TPTP Problem File: NUM926_2.p
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%------------------------------------------------------------------------------
% File : NUM926_2 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 258, 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_l258 [Bla11]
% Status : Theorem
% Rating : 0.11 v8.2.0, 0.30 v8.1.0, 0.09 v7.5.0, 0.12 v7.4.0, 0.00 v6.0.0, 0.40 v5.5.0, 0.25 v5.4.0, 0.67 v5.3.0
% Syntax : Number of formulae : 748 ( 342 unt; 49 typ; 0 def)
% Number of atoms : 1367 ( 570 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 786 ( 118 ~; 36 |; 58 &)
% ( 161 <=>; 413 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 4 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 59 ( 33 >; 26 *; 0 +; 0 <<)
% Number of predicates : 14 ( 13 usr; 0 prp; 1-3 aty)
% Number of functors : 32 ( 32 usr; 12 con; 0-2 aty)
% Number of variables : 1357 (1347 !; 10 ?;1357 :)
% SPC : TF0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:29:50
%------------------------------------------------------------------------------
%----Should-be-implicit typings (4)
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 ).
tff(ty_ty_tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
product_prod_int_int: $tType ).
%----Explicit typings (45)
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_Product__Type_OPair_000tc__Int__Oint_000tc__Int__Oint,type,
product_Pair_int_int: ( int * int ) > product_prod_int_int ).
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_c_TwoSquares__Mirabelle__xuacyfoabs_Osum2sq,type,
twoSqu2107342101sum2sq: product_prod_int_int > int ).
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_tpos,axiom,
ord_less_eq_int(one_one_int,t) ).
tff(fact_1__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06,axiom,
( ( t = one_one_int )
=> ? [X: int,Y: 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))))) = plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int) ) ).
tff(fact_2__0961_A_060_At_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06,axiom,
( ord_less_int(one_one_int,t)
=> ? [X: int,Y: 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))))) = plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int) ) ).
tff(fact_3_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_4_p,axiom,
zprime(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) ).
tff(fact_5_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_6_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_7_zadd__power2,axiom,
! [A: int,B_1: int] : power_power_int(plus_plus_int(A,B_1),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_1)),power_power_int(B_1,number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_8_zadd__power3,axiom,
! [A: int,B_1: int] : power_power_int(plus_plus_int(A,B_1),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_1)),times_times_int(times_times_int(number_number_of_int(bit1(bit1(pls))),A),power_power_int(B_1,number_number_of_nat(bit0(bit1(pls)))))),power_power_int(B_1,number_number_of_nat(bit1(bit1(pls))))) ).
tff(fact_9_power2__sum,axiom,
! [X_2: real,Y_2: real] : power_power_real(plus_plus_real(X_2,Y_2),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_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))))),times_times_real(times_times_real(number267125858f_real(bit0(bit1(pls))),X_2),Y_2)) ).
tff(fact_10_power2__sum,axiom,
! [X_2: nat,Y_2: nat] : power_power_nat(plus_plus_nat(X_2,Y_2),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_nat(plus_plus_nat(power_power_nat(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_nat(Y_2,number_number_of_nat(bit0(bit1(pls))))),times_times_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls))),X_2),Y_2)) ).
tff(fact_11_power2__sum,axiom,
! [X_2: int,Y_2: int] : power_power_int(plus_plus_int(X_2,Y_2),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_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))))),times_times_int(times_times_int(number_number_of_int(bit0(bit1(pls))),X_2),Y_2)) ).
tff(fact_12_power2__eq__square__number__of,axiom,
! [W_15: int] : power_power_nat(number_number_of_nat(W_15),number_number_of_nat(bit0(bit1(pls)))) = times_times_nat(number_number_of_nat(W_15),number_number_of_nat(W_15)) ).
tff(fact_13_power2__eq__square__number__of,axiom,
! [W_15: int] : power_power_real(number267125858f_real(W_15),number_number_of_nat(bit0(bit1(pls)))) = times_times_real(number267125858f_real(W_15),number267125858f_real(W_15)) ).
tff(fact_14_power2__eq__square__number__of,axiom,
! [W_15: int] : power_power_int(number_number_of_int(W_15),number_number_of_nat(bit0(bit1(pls)))) = times_times_int(number_number_of_int(W_15),number_number_of_int(W_15)) ).
tff(fact_15_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_16_one__power2,axiom,
power_power_real(one_one_real,number_number_of_nat(bit0(bit1(pls)))) = one_one_real ).
tff(fact_17_one__power2,axiom,
power_power_nat(one_one_nat,number_number_of_nat(bit0(bit1(pls)))) = one_one_nat ).
tff(fact_18_one__power2,axiom,
power_power_int(one_one_int,number_number_of_nat(bit0(bit1(pls)))) = one_one_int ).
tff(fact_19_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_21: nat] : times_times_nat(X_21,X_21) = power_power_nat(X_21,number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_20_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_21: real] : times_times_real(X_21,X_21) = power_power_real(X_21,number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_21_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_21: int] : times_times_int(X_21,X_21) = power_power_int(X_21,number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_22_power2__eq__square,axiom,
! [A_57: nat] : power_power_nat(A_57,number_number_of_nat(bit0(bit1(pls)))) = times_times_nat(A_57,A_57) ).
tff(fact_23_power2__eq__square,axiom,
! [A_57: real] : power_power_real(A_57,number_number_of_nat(bit0(bit1(pls)))) = times_times_real(A_57,A_57) ).
tff(fact_24_power2__eq__square,axiom,
! [A_57: int] : power_power_int(A_57,number_number_of_nat(bit0(bit1(pls)))) = times_times_int(A_57,A_57) ).
tff(fact_25_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_20: nat,N_38: nat] : power_power_nat(X_20,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_38)) = times_times_nat(power_power_nat(X_20,N_38),power_power_nat(X_20,N_38)) ).
tff(fact_26_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_20: real,N_38: nat] : power_power_real(X_20,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_38)) = times_times_real(power_power_real(X_20,N_38),power_power_real(X_20,N_38)) ).
tff(fact_27_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_20: int,N_38: nat] : power_power_int(X_20,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_38)) = times_times_int(power_power_int(X_20,N_38),power_power_int(X_20,N_38)) ).
tff(fact_28_add__special_I2_J,axiom,
! [W_14: int] : plus_plus_real(one_one_real,number267125858f_real(W_14)) = number267125858f_real(plus_plus_int(bit1(pls),W_14)) ).
tff(fact_29_add__special_I2_J,axiom,
! [W_14: int] : plus_plus_int(one_one_int,number_number_of_int(W_14)) = number_number_of_int(plus_plus_int(bit1(pls),W_14)) ).
tff(fact_30_add__special_I3_J,axiom,
! [V_17: int] : plus_plus_real(number267125858f_real(V_17),one_one_real) = number267125858f_real(plus_plus_int(V_17,bit1(pls))) ).
tff(fact_31_add__special_I3_J,axiom,
! [V_17: int] : plus_plus_int(number_number_of_int(V_17),one_one_int) = number_number_of_int(plus_plus_int(V_17,bit1(pls))) ).
tff(fact_32_one__add__one__is__two,axiom,
plus_plus_real(one_one_real,one_one_real) = number267125858f_real(bit0(bit1(pls))) ).
tff(fact_33_one__add__one__is__two,axiom,
plus_plus_int(one_one_int,one_one_int) = number_number_of_int(bit0(bit1(pls))) ).
tff(fact_34__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_35_zle__refl,axiom,
! [W: int] : ord_less_eq_int(W,W) ).
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_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_38_zless__linear,axiom,
! [X_1: int,Y_1: int] :
( ord_less_int(X_1,Y_1)
| ( X_1 = Y_1 )
| ord_less_int(Y_1,X_1) ) ).
tff(fact_39_zle__trans,axiom,
! [K: int,I: int,J: int] :
( ord_less_eq_int(I,J)
=> ( ord_less_eq_int(J,K)
=> ord_less_eq_int(I,K) ) ) ).
tff(fact_40_zle__antisym,axiom,
! [Z: int,W: int] :
( ord_less_eq_int(Z,W)
=> ( ord_less_eq_int(W,Z)
=> ( Z = W ) ) ) ).
tff(fact_41_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_19: nat,P_3: nat,Q_4: nat] : power_power_nat(power_power_nat(X_19,P_3),Q_4) = power_power_nat(X_19,times_times_nat(P_3,Q_4)) ).
tff(fact_42_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_19: real,P_3: nat,Q_4: nat] : power_power_real(power_power_real(X_19,P_3),Q_4) = power_power_real(X_19,times_times_nat(P_3,Q_4)) ).
tff(fact_43_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_19: int,P_3: nat,Q_4: nat] : power_power_int(power_power_int(X_19,P_3),Q_4) = power_power_int(X_19,times_times_nat(P_3,Q_4)) ).
tff(fact_44_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_18: nat] : power_power_nat(X_18,one_one_nat) = X_18 ).
tff(fact_45_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_18: real] : power_power_real(X_18,one_one_nat) = X_18 ).
tff(fact_46_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_18: int] : power_power_int(X_18,one_one_nat) = X_18 ).
tff(fact_47_zpower__zpower,axiom,
! [X_1: int,Y_1: nat,Z: nat] : power_power_int(power_power_int(X_1,Y_1),Z) = power_power_int(X_1,times_times_nat(Y_1,Z)) ).
tff(fact_48_le__number__of__eq__not__less,axiom,
! [V_3: int,W_1: int] :
( ord_less_eq_real(number267125858f_real(V_3),number267125858f_real(W_1))
<=> ~ ord_less_real(number267125858f_real(W_1),number267125858f_real(V_3)) ) ).
tff(fact_49_le__number__of__eq__not__less,axiom,
! [V_3: int,W_1: int] :
( ord_less_eq_nat(number_number_of_nat(V_3),number_number_of_nat(W_1))
<=> ~ ord_less_nat(number_number_of_nat(W_1),number_number_of_nat(V_3)) ) ).
tff(fact_50_le__number__of__eq__not__less,axiom,
! [V_3: int,W_1: int] :
( ord_less_eq_int(number_number_of_int(V_3),number_number_of_int(W_1))
<=> ~ ord_less_int(number_number_of_int(W_1),number_number_of_int(V_3)) ) ).
tff(fact_51_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_52_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_53_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_54_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_55_zadd__zless__mono,axiom,
! [Z_9: int,Z: int,W_13: int,W: int] :
( ord_less_int(W_13,W)
=> ( ord_less_eq_int(Z_9,Z)
=> ord_less_int(plus_plus_int(W_13,Z_9),plus_plus_int(W,Z)) ) ) ).
tff(fact_56_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_17: nat,P_2: nat,Q_3: nat] : times_times_nat(power_power_nat(X_17,P_2),power_power_nat(X_17,Q_3)) = power_power_nat(X_17,plus_plus_nat(P_2,Q_3)) ).
tff(fact_57_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_17: real,P_2: nat,Q_3: nat] : times_times_real(power_power_real(X_17,P_2),power_power_real(X_17,Q_3)) = power_power_real(X_17,plus_plus_nat(P_2,Q_3)) ).
tff(fact_58_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_17: int,P_2: nat,Q_3: nat] : times_times_int(power_power_int(X_17,P_2),power_power_int(X_17,Q_3)) = power_power_int(X_17,plus_plus_nat(P_2,Q_3)) ).
tff(fact_59_zpower__zadd__distrib,axiom,
! [X_1: int,Y_1: nat,Z: nat] : power_power_int(X_1,plus_plus_nat(Y_1,Z)) = times_times_int(power_power_int(X_1,Y_1),power_power_int(X_1,Z)) ).
tff(fact_60_nat__mult__2,axiom,
! [Z: nat] : times_times_nat(number_number_of_nat(bit0(bit1(pls))),Z) = plus_plus_nat(Z,Z) ).
tff(fact_61_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_62_nat__1__add__1,axiom,
plus_plus_nat(one_one_nat,one_one_nat) = number_number_of_nat(bit0(bit1(pls))) ).
tff(fact_63_less__int__code_I16_J,axiom,
! [K1: int,K2: int] :
( ord_less_int(bit1(K1),bit1(K2))
<=> ord_less_int(K1,K2) ) ).
tff(fact_64_rel__simps_I17_J,axiom,
! [K_1: int,L_1: int] :
( ord_less_int(bit1(K_1),bit1(L_1))
<=> ord_less_int(K_1,L_1) ) ).
tff(fact_65_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_66_rel__simps_I34_J,axiom,
! [K_1: int,L_1: int] :
( ord_less_eq_int(bit1(K_1),bit1(L_1))
<=> ord_less_eq_int(K_1,L_1) ) ).
tff(fact_67_rel__simps_I2_J,axiom,
~ ord_less_int(pls,pls) ).
tff(fact_68_less__int__code_I13_J,axiom,
! [K1: int,K2: int] :
( ord_less_int(bit0(K1),bit0(K2))
<=> ord_less_int(K1,K2) ) ).
tff(fact_69_rel__simps_I14_J,axiom,
! [K_1: int,L_1: int] :
( ord_less_int(bit0(K_1),bit0(L_1))
<=> ord_less_int(K_1,L_1) ) ).
tff(fact_70_rel__simps_I19_J,axiom,
ord_less_eq_int(pls,pls) ).
tff(fact_71_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_72_rel__simps_I31_J,axiom,
! [K_1: int,L_1: int] :
( ord_less_eq_int(bit0(K_1),bit0(L_1))
<=> ord_less_eq_int(K_1,L_1) ) ).
tff(fact_73_less__number__of__int__code,axiom,
! [K_1: int,L_1: int] :
( ord_less_int(number_number_of_int(K_1),number_number_of_int(L_1))
<=> ord_less_int(K_1,L_1) ) ).
tff(fact_74_less__eq__number__of__int__code,axiom,
! [K_1: int,L_1: int] :
( ord_less_eq_int(number_number_of_int(K_1),number_number_of_int(L_1))
<=> ord_less_eq_int(K_1,L_1) ) ).
tff(fact_75_zadd__strict__right__mono,axiom,
! [K: int,I: int,J: int] :
( ord_less_int(I,J)
=> ord_less_int(plus_plus_int(I,K),plus_plus_int(J,K)) ) ).
tff(fact_76_zadd__left__mono,axiom,
! [K: int,I: int,J: int] :
( ord_less_eq_int(I,J)
=> ord_less_eq_int(plus_plus_int(K,I),plus_plus_int(K,J)) ) ).
tff(fact_77_add__nat__number__of,axiom,
! [V_2: int,V_1: int] :
( ( ord_less_int(V_1,pls)
=> ( plus_plus_nat(number_number_of_nat(V_1),number_number_of_nat(V_2)) = number_number_of_nat(V_2) ) )
& ( ~ ord_less_int(V_1,pls)
=> ( ( ord_less_int(V_2,pls)
=> ( plus_plus_nat(number_number_of_nat(V_1),number_number_of_nat(V_2)) = number_number_of_nat(V_1) ) )
& ( ~ ord_less_int(V_2,pls)
=> ( plus_plus_nat(number_number_of_nat(V_1),number_number_of_nat(V_2)) = number_number_of_nat(plus_plus_int(V_1,V_2)) ) ) ) ) ) ).
tff(fact_78_nat__numeral__1__eq__1,axiom,
number_number_of_nat(bit1(pls)) = one_one_nat ).
tff(fact_79_Numeral1__eq1__nat,axiom,
one_one_nat = number_number_of_nat(bit1(pls)) ).
tff(fact_80_rel__simps_I29_J,axiom,
! [K_1: int] :
( ord_less_eq_int(bit1(K_1),pls)
<=> ord_less_int(K_1,pls) ) ).
tff(fact_81_rel__simps_I5_J,axiom,
! [K_1: int] :
( ord_less_int(pls,bit1(K_1))
<=> ord_less_eq_int(pls,K_1) ) ).
tff(fact_82_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_83_rel__simps_I33_J,axiom,
! [K_1: int,L_1: int] :
( ord_less_eq_int(bit1(K_1),bit0(L_1))
<=> ord_less_int(K_1,L_1) ) ).
tff(fact_84_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_85_rel__simps_I15_J,axiom,
! [K_1: int,L_1: int] :
( ord_less_int(bit0(K_1),bit1(L_1))
<=> ord_less_eq_int(K_1,L_1) ) ).
tff(fact_86_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_87_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_88_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_89_zprime__2,axiom,
zprime(number_number_of_int(bit0(bit1(pls)))) ).
tff(fact_90_is__mult__sum2sq,axiom,
! [Y_1: int,X_1: int] :
( twoSqu512355103sum2sq(X_1)
=> ( twoSqu512355103sum2sq(Y_1)
=> twoSqu512355103sum2sq(times_times_int(X_1,Y_1)) ) ) ).
tff(fact_91_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_92_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_93_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_94_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_95_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_96_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_97_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_98_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_99_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_100_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_101_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_102_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_103_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_104_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_105_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_106_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_107_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_108_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_109_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_110_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_111_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_112_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_56: real,B_17: real] : times_times_real(A_56,B_17) = times_times_real(B_17,A_56) ).
tff(fact_113_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_56: nat,B_17: nat] : times_times_nat(A_56,B_17) = times_times_nat(B_17,A_56) ).
tff(fact_114_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_56: int,B_17: int] : times_times_int(A_56,B_17) = times_times_int(B_17,A_56) ).
tff(fact_115_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_55: real,B_16: real,C_10: real,D_5: real] : plus_plus_real(plus_plus_real(A_55,B_16),plus_plus_real(C_10,D_5)) = plus_plus_real(plus_plus_real(A_55,C_10),plus_plus_real(B_16,D_5)) ).
tff(fact_116_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_55: nat,B_16: nat,C_10: nat,D_5: nat] : plus_plus_nat(plus_plus_nat(A_55,B_16),plus_plus_nat(C_10,D_5)) = plus_plus_nat(plus_plus_nat(A_55,C_10),plus_plus_nat(B_16,D_5)) ).
tff(fact_117_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_55: int,B_16: int,C_10: int,D_5: int] : plus_plus_int(plus_plus_int(A_55,B_16),plus_plus_int(C_10,D_5)) = plus_plus_int(plus_plus_int(A_55,C_10),plus_plus_int(B_16,D_5)) ).
tff(fact_118_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_54: real,B_15: real,C_9: real] : plus_plus_real(plus_plus_real(A_54,B_15),C_9) = plus_plus_real(plus_plus_real(A_54,C_9),B_15) ).
tff(fact_119_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_54: nat,B_15: nat,C_9: nat] : plus_plus_nat(plus_plus_nat(A_54,B_15),C_9) = plus_plus_nat(plus_plus_nat(A_54,C_9),B_15) ).
tff(fact_120_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_54: int,B_15: int,C_9: int] : plus_plus_int(plus_plus_int(A_54,B_15),C_9) = plus_plus_int(plus_plus_int(A_54,C_9),B_15) ).
tff(fact_121_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_53: real,B_14: real,C_8: real] : plus_plus_real(plus_plus_real(A_53,B_14),C_8) = plus_plus_real(A_53,plus_plus_real(B_14,C_8)) ).
tff(fact_122_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_53: nat,B_14: nat,C_8: nat] : plus_plus_nat(plus_plus_nat(A_53,B_14),C_8) = plus_plus_nat(A_53,plus_plus_nat(B_14,C_8)) ).
tff(fact_123_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_53: int,B_14: int,C_8: int] : plus_plus_int(plus_plus_int(A_53,B_14),C_8) = plus_plus_int(A_53,plus_plus_int(B_14,C_8)) ).
tff(fact_124_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_52: real,C_7: real,D_4: real] : plus_plus_real(A_52,plus_plus_real(C_7,D_4)) = plus_plus_real(plus_plus_real(A_52,C_7),D_4) ).
tff(fact_125_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_52: nat,C_7: nat,D_4: nat] : plus_plus_nat(A_52,plus_plus_nat(C_7,D_4)) = plus_plus_nat(plus_plus_nat(A_52,C_7),D_4) ).
tff(fact_126_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_52: int,C_7: int,D_4: int] : plus_plus_int(A_52,plus_plus_int(C_7,D_4)) = plus_plus_int(plus_plus_int(A_52,C_7),D_4) ).
tff(fact_127_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_51: real,C_6: real,D_3: real] : plus_plus_real(A_51,plus_plus_real(C_6,D_3)) = plus_plus_real(C_6,plus_plus_real(A_51,D_3)) ).
tff(fact_128_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_51: nat,C_6: nat,D_3: nat] : plus_plus_nat(A_51,plus_plus_nat(C_6,D_3)) = plus_plus_nat(C_6,plus_plus_nat(A_51,D_3)) ).
tff(fact_129_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_51: int,C_6: int,D_3: int] : plus_plus_int(A_51,plus_plus_int(C_6,D_3)) = plus_plus_int(C_6,plus_plus_int(A_51,D_3)) ).
tff(fact_130_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_50: real,C_5: real] : plus_plus_real(A_50,C_5) = plus_plus_real(C_5,A_50) ).
tff(fact_131_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_50: nat,C_5: nat] : plus_plus_nat(A_50,C_5) = plus_plus_nat(C_5,A_50) ).
tff(fact_132_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_50: int,C_5: int] : plus_plus_int(A_50,C_5) = plus_plus_int(C_5,A_50) ).
tff(fact_133_eq__number__of,axiom,
! [X_2: int,Y_2: int] :
( ( number267125858f_real(X_2) = number267125858f_real(Y_2) )
<=> ( X_2 = Y_2 ) ) ).
tff(fact_134_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_135_number__of__reorient,axiom,
! [W_1: int,X_2: real] :
( ( number267125858f_real(W_1) = X_2 )
<=> ( X_2 = number267125858f_real(W_1) ) ) ).
tff(fact_136_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_137_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_138_rel__simps_I51_J,axiom,
! [K_1: int,L_1: int] :
( ( bit1(K_1) = bit1(L_1) )
<=> ( K_1 = L_1 ) ) ).
tff(fact_139_rel__simps_I48_J,axiom,
! [K_1: int,L_1: int] :
( ( bit0(K_1) = bit0(L_1) )
<=> ( K_1 = L_1 ) ) ).
tff(fact_140_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_141_zmult__commute,axiom,
! [Z: int,W: int] : times_times_int(Z,W) = times_times_int(W,Z) ).
tff(fact_142_number__of__is__id,axiom,
! [K: int] : number_number_of_int(K) = K ).
tff(fact_143_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_144_zadd__left__commute,axiom,
! [X_1: int,Y_1: int,Z: int] : plus_plus_int(X_1,plus_plus_int(Y_1,Z)) = plus_plus_int(Y_1,plus_plus_int(X_1,Z)) ).
tff(fact_145_zadd__commute,axiom,
! [Z: int,W: int] : plus_plus_int(Z,W) = plus_plus_int(W,Z) ).
tff(fact_146_rel__simps_I12_J,axiom,
! [K_1: int] :
( ord_less_int(bit1(K_1),pls)
<=> ord_less_int(K_1,pls) ) ).
tff(fact_147_less__int__code_I15_J,axiom,
! [K1: int,K2: int] :
( ord_less_int(bit1(K1),bit0(K2))
<=> ord_less_int(K1,K2) ) ).
tff(fact_148_rel__simps_I16_J,axiom,
! [K_1: int,L_1: int] :
( ord_less_int(bit1(K_1),bit0(L_1))
<=> ord_less_int(K_1,L_1) ) ).
tff(fact_149_rel__simps_I10_J,axiom,
! [K_1: int] :
( ord_less_int(bit0(K_1),pls)
<=> ord_less_int(K_1,pls) ) ).
tff(fact_150_rel__simps_I4_J,axiom,
! [K_1: int] :
( ord_less_int(pls,bit0(K_1))
<=> ord_less_int(pls,K_1) ) ).
tff(fact_151_rel__simps_I22_J,axiom,
! [K_1: int] :
( ord_less_eq_int(pls,bit1(K_1))
<=> ord_less_eq_int(pls,K_1) ) ).
tff(fact_152_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_153_rel__simps_I32_J,axiom,
! [K_1: int,L_1: int] :
( ord_less_eq_int(bit0(K_1),bit1(L_1))
<=> ord_less_eq_int(K_1,L_1) ) ).
tff(fact_154_rel__simps_I27_J,axiom,
! [K_1: int] :
( ord_less_eq_int(bit0(K_1),pls)
<=> ord_less_eq_int(K_1,pls) ) ).
tff(fact_155_rel__simps_I21_J,axiom,
! [K_1: int] :
( ord_less_eq_int(pls,bit0(K_1))
<=> ord_less_eq_int(pls,K_1) ) ).
tff(fact_156_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_157_power__even__eq,axiom,
! [A_49: nat,N_37: nat] : power_power_nat(A_49,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_37)) = power_power_nat(power_power_nat(A_49,N_37),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_158_power__even__eq,axiom,
! [A_49: real,N_37: nat] : power_power_real(A_49,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_37)) = power_power_real(power_power_real(A_49,N_37),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_159_power__even__eq,axiom,
! [A_49: int,N_37: nat] : power_power_int(A_49,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_37)) = power_power_int(power_power_int(A_49,N_37),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_160_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_161_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_162_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_163_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_164_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_165_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_166_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_167_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_168_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_169_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_170_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_171_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_48: real,M_12: real,B_13: real] : plus_plus_real(times_times_real(A_48,M_12),times_times_real(B_13,M_12)) = times_times_real(plus_plus_real(A_48,B_13),M_12) ).
tff(fact_172_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_48: nat,M_12: nat,B_13: nat] : plus_plus_nat(times_times_nat(A_48,M_12),times_times_nat(B_13,M_12)) = times_times_nat(plus_plus_nat(A_48,B_13),M_12) ).
tff(fact_173_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_48: int,M_12: int,B_13: int] : plus_plus_int(times_times_int(A_48,M_12),times_times_int(B_13,M_12)) = times_times_int(plus_plus_int(A_48,B_13),M_12) ).
tff(fact_174_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_47: real,B_12: real,C_4: real] : times_times_real(plus_plus_real(A_47,B_12),C_4) = plus_plus_real(times_times_real(A_47,C_4),times_times_real(B_12,C_4)) ).
tff(fact_175_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_47: nat,B_12: nat,C_4: nat] : times_times_nat(plus_plus_nat(A_47,B_12),C_4) = plus_plus_nat(times_times_nat(A_47,C_4),times_times_nat(B_12,C_4)) ).
tff(fact_176_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_47: int,B_12: int,C_4: int] : times_times_int(plus_plus_int(A_47,B_12),C_4) = plus_plus_int(times_times_int(A_47,C_4),times_times_int(B_12,C_4)) ).
tff(fact_177_crossproduct__noteq,axiom,
! [C: real,D: real,A_1: real,B_2: real] :
( ( ( A_1 != B_2 )
& ( C != D ) )
<=> ( plus_plus_real(times_times_real(A_1,C),times_times_real(B_2,D)) != plus_plus_real(times_times_real(A_1,D),times_times_real(B_2,C)) ) ) ).
tff(fact_178_crossproduct__noteq,axiom,
! [C: nat,D: nat,A_1: nat,B_2: nat] :
( ( ( A_1 != B_2 )
& ( C != D ) )
<=> ( plus_plus_nat(times_times_nat(A_1,C),times_times_nat(B_2,D)) != plus_plus_nat(times_times_nat(A_1,D),times_times_nat(B_2,C)) ) ) ).
tff(fact_179_crossproduct__noteq,axiom,
! [C: int,D: int,A_1: int,B_2: int] :
( ( ( A_1 != B_2 )
& ( C != D ) )
<=> ( plus_plus_int(times_times_int(A_1,C),times_times_int(B_2,D)) != plus_plus_int(times_times_int(A_1,D),times_times_int(B_2,C)) ) ) ).
tff(fact_180_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_16: real,Y_14: real,Z_8: real] : times_times_real(X_16,plus_plus_real(Y_14,Z_8)) = plus_plus_real(times_times_real(X_16,Y_14),times_times_real(X_16,Z_8)) ).
tff(fact_181_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_16: nat,Y_14: nat,Z_8: nat] : times_times_nat(X_16,plus_plus_nat(Y_14,Z_8)) = plus_plus_nat(times_times_nat(X_16,Y_14),times_times_nat(X_16,Z_8)) ).
tff(fact_182_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_16: int,Y_14: int,Z_8: int] : times_times_int(X_16,plus_plus_int(Y_14,Z_8)) = plus_plus_int(times_times_int(X_16,Y_14),times_times_int(X_16,Z_8)) ).
tff(fact_183_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_46: real] : times_times_real(A_46,one_one_real) = A_46 ).
tff(fact_184_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_46: nat] : times_times_nat(A_46,one_one_nat) = A_46 ).
tff(fact_185_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_46: int] : times_times_int(A_46,one_one_int) = A_46 ).
tff(fact_186_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_45: real] : times_times_real(one_one_real,A_45) = A_45 ).
tff(fact_187_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_45: nat] : times_times_nat(one_one_nat,A_45) = A_45 ).
tff(fact_188_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_45: int] : times_times_int(one_one_int,A_45) = A_45 ).
tff(fact_189_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_15: nat,Y_13: nat,Q_2: nat] : power_power_nat(times_times_nat(X_15,Y_13),Q_2) = times_times_nat(power_power_nat(X_15,Q_2),power_power_nat(Y_13,Q_2)) ).
tff(fact_190_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_15: real,Y_13: real,Q_2: nat] : power_power_real(times_times_real(X_15,Y_13),Q_2) = times_times_real(power_power_real(X_15,Q_2),power_power_real(Y_13,Q_2)) ).
tff(fact_191_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_15: int,Y_13: int,Q_2: nat] : power_power_int(times_times_int(X_15,Y_13),Q_2) = times_times_int(power_power_int(X_15,Q_2),power_power_int(Y_13,Q_2)) ).
tff(fact_192_rel__simps_I46_J,axiom,
! [K: int] : bit1(K) != pls ).
tff(fact_193_rel__simps_I39_J,axiom,
! [L: int] : pls != bit1(L) ).
tff(fact_194_rel__simps_I50_J,axiom,
! [K: int,L: int] : bit1(K) != bit0(L) ).
tff(fact_195_rel__simps_I49_J,axiom,
! [K: int,L: int] : bit0(K) != bit1(L) ).
tff(fact_196_rel__simps_I44_J,axiom,
! [K_1: int] :
( ( bit0(K_1) = pls )
<=> ( K_1 = pls ) ) ).
tff(fact_197_rel__simps_I38_J,axiom,
! [L_1: int] :
( ( pls = bit0(L_1) )
<=> ( pls = L_1 ) ) ).
tff(fact_198_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_199_mult__Pls,axiom,
! [W: int] : times_times_int(pls,W) = pls ).
tff(fact_200_mult__Bit0,axiom,
! [K: int,L: int] : times_times_int(bit0(K),L) = bit0(times_times_int(K,L)) ).
tff(fact_201_add__Pls__right,axiom,
! [K: int] : plus_plus_int(K,pls) = K ).
tff(fact_202_add__Pls,axiom,
! [K: int] : plus_plus_int(pls,K) = K ).
tff(fact_203_add__Bit0__Bit0,axiom,
! [K: int,L: int] : plus_plus_int(bit0(K),bit0(L)) = bit0(plus_plus_int(K,L)) ).
tff(fact_204_Bit0__def,axiom,
! [K: int] : bit0(K) = plus_plus_int(K,K) ).
tff(fact_205_zmult__1__right,axiom,
! [Z: int] : times_times_int(Z,one_one_int) = Z ).
tff(fact_206_zmult__1,axiom,
! [Z: int] : times_times_int(one_one_int,Z) = Z ).
tff(fact_207_times__numeral__code_I5_J,axiom,
! [V_1: int,W: int] : times_times_int(number_number_of_int(V_1),number_number_of_int(W)) = number_number_of_int(times_times_int(V_1,W)) ).
tff(fact_208_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_209_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_210_plus__numeral__code_I9_J,axiom,
! [V_1: int,W: int] : plus_plus_int(number_number_of_int(V_1),number_number_of_int(W)) = number_number_of_int(plus_plus_int(V_1,W)) ).
tff(fact_211_semiring__mult__number__of,axiom,
! [V_16: int,V_15: int] :
( ord_less_eq_int(pls,V_15)
=> ( ord_less_eq_int(pls,V_16)
=> ( times_times_real(number267125858f_real(V_15),number267125858f_real(V_16)) = number267125858f_real(times_times_int(V_15,V_16)) ) ) ) ).
tff(fact_212_semiring__mult__number__of,axiom,
! [V_16: int,V_15: int] :
( ord_less_eq_int(pls,V_15)
=> ( ord_less_eq_int(pls,V_16)
=> ( times_times_nat(number_number_of_nat(V_15),number_number_of_nat(V_16)) = number_number_of_nat(times_times_int(V_15,V_16)) ) ) ) ).
tff(fact_213_semiring__mult__number__of,axiom,
! [V_16: int,V_15: int] :
( ord_less_eq_int(pls,V_15)
=> ( ord_less_eq_int(pls,V_16)
=> ( times_times_int(number_number_of_int(V_15),number_number_of_int(V_16)) = number_number_of_int(times_times_int(V_15,V_16)) ) ) ) ).
tff(fact_214_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_215_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_216_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_217_power2__ge__self,axiom,
! [X_1: int] : ord_less_eq_int(X_1,power_power_int(X_1,number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_218_left__distrib__number__of,axiom,
! [A_44: real,B_11: real,V_12: int] : times_times_real(plus_plus_real(A_44,B_11),number267125858f_real(V_12)) = plus_plus_real(times_times_real(A_44,number267125858f_real(V_12)),times_times_real(B_11,number267125858f_real(V_12))) ).
tff(fact_219_left__distrib__number__of,axiom,
! [A_44: nat,B_11: nat,V_12: int] : times_times_nat(plus_plus_nat(A_44,B_11),number_number_of_nat(V_12)) = plus_plus_nat(times_times_nat(A_44,number_number_of_nat(V_12)),times_times_nat(B_11,number_number_of_nat(V_12))) ).
tff(fact_220_left__distrib__number__of,axiom,
! [A_44: int,B_11: int,V_12: int] : times_times_int(plus_plus_int(A_44,B_11),number_number_of_int(V_12)) = plus_plus_int(times_times_int(A_44,number_number_of_int(V_12)),times_times_int(B_11,number_number_of_int(V_12))) ).
tff(fact_221_right__distrib__number__of,axiom,
! [V_11: int,B_10: real,C_3: real] : times_times_real(number267125858f_real(V_11),plus_plus_real(B_10,C_3)) = plus_plus_real(times_times_real(number267125858f_real(V_11),B_10),times_times_real(number267125858f_real(V_11),C_3)) ).
tff(fact_222_right__distrib__number__of,axiom,
! [V_11: int,B_10: nat,C_3: nat] : times_times_nat(number_number_of_nat(V_11),plus_plus_nat(B_10,C_3)) = plus_plus_nat(times_times_nat(number_number_of_nat(V_11),B_10),times_times_nat(number_number_of_nat(V_11),C_3)) ).
tff(fact_223_right__distrib__number__of,axiom,
! [V_11: int,B_10: int,C_3: int] : times_times_int(number_number_of_int(V_11),plus_plus_int(B_10,C_3)) = plus_plus_int(times_times_int(number_number_of_int(V_11),B_10),times_times_int(number_number_of_int(V_11),C_3)) ).
tff(fact_224_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_43: real,M_11: real] : plus_plus_real(times_times_real(A_43,M_11),M_11) = times_times_real(plus_plus_real(A_43,one_one_real),M_11) ).
tff(fact_225_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_43: nat,M_11: nat] : plus_plus_nat(times_times_nat(A_43,M_11),M_11) = times_times_nat(plus_plus_nat(A_43,one_one_nat),M_11) ).
tff(fact_226_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_43: int,M_11: int] : plus_plus_int(times_times_int(A_43,M_11),M_11) = times_times_int(plus_plus_int(A_43,one_one_int),M_11) ).
tff(fact_227_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_10: real,A_42: real] : plus_plus_real(M_10,times_times_real(A_42,M_10)) = times_times_real(plus_plus_real(A_42,one_one_real),M_10) ).
tff(fact_228_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_10: nat,A_42: nat] : plus_plus_nat(M_10,times_times_nat(A_42,M_10)) = times_times_nat(plus_plus_nat(A_42,one_one_nat),M_10) ).
tff(fact_229_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_10: int,A_42: int] : plus_plus_int(M_10,times_times_int(A_42,M_10)) = times_times_int(plus_plus_int(A_42,one_one_int),M_10) ).
tff(fact_230_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_9: real] : plus_plus_real(M_9,M_9) = times_times_real(plus_plus_real(one_one_real,one_one_real),M_9) ).
tff(fact_231_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_9: nat] : plus_plus_nat(M_9,M_9) = times_times_nat(plus_plus_nat(one_one_nat,one_one_nat),M_9) ).
tff(fact_232_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_9: int] : plus_plus_int(M_9,M_9) = times_times_int(plus_plus_int(one_one_int,one_one_int),M_9) ).
tff(fact_233_add__numeral__0,axiom,
! [A_41: real] : plus_plus_real(number267125858f_real(pls),A_41) = A_41 ).
tff(fact_234_add__numeral__0,axiom,
! [A_41: int] : plus_plus_int(number_number_of_int(pls),A_41) = A_41 ).
tff(fact_235_add__numeral__0__right,axiom,
! [A_40: real] : plus_plus_real(A_40,number267125858f_real(pls)) = A_40 ).
tff(fact_236_add__numeral__0__right,axiom,
! [A_40: int] : plus_plus_int(A_40,number_number_of_int(pls)) = A_40 ).
tff(fact_237_mult__number__of__left,axiom,
! [V_10: int,W_12: int,Z_7: real] : times_times_real(number267125858f_real(V_10),times_times_real(number267125858f_real(W_12),Z_7)) = times_times_real(number267125858f_real(times_times_int(V_10,W_12)),Z_7) ).
tff(fact_238_mult__number__of__left,axiom,
! [V_10: int,W_12: int,Z_7: int] : times_times_int(number_number_of_int(V_10),times_times_int(number_number_of_int(W_12),Z_7)) = times_times_int(number_number_of_int(times_times_int(V_10,W_12)),Z_7) ).
tff(fact_239_arith__simps_I32_J,axiom,
! [V_9: int,W_11: int] : times_times_real(number267125858f_real(V_9),number267125858f_real(W_11)) = number267125858f_real(times_times_int(V_9,W_11)) ).
tff(fact_240_arith__simps_I32_J,axiom,
! [V_9: int,W_11: int] : times_times_int(number_number_of_int(V_9),number_number_of_int(W_11)) = number_number_of_int(times_times_int(V_9,W_11)) ).
tff(fact_241_number__of__mult,axiom,
! [V_8: int,W_10: int] : number267125858f_real(times_times_int(V_8,W_10)) = times_times_real(number267125858f_real(V_8),number267125858f_real(W_10)) ).
tff(fact_242_number__of__mult,axiom,
! [V_8: int,W_10: int] : number_number_of_int(times_times_int(V_8,W_10)) = times_times_int(number_number_of_int(V_8),number_number_of_int(W_10)) ).
tff(fact_243_add__number__of__left,axiom,
! [V_7: int,W_9: int,Z_6: real] : plus_plus_real(number267125858f_real(V_7),plus_plus_real(number267125858f_real(W_9),Z_6)) = plus_plus_real(number267125858f_real(plus_plus_int(V_7,W_9)),Z_6) ).
tff(fact_244_add__number__of__left,axiom,
! [V_7: int,W_9: int,Z_6: int] : plus_plus_int(number_number_of_int(V_7),plus_plus_int(number_number_of_int(W_9),Z_6)) = plus_plus_int(number_number_of_int(plus_plus_int(V_7,W_9)),Z_6) ).
tff(fact_245_add__number__of__eq,axiom,
! [V_6: int,W_8: int] : plus_plus_real(number267125858f_real(V_6),number267125858f_real(W_8)) = number267125858f_real(plus_plus_int(V_6,W_8)) ).
tff(fact_246_add__number__of__eq,axiom,
! [V_6: int,W_8: int] : plus_plus_int(number_number_of_int(V_6),number_number_of_int(W_8)) = number_number_of_int(plus_plus_int(V_6,W_8)) ).
tff(fact_247_number__of__add,axiom,
! [V_5: int,W_7: int] : number267125858f_real(plus_plus_int(V_5,W_7)) = plus_plus_real(number267125858f_real(V_5),number267125858f_real(W_7)) ).
tff(fact_248_number__of__add,axiom,
! [V_5: int,W_7: int] : number_number_of_int(plus_plus_int(V_5,W_7)) = plus_plus_int(number_number_of_int(V_5),number_number_of_int(W_7)) ).
tff(fact_249_add__Bit1__Bit0,axiom,
! [K: int,L: int] : plus_plus_int(bit1(K),bit0(L)) = bit1(plus_plus_int(K,L)) ).
tff(fact_250_add__Bit0__Bit1,axiom,
! [K: int,L: int] : plus_plus_int(bit0(K),bit1(L)) = bit1(plus_plus_int(K,L)) ).
tff(fact_251_Bit1__def,axiom,
! [K: int] : bit1(K) = plus_plus_int(plus_plus_int(one_one_int,K),K) ).
tff(fact_252_number__of__Bit1,axiom,
! [W_6: int] : number267125858f_real(bit1(W_6)) = plus_plus_real(plus_plus_real(one_one_real,number267125858f_real(W_6)),number267125858f_real(W_6)) ).
tff(fact_253_number__of__Bit1,axiom,
! [W_6: int] : number_number_of_int(bit1(W_6)) = plus_plus_int(plus_plus_int(one_one_int,number_number_of_int(W_6)),number_number_of_int(W_6)) ).
tff(fact_254_mult__numeral__1,axiom,
! [A_39: real] : times_times_real(number267125858f_real(bit1(pls)),A_39) = A_39 ).
tff(fact_255_mult__numeral__1,axiom,
! [A_39: int] : times_times_int(number_number_of_int(bit1(pls)),A_39) = A_39 ).
tff(fact_256_mult__numeral__1__right,axiom,
! [A_38: real] : times_times_real(A_38,number267125858f_real(bit1(pls))) = A_38 ).
tff(fact_257_mult__numeral__1__right,axiom,
! [A_38: int] : times_times_int(A_38,number_number_of_int(bit1(pls))) = A_38 ).
tff(fact_258_semiring__numeral__1__eq__1,axiom,
number267125858f_real(bit1(pls)) = one_one_real ).
tff(fact_259_semiring__numeral__1__eq__1,axiom,
number_number_of_nat(bit1(pls)) = one_one_nat ).
tff(fact_260_semiring__numeral__1__eq__1,axiom,
number_number_of_int(bit1(pls)) = one_one_int ).
tff(fact_261_numeral__1__eq__1,axiom,
number267125858f_real(bit1(pls)) = one_one_real ).
tff(fact_262_numeral__1__eq__1,axiom,
number_number_of_int(bit1(pls)) = one_one_int ).
tff(fact_263_semiring__norm_I110_J,axiom,
one_one_real = number267125858f_real(bit1(pls)) ).
tff(fact_264_semiring__norm_I110_J,axiom,
one_one_int = number_number_of_int(bit1(pls)) ).
tff(fact_265_one__is__num__one,axiom,
one_one_int = number_number_of_int(bit1(pls)) ).
tff(fact_266_mult__Bit1,axiom,
! [K: int,L: int] : times_times_int(bit1(K),L) = plus_plus_int(bit0(times_times_int(K,L)),L) ).
tff(fact_267_double__number__of__Bit0,axiom,
! [W_5: int] : times_times_real(plus_plus_real(one_one_real,one_one_real),number267125858f_real(W_5)) = number267125858f_real(bit0(W_5)) ).
tff(fact_268_double__number__of__Bit0,axiom,
! [W_5: int] : times_times_int(plus_plus_int(one_one_int,one_one_int),number_number_of_int(W_5)) = number_number_of_int(bit0(W_5)) ).
tff(fact_269_power3__eq__cube,axiom,
! [A_37: nat] : power_power_nat(A_37,number_number_of_nat(bit1(bit1(pls)))) = times_times_nat(times_times_nat(A_37,A_37),A_37) ).
tff(fact_270_power3__eq__cube,axiom,
! [A_37: real] : power_power_real(A_37,number_number_of_nat(bit1(bit1(pls)))) = times_times_real(times_times_real(A_37,A_37),A_37) ).
tff(fact_271_power3__eq__cube,axiom,
! [A_37: int] : power_power_int(A_37,number_number_of_nat(bit1(bit1(pls)))) = times_times_int(times_times_int(A_37,A_37),A_37) ).
tff(fact_272_quartic__square__square,axiom,
! [X_1: int] : power_power_int(power_power_int(X_1,number_number_of_nat(bit0(bit1(pls)))),number_number_of_nat(bit0(bit1(pls)))) = power_power_int(X_1,number_number_of_nat(bit0(bit0(bit1(pls))))) ).
tff(fact_273_semiring__mult__2,axiom,
! [Z_5: real] : times_times_real(number267125858f_real(bit0(bit1(pls))),Z_5) = plus_plus_real(Z_5,Z_5) ).
tff(fact_274_semiring__mult__2,axiom,
! [Z_5: nat] : times_times_nat(number_number_of_nat(bit0(bit1(pls))),Z_5) = plus_plus_nat(Z_5,Z_5) ).
tff(fact_275_semiring__mult__2,axiom,
! [Z_5: int] : times_times_int(number_number_of_int(bit0(bit1(pls))),Z_5) = plus_plus_int(Z_5,Z_5) ).
tff(fact_276_mult__2,axiom,
! [Z_4: real] : times_times_real(number267125858f_real(bit0(bit1(pls))),Z_4) = plus_plus_real(Z_4,Z_4) ).
tff(fact_277_mult__2,axiom,
! [Z_4: int] : times_times_int(number_number_of_int(bit0(bit1(pls))),Z_4) = plus_plus_int(Z_4,Z_4) ).
tff(fact_278_semiring__mult__2__right,axiom,
! [Z_3: real] : times_times_real(Z_3,number267125858f_real(bit0(bit1(pls)))) = plus_plus_real(Z_3,Z_3) ).
tff(fact_279_semiring__mult__2__right,axiom,
! [Z_3: nat] : times_times_nat(Z_3,number_number_of_nat(bit0(bit1(pls)))) = plus_plus_nat(Z_3,Z_3) ).
tff(fact_280_semiring__mult__2__right,axiom,
! [Z_3: int] : times_times_int(Z_3,number_number_of_int(bit0(bit1(pls)))) = plus_plus_int(Z_3,Z_3) ).
tff(fact_281_mult__2__right,axiom,
! [Z_2: real] : times_times_real(Z_2,number267125858f_real(bit0(bit1(pls)))) = plus_plus_real(Z_2,Z_2) ).
tff(fact_282_mult__2__right,axiom,
! [Z_2: int] : times_times_int(Z_2,number_number_of_int(bit0(bit1(pls)))) = plus_plus_int(Z_2,Z_2) ).
tff(fact_283_semiring__one__add__one__is__two,axiom,
plus_plus_real(one_one_real,one_one_real) = number267125858f_real(bit0(bit1(pls))) ).
tff(fact_284_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_285_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_286_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_287__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_288_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_289__096sum2sq_A_Is_M_A1_J_A_061_A_I4_A_K_Am_A_L_A1_J_A_K_At_096,axiom,
twoSqu2107342101sum2sq(product_Pair_int_int(s,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_290_real__sum__squared__expand,axiom,
! [X_1: real,Y_1: real] : power_power_real(plus_plus_real(X_1,Y_1),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_real(plus_plus_real(power_power_real(X_1,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_1,number_number_of_nat(bit0(bit1(pls))))),times_times_real(times_times_real(number267125858f_real(bit0(bit1(pls))),X_1),Y_1)) ).
tff(fact_291_four__x__squared,axiom,
! [X_1: real] : times_times_real(number267125858f_real(bit0(bit0(bit1(pls)))),power_power_real(X_1,number_number_of_nat(bit0(bit1(pls))))) = power_power_real(times_times_real(number267125858f_real(bit0(bit1(pls))),X_1),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_292_power__less__power__Suc,axiom,
! [N_36: nat,A_36: real] :
( ord_less_real(one_one_real,A_36)
=> ord_less_real(power_power_real(A_36,N_36),times_times_real(A_36,power_power_real(A_36,N_36))) ) ).
tff(fact_293_power__less__power__Suc,axiom,
! [N_36: nat,A_36: nat] :
( ord_less_nat(one_one_nat,A_36)
=> ord_less_nat(power_power_nat(A_36,N_36),times_times_nat(A_36,power_power_nat(A_36,N_36))) ) ).
tff(fact_294_power__less__power__Suc,axiom,
! [N_36: nat,A_36: int] :
( ord_less_int(one_one_int,A_36)
=> ord_less_int(power_power_int(A_36,N_36),times_times_int(A_36,power_power_int(A_36,N_36))) ) ).
tff(fact_295_power__gt1__lemma,axiom,
! [N_35: nat,A_35: real] :
( ord_less_real(one_one_real,A_35)
=> ord_less_real(one_one_real,times_times_real(A_35,power_power_real(A_35,N_35))) ) ).
tff(fact_296_power__gt1__lemma,axiom,
! [N_35: nat,A_35: nat] :
( ord_less_nat(one_one_nat,A_35)
=> ord_less_nat(one_one_nat,times_times_nat(A_35,power_power_nat(A_35,N_35))) ) ).
tff(fact_297_power__gt1__lemma,axiom,
! [N_35: nat,A_35: int] :
( ord_less_int(one_one_int,A_35)
=> ord_less_int(one_one_int,times_times_int(A_35,power_power_int(A_35,N_35))) ) ).
tff(fact_298_power__le__imp__le__exp,axiom,
! [M_8: nat,N_34: nat,A_34: real] :
( ord_less_real(one_one_real,A_34)
=> ( ord_less_eq_real(power_power_real(A_34,M_8),power_power_real(A_34,N_34))
=> ord_less_eq_nat(M_8,N_34) ) ) ).
tff(fact_299_power__le__imp__le__exp,axiom,
! [M_8: nat,N_34: nat,A_34: nat] :
( ord_less_nat(one_one_nat,A_34)
=> ( ord_less_eq_nat(power_power_nat(A_34,M_8),power_power_nat(A_34,N_34))
=> ord_less_eq_nat(M_8,N_34) ) ) ).
tff(fact_300_power__le__imp__le__exp,axiom,
! [M_8: nat,N_34: nat,A_34: int] :
( ord_less_int(one_one_int,A_34)
=> ( ord_less_eq_int(power_power_int(A_34,M_8),power_power_int(A_34,N_34))
=> ord_less_eq_nat(M_8,N_34) ) ) ).
tff(fact_301_power__increasing__iff,axiom,
! [X_2: nat,Y_2: nat,B_2: real] :
( ord_less_real(one_one_real,B_2)
=> ( ord_less_eq_real(power_power_real(B_2,X_2),power_power_real(B_2,Y_2))
<=> ord_less_eq_nat(X_2,Y_2) ) ) ).
tff(fact_302_power__increasing__iff,axiom,
! [X_2: nat,Y_2: nat,B_2: nat] :
( ord_less_nat(one_one_nat,B_2)
=> ( ord_less_eq_nat(power_power_nat(B_2,X_2),power_power_nat(B_2,Y_2))
<=> ord_less_eq_nat(X_2,Y_2) ) ) ).
tff(fact_303_power__increasing__iff,axiom,
! [X_2: nat,Y_2: nat,B_2: int] :
( ord_less_int(one_one_int,B_2)
=> ( ord_less_eq_int(power_power_int(B_2,X_2),power_power_int(B_2,Y_2))
<=> ord_less_eq_nat(X_2,Y_2) ) ) ).
tff(fact_304__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_305_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_306__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: int] :
( ord_less_eq_int(zero_zero_int,X)
& ord_less_int(X,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))
& zcong(s1,X,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))
& ! [Y: int] :
( ( ord_less_eq_int(zero_zero_int,Y)
& ord_less_int(Y,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))
& zcong(s1,Y,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) )
=> ( Y = X ) ) ) ).
tff(fact_307__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_308_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_309_power__eq__0__iff,axiom,
! [A_1: real,N_1: nat] :
( ( power_power_real(A_1,N_1) = zero_zero_real )
<=> ( ( A_1 = zero_zero_real )
& ( N_1 != zero_zero_nat ) ) ) ).
tff(fact_310_power__eq__0__iff,axiom,
! [A_1: nat,N_1: nat] :
( ( power_power_nat(A_1,N_1) = zero_zero_nat )
<=> ( ( A_1 = zero_zero_nat )
& ( N_1 != zero_zero_nat ) ) ) ).
tff(fact_311_power__eq__0__iff,axiom,
! [A_1: int,N_1: nat] :
( ( power_power_int(A_1,N_1) = zero_zero_int )
<=> ( ( A_1 = zero_zero_int )
& ( N_1 != zero_zero_nat ) ) ) ).
tff(fact_312_le__imp__power__dvd,axiom,
! [A_33: nat,M_7: nat,N_33: nat] :
( ord_less_eq_nat(M_7,N_33)
=> dvd_dvd_nat(power_power_nat(A_33,M_7),power_power_nat(A_33,N_33)) ) ).
tff(fact_313_le__imp__power__dvd,axiom,
! [A_33: int,M_7: nat,N_33: nat] :
( ord_less_eq_nat(M_7,N_33)
=> dvd_dvd_int(power_power_int(A_33,M_7),power_power_int(A_33,N_33)) ) ).
tff(fact_314_le__imp__power__dvd,axiom,
! [A_33: real,M_7: nat,N_33: nat] :
( ord_less_eq_nat(M_7,N_33)
=> dvd_dvd_real(power_power_real(A_33,M_7),power_power_real(A_33,N_33)) ) ).
tff(fact_315_dvd__power__le,axiom,
! [N_32: nat,M_6: nat,X_14: nat,Y_12: nat] :
( dvd_dvd_nat(X_14,Y_12)
=> ( ord_less_eq_nat(N_32,M_6)
=> dvd_dvd_nat(power_power_nat(X_14,N_32),power_power_nat(Y_12,M_6)) ) ) ).
tff(fact_316_dvd__power__le,axiom,
! [N_32: nat,M_6: nat,X_14: int,Y_12: int] :
( dvd_dvd_int(X_14,Y_12)
=> ( ord_less_eq_nat(N_32,M_6)
=> dvd_dvd_int(power_power_int(X_14,N_32),power_power_int(Y_12,M_6)) ) ) ).
tff(fact_317_dvd__power__le,axiom,
! [N_32: nat,M_6: nat,X_14: real,Y_12: real] :
( dvd_dvd_real(X_14,Y_12)
=> ( ord_less_eq_nat(N_32,M_6)
=> dvd_dvd_real(power_power_real(X_14,N_32),power_power_real(Y_12,M_6)) ) ) ).
tff(fact_318_power__le__dvd,axiom,
! [M_5: nat,A_32: nat,N_31: nat,B_9: nat] :
( dvd_dvd_nat(power_power_nat(A_32,N_31),B_9)
=> ( ord_less_eq_nat(M_5,N_31)
=> dvd_dvd_nat(power_power_nat(A_32,M_5),B_9) ) ) ).
tff(fact_319_power__le__dvd,axiom,
! [M_5: nat,A_32: int,N_31: nat,B_9: int] :
( dvd_dvd_int(power_power_int(A_32,N_31),B_9)
=> ( ord_less_eq_nat(M_5,N_31)
=> dvd_dvd_int(power_power_int(A_32,M_5),B_9) ) ) ).
tff(fact_320_power__le__dvd,axiom,
! [M_5: nat,A_32: real,N_31: nat,B_9: real] :
( dvd_dvd_real(power_power_real(A_32,N_31),B_9)
=> ( ord_less_eq_nat(M_5,N_31)
=> dvd_dvd_real(power_power_real(A_32,M_5),B_9) ) ) ).
tff(fact_321_power__eq__imp__eq__base,axiom,
! [A_31: real,N_30: nat,B_8: real] :
( ( power_power_real(A_31,N_30) = power_power_real(B_8,N_30) )
=> ( ord_less_eq_real(zero_zero_real,A_31)
=> ( ord_less_eq_real(zero_zero_real,B_8)
=> ( ord_less_nat(zero_zero_nat,N_30)
=> ( A_31 = B_8 ) ) ) ) ) ).
tff(fact_322_power__eq__imp__eq__base,axiom,
! [A_31: nat,N_30: nat,B_8: nat] :
( ( power_power_nat(A_31,N_30) = power_power_nat(B_8,N_30) )
=> ( ord_less_eq_nat(zero_zero_nat,A_31)
=> ( ord_less_eq_nat(zero_zero_nat,B_8)
=> ( ord_less_nat(zero_zero_nat,N_30)
=> ( A_31 = B_8 ) ) ) ) ) ).
tff(fact_323_power__eq__imp__eq__base,axiom,
! [A_31: int,N_30: nat,B_8: int] :
( ( power_power_int(A_31,N_30) = power_power_int(B_8,N_30) )
=> ( ord_less_eq_int(zero_zero_int,A_31)
=> ( ord_less_eq_int(zero_zero_int,B_8)
=> ( ord_less_nat(zero_zero_nat,N_30)
=> ( A_31 = B_8 ) ) ) ) ) ).
tff(fact_324_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_325_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_326_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( dvd_dvd_int(times_times_int(K,M),times_times_int(K,N))
=> ( ( K != zero_zero_int )
=> dvd_dvd_int(M,N) ) ) ).
tff(fact_327_dvd__power__same,axiom,
! [N_29: nat,X_13: nat,Y_11: nat] :
( dvd_dvd_nat(X_13,Y_11)
=> dvd_dvd_nat(power_power_nat(X_13,N_29),power_power_nat(Y_11,N_29)) ) ).
tff(fact_328_dvd__power__same,axiom,
! [N_29: nat,X_13: int,Y_11: int] :
( dvd_dvd_int(X_13,Y_11)
=> dvd_dvd_int(power_power_int(X_13,N_29),power_power_int(Y_11,N_29)) ) ).
tff(fact_329_dvd__power__same,axiom,
! [N_29: nat,X_13: real,Y_11: real] :
( dvd_dvd_real(X_13,Y_11)
=> dvd_dvd_real(power_power_real(X_13,N_29),power_power_real(Y_11,N_29)) ) ).
tff(fact_330_field__power__not__zero,axiom,
! [N_28: nat,A_30: real] :
( ( A_30 != zero_zero_real )
=> ( power_power_real(A_30,N_28) != zero_zero_real ) ) ).
tff(fact_331_field__power__not__zero,axiom,
! [N_28: nat,A_30: int] :
( ( A_30 != zero_zero_int )
=> ( power_power_int(A_30,N_28) != zero_zero_int ) ) ).
tff(fact_332_power__0__left,axiom,
! [N_27: nat] :
( ( ( N_27 = zero_zero_nat )
=> ( power_power_real(zero_zero_real,N_27) = one_one_real ) )
& ( ( N_27 != zero_zero_nat )
=> ( power_power_real(zero_zero_real,N_27) = zero_zero_real ) ) ) ).
tff(fact_333_power__0__left,axiom,
! [N_27: nat] :
( ( ( N_27 = zero_zero_nat )
=> ( power_power_nat(zero_zero_nat,N_27) = one_one_nat ) )
& ( ( N_27 != zero_zero_nat )
=> ( power_power_nat(zero_zero_nat,N_27) = zero_zero_nat ) ) ) ).
tff(fact_334_power__0__left,axiom,
! [N_27: nat] :
( ( ( N_27 = zero_zero_nat )
=> ( power_power_int(zero_zero_int,N_27) = one_one_int ) )
& ( ( N_27 != zero_zero_nat )
=> ( power_power_int(zero_zero_int,N_27) = zero_zero_int ) ) ) ).
tff(fact_335_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_336_power__strict__mono,axiom,
! [N_26: nat,A_29: real,B_7: real] :
( ord_less_real(A_29,B_7)
=> ( ord_less_eq_real(zero_zero_real,A_29)
=> ( ord_less_nat(zero_zero_nat,N_26)
=> ord_less_real(power_power_real(A_29,N_26),power_power_real(B_7,N_26)) ) ) ) ).
tff(fact_337_power__strict__mono,axiom,
! [N_26: nat,A_29: nat,B_7: nat] :
( ord_less_nat(A_29,B_7)
=> ( ord_less_eq_nat(zero_zero_nat,A_29)
=> ( ord_less_nat(zero_zero_nat,N_26)
=> ord_less_nat(power_power_nat(A_29,N_26),power_power_nat(B_7,N_26)) ) ) ) ).
tff(fact_338_power__strict__mono,axiom,
! [N_26: nat,A_29: int,B_7: int] :
( ord_less_int(A_29,B_7)
=> ( ord_less_eq_int(zero_zero_int,A_29)
=> ( ord_less_nat(zero_zero_nat,N_26)
=> ord_less_int(power_power_int(A_29,N_26),power_power_int(B_7,N_26)) ) ) ) ).
tff(fact_339_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_28: real] : times_times_real(zero_zero_real,A_28) = zero_zero_real ).
tff(fact_340_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_28: nat] : times_times_nat(zero_zero_nat,A_28) = zero_zero_nat ).
tff(fact_341_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_28: int] : times_times_int(zero_zero_int,A_28) = zero_zero_int ).
tff(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_27: real] : times_times_real(A_27,zero_zero_real) = zero_zero_real ).
tff(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_27: nat] : times_times_nat(A_27,zero_zero_nat) = zero_zero_nat ).
tff(fact_344_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_27: int] : times_times_int(A_27,zero_zero_int) = zero_zero_int ).
tff(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_26: real] : plus_plus_real(zero_zero_real,A_26) = A_26 ).
tff(fact_346_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_26: nat] : plus_plus_nat(zero_zero_nat,A_26) = A_26 ).
tff(fact_347_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_26: int] : plus_plus_int(zero_zero_int,A_26) = A_26 ).
tff(fact_348_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_25: real] : plus_plus_real(A_25,zero_zero_real) = A_25 ).
tff(fact_349_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_25: nat] : plus_plus_nat(A_25,zero_zero_nat) = A_25 ).
tff(fact_350_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_25: int] : plus_plus_int(A_25,zero_zero_int) = A_25 ).
tff(fact_351_add__0__iff,axiom,
! [B_2: real,A_1: real] :
( ( B_2 = plus_plus_real(B_2,A_1) )
<=> ( A_1 = zero_zero_real ) ) ).
tff(fact_352_add__0__iff,axiom,
! [B_2: nat,A_1: nat] :
( ( B_2 = plus_plus_nat(B_2,A_1) )
<=> ( A_1 = zero_zero_nat ) ) ).
tff(fact_353_add__0__iff,axiom,
! [B_2: int,A_1: int] :
( ( B_2 = plus_plus_int(B_2,A_1) )
<=> ( A_1 = zero_zero_int ) ) ).
tff(fact_354_double__eq__0__iff,axiom,
! [A_1: real] :
( ( plus_plus_real(A_1,A_1) = zero_zero_real )
<=> ( A_1 = zero_zero_real ) ) ).
tff(fact_355_double__eq__0__iff,axiom,
! [A_1: int] :
( ( plus_plus_int(A_1,A_1) = zero_zero_int )
<=> ( A_1 = zero_zero_int ) ) ).
tff(fact_356_Pls__def,axiom,
pls = zero_zero_int ).
tff(fact_357_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
tff(fact_358_zadd__0,axiom,
! [Z: int] : plus_plus_int(zero_zero_int,Z) = Z ).
tff(fact_359_zadd__0__right,axiom,
! [Z: int] : plus_plus_int(Z,zero_zero_int) = Z ).
tff(fact_360_zero__le__power,axiom,
! [N_25: nat,A_24: real] :
( ord_less_eq_real(zero_zero_real,A_24)
=> ord_less_eq_real(zero_zero_real,power_power_real(A_24,N_25)) ) ).
tff(fact_361_zero__le__power,axiom,
! [N_25: nat,A_24: nat] :
( ord_less_eq_nat(zero_zero_nat,A_24)
=> ord_less_eq_nat(zero_zero_nat,power_power_nat(A_24,N_25)) ) ).
tff(fact_362_zero__le__power,axiom,
! [N_25: nat,A_24: int] :
( ord_less_eq_int(zero_zero_int,A_24)
=> ord_less_eq_int(zero_zero_int,power_power_int(A_24,N_25)) ) ).
tff(fact_363_power__mono,axiom,
! [N_24: nat,A_23: real,B_6: real] :
( ord_less_eq_real(A_23,B_6)
=> ( ord_less_eq_real(zero_zero_real,A_23)
=> ord_less_eq_real(power_power_real(A_23,N_24),power_power_real(B_6,N_24)) ) ) ).
tff(fact_364_power__mono,axiom,
! [N_24: nat,A_23: nat,B_6: nat] :
( ord_less_eq_nat(A_23,B_6)
=> ( ord_less_eq_nat(zero_zero_nat,A_23)
=> ord_less_eq_nat(power_power_nat(A_23,N_24),power_power_nat(B_6,N_24)) ) ) ).
tff(fact_365_power__mono,axiom,
! [N_24: nat,A_23: int,B_6: int] :
( ord_less_eq_int(A_23,B_6)
=> ( ord_less_eq_int(zero_zero_int,A_23)
=> ord_less_eq_int(power_power_int(A_23,N_24),power_power_int(B_6,N_24)) ) ) ).
tff(fact_366_zero__less__power,axiom,
! [N_23: nat,A_22: real] :
( ord_less_real(zero_zero_real,A_22)
=> ord_less_real(zero_zero_real,power_power_real(A_22,N_23)) ) ).
tff(fact_367_zero__less__power,axiom,
! [N_23: nat,A_22: nat] :
( ord_less_nat(zero_zero_nat,A_22)
=> ord_less_nat(zero_zero_nat,power_power_nat(A_22,N_23)) ) ).
tff(fact_368_zero__less__power,axiom,
! [N_23: nat,A_22: int] :
( ord_less_int(zero_zero_int,A_22)
=> ord_less_int(zero_zero_int,power_power_int(A_22,N_23)) ) ).
tff(fact_369_zcong__zpower__zmult,axiom,
! [Z: nat,X_1: int,Y_1: nat,P: int] :
( zcong(power_power_int(X_1,Y_1),one_one_int,P)
=> zcong(power_power_int(X_1,times_times_nat(Y_1,Z)),one_one_int,P) ) ).
tff(fact_370_zdvd__reduce,axiom,
! [K_1: int,N_1: int,Ma: int] :
( dvd_dvd_int(K_1,plus_plus_int(N_1,times_times_int(K_1,Ma)))
<=> dvd_dvd_int(K_1,N_1) ) ).
tff(fact_371_zdvd__period,axiom,
! [C: int,X_2: int,Ta: int,A_1: int,D: int] :
( dvd_dvd_int(A_1,D)
=> ( dvd_dvd_int(A_1,plus_plus_int(X_2,Ta))
<=> dvd_dvd_int(A_1,plus_plus_int(plus_plus_int(X_2,times_times_int(C,D)),Ta)) ) ) ).
tff(fact_372_power__less__imp__less__base,axiom,
! [A_21: real,N_22: nat,B_5: real] :
( ord_less_real(power_power_real(A_21,N_22),power_power_real(B_5,N_22))
=> ( ord_less_eq_real(zero_zero_real,B_5)
=> ord_less_real(A_21,B_5) ) ) ).
tff(fact_373_power__less__imp__less__base,axiom,
! [A_21: nat,N_22: nat,B_5: nat] :
( ord_less_nat(power_power_nat(A_21,N_22),power_power_nat(B_5,N_22))
=> ( ord_less_eq_nat(zero_zero_nat,B_5)
=> ord_less_nat(A_21,B_5) ) ) ).
tff(fact_374_power__less__imp__less__base,axiom,
! [A_21: int,N_22: nat,B_5: int] :
( ord_less_int(power_power_int(A_21,N_22),power_power_int(B_5,N_22))
=> ( ord_less_eq_int(zero_zero_int,B_5)
=> ord_less_int(A_21,B_5) ) ) ).
tff(fact_375_power__decreasing,axiom,
! [A_20: real,N_21: nat,N_20: nat] :
( ord_less_eq_nat(N_21,N_20)
=> ( ord_less_eq_real(zero_zero_real,A_20)
=> ( ord_less_eq_real(A_20,one_one_real)
=> ord_less_eq_real(power_power_real(A_20,N_20),power_power_real(A_20,N_21)) ) ) ) ).
tff(fact_376_power__decreasing,axiom,
! [A_20: nat,N_21: nat,N_20: nat] :
( ord_less_eq_nat(N_21,N_20)
=> ( ord_less_eq_nat(zero_zero_nat,A_20)
=> ( ord_less_eq_nat(A_20,one_one_nat)
=> ord_less_eq_nat(power_power_nat(A_20,N_20),power_power_nat(A_20,N_21)) ) ) ) ).
tff(fact_377_power__decreasing,axiom,
! [A_20: int,N_21: nat,N_20: nat] :
( ord_less_eq_nat(N_21,N_20)
=> ( ord_less_eq_int(zero_zero_int,A_20)
=> ( ord_less_eq_int(A_20,one_one_int)
=> ord_less_eq_int(power_power_int(A_20,N_20),power_power_int(A_20,N_21)) ) ) ) ).
tff(fact_378_power__strict__decreasing,axiom,
! [A_19: real,N_19: nat,N_18: nat] :
( ord_less_nat(N_19,N_18)
=> ( ord_less_real(zero_zero_real,A_19)
=> ( ord_less_real(A_19,one_one_real)
=> ord_less_real(power_power_real(A_19,N_18),power_power_real(A_19,N_19)) ) ) ) ).
tff(fact_379_power__strict__decreasing,axiom,
! [A_19: nat,N_19: nat,N_18: nat] :
( ord_less_nat(N_19,N_18)
=> ( ord_less_nat(zero_zero_nat,A_19)
=> ( ord_less_nat(A_19,one_one_nat)
=> ord_less_nat(power_power_nat(A_19,N_18),power_power_nat(A_19,N_19)) ) ) ) ).
tff(fact_380_power__strict__decreasing,axiom,
! [A_19: int,N_19: nat,N_18: nat] :
( ord_less_nat(N_19,N_18)
=> ( ord_less_int(zero_zero_int,A_19)
=> ( ord_less_int(A_19,one_one_int)
=> ord_less_int(power_power_int(A_19,N_18),power_power_int(A_19,N_19)) ) ) ) ).
tff(fact_381_even__less__0__iff,axiom,
! [A_1: real] :
( ord_less_real(plus_plus_real(A_1,A_1),zero_zero_real)
<=> ord_less_real(A_1,zero_zero_real) ) ).
tff(fact_382_even__less__0__iff,axiom,
! [A_1: int] :
( ord_less_int(plus_plus_int(A_1,A_1),zero_zero_int)
<=> ord_less_int(A_1,zero_zero_int) ) ).
tff(fact_383_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_384_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_385_add__scale__eq__noteq,axiom,
! [C_2: real,D_2: real,A_18: real,B_4: real,R_3: real] :
( ( R_3 != zero_zero_real )
=> ( ( ( A_18 = B_4 )
& ( C_2 != D_2 ) )
=> ( plus_plus_real(A_18,times_times_real(R_3,C_2)) != plus_plus_real(B_4,times_times_real(R_3,D_2)) ) ) ) ).
tff(fact_386_add__scale__eq__noteq,axiom,
! [C_2: nat,D_2: nat,A_18: nat,B_4: nat,R_3: nat] :
( ( R_3 != zero_zero_nat )
=> ( ( ( A_18 = B_4 )
& ( C_2 != D_2 ) )
=> ( plus_plus_nat(A_18,times_times_nat(R_3,C_2)) != plus_plus_nat(B_4,times_times_nat(R_3,D_2)) ) ) ) ).
tff(fact_387_add__scale__eq__noteq,axiom,
! [C_2: int,D_2: int,A_18: int,B_4: int,R_3: int] :
( ( R_3 != zero_zero_int )
=> ( ( ( A_18 = B_4 )
& ( C_2 != D_2 ) )
=> ( plus_plus_int(A_18,times_times_int(R_3,C_2)) != plus_plus_int(B_4,times_times_int(R_3,D_2)) ) ) ) ).
tff(fact_388_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_389_semiring__norm_I112_J,axiom,
zero_zero_real = number267125858f_real(pls) ).
tff(fact_390_semiring__norm_I112_J,axiom,
zero_zero_int = number_number_of_int(pls) ).
tff(fact_391_number__of__Pls,axiom,
number267125858f_real(pls) = zero_zero_real ).
tff(fact_392_number__of__Pls,axiom,
number_number_of_int(pls) = zero_zero_int ).
tff(fact_393_semiring__numeral__0__eq__0,axiom,
number267125858f_real(pls) = zero_zero_real ).
tff(fact_394_semiring__numeral__0__eq__0,axiom,
number_number_of_nat(pls) = zero_zero_nat ).
tff(fact_395_semiring__numeral__0__eq__0,axiom,
number_number_of_int(pls) = zero_zero_int ).
tff(fact_396_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_397_bin__less__0__simps_I1_J,axiom,
~ ord_less_int(pls,zero_zero_int) ).
tff(fact_398_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_399_zero__is__num__zero,axiom,
zero_zero_int = number_number_of_int(pls) ).
tff(fact_400_int__0__less__1,axiom,
ord_less_int(zero_zero_int,one_one_int) ).
tff(fact_401_pos__zmult__pos,axiom,
! [B_1: int,A: int] :
( ord_less_int(zero_zero_int,A)
=> ( ord_less_int(zero_zero_int,times_times_int(A,B_1))
=> ord_less_int(zero_zero_int,B_1) ) ) ).
tff(fact_402_zmult__zless__mono2,axiom,
! [K: int,I: int,J: int] :
( ord_less_int(I,J)
=> ( ord_less_int(zero_zero_int,K)
=> ord_less_int(times_times_int(K,I),times_times_int(K,J)) ) ) ).
tff(fact_403_odd__nonzero,axiom,
! [Z: int] : plus_plus_int(plus_plus_int(one_one_int,Z),Z) != zero_zero_int ).
tff(fact_404_power__Suc__less,axiom,
! [N_17: nat,A_17: real] :
( ord_less_real(zero_zero_real,A_17)
=> ( ord_less_real(A_17,one_one_real)
=> ord_less_real(times_times_real(A_17,power_power_real(A_17,N_17)),power_power_real(A_17,N_17)) ) ) ).
tff(fact_405_power__Suc__less,axiom,
! [N_17: nat,A_17: nat] :
( ord_less_nat(zero_zero_nat,A_17)
=> ( ord_less_nat(A_17,one_one_nat)
=> ord_less_nat(times_times_nat(A_17,power_power_nat(A_17,N_17)),power_power_nat(A_17,N_17)) ) ) ).
tff(fact_406_power__Suc__less,axiom,
! [N_17: nat,A_17: int] :
( ord_less_int(zero_zero_int,A_17)
=> ( ord_less_int(A_17,one_one_int)
=> ord_less_int(times_times_int(A_17,power_power_int(A_17,N_17)),power_power_int(A_17,N_17)) ) ) ).
tff(fact_407_zprime__power__zdvd__cancel__left,axiom,
! [N: nat,B_1: 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_1))
=> dvd_dvd_int(power_power_int(P,N),B_1) ) ) ) ).
tff(fact_408_zprime__power__zdvd__cancel__right,axiom,
! [N: nat,A: int,B_1: int,P: int] :
( zprime(P)
=> ( ~ dvd_dvd_int(P,B_1)
=> ( dvd_dvd_int(power_power_int(P,N),times_times_int(A,B_1))
=> dvd_dvd_int(power_power_int(P,N),A) ) ) ) ).
tff(fact_409_sum__squares__ge__zero,axiom,
! [X_12: real,Y_10: real] : ord_less_eq_real(zero_zero_real,plus_plus_real(times_times_real(X_12,X_12),times_times_real(Y_10,Y_10))) ).
tff(fact_410_sum__squares__ge__zero,axiom,
! [X_12: int,Y_10: int] : ord_less_eq_int(zero_zero_int,plus_plus_int(times_times_int(X_12,X_12),times_times_int(Y_10,Y_10))) ).
tff(fact_411_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_412_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_413_less__nat__number__of,axiom,
! [V_3: int,V_4: int] :
( ord_less_nat(number_number_of_nat(V_3),number_number_of_nat(V_4))
<=> ( ( ord_less_int(V_3,V_4)
=> ord_less_int(pls,V_4) )
& ord_less_int(V_3,V_4) ) ) ).
tff(fact_414_not__sum__squares__lt__zero,axiom,
! [X_11: real,Y_9: real] : ~ ord_less_real(plus_plus_real(times_times_real(X_11,X_11),times_times_real(Y_9,Y_9)),zero_zero_real) ).
tff(fact_415_not__sum__squares__lt__zero,axiom,
! [X_11: int,Y_9: int] : ~ ord_less_int(plus_plus_int(times_times_int(X_11,X_11),times_times_int(Y_9,Y_9)),zero_zero_int) ).
tff(fact_416_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_417_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_418_le__nat__number__of,axiom,
! [V_3: int,V_4: int] :
( ord_less_eq_nat(number_number_of_nat(V_3),number_number_of_nat(V_4))
<=> ( ~ ord_less_eq_int(V_3,V_4)
=> ord_less_eq_int(V_3,pls) ) ) ).
tff(fact_419_number__of__Bit0,axiom,
! [W_4: int] : number267125858f_real(bit0(W_4)) = plus_plus_real(plus_plus_real(zero_zero_real,number267125858f_real(W_4)),number267125858f_real(W_4)) ).
tff(fact_420_number__of__Bit0,axiom,
! [W_4: int] : number_number_of_int(bit0(W_4)) = plus_plus_int(plus_plus_int(zero_zero_int,number_number_of_int(W_4)),number_number_of_int(W_4)) ).
tff(fact_421_power__one__right,axiom,
! [A_16: nat] : power_power_nat(A_16,one_one_nat) = A_16 ).
tff(fact_422_power__one__right,axiom,
! [A_16: real] : power_power_real(A_16,one_one_nat) = A_16 ).
tff(fact_423_power__one__right,axiom,
! [A_16: int] : power_power_int(A_16,one_one_nat) = A_16 ).
tff(fact_424_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_425_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_426_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_427_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_428_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_429_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_430_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_431_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_432_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_433_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_434_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_435_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_436_zero__power2,axiom,
power_power_real(zero_zero_real,number_number_of_nat(bit0(bit1(pls)))) = zero_zero_real ).
tff(fact_437_zero__power2,axiom,
power_power_nat(zero_zero_nat,number_number_of_nat(bit0(bit1(pls)))) = zero_zero_nat ).
tff(fact_438_zero__power2,axiom,
power_power_int(zero_zero_int,number_number_of_nat(bit0(bit1(pls)))) = zero_zero_int ).
tff(fact_439_zero__eq__power2,axiom,
! [A_1: real] :
( ( power_power_real(A_1,number_number_of_nat(bit0(bit1(pls)))) = zero_zero_real )
<=> ( A_1 = zero_zero_real ) ) ).
tff(fact_440_zero__eq__power2,axiom,
! [A_1: int] :
( ( power_power_int(A_1,number_number_of_nat(bit0(bit1(pls)))) = zero_zero_int )
<=> ( A_1 = zero_zero_int ) ) ).
tff(fact_441_zero__le__power2,axiom,
! [A_15: real] : ord_less_eq_real(zero_zero_real,power_power_real(A_15,number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_442_zero__le__power2,axiom,
! [A_15: int] : ord_less_eq_int(zero_zero_int,power_power_int(A_15,number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_443_power2__le__imp__le,axiom,
! [X_10: real,Y_8: real] :
( ord_less_eq_real(power_power_real(X_10,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_8,number_number_of_nat(bit0(bit1(pls)))))
=> ( ord_less_eq_real(zero_zero_real,Y_8)
=> ord_less_eq_real(X_10,Y_8) ) ) ).
tff(fact_444_power2__le__imp__le,axiom,
! [X_10: nat,Y_8: nat] :
( ord_less_eq_nat(power_power_nat(X_10,number_number_of_nat(bit0(bit1(pls)))),power_power_nat(Y_8,number_number_of_nat(bit0(bit1(pls)))))
=> ( ord_less_eq_nat(zero_zero_nat,Y_8)
=> ord_less_eq_nat(X_10,Y_8) ) ) ).
tff(fact_445_power2__le__imp__le,axiom,
! [X_10: int,Y_8: int] :
( ord_less_eq_int(power_power_int(X_10,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_8,number_number_of_nat(bit0(bit1(pls)))))
=> ( ord_less_eq_int(zero_zero_int,Y_8)
=> ord_less_eq_int(X_10,Y_8) ) ) ).
tff(fact_446_power2__eq__imp__eq,axiom,
! [X_9: real,Y_7: real] :
( ( power_power_real(X_9,number_number_of_nat(bit0(bit1(pls)))) = power_power_real(Y_7,number_number_of_nat(bit0(bit1(pls)))) )
=> ( ord_less_eq_real(zero_zero_real,X_9)
=> ( ord_less_eq_real(zero_zero_real,Y_7)
=> ( X_9 = Y_7 ) ) ) ) ).
tff(fact_447_power2__eq__imp__eq,axiom,
! [X_9: nat,Y_7: nat] :
( ( power_power_nat(X_9,number_number_of_nat(bit0(bit1(pls)))) = power_power_nat(Y_7,number_number_of_nat(bit0(bit1(pls)))) )
=> ( ord_less_eq_nat(zero_zero_nat,X_9)
=> ( ord_less_eq_nat(zero_zero_nat,Y_7)
=> ( X_9 = Y_7 ) ) ) ) ).
tff(fact_448_power2__eq__imp__eq,axiom,
! [X_9: int,Y_7: int] :
( ( power_power_int(X_9,number_number_of_nat(bit0(bit1(pls)))) = power_power_int(Y_7,number_number_of_nat(bit0(bit1(pls)))) )
=> ( ord_less_eq_int(zero_zero_int,X_9)
=> ( ord_less_eq_int(zero_zero_int,Y_7)
=> ( X_9 = Y_7 ) ) ) ) ).
tff(fact_449_power2__less__0,axiom,
! [A_14: real] : ~ ord_less_real(power_power_real(A_14,number_number_of_nat(bit0(bit1(pls)))),zero_zero_real) ).
tff(fact_450_power2__less__0,axiom,
! [A_14: int] : ~ ord_less_int(power_power_int(A_14,number_number_of_nat(bit0(bit1(pls)))),zero_zero_int) ).
tff(fact_451_zero__less__power2,axiom,
! [A_1: real] :
( ord_less_real(zero_zero_real,power_power_real(A_1,number_number_of_nat(bit0(bit1(pls)))))
<=> ( A_1 != zero_zero_real ) ) ).
tff(fact_452_zero__less__power2,axiom,
! [A_1: int] :
( ord_less_int(zero_zero_int,power_power_int(A_1,number_number_of_nat(bit0(bit1(pls)))))
<=> ( A_1 != zero_zero_int ) ) ).
tff(fact_453_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_454_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_455_power__commutes,axiom,
! [A_13: nat,N_16: nat] : times_times_nat(power_power_nat(A_13,N_16),A_13) = times_times_nat(A_13,power_power_nat(A_13,N_16)) ).
tff(fact_456_power__commutes,axiom,
! [A_13: real,N_16: nat] : times_times_real(power_power_real(A_13,N_16),A_13) = times_times_real(A_13,power_power_real(A_13,N_16)) ).
tff(fact_457_power__commutes,axiom,
! [A_13: int,N_16: nat] : times_times_int(power_power_int(A_13,N_16),A_13) = times_times_int(A_13,power_power_int(A_13,N_16)) ).
tff(fact_458_power__mult__distrib,axiom,
! [A_12: nat,B_3: nat,N_15: nat] : power_power_nat(times_times_nat(A_12,B_3),N_15) = times_times_nat(power_power_nat(A_12,N_15),power_power_nat(B_3,N_15)) ).
tff(fact_459_power__mult__distrib,axiom,
! [A_12: real,B_3: real,N_15: nat] : power_power_real(times_times_real(A_12,B_3),N_15) = times_times_real(power_power_real(A_12,N_15),power_power_real(B_3,N_15)) ).
tff(fact_460_power__mult__distrib,axiom,
! [A_12: int,B_3: int,N_15: nat] : power_power_int(times_times_int(A_12,B_3),N_15) = times_times_int(power_power_int(A_12,N_15),power_power_int(B_3,N_15)) ).
tff(fact_461_power__add,axiom,
! [A_11: nat,M_4: nat,N_14: nat] : power_power_nat(A_11,plus_plus_nat(M_4,N_14)) = times_times_nat(power_power_nat(A_11,M_4),power_power_nat(A_11,N_14)) ).
tff(fact_462_power__add,axiom,
! [A_11: real,M_4: nat,N_14: nat] : power_power_real(A_11,plus_plus_nat(M_4,N_14)) = times_times_real(power_power_real(A_11,M_4),power_power_real(A_11,N_14)) ).
tff(fact_463_power__add,axiom,
! [A_11: int,M_4: nat,N_14: nat] : power_power_int(A_11,plus_plus_nat(M_4,N_14)) = times_times_int(power_power_int(A_11,M_4),power_power_int(A_11,N_14)) ).
tff(fact_464_power__one,axiom,
! [N_13: nat] : power_power_real(one_one_real,N_13) = one_one_real ).
tff(fact_465_power__one,axiom,
! [N_13: nat] : power_power_nat(one_one_nat,N_13) = one_one_nat ).
tff(fact_466_power__one,axiom,
! [N_13: nat] : power_power_int(one_one_int,N_13) = one_one_int ).
tff(fact_467_power__mult,axiom,
! [A_10: nat,M_3: nat,N_12: nat] : power_power_nat(A_10,times_times_nat(M_3,N_12)) = power_power_nat(power_power_nat(A_10,M_3),N_12) ).
tff(fact_468_power__mult,axiom,
! [A_10: real,M_3: nat,N_12: nat] : power_power_real(A_10,times_times_nat(M_3,N_12)) = power_power_real(power_power_real(A_10,M_3),N_12) ).
tff(fact_469_power__mult,axiom,
! [A_10: int,M_3: nat,N_12: nat] : power_power_int(A_10,times_times_nat(M_3,N_12)) = power_power_int(power_power_int(A_10,M_3),N_12) ).
tff(fact_470_power2__less__imp__less,axiom,
! [X_8: real,Y_6: real] :
( ord_less_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)))))
=> ( ord_less_eq_real(zero_zero_real,Y_6)
=> ord_less_real(X_8,Y_6) ) ) ).
tff(fact_471_power2__less__imp__less,axiom,
! [X_8: nat,Y_6: nat] :
( ord_less_nat(power_power_nat(X_8,number_number_of_nat(bit0(bit1(pls)))),power_power_nat(Y_6,number_number_of_nat(bit0(bit1(pls)))))
=> ( ord_less_eq_nat(zero_zero_nat,Y_6)
=> ord_less_nat(X_8,Y_6) ) ) ).
tff(fact_472_power2__less__imp__less,axiom,
! [X_8: int,Y_6: int] :
( ord_less_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)))))
=> ( ord_less_eq_int(zero_zero_int,Y_6)
=> ord_less_int(X_8,Y_6) ) ) ).
tff(fact_473_sum__power2__ge__zero,axiom,
! [X_7: real,Y_5: real] : ord_less_eq_real(zero_zero_real,plus_plus_real(power_power_real(X_7,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_5,number_number_of_nat(bit0(bit1(pls)))))) ).
tff(fact_474_sum__power2__ge__zero,axiom,
! [X_7: int,Y_5: int] : ord_less_eq_int(zero_zero_int,plus_plus_int(power_power_int(X_7,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_5,number_number_of_nat(bit0(bit1(pls)))))) ).
tff(fact_475_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_476_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_477_not__sum__power2__lt__zero,axiom,
! [X_6: real,Y_4: real] : ~ ord_less_real(plus_plus_real(power_power_real(X_6,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_4,number_number_of_nat(bit0(bit1(pls))))),zero_zero_real) ).
tff(fact_478_not__sum__power2__lt__zero,axiom,
! [X_6: int,Y_4: int] : ~ ord_less_int(plus_plus_int(power_power_int(X_6,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_4,number_number_of_nat(bit0(bit1(pls))))),zero_zero_int) ).
tff(fact_479_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_480_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_481_zero__le__even__power_H,axiom,
! [A_9: real,N_11: nat] : ord_less_eq_real(zero_zero_real,power_power_real(A_9,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_11))) ).
tff(fact_482_zero__le__even__power_H,axiom,
! [A_9: int,N_11: nat] : ord_less_eq_int(zero_zero_int,power_power_int(A_9,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_11))) ).
tff(fact_483_one__le__power,axiom,
! [N_10: nat,A_8: real] :
( ord_less_eq_real(one_one_real,A_8)
=> ord_less_eq_real(one_one_real,power_power_real(A_8,N_10)) ) ).
tff(fact_484_one__le__power,axiom,
! [N_10: nat,A_8: nat] :
( ord_less_eq_nat(one_one_nat,A_8)
=> ord_less_eq_nat(one_one_nat,power_power_nat(A_8,N_10)) ) ).
tff(fact_485_one__le__power,axiom,
! [N_10: nat,A_8: int] :
( ord_less_eq_int(one_one_int,A_8)
=> ord_less_eq_int(one_one_int,power_power_int(A_8,N_10)) ) ).
tff(fact_486_power__increasing,axiom,
! [A_7: real,N_9: nat,N_8: nat] :
( ord_less_eq_nat(N_9,N_8)
=> ( ord_less_eq_real(one_one_real,A_7)
=> ord_less_eq_real(power_power_real(A_7,N_9),power_power_real(A_7,N_8)) ) ) ).
tff(fact_487_power__increasing,axiom,
! [A_7: nat,N_9: nat,N_8: nat] :
( ord_less_eq_nat(N_9,N_8)
=> ( ord_less_eq_nat(one_one_nat,A_7)
=> ord_less_eq_nat(power_power_nat(A_7,N_9),power_power_nat(A_7,N_8)) ) ) ).
tff(fact_488_power__increasing,axiom,
! [A_7: int,N_9: nat,N_8: nat] :
( ord_less_eq_nat(N_9,N_8)
=> ( ord_less_eq_int(one_one_int,A_7)
=> ord_less_eq_int(power_power_int(A_7,N_9),power_power_int(A_7,N_8)) ) ) ).
tff(fact_489_power__inject__exp,axiom,
! [Ma: nat,N_1: nat,A_1: real] :
( ord_less_real(one_one_real,A_1)
=> ( ( power_power_real(A_1,Ma) = power_power_real(A_1,N_1) )
<=> ( Ma = N_1 ) ) ) ).
tff(fact_490_power__inject__exp,axiom,
! [Ma: nat,N_1: nat,A_1: nat] :
( ord_less_nat(one_one_nat,A_1)
=> ( ( power_power_nat(A_1,Ma) = power_power_nat(A_1,N_1) )
<=> ( Ma = N_1 ) ) ) ).
tff(fact_491_power__inject__exp,axiom,
! [Ma: nat,N_1: nat,A_1: int] :
( ord_less_int(one_one_int,A_1)
=> ( ( power_power_int(A_1,Ma) = power_power_int(A_1,N_1) )
<=> ( Ma = N_1 ) ) ) ).
tff(fact_492_power__strict__increasing__iff,axiom,
! [X_2: nat,Y_2: nat,B_2: real] :
( ord_less_real(one_one_real,B_2)
=> ( ord_less_real(power_power_real(B_2,X_2),power_power_real(B_2,Y_2))
<=> ord_less_nat(X_2,Y_2) ) ) ).
tff(fact_493_power__strict__increasing__iff,axiom,
! [X_2: nat,Y_2: nat,B_2: nat] :
( ord_less_nat(one_one_nat,B_2)
=> ( ord_less_nat(power_power_nat(B_2,X_2),power_power_nat(B_2,Y_2))
<=> ord_less_nat(X_2,Y_2) ) ) ).
tff(fact_494_power__strict__increasing__iff,axiom,
! [X_2: nat,Y_2: nat,B_2: int] :
( ord_less_int(one_one_int,B_2)
=> ( ord_less_int(power_power_int(B_2,X_2),power_power_int(B_2,Y_2))
<=> ord_less_nat(X_2,Y_2) ) ) ).
tff(fact_495_power__less__imp__less__exp,axiom,
! [M_2: nat,N_7: nat,A_6: real] :
( ord_less_real(one_one_real,A_6)
=> ( ord_less_real(power_power_real(A_6,M_2),power_power_real(A_6,N_7))
=> ord_less_nat(M_2,N_7) ) ) ).
tff(fact_496_power__less__imp__less__exp,axiom,
! [M_2: nat,N_7: nat,A_6: nat] :
( ord_less_nat(one_one_nat,A_6)
=> ( ord_less_nat(power_power_nat(A_6,M_2),power_power_nat(A_6,N_7))
=> ord_less_nat(M_2,N_7) ) ) ).
tff(fact_497_power__less__imp__less__exp,axiom,
! [M_2: nat,N_7: nat,A_6: int] :
( ord_less_int(one_one_int,A_6)
=> ( ord_less_int(power_power_int(A_6,M_2),power_power_int(A_6,N_7))
=> ord_less_nat(M_2,N_7) ) ) ).
tff(fact_498_power__strict__increasing,axiom,
! [A_5: real,N_6: nat,N_5: nat] :
( ord_less_nat(N_6,N_5)
=> ( ord_less_real(one_one_real,A_5)
=> ord_less_real(power_power_real(A_5,N_6),power_power_real(A_5,N_5)) ) ) ).
tff(fact_499_power__strict__increasing,axiom,
! [A_5: nat,N_6: nat,N_5: nat] :
( ord_less_nat(N_6,N_5)
=> ( ord_less_nat(one_one_nat,A_5)
=> ord_less_nat(power_power_nat(A_5,N_6),power_power_nat(A_5,N_5)) ) ) ).
tff(fact_500_power__strict__increasing,axiom,
! [A_5: int,N_6: nat,N_5: nat] :
( ord_less_nat(N_6,N_5)
=> ( ord_less_int(one_one_int,A_5)
=> ord_less_int(power_power_int(A_5,N_6),power_power_int(A_5,N_5)) ) ) ).
tff(fact_501_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_502_Euler_Oaux____1,axiom,
! [Y_1: int,X_1: int,P: int] :
( ~ zcong(X_1,zero_zero_int,P)
=> ( zcong(power_power_int(Y_1,number_number_of_nat(bit0(bit1(pls)))),X_1,P)
=> ~ dvd_dvd_int(P,Y_1) ) ) ).
tff(fact_503_int__pos__lt__two__imp__zero__or__one,axiom,
! [X_1: int] :
( ord_less_eq_int(zero_zero_int,X_1)
=> ( ord_less_int(X_1,number_number_of_int(bit0(bit1(pls))))
=> ( ( X_1 = zero_zero_int )
| ( X_1 = one_one_int ) ) ) ) ).
tff(fact_504_even__power__le__0__imp__0,axiom,
! [A_4: real,K_3: nat] :
( ord_less_eq_real(power_power_real(A_4,times_times_nat(number_number_of_nat(bit0(bit1(pls))),K_3)),zero_zero_real)
=> ( A_4 = zero_zero_real ) ) ).
tff(fact_505_even__power__le__0__imp__0,axiom,
! [A_4: int,K_3: nat] :
( ord_less_eq_int(power_power_int(A_4,times_times_nat(number_number_of_nat(bit0(bit1(pls))),K_3)),zero_zero_int)
=> ( A_4 = zero_zero_int ) ) ).
tff(fact_506_zprime__def,axiom,
! [P_1: int] :
( zprime(P_1)
<=> ( ord_less_int(one_one_int,P_1)
& ! [M_1: int] :
( ( ord_less_eq_int(zero_zero_int,M_1)
& dvd_dvd_int(M_1,P_1) )
=> ( ( M_1 = one_one_int )
| ( M_1 = P_1 ) ) ) ) ) ).
tff(fact_507__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_508__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_509_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_510_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_511_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_512_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_513_rel__simps_I47_J,axiom,
! [K_1: int] :
( ( bit1(K_1) = min )
<=> ( K_1 = min ) ) ).
tff(fact_514_rel__simps_I43_J,axiom,
! [L_1: int] :
( ( min = bit1(L_1) )
<=> ( min = L_1 ) ) ).
tff(fact_515_Bit1__Min,axiom,
bit1(min) = min ).
tff(fact_516_rel__simps_I37_J,axiom,
pls != min ).
tff(fact_517_rel__simps_I40_J,axiom,
min != pls ).
tff(fact_518_rel__simps_I45_J,axiom,
! [K: int] : bit0(K) != min ).
tff(fact_519_rel__simps_I42_J,axiom,
! [L: int] : min != bit0(L) ).
tff(fact_520_rel__simps_I7_J,axiom,
~ ord_less_int(min,min) ).
tff(fact_521_rel__simps_I24_J,axiom,
ord_less_eq_int(min,min) ).
tff(fact_522_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_523_rel__simps_I13_J,axiom,
! [K_1: int] :
( ord_less_int(bit1(K_1),min)
<=> ord_less_int(K_1,min) ) ).
tff(fact_524_rel__simps_I9_J,axiom,
! [K_1: int] :
( ord_less_int(min,bit1(K_1))
<=> ord_less_int(min,K_1) ) ).
tff(fact_525_rel__simps_I3_J,axiom,
~ ord_less_int(pls,min) ).
tff(fact_526_rel__simps_I6_J,axiom,
ord_less_int(min,pls) ).
tff(fact_527_rel__simps_I8_J,axiom,
! [K_1: int] :
( ord_less_int(min,bit0(K_1))
<=> ord_less_int(min,K_1) ) ).
tff(fact_528_bin__less__0__simps_I2_J,axiom,
ord_less_int(min,zero_zero_int) ).
tff(fact_529_rel__simps_I30_J,axiom,
! [K_1: int] :
( ord_less_eq_int(bit1(K_1),min)
<=> ord_less_eq_int(K_1,min) ) ).
tff(fact_530_rel__simps_I26_J,axiom,
! [K_1: int] :
( ord_less_eq_int(min,bit1(K_1))
<=> ord_less_eq_int(min,K_1) ) ).
tff(fact_531_rel__simps_I20_J,axiom,
~ ord_less_eq_int(pls,min) ).
tff(fact_532_rel__simps_I23_J,axiom,
ord_less_eq_int(min,pls) ).
tff(fact_533_rel__simps_I28_J,axiom,
! [K_1: int] :
( ord_less_eq_int(bit0(K_1),min)
<=> ord_less_eq_int(K_1,min) ) ).
tff(fact_534_eq__number__of__Pls__Min,axiom,
number_number_of_int(pls) != number_number_of_int(min) ).
tff(fact_535_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_536_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_5: real] : power_power_real(X_5,zero_zero_nat) = one_one_real ).
tff(fact_537_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_5: nat] : power_power_nat(X_5,zero_zero_nat) = one_one_nat ).
tff(fact_538_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_5: int] : power_power_int(X_5,zero_zero_nat) = one_one_int ).
tff(fact_539_power__0,axiom,
! [A_3: real] : power_power_real(A_3,zero_zero_nat) = one_one_real ).
tff(fact_540_power__0,axiom,
! [A_3: nat] : power_power_nat(A_3,zero_zero_nat) = one_one_nat ).
tff(fact_541_power__0,axiom,
! [A_3: int] : power_power_int(A_3,zero_zero_nat) = one_one_int ).
tff(fact_542_nat__number__of__Pls,axiom,
number_number_of_nat(pls) = zero_zero_nat ).
tff(fact_543_semiring__norm_I113_J,axiom,
zero_zero_nat = number_number_of_nat(pls) ).
tff(fact_544_rel__simps_I25_J,axiom,
! [K_1: int] :
( ord_less_eq_int(min,bit0(K_1))
<=> ord_less_int(min,K_1) ) ).
tff(fact_545_rel__simps_I11_J,axiom,
! [K_1: int] :
( ord_less_int(bit0(K_1),min)
<=> ord_less_eq_int(K_1,min) ) ).
tff(fact_546_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_547_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_548_one__less__power,axiom,
! [N_4: nat,A_2: real] :
( ord_less_real(one_one_real,A_2)
=> ( ord_less_nat(zero_zero_nat,N_4)
=> ord_less_real(one_one_real,power_power_real(A_2,N_4)) ) ) ).
tff(fact_549_one__less__power,axiom,
! [N_4: nat,A_2: nat] :
( ord_less_nat(one_one_nat,A_2)
=> ( ord_less_nat(zero_zero_nat,N_4)
=> ord_less_nat(one_one_nat,power_power_nat(A_2,N_4)) ) ) ).
tff(fact_550_one__less__power,axiom,
! [N_4: nat,A_2: int] :
( ord_less_int(one_one_int,A_2)
=> ( ord_less_nat(zero_zero_nat,N_4)
=> ord_less_int(one_one_int,power_power_int(A_2,N_4)) ) ) ).
tff(fact_551_dvd__power,axiom,
! [X_4: nat,N_3: nat] :
( ( ord_less_nat(zero_zero_nat,N_3)
| ( X_4 = one_one_nat ) )
=> dvd_dvd_nat(X_4,power_power_nat(X_4,N_3)) ) ).
tff(fact_552_dvd__power,axiom,
! [X_4: int,N_3: nat] :
( ( ord_less_nat(zero_zero_nat,N_3)
| ( X_4 = one_one_int ) )
=> dvd_dvd_int(X_4,power_power_int(X_4,N_3)) ) ).
tff(fact_553_dvd__power,axiom,
! [X_4: real,N_3: nat] :
( ( ord_less_nat(zero_zero_nat,N_3)
| ( X_4 = one_one_real ) )
=> dvd_dvd_real(X_4,power_power_real(X_4,N_3)) ) ).
tff(fact_554_less__0__number__of,axiom,
! [V_3: int] :
( ord_less_nat(zero_zero_nat,number_number_of_nat(V_3))
<=> ord_less_int(pls,V_3) ) ).
tff(fact_555_eq__number__of__0,axiom,
! [V_3: int] :
( ( number_number_of_nat(V_3) = zero_zero_nat )
<=> ord_less_eq_int(V_3,pls) ) ).
tff(fact_556_eq__0__number__of,axiom,
! [V_3: int] :
( ( zero_zero_nat = number_number_of_nat(V_3) )
<=> ord_less_eq_int(V_3,pls) ) ).
tff(fact_557_zcong__sym,axiom,
! [A_1: int,B_2: int,Ma: int] :
( zcong(A_1,B_2,Ma)
<=> zcong(B_2,A_1,Ma) ) ).
tff(fact_558_zcong__refl,axiom,
! [K: int,M: int] : zcong(K,K,M) ).
tff(fact_559_zcong__trans,axiom,
! [C_1: int,A: int,B_1: int,M: int] :
( zcong(A,B_1,M)
=> ( zcong(B_1,C_1,M)
=> zcong(A,C_1,M) ) ) ).
tff(fact_560_pos2,axiom,
ord_less_nat(zero_zero_nat,number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_561_nat__number__of__mult__left,axiom,
! [V_2: int,K: nat,V_1: int] :
( ( ord_less_int(V_1,pls)
=> ( times_times_nat(number_number_of_nat(V_1),times_times_nat(number_number_of_nat(V_2),K)) = zero_zero_nat ) )
& ( ~ ord_less_int(V_1,pls)
=> ( times_times_nat(number_number_of_nat(V_1),times_times_nat(number_number_of_nat(V_2),K)) = times_times_nat(number_number_of_nat(times_times_int(V_1,V_2)),K) ) ) ) ).
tff(fact_562_mult__nat__number__of,axiom,
! [V_2: int,V_1: int] :
( ( ord_less_int(V_1,pls)
=> ( times_times_nat(number_number_of_nat(V_1),number_number_of_nat(V_2)) = zero_zero_nat ) )
& ( ~ ord_less_int(V_1,pls)
=> ( times_times_nat(number_number_of_nat(V_1),number_number_of_nat(V_2)) = number_number_of_nat(times_times_int(V_1,V_2)) ) ) ) ).
tff(fact_563_order__le__neq__implies__less,axiom,
! [X_3: real,Y_3: real] :
( ord_less_eq_real(X_3,Y_3)
=> ( ( X_3 != Y_3 )
=> ord_less_real(X_3,Y_3) ) ) ).
tff(fact_564_order__le__neq__implies__less,axiom,
! [X_3: nat,Y_3: nat] :
( ord_less_eq_nat(X_3,Y_3)
=> ( ( X_3 != Y_3 )
=> ord_less_nat(X_3,Y_3) ) ) ).
tff(fact_565_order__le__neq__implies__less,axiom,
! [X_3: int,Y_3: int] :
( ord_less_eq_int(X_3,Y_3)
=> ( ( X_3 != Y_3 )
=> ord_less_int(X_3,Y_3) ) ) ).
tff(fact_566_Euler_Oaux2,axiom,
! [B_1: int,A: int,C_1: int] :
( ord_less_int(A,C_1)
=> ( ord_less_int(B_1,C_1)
=> ( ord_less_eq_int(A,B_1)
| ord_less_eq_int(B_1,A) ) ) ) ).
tff(fact_567_IntPrimes_Ozcong__zero,axiom,
! [A_1: int,B_2: int] :
( zcong(A_1,B_2,zero_zero_int)
<=> ( A_1 = B_2 ) ) ).
tff(fact_568_zcong__1,axiom,
! [A: int,B_1: int] : zcong(A,B_1,one_one_int) ).
tff(fact_569_zcong__zmult,axiom,
! [C_1: int,D_1: int,A: int,B_1: int,M: int] :
( zcong(A,B_1,M)
=> ( zcong(C_1,D_1,M)
=> zcong(times_times_int(A,C_1),times_times_int(B_1,D_1),M) ) ) ).
tff(fact_570_zcong__scalar2,axiom,
! [K: int,A: int,B_1: int,M: int] :
( zcong(A,B_1,M)
=> zcong(times_times_int(K,A),times_times_int(K,B_1),M) ) ).
tff(fact_571_zcong__scalar,axiom,
! [K: int,A: int,B_1: int,M: int] :
( zcong(A,B_1,M)
=> zcong(times_times_int(A,K),times_times_int(B_1,K),M) ) ).
tff(fact_572_zcong__zmult__self,axiom,
! [A: int,M: int,B_1: int] : zcong(times_times_int(A,M),times_times_int(B_1,M),M) ).
tff(fact_573_zcong__zadd,axiom,
! [C_1: int,D_1: int,A: int,B_1: int,M: int] :
( zcong(A,B_1,M)
=> ( zcong(C_1,D_1,M)
=> zcong(plus_plus_int(A,C_1),plus_plus_int(B_1,D_1),M) ) ) ).
tff(fact_574_power__m1__even,axiom,
! [N_2: nat] : power_power_real(number267125858f_real(min),times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_2)) = one_one_real ).
tff(fact_575_power__m1__even,axiom,
! [N_2: nat] : power_power_int(number_number_of_int(min),times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_2)) = one_one_int ).
tff(fact_576_power__eq__0__iff__number__of,axiom,
! [A_1: real,W_1: int] :
( ( power_power_real(A_1,number_number_of_nat(W_1)) = zero_zero_real )
<=> ( ( A_1 = zero_zero_real )
& ( number_number_of_nat(W_1) != zero_zero_nat ) ) ) ).
tff(fact_577_power__eq__0__iff__number__of,axiom,
! [A_1: nat,W_1: int] :
( ( power_power_nat(A_1,number_number_of_nat(W_1)) = zero_zero_nat )
<=> ( ( A_1 = zero_zero_nat )
& ( number_number_of_nat(W_1) != zero_zero_nat ) ) ) ).
tff(fact_578_power__eq__0__iff__number__of,axiom,
! [A_1: int,W_1: int] :
( ( power_power_int(A_1,number_number_of_nat(W_1)) = zero_zero_int )
<=> ( ( A_1 = zero_zero_int )
& ( number_number_of_nat(W_1) != zero_zero_nat ) ) ) ).
tff(fact_579_zcong__not,axiom,
! [B_1: int,M: int,A: int] :
( ord_less_int(zero_zero_int,A)
=> ( ord_less_int(A,M)
=> ( ord_less_int(zero_zero_int,B_1)
=> ( ord_less_int(B_1,A)
=> ~ zcong(A,B_1,M) ) ) ) ) ).
tff(fact_580_zcong__iff__lin,axiom,
! [A_1: int,B_2: int,Ma: int] :
( zcong(A_1,B_2,Ma)
<=> ? [K_2: int] : B_2 = plus_plus_int(A_1,times_times_int(Ma,K_2)) ) ).
tff(fact_581_power__0__left__number__of,axiom,
! [W_3: int] :
( ( ( number_number_of_nat(W_3) = zero_zero_nat )
=> ( power_power_real(zero_zero_real,number_number_of_nat(W_3)) = one_one_real ) )
& ( ( number_number_of_nat(W_3) != zero_zero_nat )
=> ( power_power_real(zero_zero_real,number_number_of_nat(W_3)) = zero_zero_real ) ) ) ).
tff(fact_582_power__0__left__number__of,axiom,
! [W_3: int] :
( ( ( number_number_of_nat(W_3) = zero_zero_nat )
=> ( power_power_nat(zero_zero_nat,number_number_of_nat(W_3)) = one_one_nat ) )
& ( ( number_number_of_nat(W_3) != zero_zero_nat )
=> ( power_power_nat(zero_zero_nat,number_number_of_nat(W_3)) = zero_zero_nat ) ) ) ).
tff(fact_583_power__0__left__number__of,axiom,
! [W_3: int] :
( ( ( number_number_of_nat(W_3) = zero_zero_nat )
=> ( power_power_int(zero_zero_int,number_number_of_nat(W_3)) = one_one_int ) )
& ( ( number_number_of_nat(W_3) != zero_zero_nat )
=> ( power_power_int(zero_zero_int,number_number_of_nat(W_3)) = zero_zero_int ) ) ) ).
tff(fact_584_zcong__zless__imp__eq,axiom,
! [B_1: 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_1)
=> ( ord_less_int(B_1,M)
=> ( zcong(A,B_1,M)
=> ( A = B_1 ) ) ) ) ) ) ).
tff(fact_585_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_586_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_587__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_588__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_589_neg__one__power__eq__mod__m,axiom,
! [J: nat,K: 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),M)
=> ( power_power_int(number_number_of_int(min),J) = power_power_int(number_number_of_int(min),K) ) ) ) ).
tff(fact_590__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_591__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_592_zcong__zdiff,axiom,
! [C_1: int,D_1: int,A: int,B_1: int,M: int] :
( zcong(A,B_1,M)
=> ( zcong(C_1,D_1,M)
=> zcong(minus_minus_int(A,C_1),minus_minus_int(B_1,D_1),M) ) ) ).
tff(fact_593_diff__bin__simps_I1_J,axiom,
! [K: int] : minus_minus_int(K,pls) = K ).
tff(fact_594_diff__bin__simps_I7_J,axiom,
! [K: int,L: int] : minus_minus_int(bit0(K),bit0(L)) = bit0(minus_minus_int(K,L)) ).
tff(fact_595_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_596_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_597_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( dvd_dvd_int(K,minus_minus_int(M,N))
=> ( dvd_dvd_int(K,N)
=> dvd_dvd_int(K,M) ) ) ).
tff(fact_598_number__of__diff,axiom,
! [V: int,W_2: int] : number_number_of_int(minus_minus_int(V,W_2)) = minus_minus_int(number_number_of_int(V),number_number_of_int(W_2)) ).
tff(fact_599_diff__bin__simps_I9_J,axiom,
! [K: int,L: int] : minus_minus_int(bit1(K),bit0(L)) = bit1(minus_minus_int(K,L)) ).
tff(fact_600_diff__bin__simps_I10_J,axiom,
! [K: int,L: int] : minus_minus_int(bit1(K),bit1(L)) = bit0(minus_minus_int(K,L)) ).
tff(fact_601_diff__bin__simps_I3_J,axiom,
! [L: int] : minus_minus_int(pls,bit0(L)) = bit0(minus_minus_int(pls,L)) ).
tff(fact_602_less__bin__lemma,axiom,
! [K_1: int,L_1: int] :
( ord_less_int(K_1,L_1)
<=> ord_less_int(minus_minus_int(K_1,L_1),zero_zero_int) ) ).
tff(fact_603_xzgcda__linear__aux1,axiom,
! [A: int,R_1: int,B_1: int,M: int,C_1: int,D_1: int,N: int] : plus_plus_int(times_times_int(minus_minus_int(A,times_times_int(R_1,B_1)),M),times_times_int(minus_minus_int(C_1,times_times_int(R_1,D_1)),N)) = minus_minus_int(plus_plus_int(times_times_int(A,M),times_times_int(C_1,N)),times_times_int(R_1,plus_plus_int(times_times_int(B_1,M),times_times_int(D_1,N)))) ).
tff(fact_604_zcong__def,axiom,
! [A_1: int,B_2: int,Ma: int] :
( zcong(A_1,B_2,Ma)
<=> dvd_dvd_int(Ma,minus_minus_int(A_1,B_2)) ) ).
tff(fact_605_Euler_Oaux1,axiom,
! [A: int,X_1: int] :
( ord_less_int(zero_zero_int,X_1)
=> ( ord_less_int(X_1,A)
=> ( ( X_1 != minus_minus_int(A,one_one_int) )
=> ord_less_int(X_1,minus_minus_int(A,one_one_int)) ) ) ) ).
tff(fact_606_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_607_diff__bin__simps_I4_J,axiom,
! [L: int] : minus_minus_int(pls,bit1(L)) = bit1(minus_minus_int(min,L)) ).
tff(fact_608_diff__bin__simps_I6_J,axiom,
! [L: int] : minus_minus_int(min,bit1(L)) = bit0(minus_minus_int(min,L)) ).
tff(fact_609_diff__bin__simps_I5_J,axiom,
! [L: int] : minus_minus_int(min,bit0(L)) = bit1(minus_minus_int(min,L)) ).
tff(fact_610_inv__not__p__minus__1__aux,axiom,
! [A_1: int,P_1: int] :
( zcong(times_times_int(A_1,minus_minus_int(P_1,one_one_int)),one_one_int,P_1)
<=> zcong(A_1,minus_minus_int(P_1,one_one_int),P_1) ) ).
tff(fact_611_mult__sum2sq,axiom,
! [A: int,B_1: int,P: int,Q: int] : times_times_int(twoSqu2107342101sum2sq(product_Pair_int_int(A,B_1)),twoSqu2107342101sum2sq(product_Pair_int_int(P,Q))) = twoSqu2107342101sum2sq(product_Pair_int_int(plus_plus_int(times_times_int(A,P),times_times_int(B_1,Q)),minus_minus_int(times_times_int(A,Q),times_times_int(B_1,P)))) ).
tff(fact_612_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_613_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_614_zspecial__product,axiom,
! [A: int,B_1: int] : times_times_int(plus_plus_int(A,B_1),minus_minus_int(A,B_1)) = minus_minus_int(power_power_int(A,number_number_of_nat(bit0(bit1(pls)))),power_power_int(B_1,number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_615_zdiff__power2,axiom,
! [A: int,B_1: int] : power_power_int(minus_minus_int(A,B_1),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_1)),power_power_int(B_1,number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_616_zdiff__power3,axiom,
! [A: int,B_1: int] : power_power_int(minus_minus_int(A,B_1),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_1)),times_times_int(times_times_int(number_number_of_int(bit1(bit1(pls))),A),power_power_int(B_1,number_number_of_nat(bit0(bit1(pls)))))),power_power_int(B_1,number_number_of_nat(bit1(bit1(pls))))) ).
tff(fact_617_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_618_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_619_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_620_zcong__neg__1__impl__ne__1,axiom,
! [X_1: int,P: int] :
( ord_less_int(number_number_of_int(bit0(bit1(pls))),P)
=> ( zcong(X_1,number_number_of_int(min),P)
=> ~ zcong(X_1,one_one_int,P) ) ) ).
tff(fact_621_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_622_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) ) ) ).
tff(fact_623_divides__antisym,axiom,
! [X_2: nat,Y_2: nat] :
( ( dvd_dvd_nat(X_2,Y_2)
& dvd_dvd_nat(Y_2,X_2) )
<=> ( X_2 = Y_2 ) ) ).
tff(fact_624_zcong__eq__trans,axiom,
! [D_1: int,C_1: int,A: int,B_1: int,M: int] :
( zcong(A,B_1,M)
=> ( ( B_1 = C_1 )
=> ( zcong(C_1,D_1,M)
=> zcong(A,D_1,M) ) ) ) ).
tff(fact_625_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_626_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_627_diff__square,axiom,
! [X_1: nat,Y_1: nat] : minus_minus_nat(power_power_nat(X_1,number_number_of_nat(bit0(bit1(pls)))),power_power_nat(Y_1,number_number_of_nat(bit0(bit1(pls))))) = times_times_nat(plus_plus_nat(X_1,Y_1),minus_minus_nat(X_1,Y_1)) ).
tff(fact_628_divides__add__revr,axiom,
! [B_1: nat,D_1: nat,A: nat] :
( dvd_dvd_nat(D_1,A)
=> ( dvd_dvd_nat(D_1,plus_plus_nat(A,B_1))
=> dvd_dvd_nat(D_1,B_1) ) ) ).
tff(fact_629_divides__mul__l,axiom,
! [C_1: nat,A: nat,B_1: nat] :
( dvd_dvd_nat(A,B_1)
=> dvd_dvd_nat(times_times_nat(C_1,A),times_times_nat(C_1,B_1)) ) ).
tff(fact_630_divides__mul__r,axiom,
! [C_1: nat,A: nat,B_1: nat] :
( dvd_dvd_nat(A,B_1)
=> dvd_dvd_nat(times_times_nat(A,C_1),times_times_nat(B_1,C_1)) ) ).
tff(fact_631_zcong__id,axiom,
! [M: int] : zcong(M,zero_zero_int,M) ).
tff(fact_632_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_633_Int2_Oaux1,axiom,
! [A: int,B_1: int,C_1: int] :
( ( minus_minus_int(A,B_1) = C_1 )
=> ( A = plus_plus_int(C_1,B_1) ) ) ).
tff(fact_634_zcong__zmult__prop2,axiom,
! [C: int,D: int,A_1: int,B_2: int,Ma: int] :
( zcong(A_1,B_2,Ma)
=> ( zcong(C,times_times_int(D,A_1),Ma)
<=> zcong(C,times_times_int(D,B_2),Ma) ) ) ).
tff(fact_635_zcong__zmult__prop1,axiom,
! [C: int,D: int,A_1: int,B_2: int,Ma: int] :
( zcong(A_1,B_2,Ma)
=> ( zcong(C,times_times_int(A_1,D),Ma)
<=> zcong(C,times_times_int(B_2,D),Ma) ) ) ).
tff(fact_636_zcong__shift,axiom,
! [C_1: int,A: int,B_1: int,M: int] :
( zcong(A,B_1,M)
=> zcong(plus_plus_int(A,C_1),plus_plus_int(B_1,C_1),M) ) ).
tff(fact_637_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_638_divides__exp,axiom,
! [N: nat,X_1: nat,Y_1: nat] :
( dvd_dvd_nat(X_1,Y_1)
=> dvd_dvd_nat(power_power_nat(X_1,N),power_power_nat(Y_1,N)) ) ).
tff(fact_639_zcong__zpower,axiom,
! [Z: nat,X_1: int,Y_1: int,M: int] :
( zcong(X_1,Y_1,M)
=> zcong(power_power_int(X_1,Z),power_power_int(Y_1,Z),M) ) ).
tff(fact_640_divides__ge,axiom,
! [A: nat,B_1: nat] :
( dvd_dvd_nat(A,B_1)
=> ( ( B_1 = zero_zero_nat )
| ord_less_eq_nat(A,B_1) ) ) ).
tff(fact_641_nat__mult__dvd__cancel__disj_H,axiom,
! [Ma: nat,K_1: nat,N_1: nat] :
( dvd_dvd_nat(times_times_nat(Ma,K_1),times_times_nat(N_1,K_1))
<=> ( ( K_1 = zero_zero_nat )
| dvd_dvd_nat(Ma,N_1) ) ) ).
tff(fact_642_zcong__not__zero,axiom,
! [M: int,X_1: int] :
( ord_less_int(zero_zero_int,X_1)
=> ( ord_less_int(X_1,M)
=> ~ zcong(X_1,zero_zero_int,M) ) ) ).
tff(fact_643_zcong__less__eq,axiom,
! [M: int,Y_1: int,X_1: int] :
( ord_less_int(zero_zero_int,X_1)
=> ( ord_less_int(zero_zero_int,Y_1)
=> ( ord_less_int(zero_zero_int,M)
=> ( zcong(X_1,Y_1,M)
=> ( ord_less_int(X_1,M)
=> ( ord_less_int(Y_1,M)
=> ( X_1 = Y_1 ) ) ) ) ) ) ) ).
tff(fact_644_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_645_divides__exp2,axiom,
! [X_1: nat,Y_1: nat,N: nat] :
( ( N != zero_zero_nat )
=> ( dvd_dvd_nat(power_power_nat(X_1,N),Y_1)
=> dvd_dvd_nat(X_1,Y_1) ) ) ).
tff(fact_646_divides__rev,axiom,
! [A: nat,N: nat,B_1: nat] :
( dvd_dvd_nat(power_power_nat(A,N),power_power_nat(B_1,N))
=> ( ( N != zero_zero_nat )
=> dvd_dvd_nat(A,B_1) ) ) ).
tff(fact_647_zcong__zero__equiv__div,axiom,
! [A_1: int,Ma: int] :
( zcong(A_1,zero_zero_int,Ma)
<=> dvd_dvd_int(Ma,A_1) ) ).
tff(fact_648_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_649_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_650_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_651_Int2_Ozcong__zero,axiom,
! [M: int,X_1: int] :
( ord_less_eq_int(zero_zero_int,X_1)
=> ( ord_less_int(X_1,M)
=> ( zcong(X_1,zero_zero_int,M)
=> ( X_1 = zero_zero_int ) ) ) ) ).
tff(fact_652_zpower__zdvd__prop1,axiom,
! [P: int,Y_1: int,N: nat] :
( ord_less_nat(zero_zero_nat,N)
=> ( dvd_dvd_int(P,Y_1)
=> dvd_dvd_int(P,power_power_int(Y_1,N)) ) ) ).
tff(fact_653_zcong__zmult__prop3,axiom,
! [Y_1: int,X_1: int,P: int] :
( zprime(P)
=> ( ~ zcong(X_1,zero_zero_int,P)
=> ( ~ zcong(Y_1,zero_zero_int,P)
=> ~ zcong(times_times_int(X_1,Y_1),zero_zero_int,P) ) ) ) ).
tff(fact_654_divides__div__not,axiom,
! [X_1: nat,Q: nat,N: nat,R_1: nat] :
( ( X_1 = plus_plus_nat(times_times_nat(Q,N),R_1) )
=> ( ord_less_nat(zero_zero_nat,R_1)
=> ( ord_less_nat(R_1,N)
=> ~ dvd_dvd_nat(N,X_1) ) ) ) ).
tff(fact_655_zcong__zprime__prod__zero__contra,axiom,
! [B_1: int,A: int,P: int] :
( zprime(P)
=> ( ord_less_int(zero_zero_int,A)
=> ( ( ~ zcong(A,zero_zero_int,P)
& ~ zcong(B_1,zero_zero_int,P) )
=> ~ zcong(times_times_int(A,B_1),zero_zero_int,P) ) ) ) ).
tff(fact_656_zcong__zprime__prod__zero,axiom,
! [B_1: int,A: int,P: int] :
( zprime(P)
=> ( ord_less_int(zero_zero_int,A)
=> ( zcong(times_times_int(A,B_1),zero_zero_int,P)
=> ( zcong(A,zero_zero_int,P)
| zcong(B_1,zero_zero_int,P) ) ) ) ) ).
tff(fact_657_zpower__zdvd__prop2,axiom,
! [Y_1: int,N: nat,P: int] :
( zprime(P)
=> ( dvd_dvd_int(P,power_power_int(Y_1,N))
=> ( ord_less_nat(zero_zero_nat,N)
=> dvd_dvd_int(P,Y_1) ) ) ) ).
tff(fact_658_QuadRes__def,axiom,
! [Ma: int,X_2: int] :
( quadRes(Ma,X_2)
<=> ? [Y: int] : zcong(power_power_int(Y,number_number_of_nat(bit0(bit1(pls)))),X_2,Ma) ) ).
tff(fact_659_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_660_self__quotient__aux1,axiom,
! [R_1: int,Q: int,A: int] :
( ord_less_int(zero_zero_int,A)
=> ( ( A = plus_plus_int(R_1,times_times_int(A,Q)) )
=> ( ord_less_int(R_1,A)
=> ord_less_eq_int(one_one_int,Q) ) ) ) ).
tff(fact_661_real__zero__not__eq__one,axiom,
zero_zero_real != one_one_real ).
tff(fact_662_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_663_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_664_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_665_real__mult__1,axiom,
! [Z: real] : times_times_real(one_one_real,Z) = Z ).
tff(fact_666_real__mult__commute,axiom,
! [Z: real,W: real] : times_times_real(Z,W) = times_times_real(W,Z) ).
tff(fact_667_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_668_real__add__left__mono,axiom,
! [Z: real,X_1: real,Y_1: real] :
( ord_less_eq_real(X_1,Y_1)
=> ord_less_eq_real(plus_plus_real(Z,X_1),plus_plus_real(Z,Y_1)) ) ).
tff(fact_669_real__mult__left__cancel,axiom,
! [A_1: real,B_2: real,C: real] :
( ( C != zero_zero_real )
=> ( ( times_times_real(C,A_1) = times_times_real(C,B_2) )
<=> ( A_1 = B_2 ) ) ) ).
tff(fact_670_real__mult__right__cancel,axiom,
! [A_1: real,B_2: real,C: real] :
( ( C != zero_zero_real )
=> ( ( times_times_real(A_1,C) = times_times_real(B_2,C) )
<=> ( A_1 = B_2 ) ) ) ).
tff(fact_671_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_672_real__mult__less__mono2,axiom,
! [X_1: real,Y_1: real,Z: real] :
( ord_less_real(zero_zero_real,Z)
=> ( ord_less_real(X_1,Y_1)
=> ord_less_real(times_times_real(Z,X_1),times_times_real(Z,Y_1)) ) ) ).
tff(fact_673_real__mult__order,axiom,
! [Y_1: real,X_1: real] :
( ord_less_real(zero_zero_real,X_1)
=> ( ord_less_real(zero_zero_real,Y_1)
=> ord_less_real(zero_zero_real,times_times_real(X_1,Y_1)) ) ) ).
tff(fact_674_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_675_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_676_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_677_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_678_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_679_q__pos__lemma,axiom,
! [B: int,Q_1: int,R_2: int] :
( ord_less_eq_int(zero_zero_int,plus_plus_int(times_times_int(B,Q_1),R_2))
=> ( ord_less_int(R_2,B)
=> ( ord_less_int(zero_zero_int,B)
=> ord_less_eq_int(zero_zero_int,Q_1) ) ) ) ).
tff(fact_680_q__neg__lemma,axiom,
! [B: int,Q_1: int,R_2: int] :
( ord_less_int(plus_plus_int(times_times_int(B,Q_1),R_2),zero_zero_int)
=> ( ord_less_eq_int(zero_zero_int,R_2)
=> ( ord_less_int(zero_zero_int,B)
=> ord_less_eq_int(Q_1,zero_zero_int) ) ) ) ).
tff(fact_681_unique__quotient__lemma,axiom,
! [B_1: int,Q_1: int,R_2: int,Q: int,R_1: int] :
( ord_less_eq_int(plus_plus_int(times_times_int(B_1,Q_1),R_2),plus_plus_int(times_times_int(B_1,Q),R_1))
=> ( ord_less_eq_int(zero_zero_int,R_2)
=> ( ord_less_int(R_2,B_1)
=> ( ord_less_int(R_1,B_1)
=> ord_less_eq_int(Q_1,Q) ) ) ) ) ).
tff(fact_682_zdiv__mono2__lemma,axiom,
! [B_1: int,Q: int,R_1: int,B: int,Q_1: int,R_2: int] :
( ( plus_plus_int(times_times_int(B_1,Q),R_1) = plus_plus_int(times_times_int(B,Q_1),R_2) )
=> ( ord_less_eq_int(zero_zero_int,plus_plus_int(times_times_int(B,Q_1),R_2))
=> ( ord_less_int(R_2,B)
=> ( ord_less_eq_int(zero_zero_int,R_1)
=> ( ord_less_int(zero_zero_int,B)
=> ( ord_less_eq_int(B,B_1)
=> ord_less_eq_int(Q,Q_1) ) ) ) ) ) ) ).
tff(fact_683_unique__quotient__lemma__neg,axiom,
! [B_1: int,Q_1: int,R_2: int,Q: int,R_1: int] :
( ord_less_eq_int(plus_plus_int(times_times_int(B_1,Q_1),R_2),plus_plus_int(times_times_int(B_1,Q),R_1))
=> ( ord_less_eq_int(R_1,zero_zero_int)
=> ( ord_less_int(B_1,R_1)
=> ( ord_less_int(B_1,R_2)
=> ord_less_eq_int(Q,Q_1) ) ) ) ) ).
tff(fact_684_zdiv__mono2__neg__lemma,axiom,
! [B_1: int,Q: int,R_1: int,B: int,Q_1: int,R_2: int] :
( ( plus_plus_int(times_times_int(B_1,Q),R_1) = plus_plus_int(times_times_int(B,Q_1),R_2) )
=> ( ord_less_int(plus_plus_int(times_times_int(B,Q_1),R_2),zero_zero_int)
=> ( ord_less_int(R_1,B_1)
=> ( ord_less_eq_int(zero_zero_int,R_2)
=> ( ord_less_int(zero_zero_int,B)
=> ( ord_less_eq_int(B,B_1)
=> ord_less_eq_int(Q_1,Q) ) ) ) ) ) ) ).
tff(fact_685_self__quotient__aux2,axiom,
! [R_1: int,Q: int,A: int] :
( ord_less_int(zero_zero_int,A)
=> ( ( A = plus_plus_int(R_1,times_times_int(A,Q)) )
=> ( ord_less_eq_int(zero_zero_int,R_1)
=> ord_less_eq_int(Q,one_one_int) ) ) ) ).
tff(fact_686_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_687_real__le__antisym,axiom,
! [Z: real,W: real] :
( ord_less_eq_real(Z,W)
=> ( ord_less_eq_real(W,Z)
=> ( Z = W ) ) ) ).
tff(fact_688_real__le__trans,axiom,
! [K: real,I: real,J: real] :
( ord_less_eq_real(I,J)
=> ( ord_less_eq_real(J,K)
=> ord_less_eq_real(I,K) ) ) ).
tff(fact_689_real__le__linear,axiom,
! [Z: real,W: real] :
( ord_less_eq_real(Z,W)
| ord_less_eq_real(W,Z) ) ).
tff(fact_690_real__le__refl,axiom,
! [W: real] : ord_less_eq_real(W,W) ).
tff(fact_691_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq_int(zero_zero_int,zero_zero_int) ).
tff(fact_692_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
ord_less_eq_int(zero_zero_int,one_one_int) ).
tff(fact_693_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y_1: int,X_1: int] :
( ord_less_eq_int(zero_zero_int,X_1)
=> ( ord_less_eq_int(zero_zero_int,Y_1)
=> ord_less_eq_int(zero_zero_int,times_times_int(X_1,Y_1)) ) ) ).
tff(fact_694_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y_1: int,X_1: int] :
( ord_less_eq_int(zero_zero_int,X_1)
=> ( ord_less_eq_int(zero_zero_int,Y_1)
=> ord_less_eq_int(zero_zero_int,plus_plus_int(X_1,Y_1)) ) ) ).
tff(fact_695_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [N: nat,X_1: int] :
( ord_less_eq_int(zero_zero_int,X_1)
=> ord_less_eq_int(zero_zero_int,power_power_int(X_1,N)) ) ).
tff(fact_696_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_697_realpow__pos__nth,axiom,
! [A: real,N: nat] :
( ord_less_nat(zero_zero_nat,N)
=> ( ord_less_real(zero_zero_real,A)
=> ? [R: real] :
( ord_less_real(zero_zero_real,R)
& ( power_power_real(R,N) = A ) ) ) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
? [X: int,Y: 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))))) = plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int) ).
%------------------------------------------------------------------------------