( ord_less_real(one_one_real,number267125858f_real(Y_2))
<=> ord_less_int(bit1(pls),Y_2) ) ).
tff(fact_267_less__special_I2_J,axiom,
! [Y_2: int] :
( ord_less_int(one_one_int,number_number_of_int(Y_2))
<=> ord_less_int(bit1(pls),Y_2) ) ).
tff(fact_268__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_L_A1_096,axiom,
dvd_dvd_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),plus_plus_int(power_power_int(s,number_number_of_nat(bit0(bit1(pls)))),one_one_int)) ).
tff(fact_269_zadd__power3,axiom,
! [A: int,B: int] : power_power_int(plus_plus_int(A,B),number_number_of_nat(bit1(bit1(pls)))) = plus_plus_int(plus_plus_int(plus_plus_int(power_power_int(A,number_number_of_nat(bit1(bit1(pls)))),times_times_int(times_times_int(number_number_of_int(bit1(bit1(pls))),power_power_int(A,number_number_of_nat(bit0(bit1(pls))))),B)),times_times_int(times_times_int(number_number_of_int(bit1(bit1(pls))),A),power_power_int(B,number_number_of_nat(bit0(bit1(pls)))))),power_power_int(B,number_number_of_nat(bit1(bit1(pls))))) ).
tff(fact_270_zadd__power2,axiom,
! [A: int,B: int] : power_power_int(plus_plus_int(A,B),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_int(plus_plus_int(power_power_int(A,number_number_of_nat(bit0(bit1(pls)))),times_times_int(times_times_int(number_number_of_int(bit0(bit1(pls))),A),B)),power_power_int(B,number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_271_int__pos__lt__two__imp__zero__or__one,axiom,
! [X: int] :
( ord_less_eq_int(zero_zero_int,X)
=> ( ord_less_int(X,number_number_of_int(bit0(bit1(pls))))
=> ( ( X = zero_zero_int )
| ( X = one_one_int ) ) ) ) ).
tff(fact_272_s0p,axiom,
( ord_less_eq_int(zero_zero_int,s)
& ord_less_int(s,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))
& zcong(s1,s,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) ) ).
tff(fact_273_cube__square,axiom,
! [A: int] : times_times_int(A,power_power_int(A,number_number_of_nat(bit0(bit1(pls))))) = power_power_int(A,number_number_of_nat(bit1(bit1(pls)))) ).
tff(fact_274_power2__ge__self,axiom,
! [X: int] : ord_less_eq_int(X,power_power_int(X,number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_275_self__quotient__aux1,axiom,
! [R: int,Q: int,A: int] :
( ord_less_int(zero_zero_int,A)
=> ( ( A = plus_plus_int(R,times_times_int(A,Q)) )
=> ( ord_less_int(R,A)
=> ord_less_eq_int(one_one_int,Q) ) ) ) ).
tff(fact_276_self__quotient__aux2,axiom,
! [R: int,Q: int,A: int] :
( ord_less_int(zero_zero_int,A)
=> ( ( A = plus_plus_int(R,times_times_int(A,Q)) )
=> ( ord_less_eq_int(zero_zero_int,R)
=> ord_less_eq_int(Q,one_one_int) ) ) ) ).
tff(fact_277_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J,axiom,
ord_less_eq_int(zero_zero_int,number_number_of_int(bit0(bit1(pls)))) ).
tff(fact_278_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_17: real,N_34: nat] : power_power_real(X_17,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_34)) = times_times_real(power_power_real(X_17,N_34),power_power_real(X_17,N_34)) ).
tff(fact_279_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_17: nat,N_34: nat] : power_power_nat(X_17,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_34)) = times_times_nat(power_power_nat(X_17,N_34),power_power_nat(X_17,N_34)) ).
tff(fact_280_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_17: int,N_34: nat] : power_power_int(X_17,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_34)) = times_times_int(power_power_int(X_17,N_34),power_power_int(X_17,N_34)) ).
tff(fact_281__096_091s_A_094_A2_A_061_As1_A_094_A2_093_A_Imod_A4_A_K_Am_A_L_A1_J_096,axiom,
zcong(power_power_int(s,number_number_of_nat(bit0(bit1(pls)))),power_power_int(s1,number_number_of_nat(bit0(bit1(pls)))),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) ).
tff(fact_282__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s1,axiom,
? [X_1: int] :
( ord_less_eq_int(zero_zero_int,X_1)
& ord_less_int(X_1,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))
& zcong(s1,X_1,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))
& ! [Y_1: int] :
( ( ord_less_eq_int(zero_zero_int,Y_1)
& ord_less_int(Y_1,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))
& zcong(s1,Y_1,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) )
=> ( Y_1 = X_1 ) ) ) ).
tff(fact_283__096_B_Bthesis_O_A_I_B_Bs_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_,axiom,
~ ! [S: int] :
~ ( ord_less_eq_int(zero_zero_int,S)
& ord_less_int(S,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))
& zcong(s1,S,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) ) ).
tff(fact_284_s1,axiom,
zcong(power_power_int(s1,number_number_of_nat(bit0(bit1(pls)))),number_number_of_int(min),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) ).
tff(fact_285_zero__less__power__nat__eq,axiom,
! [X_2: nat,N_1: nat] :
( ord_less_nat(zero_zero_nat,power_power_nat(X_2,N_1))
<=> ( ( N_1 = zero_zero_nat )
| ord_less_nat(zero_zero_nat,X_2) ) ) ).
tff(fact_286_zprime__zdvd__power,axiom,
! [A: int,N: nat,P: int] :
( zprime(P)
=> ( dvd_dvd_int(P,power_power_int(A,N))
=> dvd_dvd_int(P,A) ) ) ).
tff(fact_287_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_16: real,P_3: nat,Q_4: nat] : power_power_real(power_power_real(X_16,P_3),Q_4) = power_power_real(X_16,times_times_nat(P_3,Q_4)) ).
tff(fact_288_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_16: nat,P_3: nat,Q_4: nat] : power_power_nat(power_power_nat(X_16,P_3),Q_4) = power_power_nat(X_16,times_times_nat(P_3,Q_4)) ).
tff(fact_289_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_16: int,P_3: nat,Q_4: nat] : power_power_int(power_power_int(X_16,P_3),Q_4) = power_power_int(X_16,times_times_nat(P_3,Q_4)) ).
tff(fact_290_zprime__power__zdvd__cancel__right,axiom,
! [N: nat,A: int,B: int,P: int] :
( zprime(P)
=> ( ~ dvd_dvd_int(P,B)
=> ( dvd_dvd_int(power_power_int(P,N),times_times_int(A,B))
=> dvd_dvd_int(power_power_int(P,N),A) ) ) ) ).
tff(fact_291_zprime__power__zdvd__cancel__left,axiom,
! [N: nat,B: int,A: int,P: int] :
( zprime(P)
=> ( ~ dvd_dvd_int(P,A)
=> ( dvd_dvd_int(power_power_int(P,N),times_times_int(A,B))
=> dvd_dvd_int(power_power_int(P,N),B) ) ) ) ).
tff(fact_292_zpower__zpower,axiom,
! [X: int,Y: nat,Z: nat] : power_power_int(power_power_int(X,Y),Z) = power_power_int(X,times_times_nat(Y,Z)) ).
tff(fact_293_zdvd__not__zless,axiom,
! [N: int,M: int] :
( ord_less_int(zero_zero_int,M)
=> ( ord_less_int(M,N)
=> ~ dvd_dvd_int(N,M) ) ) ).
tff(fact_294_zdvd__antisym__nonneg,axiom,
! [N: int,M: int] :
( ord_less_eq_int(zero_zero_int,M)
=> ( ord_less_eq_int(zero_zero_int,N)
=> ( dvd_dvd_int(M,N)
=> ( dvd_dvd_int(N,M)
=> ( M = N ) ) ) ) ) ).
tff(fact_295_zdvd__mult__cancel,axiom,
! [K_1: int,M: int,N: int] :
( dvd_dvd_int(times_times_int(K_1,M),times_times_int(K_1,N))
=> ( ( K_1 != zero_zero_int )
=> dvd_dvd_int(M,N) ) ) ).
tff(fact_296_zdvd__reduce,axiom,
! [K: int,N_1: int,Ma: int] :
( dvd_dvd_int(K,plus_plus_int(N_1,times_times_int(K,Ma)))
<=> dvd_dvd_int(K,N_1) ) ).
tff(fact_297_zdvd__period,axiom,
! [C_1: int,X_2: int,Ta: int,A_13: int,D_2: int] :
( dvd_dvd_int(A_13,D_2)
=> ( dvd_dvd_int(A_13,plus_plus_int(X_2,Ta))
<=> dvd_dvd_int(A_13,plus_plus_int(plus_plus_int(X_2,times_times_int(C_1,D_2)),Ta)) ) ) ).
tff(fact_298_zprime__2,axiom,
zprime(number_number_of_int(bit0(bit1(pls)))) ).
tff(fact_299_zdvd__imp__le,axiom,
! [Z: int,N: int] :
( dvd_dvd_int(Z,N)
=> ( ord_less_int(zero_zero_int,N)
=> ord_less_eq_int(Z,N) ) ) ).
tff(fact_300_is__mult__sum2sq,axiom,
! [Y: int,X: int] :
( twoSqu512355103sum2sq(X)
=> ( twoSqu512355103sum2sq(Y)
=> twoSqu512355103sum2sq(times_times_int(X,Y)) ) ) ).
tff(fact_301_le__nat__number__of,axiom,
! [V_7: int,V_6: int] :
( ord_less_eq_nat(number_number_of_nat(V_7),number_number_of_nat(V_6))
<=> ( ~ ord_less_eq_int(V_7,V_6)
=> ord_less_eq_int(V_7,pls) ) ) ).
tff(fact_302_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: real,Ly_4: real,Rx_6: real,Ry_4: real] : times_times_real(times_times_real(Lx_6,Ly_4),times_times_real(Rx_6,Ry_4)) = times_times_real(times_times_real(Lx_6,Rx_6),times_times_real(Ly_4,Ry_4)) ).
tff(fact_303_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: nat,Ly_4: nat,Rx_6: nat,Ry_4: nat] : times_times_nat(times_times_nat(Lx_6,Ly_4),times_times_nat(Rx_6,Ry_4)) = times_times_nat(times_times_nat(Lx_6,Rx_6),times_times_nat(Ly_4,Ry_4)) ).
tff(fact_304_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: int,Ly_4: int,Rx_6: int,Ry_4: int] : times_times_int(times_times_int(Lx_6,Ly_4),times_times_int(Rx_6,Ry_4)) = times_times_int(times_times_int(Lx_6,Rx_6),times_times_int(Ly_4,Ry_4)) ).
tff(fact_305_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: real,Ly_3: real,Rx_5: real,Ry_3: real] : times_times_real(times_times_real(Lx_5,Ly_3),times_times_real(Rx_5,Ry_3)) = times_times_real(Rx_5,times_times_real(times_times_real(Lx_5,Ly_3),Ry_3)) ).
tff(fact_306_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: nat,Ly_3: nat,Rx_5: nat,Ry_3: nat] : times_times_nat(times_times_nat(Lx_5,Ly_3),times_times_nat(Rx_5,Ry_3)) = times_times_nat(Rx_5,times_times_nat(times_times_nat(Lx_5,Ly_3),Ry_3)) ).
tff(fact_307_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: int,Ly_3: int,Rx_5: int,Ry_3: int] : times_times_int(times_times_int(Lx_5,Ly_3),times_times_int(Rx_5,Ry_3)) = times_times_int(Rx_5,times_times_int(times_times_int(Lx_5,Ly_3),Ry_3)) ).
tff(fact_308_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: real,Ly_2: real,Rx_4: real,Ry_2: real] : times_times_real(times_times_real(Lx_4,Ly_2),times_times_real(Rx_4,Ry_2)) = times_times_real(Lx_4,times_times_real(Ly_2,times_times_real(Rx_4,Ry_2))) ).
tff(fact_309_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: nat,Ly_2: nat,Rx_4: nat,Ry_2: nat] : times_times_nat(times_times_nat(Lx_4,Ly_2),times_times_nat(Rx_4,Ry_2)) = times_times_nat(Lx_4,times_times_nat(Ly_2,times_times_nat(Rx_4,Ry_2))) ).
tff(fact_310_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: int,Ly_2: int,Rx_4: int,Ry_2: int] : times_times_int(times_times_int(Lx_4,Ly_2),times_times_int(Rx_4,Ry_2)) = times_times_int(Lx_4,times_times_int(Ly_2,times_times_int(Rx_4,Ry_2))) ).
tff(fact_311_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: real,Ly_1: real,Rx_3: real] : times_times_real(times_times_real(Lx_3,Ly_1),Rx_3) = times_times_real(times_times_real(Lx_3,Rx_3),Ly_1) ).
tff(fact_312_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: nat,Ly_1: nat,Rx_3: nat] : times_times_nat(times_times_nat(Lx_3,Ly_1),Rx_3) = times_times_nat(times_times_nat(Lx_3,Rx_3),Ly_1) ).
tff(fact_313_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: int,Ly_1: int,Rx_3: int] : times_times_int(times_times_int(Lx_3,Ly_1),Rx_3) = times_times_int(times_times_int(Lx_3,Rx_3),Ly_1) ).
tff(fact_314_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: real,Ly: real,Rx_2: real] : times_times_real(times_times_real(Lx_2,Ly),Rx_2) = times_times_real(Lx_2,times_times_real(Ly,Rx_2)) ).
tff(fact_315_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: nat,Ly: nat,Rx_2: nat] : times_times_nat(times_times_nat(Lx_2,Ly),Rx_2) = times_times_nat(Lx_2,times_times_nat(Ly,Rx_2)) ).
tff(fact_316_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: int,Ly: int,Rx_2: int] : times_times_int(times_times_int(Lx_2,Ly),Rx_2) = times_times_int(Lx_2,times_times_int(Ly,Rx_2)) ).
tff(fact_317_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: real,Rx_1: real,Ry_1: real] : times_times_real(Lx_1,times_times_real(Rx_1,Ry_1)) = times_times_real(times_times_real(Lx_1,Rx_1),Ry_1) ).
tff(fact_318_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: nat,Rx_1: nat,Ry_1: nat] : times_times_nat(Lx_1,times_times_nat(Rx_1,Ry_1)) = times_times_nat(times_times_nat(Lx_1,Rx_1),Ry_1) ).
tff(fact_319_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: int,Rx_1: int,Ry_1: int] : times_times_int(Lx_1,times_times_int(Rx_1,Ry_1)) = times_times_int(times_times_int(Lx_1,Rx_1),Ry_1) ).
tff(fact_320_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: real,Rx: real,Ry: real] : times_times_real(Lx,times_times_real(Rx,Ry)) = times_times_real(Rx,times_times_real(Lx,Ry)) ).
tff(fact_321_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: nat,Rx: nat,Ry: nat] : times_times_nat(Lx,times_times_nat(Rx,Ry)) = times_times_nat(Rx,times_times_nat(Lx,Ry)) ).
tff(fact_322_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: int,Rx: int,Ry: int] : times_times_int(Lx,times_times_int(Rx,Ry)) = times_times_int(Rx,times_times_int(Lx,Ry)) ).
tff(fact_323_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_47: real,B_18: real] : times_times_real(A_47,B_18) = times_times_real(B_18,A_47) ).
tff(fact_324_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_47: nat,B_18: nat] : times_times_nat(A_47,B_18) = times_times_nat(B_18,A_47) ).
tff(fact_325_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_47: int,B_18: int] : times_times_int(A_47,B_18) = times_times_int(B_18,A_47) ).
tff(fact_326_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_46: real,C_12: real] : plus_plus_real(A_46,C_12) = plus_plus_real(C_12,A_46) ).
tff(fact_327_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_46: nat,C_12: nat] : plus_plus_nat(A_46,C_12) = plus_plus_nat(C_12,A_46) ).
tff(fact_328_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_46: int,C_12: int] : plus_plus_int(A_46,C_12) = plus_plus_int(C_12,A_46) ).
tff(fact_329_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_45: real,C_11: real,D_6: real] : plus_plus_real(A_45,plus_plus_real(C_11,D_6)) = plus_plus_real(C_11,plus_plus_real(A_45,D_6)) ).
tff(fact_330_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_45: nat,C_11: nat,D_6: nat] : plus_plus_nat(A_45,plus_plus_nat(C_11,D_6)) = plus_plus_nat(C_11,plus_plus_nat(A_45,D_6)) ).
tff(fact_331_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_45: int,C_11: int,D_6: int] : plus_plus_int(A_45,plus_plus_int(C_11,D_6)) = plus_plus_int(C_11,plus_plus_int(A_45,D_6)) ).
tff(fact_332_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_44: real,C_10: real,D_5: real] : plus_plus_real(A_44,plus_plus_real(C_10,D_5)) = plus_plus_real(plus_plus_real(A_44,C_10),D_5) ).
tff(fact_333_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_44: nat,C_10: nat,D_5: nat] : plus_plus_nat(A_44,plus_plus_nat(C_10,D_5)) = plus_plus_nat(plus_plus_nat(A_44,C_10),D_5) ).
tff(fact_334_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_44: int,C_10: int,D_5: int] : plus_plus_int(A_44,plus_plus_int(C_10,D_5)) = plus_plus_int(plus_plus_int(A_44,C_10),D_5) ).
tff(fact_335_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_43: real,B_17: real,C_9: real] : plus_plus_real(plus_plus_real(A_43,B_17),C_9) = plus_plus_real(A_43,plus_plus_real(B_17,C_9)) ).
tff(fact_336_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_43: nat,B_17: nat,C_9: nat] : plus_plus_nat(plus_plus_nat(A_43,B_17),C_9) = plus_plus_nat(A_43,plus_plus_nat(B_17,C_9)) ).
tff(fact_337_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_43: int,B_17: int,C_9: int] : plus_plus_int(plus_plus_int(A_43,B_17),C_9) = plus_plus_int(A_43,plus_plus_int(B_17,C_9)) ).
tff(fact_338_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_42: real,B_16: real,C_8: real] : plus_plus_real(plus_plus_real(A_42,B_16),C_8) = plus_plus_real(plus_plus_real(A_42,C_8),B_16) ).
tff(fact_339_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_42: nat,B_16: nat,C_8: nat] : plus_plus_nat(plus_plus_nat(A_42,B_16),C_8) = plus_plus_nat(plus_plus_nat(A_42,C_8),B_16) ).
tff(fact_340_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_42: int,B_16: int,C_8: int] : plus_plus_int(plus_plus_int(A_42,B_16),C_8) = plus_plus_int(plus_plus_int(A_42,C_8),B_16) ).
tff(fact_341_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_41: real,B_15: real,C_7: real,D_4: real] : plus_plus_real(plus_plus_real(A_41,B_15),plus_plus_real(C_7,D_4)) = plus_plus_real(plus_plus_real(A_41,C_7),plus_plus_real(B_15,D_4)) ).
tff(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_41: nat,B_15: nat,C_7: nat,D_4: nat] : plus_plus_nat(plus_plus_nat(A_41,B_15),plus_plus_nat(C_7,D_4)) = plus_plus_nat(plus_plus_nat(A_41,C_7),plus_plus_nat(B_15,D_4)) ).
tff(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_41: int,B_15: int,C_7: int,D_4: int] : plus_plus_int(plus_plus_int(A_41,B_15),plus_plus_int(C_7,D_4)) = plus_plus_int(plus_plus_int(A_41,C_7),plus_plus_int(B_15,D_4)) ).
tff(fact_344_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_15: real] : power_power_real(X_15,one_one_nat) = X_15 ).
tff(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_15: nat] : power_power_nat(X_15,one_one_nat) = X_15 ).
tff(fact_346_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_15: int] : power_power_int(X_15,one_one_nat) = X_15 ).
tff(fact_347_zero__less__power__nat__eq__number__of,axiom,
! [X_2: nat,W_1: int] :
( ord_less_nat(zero_zero_nat,power_power_nat(X_2,number_number_of_nat(W_1)))
<=> ( ( number_number_of_nat(W_1) = zero_zero_nat )
| ord_less_nat(zero_zero_nat,X_2) ) ) ).
tff(fact_348_nat__mult__2__right,axiom,
! [Z: nat] : times_times_nat(Z,number_number_of_nat(bit0(bit1(pls)))) = plus_plus_nat(Z,Z) ).
tff(fact_349_nat__mult__2,axiom,
! [Z: nat] : times_times_nat(number_number_of_nat(bit0(bit1(pls))),Z) = plus_plus_nat(Z,Z) ).
tff(fact_350_mult__nat__number__of,axiom,
! [V_5: int,V_4: int] :
( ( ord_less_int(V_4,pls)
=> ( times_times_nat(number_number_of_nat(V_4),number_number_of_nat(V_5)) = zero_zero_nat ) )
& ( ~ ord_less_int(V_4,pls)
=> ( times_times_nat(number_number_of_nat(V_4),number_number_of_nat(V_5)) = number_number_of_nat(times_times_int(V_4,V_5)) ) ) ) ).
tff(fact_351_nat__number__of__mult__left,axiom,
! [V_5: int,K_1: nat,V_4: int] :
( ( ord_less_int(V_4,pls)
=> ( times_times_nat(number_number_of_nat(V_4),times_times_nat(number_number_of_nat(V_5),K_1)) = zero_zero_nat ) )
& ( ~ ord_less_int(V_4,pls)
=> ( times_times_nat(number_number_of_nat(V_4),times_times_nat(number_number_of_nat(V_5),K_1)) = times_times_nat(number_number_of_nat(times_times_int(V_4,V_5)),K_1) ) ) ) ).
tff(fact_352_power__even__eq,axiom,
! [A_40: real,N_33: nat] : power_power_real(A_40,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_33)) = power_power_real(power_power_real(A_40,N_33),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_353_power__even__eq,axiom,
! [A_40: nat,N_33: nat] : power_power_nat(A_40,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_33)) = power_power_nat(power_power_nat(A_40,N_33),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_354_power__even__eq,axiom,
! [A_40: int,N_33: nat] : power_power_int(A_40,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_33)) = power_power_int(power_power_int(A_40,N_33),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_355_even__power__le__0__imp__0,axiom,
! [A_39: real,K_3: nat] :
( ord_less_eq_real(power_power_real(A_39,times_times_nat(number_number_of_nat(bit0(bit1(pls))),K_3)),zero_zero_real)
=> ( A_39 = zero_zero_real ) ) ).
tff(fact_356_even__power__le__0__imp__0,axiom,
! [A_39: int,K_3: nat] :
( ord_less_eq_int(power_power_int(A_39,times_times_nat(number_number_of_nat(bit0(bit1(pls))),K_3)),zero_zero_int)
=> ( A_39 = zero_zero_int ) ) ).
tff(fact_357_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_38: real] : times_times_real(zero_zero_real,A_38) = zero_zero_real ).
tff(fact_358_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_38: nat] : times_times_nat(zero_zero_nat,A_38) = zero_zero_nat ).
tff(fact_359_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_38: int] : times_times_int(zero_zero_int,A_38) = zero_zero_int ).
tff(fact_360_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_37: real] : times_times_real(A_37,zero_zero_real) = zero_zero_real ).
tff(fact_361_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_37: nat] : times_times_nat(A_37,zero_zero_nat) = zero_zero_nat ).
tff(fact_362_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_37: int] : times_times_int(A_37,zero_zero_int) = zero_zero_int ).
tff(fact_363_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_36: real] : plus_plus_real(zero_zero_real,A_36) = A_36 ).
tff(fact_364_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_36: nat] : plus_plus_nat(zero_zero_nat,A_36) = A_36 ).
tff(fact_365_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_36: int] : plus_plus_int(zero_zero_int,A_36) = A_36 ).
tff(fact_366_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_35: real] : plus_plus_real(A_35,zero_zero_real) = A_35 ).
tff(fact_367_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_35: nat] : plus_plus_nat(A_35,zero_zero_nat) = A_35 ).
tff(fact_368_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_35: int] : plus_plus_int(A_35,zero_zero_int) = A_35 ).
tff(fact_369_add__0__iff,axiom,
! [B_1: real,A_13: real] :
( ( B_1 = plus_plus_real(B_1,A_13) )
<=> ( A_13 = zero_zero_real ) ) ).
tff(fact_370_add__0__iff,axiom,
! [B_1: nat,A_13: nat] :
( ( B_1 = plus_plus_nat(B_1,A_13) )
<=> ( A_13 = zero_zero_nat ) ) ).
tff(fact_371_add__0__iff,axiom,
! [B_1: int,A_13: int] :
( ( B_1 = plus_plus_int(B_1,A_13) )
<=> ( A_13 = zero_zero_int ) ) ).
tff(fact_372_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_14: real,Y_9: real,Z_2: real] : times_times_real(X_14,plus_plus_real(Y_9,Z_2)) = plus_plus_real(times_times_real(X_14,Y_9),times_times_real(X_14,Z_2)) ).
tff(fact_373_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_14: nat,Y_9: nat,Z_2: nat] : times_times_nat(X_14,plus_plus_nat(Y_9,Z_2)) = plus_plus_nat(times_times_nat(X_14,Y_9),times_times_nat(X_14,Z_2)) ).
tff(fact_374_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_14: int,Y_9: int,Z_2: int] : times_times_int(X_14,plus_plus_int(Y_9,Z_2)) = plus_plus_int(times_times_int(X_14,Y_9),times_times_int(X_14,Z_2)) ).
tff(fact_375_crossproduct__noteq,axiom,
! [C_1: real,D_2: real,A_13: real,B_1: real] :
( ( ( A_13 != B_1 )
& ( C_1 != D_2 ) )
<=> ( plus_plus_real(times_times_real(A_13,C_1),times_times_real(B_1,D_2)) != plus_plus_real(times_times_real(A_13,D_2),times_times_real(B_1,C_1)) ) ) ).
tff(fact_376_crossproduct__noteq,axiom,
! [C_1: nat,D_2: nat,A_13: nat,B_1: nat] :
( ( ( A_13 != B_1 )
& ( C_1 != D_2 ) )
<=> ( plus_plus_nat(times_times_nat(A_13,C_1),times_times_nat(B_1,D_2)) != plus_plus_nat(times_times_nat(A_13,D_2),times_times_nat(B_1,C_1)) ) ) ).
tff(fact_377_crossproduct__noteq,axiom,
! [C_1: int,D_2: int,A_13: int,B_1: int] :
( ( ( A_13 != B_1 )
& ( C_1 != D_2 ) )
<=> ( plus_plus_int(times_times_int(A_13,C_1),times_times_int(B_1,D_2)) != plus_plus_int(times_times_int(A_13,D_2),times_times_int(B_1,C_1)) ) ) ).
tff(fact_378_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_34: real,B_14: real,C_6: real] : times_times_real(plus_plus_real(A_34,B_14),C_6) = plus_plus_real(times_times_real(A_34,C_6),times_times_real(B_14,C_6)) ).
tff(fact_379_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_34: nat,B_14: nat,C_6: nat] : times_times_nat(plus_plus_nat(A_34,B_14),C_6) = plus_plus_nat(times_times_nat(A_34,C_6),times_times_nat(B_14,C_6)) ).
tff(fact_380_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_34: int,B_14: int,C_6: int] : times_times_int(plus_plus_int(A_34,B_14),C_6) = plus_plus_int(times_times_int(A_34,C_6),times_times_int(B_14,C_6)) ).
tff(fact_381_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_33: real,M_12: real,B_13: real] : plus_plus_real(times_times_real(A_33,M_12),times_times_real(B_13,M_12)) = times_times_real(plus_plus_real(A_33,B_13),M_12) ).
tff(fact_382_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_33: nat,M_12: nat,B_13: nat] : plus_plus_nat(times_times_nat(A_33,M_12),times_times_nat(B_13,M_12)) = times_times_nat(plus_plus_nat(A_33,B_13),M_12) ).
tff(fact_383_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_33: int,M_12: int,B_13: int] : plus_plus_int(times_times_int(A_33,M_12),times_times_int(B_13,M_12)) = times_times_int(plus_plus_int(A_33,B_13),M_12) ).
tff(fact_384_crossproduct__eq,axiom,
! [W_1: real,Y_2: real,X_2: real,Z_1: real] :
( ( plus_plus_real(times_times_real(W_1,Y_2),times_times_real(X_2,Z_1)) = plus_plus_real(times_times_real(W_1,Z_1),times_times_real(X_2,Y_2)) )
<=> ( ( W_1 = X_2 )
| ( Y_2 = Z_1 ) ) ) ).
tff(fact_385_crossproduct__eq,axiom,
! [W_1: nat,Y_2: nat,X_2: nat,Z_1: nat] :
( ( plus_plus_nat(times_times_nat(W_1,Y_2),times_times_nat(X_2,Z_1)) = plus_plus_nat(times_times_nat(W_1,Z_1),times_times_nat(X_2,Y_2)) )
<=> ( ( W_1 = X_2 )
| ( Y_2 = Z_1 ) ) ) ).
tff(fact_386_crossproduct__eq,axiom,
! [W_1: int,Y_2: int,X_2: int,Z_1: int] :
( ( plus_plus_int(times_times_int(W_1,Y_2),times_times_int(X_2,Z_1)) = plus_plus_int(times_times_int(W_1,Z_1),times_times_int(X_2,Y_2)) )
<=> ( ( W_1 = X_2 )
| ( Y_2 = Z_1 ) ) ) ).
tff(fact_387_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_32: real] : times_times_real(one_one_real,A_32) = A_32 ).
tff(fact_388_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_32: nat] : times_times_nat(one_one_nat,A_32) = A_32 ).
tff(fact_389_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_32: int] : times_times_int(one_one_int,A_32) = A_32 ).
tff(fact_390_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_31: real] : times_times_real(A_31,one_one_real) = A_31 ).
tff(fact_391_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_31: nat] : times_times_nat(A_31,one_one_nat) = A_31 ).
tff(fact_392_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_31: int] : times_times_int(A_31,one_one_int) = A_31 ).
tff(fact_393_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_13: real,Y_8: real,Q_3: nat] : power_power_real(times_times_real(X_13,Y_8),Q_3) = times_times_real(power_power_real(X_13,Q_3),power_power_real(Y_8,Q_3)) ).
tff(fact_394_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_13: nat,Y_8: nat,Q_3: nat] : power_power_nat(times_times_nat(X_13,Y_8),Q_3) = times_times_nat(power_power_nat(X_13,Q_3),power_power_nat(Y_8,Q_3)) ).
tff(fact_395_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_13: int,Y_8: int,Q_3: nat] : power_power_int(times_times_int(X_13,Y_8),Q_3) = times_times_int(power_power_int(X_13,Q_3),power_power_int(Y_8,Q_3)) ).
tff(fact_396_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_12: real,P_2: nat,Q_2: nat] : times_times_real(power_power_real(X_12,P_2),power_power_real(X_12,Q_2)) = power_power_real(X_12,plus_plus_nat(P_2,Q_2)) ).
tff(fact_397_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_12: nat,P_2: nat,Q_2: nat] : times_times_nat(power_power_nat(X_12,P_2),power_power_nat(X_12,Q_2)) = power_power_nat(X_12,plus_plus_nat(P_2,Q_2)) ).
tff(fact_398_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_12: int,P_2: nat,Q_2: nat] : times_times_int(power_power_int(X_12,P_2),power_power_int(X_12,Q_2)) = power_power_int(X_12,plus_plus_nat(P_2,Q_2)) ).
tff(fact_399_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_11: real] : power_power_real(X_11,zero_zero_nat) = one_one_real ).
tff(fact_400_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_11: nat] : power_power_nat(X_11,zero_zero_nat) = one_one_nat ).
tff(fact_401_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_11: int] : power_power_int(X_11,zero_zero_nat) = one_one_int ).
tff(fact_402_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq_int(zero_zero_int,zero_zero_int) ).
tff(fact_403_zero__le__even__power_H,axiom,
! [A_30: real,N_32: nat] : ord_less_eq_real(zero_zero_real,power_power_real(A_30,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_32))) ).
tff(fact_404_zero__le__even__power_H,axiom,
! [A_30: int,N_32: nat] : ord_less_eq_int(zero_zero_int,power_power_int(A_30,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_32))) ).
tff(fact_405_add__scale__eq__noteq,axiom,
! [C_5: real,D_3: real,A_29: real,B_12: real,R_2: real] :
( ( R_2 != zero_zero_real )
=> ( ( ( A_29 = B_12 )
& ( C_5 != D_3 ) )
=> ( plus_plus_real(A_29,times_times_real(R_2,C_5)) != plus_plus_real(B_12,times_times_real(R_2,D_3)) ) ) ) ).
tff(fact_406_add__scale__eq__noteq,axiom,
! [C_5: nat,D_3: nat,A_29: nat,B_12: nat,R_2: nat] :
( ( R_2 != zero_zero_nat )
=> ( ( ( A_29 = B_12 )
& ( C_5 != D_3 ) )
=> ( plus_plus_nat(A_29,times_times_nat(R_2,C_5)) != plus_plus_nat(B_12,times_times_nat(R_2,D_3)) ) ) ) ).
tff(fact_407_add__scale__eq__noteq,axiom,
! [C_5: int,D_3: int,A_29: int,B_12: int,R_2: int] :
( ( R_2 != zero_zero_int )
=> ( ( ( A_29 = B_12 )
& ( C_5 != D_3 ) )
=> ( plus_plus_int(A_29,times_times_int(R_2,C_5)) != plus_plus_int(B_12,times_times_int(R_2,D_3)) ) ) ) ).
tff(fact_408_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_11: real] : plus_plus_real(M_11,M_11) = times_times_real(plus_plus_real(one_one_real,one_one_real),M_11) ).
tff(fact_409_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_11: nat] : plus_plus_nat(M_11,M_11) = times_times_nat(plus_plus_nat(one_one_nat,one_one_nat),M_11) ).
tff(fact_410_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_11: int] : plus_plus_int(M_11,M_11) = times_times_int(plus_plus_int(one_one_int,one_one_int),M_11) ).
tff(fact_411_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_10: real,A_28: real] : plus_plus_real(M_10,times_times_real(A_28,M_10)) = times_times_real(plus_plus_real(A_28,one_one_real),M_10) ).
tff(fact_412_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_10: nat,A_28: nat] : plus_plus_nat(M_10,times_times_nat(A_28,M_10)) = times_times_nat(plus_plus_nat(A_28,one_one_nat),M_10) ).
tff(fact_413_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_10: int,A_28: int] : plus_plus_int(M_10,times_times_int(A_28,M_10)) = times_times_int(plus_plus_int(A_28,one_one_int),M_10) ).
tff(fact_414_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_27: real,M_9: real] : plus_plus_real(times_times_real(A_27,M_9),M_9) = times_times_real(plus_plus_real(A_27,one_one_real),M_9) ).
tff(fact_415_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_27: nat,M_9: nat] : plus_plus_nat(times_times_nat(A_27,M_9),M_9) = times_times_nat(plus_plus_nat(A_27,one_one_nat),M_9) ).
tff(fact_416_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_27: int,M_9: int] : plus_plus_int(times_times_int(A_27,M_9),M_9) = times_times_int(plus_plus_int(A_27,one_one_int),M_9) ).
tff(fact_417_power__eq__0__iff__number__of,axiom,
! [A_13: real,W_1: int] :
( ( power_power_real(A_13,number_number_of_nat(W_1)) = zero_zero_real )
<=> ( ( A_13 = zero_zero_real )
& ( number_number_of_nat(W_1) != zero_zero_nat ) ) ) ).
tff(fact_418_power__eq__0__iff__number__of,axiom,
! [A_13: nat,W_1: int] :
( ( power_power_nat(A_13,number_number_of_nat(W_1)) = zero_zero_nat )
<=> ( ( A_13 = zero_zero_nat )
& ( number_number_of_nat(W_1) != zero_zero_nat ) ) ) ).
tff(fact_419_power__eq__0__iff__number__of,axiom,
! [A_13: int,W_1: int] :
( ( power_power_int(A_13,number_number_of_nat(W_1)) = zero_zero_int )
<=> ( ( A_13 = zero_zero_int )
& ( number_number_of_nat(W_1) != zero_zero_nat ) ) ) ).
tff(fact_420_pos__zmult__pos,axiom,
! [B: int,A: int] :
( ord_less_int(zero_zero_int,A)
=> ( ord_less_int(zero_zero_int,times_times_int(A,B))
=> ord_less_int(zero_zero_int,B) ) ) ).
tff(fact_421_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
ord_less_eq_int(zero_zero_int,one_one_int) ).
tff(fact_422_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y: int,X: int] :
( ord_less_eq_int(zero_zero_int,X)
=> ( ord_less_eq_int(zero_zero_int,Y)
=> ord_less_eq_int(zero_zero_int,times_times_int(X,Y)) ) ) ).
tff(fact_423_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y: int,X: int] :
( ord_less_eq_int(zero_zero_int,X)
=> ( ord_less_eq_int(zero_zero_int,Y)
=> ord_less_eq_int(zero_zero_int,plus_plus_int(X,Y)) ) ) ).
tff(fact_424_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [N: nat,X: int] :
( ord_less_eq_int(zero_zero_int,X)
=> ord_less_eq_int(zero_zero_int,power_power_int(X,N)) ) ).
tff(fact_425_power__0__left__number__of,axiom,
! [W_4: int] :
( ( ( number_number_of_nat(W_4) = zero_zero_nat )
=> ( power_power_real(zero_zero_real,number_number_of_nat(W_4)) = one_one_real ) )
& ( ( number_number_of_nat(W_4) != zero_zero_nat )
=> ( power_power_real(zero_zero_real,number_number_of_nat(W_4)) = zero_zero_real ) ) ) ).
tff(fact_426_power__0__left__number__of,axiom,
! [W_4: int] :
( ( ( number_number_of_nat(W_4) = zero_zero_nat )
=> ( power_power_nat(zero_zero_nat,number_number_of_nat(W_4)) = one_one_nat ) )
& ( ( number_number_of_nat(W_4) != zero_zero_nat )
=> ( power_power_nat(zero_zero_nat,number_number_of_nat(W_4)) = zero_zero_nat ) ) ) ).
tff(fact_427_power__0__left__number__of,axiom,
! [W_4: int] :
( ( ( number_number_of_nat(W_4) = zero_zero_nat )
=> ( power_power_int(zero_zero_int,number_number_of_nat(W_4)) = one_one_int ) )
& ( ( number_number_of_nat(W_4) != zero_zero_nat )
=> ( power_power_int(zero_zero_int,number_number_of_nat(W_4)) = zero_zero_int ) ) ) ).
tff(fact_428_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,
ord_less_eq_int(zero_zero_int,number_number_of_int(bit1(bit1(pls)))) ).
tff(fact_429_q__pos__lemma,axiom,
! [B_11: int,Q_1: int,R_1: int] :
( ord_less_eq_int(zero_zero_int,plus_plus_int(times_times_int(B_11,Q_1),R_1))
=> ( ord_less_int(R_1,B_11)
=> ( ord_less_int(zero_zero_int,B_11)
=> ord_less_eq_int(zero_zero_int,Q_1) ) ) ) ).
tff(fact_430_q__neg__lemma,axiom,
! [B_11: int,Q_1: int,R_1: int] :
( ord_less_int(plus_plus_int(times_times_int(B_11,Q_1),R_1),zero_zero_int)
=> ( ord_less_eq_int(zero_zero_int,R_1)
=> ( ord_less_int(zero_zero_int,B_11)
=> ord_less_eq_int(Q_1,zero_zero_int) ) ) ) ).
tff(fact_431_unique__quotient__lemma,axiom,
! [B: int,Q_1: int,R_1: int,Q: int,R: int] :
( ord_less_eq_int(plus_plus_int(times_times_int(B,Q_1),R_1),plus_plus_int(times_times_int(B,Q),R))
=> ( ord_less_eq_int(zero_zero_int,R_1)
=> ( ord_less_int(R_1,B)
=> ( ord_less_int(R,B)
=> ord_less_eq_int(Q_1,Q) ) ) ) ) ).
tff(fact_432_zdiv__mono2__lemma,axiom,
! [B: int,Q: int,R: int,B_11: int,Q_1: int,R_1: int] :
( ( plus_plus_int(times_times_int(B,Q),R) = plus_plus_int(times_times_int(B_11,Q_1),R_1) )
=> ( ord_less_eq_int(zero_zero_int,plus_plus_int(times_times_int(B_11,Q_1),R_1))
=> ( ord_less_int(R_1,B_11)
=> ( ord_less_eq_int(zero_zero_int,R)
=> ( ord_less_int(zero_zero_int,B_11)
=> ( ord_less_eq_int(B_11,B)
=> ord_less_eq_int(Q,Q_1) ) ) ) ) ) ) ).
tff(fact_433_unique__quotient__lemma__neg,axiom,
! [B: int,Q_1: int,R_1: int,Q: int,R: int] :
( ord_less_eq_int(plus_plus_int(times_times_int(B,Q_1),R_1),plus_plus_int(times_times_int(B,Q),R))
=> ( ord_less_eq_int(R,zero_zero_int)
=> ( ord_less_int(B,R)
=> ( ord_less_int(B,R_1)
=> ord_less_eq_int(Q,Q_1) ) ) ) ) ).
tff(fact_434_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q: int,R: int,B_11: int,Q_1: int,R_1: int] :
( ( plus_plus_int(times_times_int(B,Q),R) = plus_plus_int(times_times_int(B_11,Q_1),R_1) )
=> ( ord_less_int(plus_plus_int(times_times_int(B_11,Q_1),R_1),zero_zero_int)
=> ( ord_less_int(R,B)
=> ( ord_less_eq_int(zero_zero_int,R_1)
=> ( ord_less_int(zero_zero_int,B_11)
=> ( ord_less_eq_int(B_11,B)
=> ord_less_eq_int(Q_1,Q) ) ) ) ) ) ) ).
tff(fact_435_quartic__square__square,axiom,
! [X: int] : power_power_int(power_power_int(X,number_number_of_nat(bit0(bit1(pls)))),number_number_of_nat(bit0(bit1(pls)))) = power_power_int(X,number_number_of_nat(bit0(bit0(bit1(pls))))) ).
tff(fact_436_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_10: real] : times_times_real(X_10,X_10) = power_power_real(X_10,number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_437_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_10: nat] : times_times_nat(X_10,X_10) = power_power_nat(X_10,number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_438_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_10: int] : times_times_int(X_10,X_10) = power_power_int(X_10,number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_439_s,axiom,
zcong(power_power_int(s,number_number_of_nat(bit0(bit1(pls)))),number_number_of_int(min),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) ).
tff(fact_440_Euler_Oaux____1,axiom,
! [Y: int,X: int,P: int] :
( ~ zcong(X,zero_zero_int,P)
=> ( zcong(power_power_int(Y,number_number_of_nat(bit0(bit1(pls)))),X,P)
=> ~ dvd_dvd_int(P,Y) ) ) ).
tff(fact_441_zprime__def,axiom,
! [P_1: int] :
( zprime(P_1)
<=> ( ord_less_int(one_one_int,P_1)
& ! [M_8: int] :
( ( ord_less_eq_int(zero_zero_int,M_8)
& dvd_dvd_int(M_8,P_1) )
=> ( ( M_8 = one_one_int )
| ( M_8 = P_1 ) ) ) ) ) ).
tff(fact_442_prime__g__5,axiom,
! [P: int] :
( zprime(P)
=> ( ( P != number_number_of_int(bit0(bit1(pls))) )
=> ( ( P != number_number_of_int(bit1(bit1(pls))) )
=> ord_less_eq_int(number_number_of_int(bit1(bit0(bit1(pls)))),P) ) ) ) ).
tff(fact_443_pos2,axiom,
ord_less_nat(zero_zero_nat,number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_444__096_B_Bthesis_O_A_I_B_Bs1_O_A_091s1_A_094_A2_A_061_A_N1_093_A_Imod_A4_,axiom,
~ ! [S1: int] : ~ zcong(power_power_int(S1,number_number_of_nat(bit0(bit1(pls)))),number_number_of_int(min),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) ).
tff(fact_445__096Legendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_061_A1_096,axiom,
legendre(number_number_of_int(min),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) = one_one_int ).
tff(fact_446_Bit1__Min,axiom,
bit1(min) = min ).
tff(fact_447_rel__simps_I43_J,axiom,
! [L_1: int] :
( ( min = bit1(L_1) )
<=> ( min = L_1 ) ) ).
tff(fact_448_rel__simps_I47_J,axiom,
! [K: int] :
( ( bit1(K) = min )
<=> ( K = min ) ) ).
tff(fact_449_rel__simps_I40_J,axiom,
min != pls ).
tff(fact_450_rel__simps_I37_J,axiom,
pls != min ).
tff(fact_451_rel__simps_I42_J,axiom,
! [L: int] : min != bit0(L) ).
tff(fact_452_rel__simps_I45_J,axiom,
! [K_1: int] : bit0(K_1) != min ).
tff(fact_453_rel__simps_I7_J,axiom,
~ ord_less_int(min,min) ).
tff(fact_454_rel__simps_I24_J,axiom,
ord_less_eq_int(min,min) ).
tff(fact_455_rel__simps_I13_J,axiom,
! [K: int] :
( ord_less_int(bit1(K),min)
<=> ord_less_int(K,min) ) ).
tff(fact_456_rel__simps_I9_J,axiom,
! [K: int] :
( ord_less_int(min,bit1(K))
<=> ord_less_int(min,K) ) ).
tff(fact_457_rel__simps_I3_J,axiom,
~ ord_less_int(pls,min) ).
tff(fact_458_rel__simps_I6_J,axiom,
ord_less_int(min,pls) ).
tff(fact_459_rel__simps_I8_J,axiom,
! [K: int] :
( ord_less_int(min,bit0(K))
<=> ord_less_int(min,K) ) ).
tff(fact_460_bin__less__0__simps_I2_J,axiom,
ord_less_int(min,zero_zero_int) ).
tff(fact_461_rel__simps_I26_J,axiom,
! [K: int] :
( ord_less_eq_int(min,bit1(K))
<=> ord_less_eq_int(min,K) ) ).
tff(fact_462_rel__simps_I30_J,axiom,
! [K: int] :
( ord_less_eq_int(bit1(K),min)
<=> ord_less_eq_int(K,min) ) ).
tff(fact_463_rel__simps_I23_J,axiom,
ord_less_eq_int(min,pls) ).
tff(fact_464_rel__simps_I20_J,axiom,
~ ord_less_eq_int(pls,min) ).
tff(fact_465_rel__simps_I28_J,axiom,
! [K: int] :
( ord_less_eq_int(bit0(K),min)
<=> ord_less_eq_int(K,min) ) ).
tff(fact_466_eq__number__of__Pls__Min,axiom,
number_number_of_int(pls) != number_number_of_int(min) ).
tff(fact_467_rel__simps_I11_J,axiom,
! [K: int] :
( ord_less_int(bit0(K),min)
<=> ord_less_eq_int(K,min) ) ).
tff(fact_468_rel__simps_I25_J,axiom,
! [K: int] :
( ord_less_eq_int(min,bit0(K))
<=> ord_less_int(min,K) ) ).
tff(fact_469_zmult__eq__1__iff,axiom,
! [Ma: int,N_1: int] :
( ( times_times_int(Ma,N_1) = one_one_int )
<=> ( ( ( Ma = one_one_int )
& ( N_1 = one_one_int ) )
| ( ( Ma = number_number_of_int(min) )
& ( N_1 = number_number_of_int(min) ) ) ) ) ).
tff(fact_470_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( times_times_int(M,N) = one_one_int )
=> ( ( M = one_one_int )
| ( M = number_number_of_int(min) ) ) ) ).
tff(fact_471_zcong__sym,axiom,
! [A_13: int,B_1: int,Ma: int] :
( zcong(A_13,B_1,Ma)
<=> zcong(B_1,A_13,Ma) ) ).
tff(fact_472_zcong__refl,axiom,
! [K_1: int,M: int] : zcong(K_1,K_1,M) ).
tff(fact_473_zcong__trans,axiom,
! [C: int,A: int,B: int,M: int] :
( zcong(A,B,M)
=> ( zcong(B,C,M)
=> zcong(A,C,M) ) ) ).
tff(fact_474_order__le__neq__implies__less,axiom,
! [X_9: real,Y_7: real] :
( ord_less_eq_real(X_9,Y_7)
=> ( ( X_9 != Y_7 )
=> ord_less_real(X_9,Y_7) ) ) ).
tff(fact_475_order__le__neq__implies__less,axiom,
! [X_9: nat,Y_7: nat] :
( ord_less_eq_nat(X_9,Y_7)
=> ( ( X_9 != Y_7 )
=> ord_less_nat(X_9,Y_7) ) ) ).
tff(fact_476_order__le__neq__implies__less,axiom,
! [X_9: int,Y_7: int] :
( ord_less_eq_int(X_9,Y_7)
=> ( ( X_9 != Y_7 )
=> ord_less_int(X_9,Y_7) ) ) ).
tff(fact_477_Euler_Oaux2,axiom,
! [B: int,A: int,C: int] :
( ord_less_int(A,C)
=> ( ord_less_int(B,C)
=> ( ord_less_eq_int(A,B)
| ord_less_eq_int(B,A) ) ) ) ).
tff(fact_478_IntPrimes_Ozcong__zero,axiom,
! [A_13: int,B_1: int] :
( zcong(A_13,B_1,zero_zero_int)
<=> ( A_13 = B_1 ) ) ).
tff(fact_479_zcong__1,axiom,
! [A: int,B: int] : zcong(A,B,one_one_int) ).
tff(fact_480_zcong__zmult__self,axiom,
! [A: int,M: int,B: int] : zcong(times_times_int(A,M),times_times_int(B,M),M) ).
tff(fact_481_zcong__scalar,axiom,
! [K_1: int,A: int,B: int,M: int] :
( zcong(A,B,M)
=> zcong(times_times_int(A,K_1),times_times_int(B,K_1),M) ) ).
tff(fact_482_zcong__scalar2,axiom,
! [K_1: int,A: int,B: int,M: int] :
( zcong(A,B,M)
=> zcong(times_times_int(K_1,A),times_times_int(K_1,B),M) ) ).
tff(fact_483_zcong__zmult,axiom,
! [C: int,D: int,A: int,B: int,M: int] :
( zcong(A,B,M)
=> ( zcong(C,D,M)
=> zcong(times_times_int(A,C),times_times_int(B,D),M) ) ) ).
tff(fact_484_zcong__zadd,axiom,
! [C: int,D: int,A: int,B: int,M: int] :
( zcong(A,B,M)
=> ( zcong(C,D,M)
=> zcong(plus_plus_int(A,C),plus_plus_int(B,D),M) ) ) ).
tff(fact_485_power__m1__even,axiom,
! [N_31: nat] : power_power_real(number267125858f_real(min),times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_31)) = one_one_real ).
tff(fact_486_power__m1__even,axiom,
! [N_31: nat] : power_power_int(number_number_of_int(min),times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_31)) = one_one_int ).
tff(fact_487_zcong__not,axiom,
! [B: int,M: int,A: int] :
( ord_less_int(zero_zero_int,A)
=> ( ord_less_int(A,M)
=> ( ord_less_int(zero_zero_int,B)
=> ( ord_less_int(B,A)
=> ~ zcong(A,B,M) ) ) ) ) ).
tff(fact_488_zcong__iff__lin,axiom,
! [A_13: int,B_1: int,Ma: int] :
( zcong(A_13,B_1,Ma)
<=> ? [K_2: int] : B_1 = plus_plus_int(A_13,times_times_int(Ma,K_2)) ) ).
tff(fact_489_four__x__squared,axiom,
! [X: real] : times_times_real(number267125858f_real(bit0(bit0(bit1(pls)))),power_power_real(X,number_number_of_nat(bit0(bit1(pls))))) = power_power_real(times_times_real(number267125858f_real(bit0(bit1(pls))),X),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_490_zcong__zless__0,axiom,
! [M: int,A: int] :
( ord_less_eq_int(zero_zero_int,A)
=> ( ord_less_int(A,M)
=> ( zcong(A,zero_zero_int,M)
=> ( A = zero_zero_int ) ) ) ) ).
tff(fact_491_zcong__zless__imp__eq,axiom,
! [B: int,M: int,A: int] :
( ord_less_eq_int(zero_zero_int,A)
=> ( ord_less_int(A,M)
=> ( ord_less_eq_int(zero_zero_int,B)
=> ( ord_less_int(B,M)
=> ( zcong(A,B,M)
=> ( A = B ) ) ) ) ) ) ).
tff(fact_492_zcong__zpower__zmult,axiom,
! [Z: nat,X: int,Y: nat,P: int] :
( zcong(power_power_int(X,Y),one_one_int,P)
=> zcong(power_power_int(X,times_times_nat(Y,Z)),one_one_int,P) ) ).
tff(fact_493_zprime__zdvd__zmult,axiom,
! [N: int,P: int,M: int] :
( ord_less_eq_int(zero_zero_int,M)
=> ( zprime(P)
=> ( dvd_dvd_int(P,times_times_int(M,N))
=> ( dvd_dvd_int(P,M)
| dvd_dvd_int(P,N) ) ) ) ) ).
tff(fact_494__096QuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_096,axiom,
quadRes(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),number_number_of_int(min)) ).
tff(fact_495__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_N_A_N1_096,axiom,
dvd_dvd_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),minus_minus_int(power_power_int(s,number_number_of_nat(bit0(bit1(pls)))),number_number_of_int(min))) ).
tff(fact_496_neg__one__power__eq__mod__m,axiom,
! [J: nat,K_1: nat,M: int] :
( ord_less_int(number_number_of_int(bit0(bit1(pls))),M)
=> ( zcong(power_power_int(number_number_of_int(min),J),power_power_int(number_number_of_int(min),K_1),M)
=> ( power_power_int(number_number_of_int(min),J) = power_power_int(number_number_of_int(min),K_1) ) ) ) ).
tff(fact_497_zcong__neg__1__impl__ne__1,axiom,
! [X: int,P: int] :
( ord_less_int(number_number_of_int(bit0(bit1(pls))),P)
=> ( zcong(X,number_number_of_int(min),P)
=> ~ zcong(X,one_one_int,P) ) ) ).
tff(fact_498__096s_A_094_A2_A_N_A_N1_A_061_As_A_094_A2_A_L_A1_096,axiom,
minus_minus_int(power_power_int(s,number_number_of_nat(bit0(bit1(pls)))),number_number_of_int(min)) = plus_plus_int(power_power_int(s,number_number_of_nat(bit0(bit1(pls)))),one_one_int) ).
tff(fact_499__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_N,axiom,
( ~ quadRes(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),number_number_of_int(min))
=> ( legendre(number_number_of_int(min),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) != one_one_int ) ) ).
tff(fact_500_not__real__square__gt__zero,axiom,
! [X_2: real] :
( ~ ord_less_real(zero_zero_real,times_times_real(X_2,X_2))
<=> ( X_2 = zero_zero_real ) ) ).
tff(fact_501_zcong__zdiff,axiom,
! [C: int,D: int,A: int,B: int,M: int] :
( zcong(A,B,M)
=> ( zcong(C,D,M)
=> zcong(minus_minus_int(A,C),minus_minus_int(B,D),M) ) ) ).
tff(fact_502_zdvd__zdiffD,axiom,
! [K_1: int,M: int,N: int] :
( dvd_dvd_int(K_1,minus_minus_int(M,N))
=> ( dvd_dvd_int(K_1,N)
=> dvd_dvd_int(K_1,M) ) ) ).
tff(fact_503_zdiff__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] : times_times_int(W,minus_minus_int(Z1,Z2)) = minus_minus_int(times_times_int(W,Z1),times_times_int(W,Z2)) ).
tff(fact_504_zdiff__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] : times_times_int(minus_minus_int(Z1,Z2),W) = minus_minus_int(times_times_int(Z1,W),times_times_int(Z2,W)) ).
tff(fact_505_diff__bin__simps_I7_J,axiom,
! [K_1: int,L: int] : minus_minus_int(bit0(K_1),bit0(L)) = bit0(minus_minus_int(K_1,L)) ).
tff(fact_506_diff__bin__simps_I1_J,axiom,
! [K_1: int] : minus_minus_int(K_1,pls) = K_1 ).
tff(fact_507_Int2_Oaux1,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int(A,B) = C )
=> ( A = plus_plus_int(C,B) ) ) ).
tff(fact_508_number__of__diff,axiom,
! [V_3: int,W_3: int] : number267125858f_real(minus_minus_int(V_3,W_3)) = minus_minus_real(number267125858f_real(V_3),number267125858f_real(W_3)) ).
tff(fact_509_number__of__diff,axiom,
! [V_3: int,W_3: int] : number_number_of_int(minus_minus_int(V_3,W_3)) = minus_minus_int(number_number_of_int(V_3),number_number_of_int(W_3)) ).
tff(fact_510_right__diff__distrib__number__of,axiom,
! [V_2: int,B_10: real,C_4: real] : times_times_real(number267125858f_real(V_2),minus_minus_real(B_10,C_4)) = minus_minus_real(times_times_real(number267125858f_real(V_2),B_10),times_times_real(number267125858f_real(V_2),C_4)) ).
tff(fact_511_right__diff__distrib__number__of,axiom,
! [V_2: int,B_10: int,C_4: int] : times_times_int(number_number_of_int(V_2),minus_minus_int(B_10,C_4)) = minus_minus_int(times_times_int(number_number_of_int(V_2),B_10),times_times_int(number_number_of_int(V_2),C_4)) ).
tff(fact_512_left__diff__distrib__number__of,axiom,
! [A_26: real,B_9: real,V_1: int] : times_times_real(minus_minus_real(A_26,B_9),number267125858f_real(V_1)) = minus_minus_real(times_times_real(A_26,number267125858f_real(V_1)),times_times_real(B_9,number267125858f_real(V_1))) ).
tff(fact_513_left__diff__distrib__number__of,axiom,
! [A_26: int,B_9: int,V_1: int] : times_times_int(minus_minus_int(A_26,B_9),number_number_of_int(V_1)) = minus_minus_int(times_times_int(A_26,number_number_of_int(V_1)),times_times_int(B_9,number_number_of_int(V_1))) ).
tff(fact_514_diff__bin__simps_I9_J,axiom,
! [K_1: int,L: int] : minus_minus_int(bit1(K_1),bit0(L)) = bit1(minus_minus_int(K_1,L)) ).
tff(fact_515_diff__bin__simps_I10_J,axiom,
! [K_1: int,L: int] : minus_minus_int(bit1(K_1),bit1(L)) = bit0(minus_minus_int(K_1,L)) ).
tff(fact_516_diff__bin__simps_I3_J,axiom,
! [L: int] : minus_minus_int(pls,bit0(L)) = bit0(minus_minus_int(pls,L)) ).
tff(fact_517_less__bin__lemma,axiom,
! [K: int,L_1: int] :
( ord_less_int(K,L_1)
<=> ord_less_int(minus_minus_int(K,L_1),zero_zero_int) ) ).
tff(fact_518_xzgcda__linear__aux1,axiom,
! [A: int,R: int,B: int,M: int,C: int,D: int,N: int] : plus_plus_int(times_times_int(minus_minus_int(A,times_times_int(R,B)),M),times_times_int(minus_minus_int(C,times_times_int(R,D)),N)) = minus_minus_int(plus_plus_int(times_times_int(A,M),times_times_int(C,N)),times_times_int(R,plus_plus_int(times_times_int(B,M),times_times_int(D,N)))) ).
tff(fact_519_zcong__def,axiom,
! [A_13: int,B_1: int,Ma: int] :
( zcong(A_13,B_1,Ma)
<=> dvd_dvd_int(Ma,minus_minus_int(A_13,B_1)) ) ).
tff(fact_520_add__number__of__diff1,axiom,
! [V: int,W_2: int,C_3: real] : plus_plus_real(number267125858f_real(V),minus_minus_real(number267125858f_real(W_2),C_3)) = minus_minus_real(number267125858f_real(plus_plus_int(V,W_2)),C_3) ).
tff(fact_521_add__number__of__diff1,axiom,
! [V: int,W_2: int,C_3: int] : plus_plus_int(number_number_of_int(V),minus_minus_int(number_number_of_int(W_2),C_3)) = minus_minus_int(number_number_of_int(plus_plus_int(V,W_2)),C_3) ).
tff(fact_522_Euler_Oaux1,axiom,
! [A: int,X: int] :
( ord_less_int(zero_zero_int,X)
=> ( ord_less_int(X,A)
=> ( ( X != minus_minus_int(A,one_one_int) )
=> ord_less_int(X,minus_minus_int(A,one_one_int)) ) ) ) ).
tff(fact_523_zle__diff1__eq,axiom,
! [W_1: int,Z_1: int] :
( ord_less_eq_int(W_1,minus_minus_int(Z_1,one_one_int))
<=> ord_less_int(W_1,Z_1) ) ).
tff(fact_524_diff__bin__simps_I4_J,axiom,
! [L: int] : minus_minus_int(pls,bit1(L)) = bit1(minus_minus_int(min,L)) ).
tff(fact_525_diff__bin__simps_I6_J,axiom,
! [L: int] : minus_minus_int(min,bit1(L)) = bit0(minus_minus_int(min,L)) ).
tff(fact_526_diff__bin__simps_I5_J,axiom,
! [L: int] : minus_minus_int(min,bit0(L)) = bit1(minus_minus_int(min,L)) ).
tff(fact_527_inv__not__p__minus__1__aux,axiom,
! [A_13: int,P_1: int] :
( zcong(times_times_int(A_13,minus_minus_int(P_1,one_one_int)),one_one_int,P_1)
<=> zcong(A_13,minus_minus_int(P_1,one_one_int),P_1) ) ).
tff(fact_528_zcong__eq__trans,axiom,
! [D: int,C: int,A: int,B: int,M: int] :
( zcong(A,B,M)
=> ( ( B = C )
=> ( zcong(C,D,M)
=> zcong(A,D,M) ) ) ) ).
tff(fact_529_zcong__square__zless,axiom,
! [A: int,P: int] :
( zprime(P)
=> ( ord_less_int(zero_zero_int,A)
=> ( ord_less_int(A,P)
=> ( zcong(times_times_int(A,A),one_one_int,P)
=> ( ( A = one_one_int )
| ( A = minus_minus_int(P,one_one_int) ) ) ) ) ) ) ).
tff(fact_530_zcong__square,axiom,
! [A: int,P: int] :
( zprime(P)
=> ( ord_less_int(zero_zero_int,A)
=> ( zcong(times_times_int(A,A),one_one_int,P)
=> ( zcong(A,one_one_int,P)
| zcong(A,minus_minus_int(P,one_one_int),P) ) ) ) ) ).
tff(fact_531_zspecial__product,axiom,
! [A: int,B: int] : times_times_int(plus_plus_int(A,B),minus_minus_int(A,B)) = minus_minus_int(power_power_int(A,number_number_of_nat(bit0(bit1(pls)))),power_power_int(B,number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_532_zcong__id,axiom,
! [M: int] : zcong(M,zero_zero_int,M) ).
tff(fact_533_zcong__zmult__prop2,axiom,
! [C_1: int,D_2: int,A_13: int,B_1: int,Ma: int] :
( zcong(A_13,B_1,Ma)
=> ( zcong(C_1,times_times_int(D_2,A_13),Ma)
<=> zcong(C_1,times_times_int(D_2,B_1),Ma) ) ) ).
tff(fact_534_zcong__zmult__prop1,axiom,
! [C_1: int,D_2: int,A_13: int,B_1: int,Ma: int] :
( zcong(A_13,B_1,Ma)
=> ( zcong(C_1,times_times_int(A_13,D_2),Ma)
<=> zcong(C_1,times_times_int(B_1,D_2),Ma) ) ) ).
tff(fact_535_zcong__shift,axiom,
! [C: int,A: int,B: int,M: int] :
( zcong(A,B,M)
=> zcong(plus_plus_int(A,C),plus_plus_int(B,C),M) ) ).
tff(fact_536_zcong__zpower,axiom,
! [Z: nat,X: int,Y: int,M: int] :
( zcong(X,Y,M)
=> zcong(power_power_int(X,Z),power_power_int(Y,Z),M) ) ).
tff(fact_537_power2__diff,axiom,
! [X_8: real,Y_6: real] : power_power_real(minus_minus_real(X_8,Y_6),number_number_of_nat(bit0(bit1(pls)))) = minus_minus_real(plus_plus_real(power_power_real(X_8,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_6,number_number_of_nat(bit0(bit1(pls))))),times_times_real(times_times_real(number267125858f_real(bit0(bit1(pls))),X_8),Y_6)) ).
tff(fact_538_power2__diff,axiom,
! [X_8: int,Y_6: int] : power_power_int(minus_minus_int(X_8,Y_6),number_number_of_nat(bit0(bit1(pls)))) = minus_minus_int(plus_plus_int(power_power_int(X_8,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_6,number_number_of_nat(bit0(bit1(pls))))),times_times_int(times_times_int(number_number_of_int(bit0(bit1(pls))),X_8),Y_6)) ).
tff(fact_539_zdiff__power2,axiom,
! [A: int,B: int] : power_power_int(minus_minus_int(A,B),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_int(minus_minus_int(power_power_int(A,number_number_of_nat(bit0(bit1(pls)))),times_times_int(times_times_int(number_number_of_int(bit0(bit1(pls))),A),B)),power_power_int(B,number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_540_zdiff__power3,axiom,
! [A: int,B: int] : power_power_int(minus_minus_int(A,B),number_number_of_nat(bit1(bit1(pls)))) = minus_minus_int(plus_plus_int(minus_minus_int(power_power_int(A,number_number_of_nat(bit1(bit1(pls)))),times_times_int(times_times_int(number_number_of_int(bit1(bit1(pls))),power_power_int(A,number_number_of_nat(bit0(bit1(pls))))),B)),times_times_int(times_times_int(number_number_of_int(bit1(bit1(pls))),A),power_power_int(B,number_number_of_nat(bit0(bit1(pls)))))),power_power_int(B,number_number_of_nat(bit1(bit1(pls))))) ).
tff(fact_541_zcong__less__eq,axiom,
! [M: int,Y: int,X: int] :
( ord_less_int(zero_zero_int,X)
=> ( ord_less_int(zero_zero_int,Y)
=> ( ord_less_int(zero_zero_int,M)
=> ( zcong(X,Y,M)
=> ( ord_less_int(X,M)
=> ( ord_less_int(Y,M)
=> ( X = Y ) ) ) ) ) ) ) ).
tff(fact_542_zcong__not__zero,axiom,
! [M: int,X: int] :
( ord_less_int(zero_zero_int,X)
=> ( ord_less_int(X,M)
=> ~ zcong(X,zero_zero_int,M) ) ) ).
tff(fact_543_zdvd__bounds,axiom,
! [N: int,M: int] :
( dvd_dvd_int(N,M)
=> ( ord_less_eq_int(M,zero_zero_int)
| ord_less_eq_int(N,M) ) ) ).
tff(fact_544_zcong__zero__equiv__div,axiom,
! [A_13: int,Ma: int] :
( zcong(A_13,zero_zero_int,Ma)
<=> dvd_dvd_int(Ma,A_13) ) ).
tff(fact_545_zcong__eq__zdvd__prop,axiom,
! [X_2: int,P_1: int] :
( zcong(X_2,zero_zero_int,P_1)
<=> dvd_dvd_int(P_1,X_2) ) ).
tff(fact_546_zprime__zdvd__zmult__better,axiom,
! [M: int,N: int,P: int] :
( zprime(P)
=> ( dvd_dvd_int(P,times_times_int(M,N))
=> ( dvd_dvd_int(P,M)
| dvd_dvd_int(P,N) ) ) ) ).
tff(fact_547_Int2_Ozcong__zero,axiom,
! [M: int,X: int] :
( ord_less_eq_int(zero_zero_int,X)
=> ( ord_less_int(X,M)
=> ( zcong(X,zero_zero_int,M)
=> ( X = zero_zero_int ) ) ) ) ).
tff(fact_548_zpower__zdvd__prop1,axiom,
! [P: int,Y: int,N: nat] :
( ord_less_nat(zero_zero_nat,N)
=> ( dvd_dvd_int(P,Y)
=> dvd_dvd_int(P,power_power_int(Y,N)) ) ) ).
tff(fact_549_neg__one__power,axiom,
! [N: nat] :
( ( power_power_int(number_number_of_int(min),N) = one_one_int )
| ( power_power_int(number_number_of_int(min),N) = number_number_of_int(min) ) ) ).
tff(fact_550_zcong__zmult__prop3,axiom,
! [Y: int,X: int,P: int] :
( zprime(P)
=> ( ~ zcong(X,zero_zero_int,P)
=> ( ~ zcong(Y,zero_zero_int,P)
=> ~ zcong(times_times_int(X,Y),zero_zero_int,P) ) ) ) ).
tff(fact_551_zcong__zprime__prod__zero__contra,axiom,
! [B: int,A: int,P: int] :
( zprime(P)
=> ( ord_less_int(zero_zero_int,A)
=> ( ( ~ zcong(A,zero_zero_int,P)
& ~ zcong(B,zero_zero_int,P) )
=> ~ zcong(times_times_int(A,B),zero_zero_int,P) ) ) ) ).
tff(fact_552_zcong__zprime__prod__zero,axiom,
! [B: int,A: int,P: int] :
( zprime(P)
=> ( ord_less_int(zero_zero_int,A)
=> ( zcong(times_times_int(A,B),zero_zero_int,P)
=> ( zcong(A,zero_zero_int,P)
| zcong(B,zero_zero_int,P) ) ) ) ) ).
tff(fact_553_zpower__zdvd__prop2,axiom,
! [Y: int,N: nat,P: int] :
( zprime(P)
=> ( dvd_dvd_int(P,power_power_int(Y,N))
=> ( ord_less_nat(zero_zero_nat,N)
=> dvd_dvd_int(P,Y) ) ) ) ).
tff(fact_554_Legendre__1mod4,axiom,
! [M: int] :
( zprime(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),M),one_one_int))
=> ( legendre(number_number_of_int(min),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),M),one_one_int)) = one_one_int ) ) ).
tff(fact_555_one__not__neg__one__mod__m,axiom,
! [M: int] :
( ord_less_int(number_number_of_int(bit0(bit1(pls))),M)
=> ~ zcong(one_one_int,number_number_of_int(min),M) ) ).
tff(fact_556_Legendre__def,axiom,
! [A: int,P: int] :
( ( zcong(A,zero_zero_int,P)
=> ( legendre(A,P) = zero_zero_int ) )
& ( ~ zcong(A,zero_zero_int,P)
=> ( ( quadRes(P,A)
=> ( legendre(A,P) = one_one_int ) )
& ( ~ quadRes(P,A)
=> ( legendre(A,P) = number_number_of_int(min) ) ) ) ) ) ).
tff(fact_557_QuadRes__def,axiom,
! [Ma: int,X_2: int] :
( quadRes(Ma,X_2)
<=> ? [Y_1: int] : zcong(power_power_int(Y_1,number_number_of_nat(bit0(bit1(pls)))),X_2,Ma) ) ).
tff(fact_558_real__sum__squared__expand,axiom,
! [X: real,Y: real] : power_power_real(plus_plus_real(X,Y),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_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))))),times_times_real(times_times_real(number267125858f_real(bit0(bit1(pls))),X),Y)) ).
tff(fact_559_mult__eq__if,axiom,
! [N: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( times_times_nat(M,N) = zero_zero_nat ) )
& ( ( M != zero_zero_nat )
=> ( times_times_nat(M,N) = plus_plus_nat(N,times_times_nat(minus_minus_nat(M,one_one_nat),N)) ) ) ) ).
tff(fact_560_power__eq__if,axiom,
! [P: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( power_power_nat(P,M) = one_one_nat ) )
& ( ( M != zero_zero_nat )
=> ( power_power_nat(P,M) = times_times_nat(P,power_power_nat(P,minus_minus_nat(M,one_one_nat))) ) ) ) ).
tff(fact_561_diff__square,axiom,
! [X: nat,Y: nat] : minus_minus_nat(power_power_nat(X,number_number_of_nat(bit0(bit1(pls)))),power_power_nat(Y,number_number_of_nat(bit0(bit1(pls))))) = times_times_nat(plus_plus_nat(X,Y),minus_minus_nat(X,Y)) ).
tff(fact_562_realpow__two__sum__zero__iff,axiom,
! [X_2: real,Y_2: real] :
( ( plus_plus_real(power_power_real(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_2,number_number_of_nat(bit0(bit1(pls))))) = zero_zero_real )
<=> ( ( X_2 = zero_zero_real )
& ( Y_2 = zero_zero_real ) ) ) ).
tff(fact_563_real__zero__not__eq__one,axiom,
zero_zero_real != one_one_real ).
tff(fact_564_real__le__eq__diff,axiom,
! [X_2: real,Y_2: real] :
( ord_less_eq_real(X_2,Y_2)
<=> ord_less_eq_real(minus_minus_real(X_2,Y_2),zero_zero_real) ) ).
tff(fact_565_real__less__def,axiom,
! [X_2: real,Y_2: real] :
( ord_less_real(X_2,Y_2)
<=> ( ord_less_eq_real(X_2,Y_2)
& ( X_2 != Y_2 ) ) ) ).
tff(fact_566_less__eq__real__def,axiom,
! [X_2: real,Y_2: real] :
( ord_less_eq_real(X_2,Y_2)
<=> ( ord_less_real(X_2,Y_2)
| ( X_2 = Y_2 ) ) ) ).
tff(fact_567_real__mult__assoc,axiom,
! [Z1: real,Z2: real,Z3: real] : times_times_real(times_times_real(Z1,Z2),Z3) = times_times_real(Z1,times_times_real(Z2,Z3)) ).
tff(fact_568_real__mult__commute,axiom,
! [Z: real,W: real] : times_times_real(Z,W) = times_times_real(W,Z) ).
tff(fact_569_real__mult__1,axiom,
! [Z: real] : times_times_real(one_one_real,Z) = Z ).
tff(fact_570_real__add__left__mono,axiom,
! [Z: real,X: real,Y: real] :
( ord_less_eq_real(X,Y)
=> ord_less_eq_real(plus_plus_real(Z,X),plus_plus_real(Z,Y)) ) ).
tff(fact_571_add__diff__add,axiom,
! [A_25: real,C_2: real,B_8: real,D_1: real] : minus_minus_real(plus_plus_real(A_25,C_2),plus_plus_real(B_8,D_1)) = plus_plus_real(minus_minus_real(A_25,B_8),minus_minus_real(C_2,D_1)) ).
tff(fact_572_add__diff__add,axiom,
! [A_25: int,C_2: int,B_8: int,D_1: int] : minus_minus_int(plus_plus_int(A_25,C_2),plus_plus_int(B_8,D_1)) = plus_plus_int(minus_minus_int(A_25,B_8),minus_minus_int(C_2,D_1)) ).
tff(fact_573_real__mult__left__cancel,axiom,
! [A_13: real,B_1: real,C_1: real] :
( ( C_1 != zero_zero_real )
=> ( ( times_times_real(C_1,A_13) = times_times_real(C_1,B_1) )
<=> ( A_13 = B_1 ) ) ) ).
tff(fact_574_real__mult__right__cancel,axiom,
! [A_13: real,B_1: real,C_1: real] :
( ( C_1 != zero_zero_real )
=> ( ( times_times_real(A_13,C_1) = times_times_real(B_1,C_1) )
<=> ( A_13 = B_1 ) ) ) ).
tff(fact_575_nat__mult__eq__one,axiom,
! [N_1: nat,Ma: nat] :
( ( times_times_nat(N_1,Ma) = one_one_nat )
<=> ( ( N_1 = one_one_nat )
& ( Ma = one_one_nat ) ) ) ).
tff(fact_576_real__add__mult__distrib,axiom,
! [Z1: real,Z2: real,W: real] : times_times_real(plus_plus_real(Z1,Z2),W) = plus_plus_real(times_times_real(Z1,W),times_times_real(Z2,W)) ).
tff(fact_577_nat__power__eq__0__iff,axiom,
! [Ma: nat,N_1: nat] :
( ( power_power_nat(Ma,N_1) = zero_zero_nat )
<=> ( ( N_1 != zero_zero_nat )
& ( Ma = zero_zero_nat ) ) ) ).
tff(fact_578_mult__diff__mult,axiom,
! [X_7: real,Y_5: real,A_24: real,B_7: real] : minus_minus_real(times_times_real(X_7,Y_5),times_times_real(A_24,B_7)) = plus_plus_real(times_times_real(X_7,minus_minus_real(Y_5,B_7)),times_times_real(minus_minus_real(X_7,A_24),B_7)) ).
tff(fact_579_mult__diff__mult,axiom,
! [X_7: int,Y_5: int,A_24: int,B_7: int] : minus_minus_int(times_times_int(X_7,Y_5),times_times_int(A_24,B_7)) = plus_plus_int(times_times_int(X_7,minus_minus_int(Y_5,B_7)),times_times_int(minus_minus_int(X_7,A_24),B_7)) ).
tff(fact_580_real__mult__le__cancel__iff1,axiom,
! [X_2: real,Y_2: real,Z_1: real] :
( ord_less_real(zero_zero_real,Z_1)
=> ( ord_less_eq_real(times_times_real(X_2,Z_1),times_times_real(Y_2,Z_1))
<=> ord_less_eq_real(X_2,Y_2) ) ) ).
tff(fact_581_real__mult__le__cancel__iff2,axiom,
! [X_2: real,Y_2: real,Z_1: real] :
( ord_less_real(zero_zero_real,Z_1)
=> ( ord_less_eq_real(times_times_real(Z_1,X_2),times_times_real(Z_1,Y_2))
<=> ord_less_eq_real(X_2,Y_2) ) ) ).
tff(fact_582_real__mult__less__mono2,axiom,
! [X: real,Y: real,Z: real] :
( ord_less_real(zero_zero_real,Z)
=> ( ord_less_real(X,Y)
=> ord_less_real(times_times_real(Z,X),times_times_real(Z,Y)) ) ) ).
tff(fact_583_real__mult__order,axiom,
! [Y: real,X: real] :
( ord_less_real(zero_zero_real,X)
=> ( ord_less_real(zero_zero_real,Y)
=> ord_less_real(zero_zero_real,times_times_real(X,Y)) ) ) ).
tff(fact_584_real__mult__less__iff1,axiom,
! [X_2: real,Y_2: real,Z_1: real] :
( ord_less_real(zero_zero_real,Z_1)
=> ( ord_less_real(times_times_real(X_2,Z_1),times_times_real(Y_2,Z_1))
<=> ord_less_real(X_2,Y_2) ) ) ).
tff(fact_585_real__two__squares__add__zero__iff,axiom,
! [X_2: real,Y_2: real] :
( ( plus_plus_real(times_times_real(X_2,X_2),times_times_real(Y_2,Y_2)) = zero_zero_real )
<=> ( ( X_2 = zero_zero_real )
& ( Y_2 = zero_zero_real ) ) ) ).
tff(fact_586_exp__eq__1,axiom,
! [X_2: nat,N_1: nat] :
( ( power_power_nat(X_2,N_1) = one_one_nat )
<=> ( ( X_2 = one_one_nat )
| ( N_1 = zero_zero_nat ) ) ) ).
tff(fact_587_real__squared__diff__one__factored,axiom,
! [X_6: real] : minus_minus_real(times_times_real(X_6,X_6),one_one_real) = times_times_real(plus_plus_real(X_6,one_one_real),minus_minus_real(X_6,one_one_real)) ).
tff(fact_588_real__squared__diff__one__factored,axiom,
! [X_6: int] : minus_minus_int(times_times_int(X_6,X_6),one_one_int) = times_times_int(plus_plus_int(X_6,one_one_int),minus_minus_int(X_6,one_one_int)) ).
tff(fact_589_realpow__minus__mult,axiom,
! [X_5: real,N_30: nat] :
( ord_less_nat(zero_zero_nat,N_30)
=> ( times_times_real(power_power_real(X_5,minus_minus_nat(N_30,one_one_nat)),X_5) = power_power_real(X_5,N_30) ) ) ).
tff(fact_590_realpow__minus__mult,axiom,
! [X_5: nat,N_30: nat] :
( ord_less_nat(zero_zero_nat,N_30)
=> ( times_times_nat(power_power_nat(X_5,minus_minus_nat(N_30,one_one_nat)),X_5) = power_power_nat(X_5,N_30) ) ) ).
tff(fact_591_realpow__minus__mult,axiom,
! [X_5: int,N_30: nat] :
( ord_less_nat(zero_zero_nat,N_30)
=> ( times_times_int(power_power_int(X_5,minus_minus_nat(N_30,one_one_nat)),X_5) = power_power_int(X_5,N_30) ) ) ).
tff(fact_592_two__realpow__ge__one,axiom,
! [N: nat] : ord_less_eq_real(one_one_real,power_power_real(number267125858f_real(bit0(bit1(pls))),N)) ).
tff(fact_593_realpow__num__eq__if,axiom,
! [M_7: real,N_29: nat] :
( ( ( N_29 = zero_zero_nat )
=> ( power_power_real(M_7,N_29) = one_one_real ) )
& ( ( N_29 != zero_zero_nat )
=> ( power_power_real(M_7,N_29) = times_times_real(M_7,power_power_real(M_7,minus_minus_nat(N_29,one_one_nat))) ) ) ) ).
tff(fact_594_realpow__num__eq__if,axiom,
! [M_7: nat,N_29: nat] :
( ( ( N_29 = zero_zero_nat )
=> ( power_power_nat(M_7,N_29) = one_one_nat ) )
& ( ( N_29 != zero_zero_nat )
=> ( power_power_nat(M_7,N_29) = times_times_nat(M_7,power_power_nat(M_7,minus_minus_nat(N_29,one_one_nat))) ) ) ) ).
tff(fact_595_realpow__num__eq__if,axiom,
! [M_7: int,N_29: nat] :
( ( ( N_29 = zero_zero_nat )
=> ( power_power_int(M_7,N_29) = one_one_int ) )
& ( ( N_29 != zero_zero_nat )
=> ( power_power_int(M_7,N_29) = times_times_int(M_7,power_power_int(M_7,minus_minus_nat(N_29,one_one_nat))) ) ) ) ).
tff(fact_596_power__strict__mono,axiom,
! [N_28: nat,A_23: real,B_6: real] :
( ord_less_real(A_23,B_6)
=> ( ord_less_eq_real(zero_zero_real,A_23)
=> ( ord_less_nat(zero_zero_nat,N_28)
=> ord_less_real(power_power_real(A_23,N_28),power_power_real(B_6,N_28)) ) ) ) ).
tff(fact_597_power__strict__mono,axiom,
! [N_28: nat,A_23: nat,B_6: nat] :
( ord_less_nat(A_23,B_6)
=> ( ord_less_eq_nat(zero_zero_nat,A_23)
=> ( ord_less_nat(zero_zero_nat,N_28)
=> ord_less_nat(power_power_nat(A_23,N_28),power_power_nat(B_6,N_28)) ) ) ) ).
tff(fact_598_power__strict__mono,axiom,
! [N_28: nat,A_23: int,B_6: int] :
( ord_less_int(A_23,B_6)
=> ( ord_less_eq_int(zero_zero_int,A_23)
=> ( ord_less_nat(zero_zero_nat,N_28)
=> ord_less_int(power_power_int(A_23,N_28),power_power_int(B_6,N_28)) ) ) ) ).
tff(fact_599_real__le__refl,axiom,
! [W: real] : ord_less_eq_real(W,W) ).
tff(fact_600_real__le__linear,axiom,
! [Z: real,W: real] :
( ord_less_eq_real(Z,W)
| ord_less_eq_real(W,Z) ) ).
tff(fact_601_real__le__trans,axiom,
! [K_1: real,I: real,J: real] :
( ord_less_eq_real(I,J)
=> ( ord_less_eq_real(J,K_1)
=> ord_less_eq_real(I,K_1) ) ) ).
tff(fact_602_real__le__antisym,axiom,
! [Z: real,W: real] :
( ord_less_eq_real(Z,W)
=> ( ord_less_eq_real(W,Z)
=> ( Z = W ) ) ) ).
tff(fact_603_field__power__not__zero,axiom,
! [N_27: nat,A_22: real] :
( ( A_22 != zero_zero_real )
=> ( power_power_real(A_22,N_27) != zero_zero_real ) ) ).
tff(fact_604_field__power__not__zero,axiom,
! [N_27: nat,A_22: int] :
( ( A_22 != zero_zero_int )
=> ( power_power_int(A_22,N_27) != zero_zero_int ) ) ).
tff(fact_605_power__mult__distrib,axiom,
! [A_21: real,B_5: real,N_26: nat] : power_power_real(times_times_real(A_21,B_5),N_26) = times_times_real(power_power_real(A_21,N_26),power_power_real(B_5,N_26)) ).
tff(fact_606_power__mult__distrib,axiom,
! [A_21: nat,B_5: nat,N_26: nat] : power_power_nat(times_times_nat(A_21,B_5),N_26) = times_times_nat(power_power_nat(A_21,N_26),power_power_nat(B_5,N_26)) ).
tff(fact_607_power__mult__distrib,axiom,
! [A_21: int,B_5: int,N_26: nat] : power_power_int(times_times_int(A_21,B_5),N_26) = times_times_int(power_power_int(A_21,N_26),power_power_int(B_5,N_26)) ).
tff(fact_608_power__commutes,axiom,
! [A_20: real,N_25: nat] : times_times_real(power_power_real(A_20,N_25),A_20) = times_times_real(A_20,power_power_real(A_20,N_25)) ).
tff(fact_609_power__commutes,axiom,
! [A_20: nat,N_25: nat] : times_times_nat(power_power_nat(A_20,N_25),A_20) = times_times_nat(A_20,power_power_nat(A_20,N_25)) ).
tff(fact_610_power__commutes,axiom,
! [A_20: int,N_25: nat] : times_times_int(power_power_int(A_20,N_25),A_20) = times_times_int(A_20,power_power_int(A_20,N_25)) ).
tff(fact_611_power__one,axiom,
! [N_24: nat] : power_power_real(one_one_real,N_24) = one_one_real ).
tff(fact_612_power__one,axiom,
! [N_24: nat] : power_power_nat(one_one_nat,N_24) = one_one_nat ).
tff(fact_613_power__one,axiom,
! [N_24: nat] : power_power_int(one_one_int,N_24) = one_one_int ).
tff(fact_614_dvd__power__same,axiom,
! [N_23: nat,X_4: nat,Y_4: nat] :
( dvd_dvd_nat(X_4,Y_4)
=> dvd_dvd_nat(power_power_nat(X_4,N_23),power_power_nat(Y_4,N_23)) ) ).
tff(fact_615_dvd__power__same,axiom,
! [N_23: nat,X_4: int,Y_4: int] :
( dvd_dvd_int(X_4,Y_4)
=> dvd_dvd_int(power_power_int(X_4,N_23),power_power_int(Y_4,N_23)) ) ).
tff(fact_616_dvd__power__same,axiom,
! [N_23: nat,X_4: real,Y_4: real] :
( dvd_dvd_real(X_4,Y_4)
=> dvd_dvd_real(power_power_real(X_4,N_23),power_power_real(Y_4,N_23)) ) ).
tff(fact_617_power__mult,axiom,
! [A_19: real,M_6: nat,N_22: nat] : power_power_real(A_19,times_times_nat(M_6,N_22)) = power_power_real(power_power_real(A_19,M_6),N_22) ).
tff(fact_618_power__mult,axiom,
! [A_19: nat,M_6: nat,N_22: nat] : power_power_nat(A_19,times_times_nat(M_6,N_22)) = power_power_nat(power_power_nat(A_19,M_6),N_22) ).
tff(fact_619_power__mult,axiom,
! [A_19: int,M_6: nat,N_22: nat] : power_power_int(A_19,times_times_nat(M_6,N_22)) = power_power_int(power_power_int(A_19,M_6),N_22) ).
tff(fact_620_power__one__right,axiom,
! [A_18: real] : power_power_real(A_18,one_one_nat) = A_18 ).
tff(fact_621_power__one__right,axiom,
! [A_18: nat] : power_power_nat(A_18,one_one_nat) = A_18 ).
tff(fact_622_power__one__right,axiom,
! [A_18: int] : power_power_int(A_18,one_one_nat) = A_18 ).
tff(fact_623_power__mono,axiom,
! [N_21: nat,A_17: real,B_4: real] :
( ord_less_eq_real(A_17,B_4)
=> ( ord_less_eq_real(zero_zero_real,A_17)
=> ord_less_eq_real(power_power_real(A_17,N_21),power_power_real(B_4,N_21)) ) ) ).
tff(fact_624_power__mono,axiom,
! [N_21: nat,A_17: nat,B_4: nat] :
( ord_less_eq_nat(A_17,B_4)
=> ( ord_less_eq_nat(zero_zero_nat,A_17)
=> ord_less_eq_nat(power_power_nat(A_17,N_21),power_power_nat(B_4,N_21)) ) ) ).
tff(fact_625_power__mono,axiom,
! [N_21: nat,A_17: int,B_4: int] :
( ord_less_eq_int(A_17,B_4)
=> ( ord_less_eq_int(zero_zero_int,A_17)
=> ord_less_eq_int(power_power_int(A_17,N_21),power_power_int(B_4,N_21)) ) ) ).
tff(fact_626_zero__le__power,axiom,
! [N_20: nat,A_16: real] :
( ord_less_eq_real(zero_zero_real,A_16)
=> ord_less_eq_real(zero_zero_real,power_power_real(A_16,N_20)) ) ).
tff(fact_627_zero__le__power,axiom,
! [N_20: nat,A_16: nat] :
( ord_less_eq_nat(zero_zero_nat,A_16)
=> ord_less_eq_nat(zero_zero_nat,power_power_nat(A_16,N_20)) ) ).
tff(fact_628_zero__le__power,axiom,
! [N_20: nat,A_16: int] :
( ord_less_eq_int(zero_zero_int,A_16)
=> ord_less_eq_int(zero_zero_int,power_power_int(A_16,N_20)) ) ).
tff(fact_629_zero__less__power,axiom,
! [N_19: nat,A_15: real] :
( ord_less_real(zero_zero_real,A_15)
=> ord_less_real(zero_zero_real,power_power_real(A_15,N_19)) ) ).
tff(fact_630_zero__less__power,axiom,
! [N_19: nat,A_15: nat] :
( ord_less_nat(zero_zero_nat,A_15)
=> ord_less_nat(zero_zero_nat,power_power_nat(A_15,N_19)) ) ).
tff(fact_631_zero__less__power,axiom,
! [N_19: nat,A_15: int] :
( ord_less_int(zero_zero_int,A_15)
=> ord_less_int(zero_zero_int,power_power_int(A_15,N_19)) ) ).
tff(fact_632_one__le__power,axiom,
! [N_18: nat,A_14: real] :
( ord_less_eq_real(one_one_real,A_14)
=> ord_less_eq_real(one_one_real,power_power_real(A_14,N_18)) ) ).
tff(fact_633_one__le__power,axiom,
! [N_18: nat,A_14: nat] :
( ord_less_eq_nat(one_one_nat,A_14)
=> ord_less_eq_nat(one_one_nat,power_power_nat(A_14,N_18)) ) ).
tff(fact_634_one__le__power,axiom,
! [N_18: nat,A_14: int] :
( ord_less_eq_int(one_one_int,A_14)
=> ord_less_eq_int(one_one_int,power_power_int(A_14,N_18)) ) ).
tff(fact_635_power__inject__exp,axiom,
! [Ma: nat,N_1: nat,A_13: real] :
( ord_less_real(one_one_real,A_13)
=> ( ( power_power_real(A_13,Ma) = power_power_real(A_13,N_1) )
<=> ( Ma = N_1 ) ) ) ).
tff(fact_636_power__inject__exp,axiom,
! [Ma: nat,N_1: nat,A_13: nat] :
( ord_less_nat(one_one_nat,A_13)
=> ( ( power_power_nat(A_13,Ma) = power_power_nat(A_13,N_1) )
<=> ( Ma = N_1 ) ) ) ).
tff(fact_637_power__inject__exp,axiom,
! [Ma: nat,N_1: nat,A_13: int] :
( ord_less_int(one_one_int,A_13)
=> ( ( power_power_int(A_13,Ma) = power_power_int(A_13,N_1) )
<=> ( Ma = N_1 ) ) ) ).
tff(fact_638_power__eq__0__iff,axiom,
! [A_13: real,N_1: nat] :
( ( power_power_real(A_13,N_1) = zero_zero_real )
<=> ( ( A_13 = zero_zero_real )
& ( N_1 != zero_zero_nat ) ) ) ).
tff(fact_639_power__eq__0__iff,axiom,
! [A_13: nat,N_1: nat] :
( ( power_power_nat(A_13,N_1) = zero_zero_nat )
<=> ( ( A_13 = zero_zero_nat )
& ( N_1 != zero_zero_nat ) ) ) ).
tff(fact_640_power__eq__0__iff,axiom,
! [A_13: int,N_1: nat] :
( ( power_power_int(A_13,N_1) = zero_zero_int )
<=> ( ( A_13 = zero_zero_int )
& ( N_1 != zero_zero_nat ) ) ) ).
tff(fact_641_power__0,axiom,
! [A_12: real] : power_power_real(A_12,zero_zero_nat) = one_one_real ).
tff(fact_642_power__0,axiom,
! [A_12: nat] : power_power_nat(A_12,zero_zero_nat) = one_one_nat ).
tff(fact_643_power__0,axiom,
! [A_12: int] : power_power_int(A_12,zero_zero_nat) = one_one_int ).
tff(fact_644_power__add,axiom,
! [A_11: real,M_5: nat,N_17: nat] : power_power_real(A_11,plus_plus_nat(M_5,N_17)) = times_times_real(power_power_real(A_11,M_5),power_power_real(A_11,N_17)) ).
tff(fact_645_power__add,axiom,
! [A_11: nat,M_5: nat,N_17: nat] : power_power_nat(A_11,plus_plus_nat(M_5,N_17)) = times_times_nat(power_power_nat(A_11,M_5),power_power_nat(A_11,N_17)) ).
tff(fact_646_power__add,axiom,
! [A_11: int,M_5: nat,N_17: nat] : power_power_int(A_11,plus_plus_nat(M_5,N_17)) = times_times_int(power_power_int(A_11,M_5),power_power_int(A_11,N_17)) ).
tff(fact_647_power__le__dvd,axiom,
! [M_4: nat,A_10: nat,N_16: nat,B_3: nat] :
( dvd_dvd_nat(power_power_nat(A_10,N_16),B_3)
=> ( ord_less_eq_nat(M_4,N_16)
=> dvd_dvd_nat(power_power_nat(A_10,M_4),B_3) ) ) ).
tff(fact_648_power__le__dvd,axiom,
! [M_4: nat,A_10: int,N_16: nat,B_3: int] :
( dvd_dvd_int(power_power_int(A_10,N_16),B_3)
=> ( ord_less_eq_nat(M_4,N_16)
=> dvd_dvd_int(power_power_int(A_10,M_4),B_3) ) ) ).
tff(fact_649_power__le__dvd,axiom,
! [M_4: nat,A_10: real,N_16: nat,B_3: real] :
( dvd_dvd_real(power_power_real(A_10,N_16),B_3)
=> ( ord_less_eq_nat(M_4,N_16)
=> dvd_dvd_real(power_power_real(A_10,M_4),B_3) ) ) ).
tff(fact_650_dvd__power__le,axiom,
! [N_15: nat,M_3: nat,X_3: nat,Y_3: nat] :
( dvd_dvd_nat(X_3,Y_3)
=> ( ord_less_eq_nat(N_15,M_3)
=> dvd_dvd_nat(power_power_nat(X_3,N_15),power_power_nat(Y_3,M_3)) ) ) ).
tff(fact_651_dvd__power__le,axiom,
! [N_15: nat,M_3: nat,X_3: int,Y_3: int] :
( dvd_dvd_int(X_3,Y_3)
=> ( ord_less_eq_nat(N_15,M_3)
=> dvd_dvd_int(power_power_int(X_3,N_15),power_power_int(Y_3,M_3)) ) ) ).
tff(fact_652_dvd__power__le,axiom,
! [N_15: nat,M_3: nat,X_3: real,Y_3: real] :
( dvd_dvd_real(X_3,Y_3)
=> ( ord_less_eq_nat(N_15,M_3)
=> dvd_dvd_real(power_power_real(X_3,N_15),power_power_real(Y_3,M_3)) ) ) ).
tff(fact_653_le__imp__power__dvd,axiom,
! [A_9: nat,M_2: nat,N_14: nat] :
( ord_less_eq_nat(M_2,N_14)
=> dvd_dvd_nat(power_power_nat(A_9,M_2),power_power_nat(A_9,N_14)) ) ).
tff(fact_654_le__imp__power__dvd,axiom,
! [A_9: int,M_2: nat,N_14: nat] :
( ord_less_eq_nat(M_2,N_14)
=> dvd_dvd_int(power_power_int(A_9,M_2),power_power_int(A_9,N_14)) ) ).
tff(fact_655_le__imp__power__dvd,axiom,
! [A_9: real,M_2: nat,N_14: nat] :
( ord_less_eq_nat(M_2,N_14)
=> dvd_dvd_real(power_power_real(A_9,M_2),power_power_real(A_9,N_14)) ) ).
tff(fact_656_nat__power__less__imp__less,axiom,
! [M: nat,N: 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_657_nat__zero__less__power__iff,axiom,
! [X_2: nat,N_1: nat] :
( ord_less_nat(zero_zero_nat,power_power_nat(X_2,N_1))
<=> ( ord_less_nat(zero_zero_nat,X_2)
| ( N_1 = zero_zero_nat ) ) ) ).
tff(fact_658_power__less__imp__less__base,axiom,
! [A_8: real,N_13: nat,B_2: real] :
( ord_less_real(power_power_real(A_8,N_13),power_power_real(B_2,N_13))
=> ( ord_less_eq_real(zero_zero_real,B_2)
=> ord_less_real(A_8,B_2) ) ) ).
tff(fact_659_power__less__imp__less__base,axiom,
! [A_8: nat,N_13: nat,B_2: nat] :
( ord_less_nat(power_power_nat(A_8,N_13),power_power_nat(B_2,N_13))
=> ( ord_less_eq_nat(zero_zero_nat,B_2)
=> ord_less_nat(A_8,B_2) ) ) ).
tff(fact_660_power__less__imp__less__base,axiom,
! [A_8: int,N_13: nat,B_2: int] :
( ord_less_int(power_power_int(A_8,N_13),power_power_int(B_2,N_13))
=> ( ord_less_eq_int(zero_zero_int,B_2)
=> ord_less_int(A_8,B_2) ) ) ).
tff(fact_661_power__less__power__Suc,axiom,
! [N_12: nat,A_7: real] :
( ord_less_real(one_one_real,A_7)
=> ord_less_real(power_power_real(A_7,N_12),times_times_real(A_7,power_power_real(A_7,N_12))) ) ).
tff(fact_662_power__less__power__Suc,axiom,
! [N_12: nat,A_7: nat] :
( ord_less_nat(one_one_nat,A_7)
=> ord_less_nat(power_power_nat(A_7,N_12),times_times_nat(A_7,power_power_nat(A_7,N_12))) ) ).
tff(fact_663_power__less__power__Suc,axiom,
! [N_12: nat,A_7: int] :
( ord_less_int(one_one_int,A_7)
=> ord_less_int(power_power_int(A_7,N_12),times_times_int(A_7,power_power_int(A_7,N_12))) ) ).
tff(fact_664_power__gt1__lemma,axiom,
! [N_11: nat,A_6: real] :
( ord_less_real(one_one_real,A_6)
=> ord_less_real(one_one_real,times_times_real(A_6,power_power_real(A_6,N_11))) ) ).
tff(fact_665_power__gt1__lemma,axiom,
! [N_11: nat,A_6: nat] :
( ord_less_nat(one_one_nat,A_6)
=> ord_less_nat(one_one_nat,times_times_nat(A_6,power_power_nat(A_6,N_11))) ) ).
tff(fact_666_power__gt1__lemma,axiom,
! [N_11: nat,A_6: int] :
( ord_less_int(one_one_int,A_6)
=> ord_less_int(one_one_int,times_times_int(A_6,power_power_int(A_6,N_11))) ) ).
tff(fact_667_power__0__left,axiom,
! [N_10: nat] :
( ( ( N_10 = zero_zero_nat )
=> ( power_power_real(zero_zero_real,N_10) = one_one_real ) )
& ( ( N_10 != zero_zero_nat )
=> ( power_power_real(zero_zero_real,N_10) = zero_zero_real ) ) ) ).
tff(fact_668_power__0__left,axiom,
! [N_10: nat] :
( ( ( N_10 = zero_zero_nat )
=> ( power_power_nat(zero_zero_nat,N_10) = one_one_nat ) )
& ( ( N_10 != zero_zero_nat )
=> ( power_power_nat(zero_zero_nat,N_10) = zero_zero_nat ) ) ) ).
tff(fact_669_power__0__left,axiom,
! [N_10: nat] :
( ( ( N_10 = zero_zero_nat )
=> ( power_power_int(zero_zero_int,N_10) = one_one_int ) )
& ( ( N_10 != zero_zero_nat )
=> ( power_power_int(zero_zero_int,N_10) = zero_zero_int ) ) ) ).
tff(fact_670_power__strict__increasing,axiom,
! [A_5: real,N_9: nat,N_8: nat] :
( ord_less_nat(N_9,N_8)
=> ( ord_less_real(one_one_real,A_5)
=> ord_less_real(power_power_real(A_5,N_9),power_power_real(A_5,N_8)) ) ) ).
tff(fact_671_power__strict__increasing,axiom,
! [A_5: nat,N_9: nat,N_8: nat] :
( ord_less_nat(N_9,N_8)
=> ( ord_less_nat(one_one_nat,A_5)
=> ord_less_nat(power_power_nat(A_5,N_9),power_power_nat(A_5,N_8)) ) ) ).
tff(fact_672_power__strict__increasing,axiom,
! [A_5: int,N_9: nat,N_8: nat] :
( ord_less_nat(N_9,N_8)
=> ( ord_less_int(one_one_int,A_5)
=> ord_less_int(power_power_int(A_5,N_9),power_power_int(A_5,N_8)) ) ) ).
tff(fact_673_power__less__imp__less__exp,axiom,
! [M_1: nat,N_7: nat,A_4: real] :
( ord_less_real(one_one_real,A_4)
=> ( ord_less_real(power_power_real(A_4,M_1),power_power_real(A_4,N_7))
=> ord_less_nat(M_1,N_7) ) ) ).
tff(fact_674_power__less__imp__less__exp,axiom,
! [M_1: nat,N_7: nat,A_4: nat] :
( ord_less_nat(one_one_nat,A_4)
=> ( ord_less_nat(power_power_nat(A_4,M_1),power_power_nat(A_4,N_7))
=> ord_less_nat(M_1,N_7) ) ) ).
tff(fact_675_power__less__imp__less__exp,axiom,
! [M_1: nat,N_7: nat,A_4: int] :
( ord_less_int(one_one_int,A_4)
=> ( ord_less_int(power_power_int(A_4,M_1),power_power_int(A_4,N_7))
=> ord_less_nat(M_1,N_7) ) ) ).
tff(fact_676_power__strict__increasing__iff,axiom,
! [X_2: nat,Y_2: nat,B_1: real] :
( ord_less_real(one_one_real,B_1)
=> ( ord_less_real(power_power_real(B_1,X_2),power_power_real(B_1,Y_2))
<=> ord_less_nat(X_2,Y_2) ) ) ).
tff(fact_677_power__strict__increasing__iff,axiom,
! [X_2: nat,Y_2: nat,B_1: nat] :
( ord_less_nat(one_one_nat,B_1)
=> ( ord_less_nat(power_power_nat(B_1,X_2),power_power_nat(B_1,Y_2))
<=> ord_less_nat(X_2,Y_2) ) ) ).
tff(fact_678_power__strict__increasing__iff,axiom,
! [X_2: nat,Y_2: nat,B_1: int] :
( ord_less_int(one_one_int,B_1)
=> ( ord_less_int(power_power_int(B_1,X_2),power_power_int(B_1,Y_2))
<=> ord_less_nat(X_2,Y_2) ) ) ).
tff(fact_679_power__increasing,axiom,
! [A_3: real,N_6: nat,N_5: nat] :
( ord_less_eq_nat(N_6,N_5)
=> ( ord_less_eq_real(one_one_real,A_3)
=> ord_less_eq_real(power_power_real(A_3,N_6),power_power_real(A_3,N_5)) ) ) ).
tff(fact_680_power__increasing,axiom,
! [A_3: nat,N_6: nat,N_5: nat] :
( ord_less_eq_nat(N_6,N_5)
=> ( ord_less_eq_nat(one_one_nat,A_3)
=> ord_less_eq_nat(power_power_nat(A_3,N_6),power_power_nat(A_3,N_5)) ) ) ).
tff(fact_681_power__increasing,axiom,
! [A_3: int,N_6: nat,N_5: nat] :
( ord_less_eq_nat(N_6,N_5)
=> ( ord_less_eq_int(one_one_int,A_3)
=> ord_less_eq_int(power_power_int(A_3,N_6),power_power_int(A_3,N_5)) ) ) ).
tff(fact_682_power__Suc__less,axiom,
! [N_4: nat,A_2: real] :
( ord_less_real(zero_zero_real,A_2)
=> ( ord_less_real(A_2,one_one_real)
=> ord_less_real(times_times_real(A_2,power_power_real(A_2,N_4)),power_power_real(A_2,N_4)) ) ) ).
tff(fact_683_power__Suc__less,axiom,
! [N_4: nat,A_2: nat] :
( ord_less_nat(zero_zero_nat,A_2)
=> ( ord_less_nat(A_2,one_one_nat)
=> ord_less_nat(times_times_nat(A_2,power_power_nat(A_2,N_4)),power_power_nat(A_2,N_4)) ) ) ).
tff(fact_684_power__Suc__less,axiom,
! [N_4: nat,A_2: int] :
( ord_less_int(zero_zero_int,A_2)
=> ( ord_less_int(A_2,one_one_int)
=> ord_less_int(times_times_int(A_2,power_power_int(A_2,N_4)),power_power_int(A_2,N_4)) ) ) ).
tff(fact_685_power__strict__decreasing,axiom,
! [A_1: int,N_3: nat,N_2: nat] :
( ord_less_nat(N_3,N_2)
=> ( ord_less_int(zero_zero_int,A_1)
=> ( ord_less_int(A_1,one_one_int)
=> ord_less_int(power_power_int(A_1,N_2),power_power_int(A_1,N_3)) ) ) ) ).
tff(fact_686_realpow__pos__nth__unique,axiom,
! [A: real,N: nat] :
( ord_less_nat(zero_zero_nat,N)
=> ( ord_less_real(zero_zero_real,A)
=> ? [X_1: real] :
( ord_less_real(zero_zero_real,X_1)
& ( power_power_real(X_1,N) = A )
& ! [Y_1: real] :
( ( ord_less_real(zero_zero_real,Y_1)
& ( power_power_real(Y_1,N) = A ) )
=> ( Y_1 = X_1 ) ) ) ) ) ).
tff(fact_687_divides__ge,axiom,
! [A: nat,B: nat] :
( dvd_dvd_nat(A,B)
=> ( ( B = zero_zero_nat )
| ord_less_eq_nat(A,B) ) ) ).
tff(fact_688_divides__exp,axiom,
! [N: nat,X: nat,Y: nat] :
( dvd_dvd_nat(X,Y)
=> dvd_dvd_nat(power_power_nat(X,N),power_power_nat(Y,N)) ) ).
tff(fact_689_divides__mul__l,axiom,
! [C: nat,A: nat,B: nat] :
( dvd_dvd_nat(A,B)
=> dvd_dvd_nat(times_times_nat(C,A),times_times_nat(C,B)) ) ).
tff(fact_690_divides__mul__r,axiom,
! [C: nat,A: nat,B: nat] :
( dvd_dvd_nat(A,B)
=> dvd_dvd_nat(times_times_nat(A,C),times_times_nat(B,C)) ) ).
tff(fact_691_divides__add__revr,axiom,
! [B: nat,D: nat,A: nat] :
( dvd_dvd_nat(D,A)
=> ( dvd_dvd_nat(D,plus_plus_nat(A,B))
=> dvd_dvd_nat(D,B) ) ) ).
tff(fact_692_nat__mult__dvd__cancel__disj_H,axiom,
! [Ma: nat,K: nat,N_1: nat] :
( dvd_dvd_nat(times_times_nat(Ma,K),times_times_nat(N_1,K))
<=> ( ( K = zero_zero_nat )
| dvd_dvd_nat(Ma,N_1) ) ) ).
tff(fact_693_divides__rev,axiom,
! [A: nat,N: nat,B: nat] :
( dvd_dvd_nat(power_power_nat(A,N),power_power_nat(B,N))
=> ( ( N != zero_zero_nat )
=> dvd_dvd_nat(A,B) ) ) ).
tff(fact_694_divides__exp2,axiom,
! [X: nat,Y: nat,N: nat] :
( ( N != zero_zero_nat )
=> ( dvd_dvd_nat(power_power_nat(X,N),Y)
=> dvd_dvd_nat(X,Y) ) ) ).
tff(fact_695_divides__div__not,axiom,
! [X: nat,Q: nat,N: nat,R: nat] :
( ( X = plus_plus_nat(times_times_nat(Q,N),R) )
=> ( ord_less_nat(zero_zero_nat,R)
=> ( ord_less_nat(R,N)
=> ~ dvd_dvd_nat(N,X) ) ) ) ).
tff(fact_696_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( dvd_dvd_nat(power_power_nat(I,M),power_power_nat(I,N))
=> ( ord_less_nat(one_one_nat,I)
=> ord_less_eq_nat(M,N) ) ) ).
tff(fact_697_divides__cases,axiom,
! [N: nat,M: nat] :
( dvd_dvd_nat(N,M)
=> ( ( M = zero_zero_nat )
| ( M = N )
| ord_less_eq_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls))),N),M) ) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
ord_less_int(plus_plus_int(power_power_int(s,number_number_of_nat(bit0(bit1(pls)))),one_one_int),zero_zero_int) ).
%------------------------------------------------------------------------------