TPTP Problem File: NUM926_3.p
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%------------------------------------------------------------------------------
% File : NUM926_3 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 258, 1000 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : s2s_1000_tff_l258 [Bla11]
% Status : Theorem
% Rating : 0.22 v8.2.0, 0.30 v8.1.0, 0.18 v7.5.0, 0.25 v7.4.0, 0.00 v6.0.0, 0.60 v5.5.0, 0.50 v5.4.0, 0.67 v5.3.0
% Syntax : Number of formulae : 1299 ( 456 unt; 94 typ; 0 def)
% Number of atoms : 2733 ( 845 equ)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 1791 ( 263 ~; 74 |; 165 &)
% ( 254 <=>;1035 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 5 avg)
% Maximal term depth : 13 ( 2 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 76 ( 51 >; 25 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 74 ( 74 usr; 24 con; 0-2 aty)
% Number of variables : 2605 (2582 !; 23 ?;2605 :)
% SPC : TF0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:59:57
%------------------------------------------------------------------------------
%----Should-be-implicit typings (19)
tff(ty_ty_tc__HOL__Obool,type,
bool: $tType ).
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__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
fun_bool_bool: $tType ).
tff(ty_ty_tc__fun_Itc__HOL__Obool_Mtc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_J,type,
fun_bo1549164019l_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
fun_int_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J,type,
fun_int_int: $tType ).
tff(ty_ty_tc__fun_Itc__Int__Oint_Mtc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_J,type,
fun_in531499254l_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_J,type,
fun_int_fun_int_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
fun_nat_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__Int__Oint_J,type,
fun_nat_int: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
fun_nat_nat: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J,type,
fun_nat_real: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_J,type,
fun_nat_fun_nat_bool: $tType ).
tff(ty_ty_tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J,type,
fun_real_bool: $tType ).
tff(ty_ty_tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J,type,
fun_real_real: $tType ).
tff(ty_ty_tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J,type,
fun_re413263731l_bool: $tType ).
tff(ty_ty_tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
product_prod_int_int: $tType ).
%----Explicit typings (75)
tff(sy_c_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc_,type,
cOMBB_1652995168ol_int: ( fun_bo1549164019l_bool * fun_int_bool ) > fun_in531499254l_bool ).
tff(sy_c_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__HOL__Obool,type,
cOMBC_int_int_bool: ( fun_int_fun_int_bool * int ) > fun_int_bool ).
tff(sy_c_COMBS_000tc__Int__Oint_000tc__HOL__Obool_000tc__HOL__Obool,type,
cOMBS_int_bool_bool: ( fun_in531499254l_bool * fun_int_bool ) > fun_int_bool ).
tff(sy_c_Divides_Odiv__class_Omod_000tc__Int__Oint,type,
div_mod_int: int > fun_int_int ).
tff(sy_c_Divides_Odiv__class_Omod_000tc__Nat__Onat,type,
div_mod_nat: nat > fun_nat_nat ).
tff(sy_c_Groups_Ominus__class_Ominus_000tc__Int__Oint,type,
minus_minus_int: int > fun_int_int ).
tff(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
minus_minus_nat: nat > fun_nat_nat ).
tff(sy_c_Groups_Ominus__class_Ominus_000tc__RealDef__Oreal,type,
minus_minus_real: real > fun_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 > fun_int_int ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
plus_plus_nat: nat > fun_nat_nat ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__RealDef__Oreal,type,
plus_plus_real: real > fun_real_real ).
tff(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
times_times_int: int > fun_int_int ).
tff(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
times_times_nat: nat > fun_nat_nat ).
tff(sy_c_Groups_Otimes__class_Otimes_000tc__RealDef__Oreal,type,
times_times_real: real > fun_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_Int2_OMultInv,type,
multInv: ( int * int ) > int ).
tff(sy_c_IntFact_Od22set,type,
d22set: int > fun_int_bool ).
tff(sy_c_IntFact_Ozfact,type,
zfact: int > int ).
tff(sy_c_IntPrimes_Ozcong,type,
zcong: ( int * int ) > fun_int_bool ).
tff(sy_c_IntPrimes_Ozprime,type,
zprime: fun_int_bool ).
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: fun_int_fun_int_bool ).
tff(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
ord_less_nat: fun_nat_fun_nat_bool ).
tff(sy_c_Orderings_Oord__class_Oless_000tc__RealDef__Oreal,type,
ord_less_real: fun_re413263731l_bool ).
tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Int__Oint,type,
ord_less_eq_int: fun_int_fun_int_bool ).
tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
ord_less_eq_nat: fun_nat_fun_nat_bool ).
tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__RealDef__Oreal,type,
ord_less_eq_real: fun_re413263731l_bool ).
tff(sy_c_Power_Opower__class_Opower_000tc__Int__Oint,type,
power_power_int: int > fun_nat_int ).
tff(sy_c_Power_Opower__class_Opower_000tc__Nat__Onat,type,
power_power_nat: nat > fun_nat_nat ).
tff(sy_c_Power_Opower__class_Opower_000tc__RealDef__Oreal,type,
power_power_real: real > fun_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 > fun_int_bool ).
tff(sy_c_Residues_OSR,type,
sr: int > fun_int_bool ).
tff(sy_c_Residues_OStandardRes,type,
standardRes: ( int * int ) > int ).
tff(sy_c_Rings_Odvd__class_Odvd_000tc__Int__Oint,type,
dvd_dvd_int: fun_int_fun_int_bool ).
tff(sy_c_Rings_Odvd__class_Odvd_000tc__Nat__Onat,type,
dvd_dvd_nat: fun_nat_fun_nat_bool ).
tff(sy_c_Rings_Odvd__class_Odvd_000tc__RealDef__Oreal,type,
dvd_dvd_real: fun_re413263731l_bool ).
tff(sy_c_Set_OCollect_000tc__Int__Oint,type,
collect_int: fun_int_bool > fun_int_bool ).
tff(sy_c_TwoSquares__Mirabelle__klrirdmhmq_Ois__sum2sq,type,
twoSqu820444569sum2sq: fun_int_bool ).
tff(sy_c_TwoSquares__Mirabelle__klrirdmhmq_Osum2sq,type,
twoSqu949963151sum2sq: product_prod_int_int > int ).
tff(sy_c_WilsonRuss_Oinv,type,
inv: ( int * int ) > int ).
tff(sy_c_WilsonRuss_Owset,type,
wset: ( int * int ) > fun_int_bool ).
tff(sy_c_fconj,type,
fconj: fun_bo1549164019l_bool ).
tff(sy_c_hAPP_000tc__HOL__Obool_000tc__HOL__Obool,type,
hAPP_bool_bool: ( fun_bool_bool * bool ) > bool ).
tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
hAPP_b589554111l_bool: ( fun_bo1549164019l_bool * bool ) > fun_bool_bool ).
tff(sy_c_hAPP_000tc__Int__Oint_000tc__HOL__Obool,type,
hAPP_int_bool: ( fun_int_bool * int ) > bool ).
tff(sy_c_hAPP_000tc__Int__Oint_000tc__Int__Oint,type,
hAPP_int_int: ( fun_int_int * int ) > int ).
tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
hAPP_i68813070l_bool: ( fun_in531499254l_bool * int ) > fun_bool_bool ).
tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
hAPP_i1948725293t_bool: ( fun_int_fun_int_bool * int ) > fun_int_bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool,type,
hAPP_nat_bool: ( fun_nat_bool * nat ) > bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Int__Oint,type,
hAPP_nat_int: ( fun_nat_int * nat ) > int ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Nat__Onat,type,
hAPP_nat_nat: ( fun_nat_nat * nat ) > nat ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__RealDef__Oreal,type,
hAPP_nat_real: ( fun_nat_real * nat ) > real ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
hAPP_n1699378549t_bool: ( fun_nat_fun_nat_bool * nat ) > fun_nat_bool ).
tff(sy_c_hAPP_000tc__RealDef__Oreal_000tc__HOL__Obool,type,
hAPP_real_bool: ( fun_real_bool * real ) > bool ).
tff(sy_c_hAPP_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
hAPP_real_real: ( fun_real_real * real ) > real ).
tff(sy_c_hAPP_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J,type,
hAPP_r1134773055l_bool: ( fun_re413263731l_bool * real ) > fun_real_bool ).
tff(sy_c_hBOOL,type,
hBOOL: bool > $o ).
tff(sy_c_member_000tc__Int__Oint,type,
member_int: ( int * fun_int_bool ) > bool ).
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 (1198)
tff(fact_0_tpos,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(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] : hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y),number_number_of_nat(bit0(bit1(pls))))) = hAPP_int_int(plus_plus_int(hAPP_int_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,
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),t))
=> ? [X: int,Y: int] : hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y),number_number_of_nat(bit0(bit1(pls))))) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int) ) ).
tff(fact_3_t__l__p,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,t),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int))) ).
tff(fact_4_p,axiom,
hBOOL(hAPP_int_bool(zprime,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int))) ).
tff(fact_5_t,axiom,
hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls))))),one_one_int) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int)),t) ).
tff(fact_6_qf1pt,axiom,
hBOOL(hAPP_int_bool(twoSqu820444569sum2sq,hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(hAPP_int_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: int] : hAPP_nat_int(power_power_int(hAPP_int_int(plus_plus_int(A),B)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls))))),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),A)),B))),hAPP_nat_int(power_power_int(B),number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_8_zadd__power3,axiom,
! [A: int,B: int] : hAPP_nat_int(power_power_int(hAPP_int_int(plus_plus_int(A),B)),number_number_of_nat(bit1(bit1(pls)))) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit1(bit1(pls))))),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(number_number_of_int(bit1(bit1(pls)))),hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))))),B))),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(number_number_of_int(bit1(bit1(pls)))),A)),hAPP_nat_int(power_power_int(B),number_number_of_nat(bit0(bit1(pls))))))),hAPP_nat_int(power_power_int(B),number_number_of_nat(bit1(bit1(pls))))) ).
tff(fact_9_power2__sum,axiom,
! [X_2: int,Y_2: int] : hAPP_nat_int(power_power_int(hAPP_int_int(plus_plus_int(X_2),Y_2)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_2),number_number_of_nat(bit0(bit1(pls)))))),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),X_2)),Y_2)) ).
tff(fact_10_power2__sum,axiom,
! [X_2: nat,Y_2: nat] : hAPP_nat_nat(power_power_nat(hAPP_nat_nat(plus_plus_nat(X_2),Y_2)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(power_power_nat(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_nat(power_power_nat(Y_2),number_number_of_nat(bit0(bit1(pls)))))),hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),X_2)),Y_2)) ).
tff(fact_11_power2__sum,axiom,
! [X_2: real,Y_2: real] : hAPP_nat_real(power_power_real(hAPP_real_real(plus_plus_real(X_2),Y_2)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_2),number_number_of_nat(bit0(bit1(pls)))))),hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(number267125858f_real(bit0(bit1(pls)))),X_2)),Y_2)) ).
tff(fact_12_power2__eq__square__number__of,axiom,
! [W_16: int] : hAPP_nat_int(power_power_int(number_number_of_int(W_16)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_int_int(times_times_int(number_number_of_int(W_16)),number_number_of_int(W_16)) ).
tff(fact_13_power2__eq__square__number__of,axiom,
! [W_16: int] : hAPP_nat_real(power_power_real(number267125858f_real(W_16)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_real_real(times_times_real(number267125858f_real(W_16)),number267125858f_real(W_16)) ).
tff(fact_14_power2__eq__square__number__of,axiom,
! [W_16: int] : hAPP_nat_nat(power_power_nat(number_number_of_nat(W_16)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_nat(times_times_nat(number_number_of_nat(W_16)),number_number_of_nat(W_16)) ).
tff(fact_15_cube__square,axiom,
! [A: int] : hAPP_int_int(times_times_int(A),hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls))))) = hAPP_nat_int(power_power_int(A),number_number_of_nat(bit1(bit1(pls)))) ).
tff(fact_16_one__power2,axiom,
hAPP_nat_int(power_power_int(one_one_int),number_number_of_nat(bit0(bit1(pls)))) = one_one_int ).
tff(fact_17_one__power2,axiom,
hAPP_nat_nat(power_power_nat(one_one_nat),number_number_of_nat(bit0(bit1(pls)))) = one_one_nat ).
tff(fact_18_one__power2,axiom,
hAPP_nat_real(power_power_real(one_one_real),number_number_of_nat(bit0(bit1(pls)))) = one_one_real ).
tff(fact_19_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_26: int] : hAPP_int_int(times_times_int(X_26),X_26) = hAPP_nat_int(power_power_int(X_26),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_20_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_26: real] : hAPP_real_real(times_times_real(X_26),X_26) = hAPP_nat_real(power_power_real(X_26),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_21_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_26: nat] : hAPP_nat_nat(times_times_nat(X_26),X_26) = hAPP_nat_nat(power_power_nat(X_26),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_22_power2__eq__square,axiom,
! [A_112: int] : hAPP_nat_int(power_power_int(A_112),number_number_of_nat(bit0(bit1(pls)))) = hAPP_int_int(times_times_int(A_112),A_112) ).
tff(fact_23_power2__eq__square,axiom,
! [A_112: real] : hAPP_nat_real(power_power_real(A_112),number_number_of_nat(bit0(bit1(pls)))) = hAPP_real_real(times_times_real(A_112),A_112) ).
tff(fact_24_power2__eq__square,axiom,
! [A_112: nat] : hAPP_nat_nat(power_power_nat(A_112),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_nat(times_times_nat(A_112),A_112) ).
tff(fact_25_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_25: int,N_40: nat] : hAPP_nat_int(power_power_int(X_25),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_40)) = hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(X_25),N_40)),hAPP_nat_int(power_power_int(X_25),N_40)) ).
tff(fact_26_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_25: real,N_40: nat] : hAPP_nat_real(power_power_real(X_25),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_40)) = hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(X_25),N_40)),hAPP_nat_real(power_power_real(X_25),N_40)) ).
tff(fact_27_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_25: nat,N_40: nat] : hAPP_nat_nat(power_power_nat(X_25),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_40)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(X_25),N_40)),hAPP_nat_nat(power_power_nat(X_25),N_40)) ).
tff(fact_28_add__special_I2_J,axiom,
! [W_15: int] : hAPP_int_int(plus_plus_int(one_one_int),number_number_of_int(W_15)) = number_number_of_int(hAPP_int_int(plus_plus_int(bit1(pls)),W_15)) ).
tff(fact_29_add__special_I2_J,axiom,
! [W_15: int] : hAPP_real_real(plus_plus_real(one_one_real),number267125858f_real(W_15)) = number267125858f_real(hAPP_int_int(plus_plus_int(bit1(pls)),W_15)) ).
tff(fact_30_add__special_I3_J,axiom,
! [V_21: int] : hAPP_int_int(plus_plus_int(number_number_of_int(V_21)),one_one_int) = number_number_of_int(hAPP_int_int(plus_plus_int(V_21),bit1(pls))) ).
tff(fact_31_add__special_I3_J,axiom,
! [V_21: int] : hAPP_real_real(plus_plus_real(number267125858f_real(V_21)),one_one_real) = number267125858f_real(hAPP_int_int(plus_plus_int(V_21),bit1(pls))) ).
tff(fact_32_one__add__one__is__two,axiom,
hAPP_int_int(plus_plus_int(one_one_int),one_one_int) = number_number_of_int(bit0(bit1(pls))) ).
tff(fact_33_one__add__one__is__two,axiom,
hAPP_real_real(plus_plus_real(one_one_real),one_one_real) = number267125858f_real(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_1: int] : hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls))))),one_one_int) != hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int)),T_1) ).
tff(fact_35_zle__refl,axiom,
! [W: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),W)) ).
tff(fact_36_zle__linear,axiom,
! [Z: int,W: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z)) ) ).
tff(fact_37_zless__le,axiom,
! [Z_1: int,W_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),W_1))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W_1))
& ( Z_1 != W_1 ) ) ) ).
tff(fact_38_zless__linear,axiom,
! [X_1: int,Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
| ( X_1 = Y_1 )
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),X_1)) ) ).
tff(fact_39_zle__trans,axiom,
! [K_1: int,I_1: int,J_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),J_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),K_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),K_1)) ) ) ).
tff(fact_40_zle__antisym,axiom,
! [Z: int,W: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z))
=> ( Z = W ) ) ) ).
tff(fact_41_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_24: int,P_7: nat,Q_7: nat] : hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(X_24),P_7)),Q_7) = hAPP_nat_int(power_power_int(X_24),hAPP_nat_nat(times_times_nat(P_7),Q_7)) ).
tff(fact_42_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_24: real,P_7: nat,Q_7: nat] : hAPP_nat_real(power_power_real(hAPP_nat_real(power_power_real(X_24),P_7)),Q_7) = hAPP_nat_real(power_power_real(X_24),hAPP_nat_nat(times_times_nat(P_7),Q_7)) ).
tff(fact_43_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_24: nat,P_7: nat,Q_7: nat] : hAPP_nat_nat(power_power_nat(hAPP_nat_nat(power_power_nat(X_24),P_7)),Q_7) = hAPP_nat_nat(power_power_nat(X_24),hAPP_nat_nat(times_times_nat(P_7),Q_7)) ).
tff(fact_44_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_23: int] : hAPP_nat_int(power_power_int(X_23),one_one_nat) = X_23 ).
tff(fact_45_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_23: real] : hAPP_nat_real(power_power_real(X_23),one_one_nat) = X_23 ).
tff(fact_46_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_23: nat] : hAPP_nat_nat(power_power_nat(X_23),one_one_nat) = X_23 ).
tff(fact_47_zpower__zpower,axiom,
! [X_1: int,Y_1: nat,Z: nat] : hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(X_1),Y_1)),Z) = hAPP_nat_int(power_power_int(X_1),hAPP_nat_nat(times_times_nat(Y_1),Z)) ).
tff(fact_48_le__number__of__eq__not__less,axiom,
! [V_7: int,W_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(V_7)),number_number_of_int(W_1)))
<=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(W_1)),number_number_of_int(V_7))) ) ).
tff(fact_49_le__number__of__eq__not__less,axiom,
! [V_7: int,W_1: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,number_number_of_nat(V_7)),number_number_of_nat(W_1)))
<=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,number_number_of_nat(W_1)),number_number_of_nat(V_7))) ) ).
tff(fact_50_le__number__of__eq__not__less,axiom,
! [V_7: int,W_1: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(V_7)),number267125858f_real(W_1)))
<=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(W_1)),number267125858f_real(V_7))) ) ).
tff(fact_51_less__number__of,axiom,
! [X_2: int,Y_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(X_2)),number_number_of_int(Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Y_2)) ) ).
tff(fact_52_less__number__of,axiom,
! [X_2: int,Y_2: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(X_2)),number267125858f_real(Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Y_2)) ) ).
tff(fact_53_le__number__of,axiom,
! [X_2: int,Y_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(X_2)),number_number_of_int(Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_2)) ) ).
tff(fact_54_le__number__of,axiom,
! [X_2: int,Y_2: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(X_2)),number267125858f_real(Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_2)) ) ).
tff(fact_55_zadd__zless__mono,axiom,
! [Z_10: int,Z: int,W_14: int,W: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_14),W))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_10),Z))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(W_14),Z_10)),hAPP_int_int(plus_plus_int(W),Z))) ) ) ).
tff(fact_56_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_22: int,P_6: nat,Q_6: nat] : hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(X_22),P_6)),hAPP_nat_int(power_power_int(X_22),Q_6)) = hAPP_nat_int(power_power_int(X_22),hAPP_nat_nat(plus_plus_nat(P_6),Q_6)) ).
tff(fact_57_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_22: real,P_6: nat,Q_6: nat] : hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(X_22),P_6)),hAPP_nat_real(power_power_real(X_22),Q_6)) = hAPP_nat_real(power_power_real(X_22),hAPP_nat_nat(plus_plus_nat(P_6),Q_6)) ).
tff(fact_58_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_22: nat,P_6: nat,Q_6: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(X_22),P_6)),hAPP_nat_nat(power_power_nat(X_22),Q_6)) = hAPP_nat_nat(power_power_nat(X_22),hAPP_nat_nat(plus_plus_nat(P_6),Q_6)) ).
tff(fact_59_zpower__zadd__distrib,axiom,
! [X_1: int,Y_1: nat,Z: nat] : hAPP_nat_int(power_power_int(X_1),hAPP_nat_nat(plus_plus_nat(Y_1),Z)) = hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(X_1),Y_1)),hAPP_nat_int(power_power_int(X_1),Z)) ).
tff(fact_60_nat__mult__2,axiom,
! [Z: nat] : hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),Z) = hAPP_nat_nat(plus_plus_nat(Z),Z) ).
tff(fact_61_nat__mult__2__right,axiom,
! [Z: nat] : hAPP_nat_nat(times_times_nat(Z),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_nat(plus_plus_nat(Z),Z) ).
tff(fact_62_nat__1__add__1,axiom,
hAPP_nat_nat(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] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K1)),bit1(K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).
tff(fact_64_rel__simps_I17_J,axiom,
! [K: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K)),bit1(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L_1)) ) ).
tff(fact_65_less__eq__int__code_I16_J,axiom,
! [K1: int,K2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K1)),bit1(K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).
tff(fact_66_rel__simps_I34_J,axiom,
! [K: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K)),bit1(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L_1)) ) ).
tff(fact_67_rel__simps_I2_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),pls)) ).
tff(fact_68_less__int__code_I13_J,axiom,
! [K1: int,K2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K1)),bit0(K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).
tff(fact_69_rel__simps_I14_J,axiom,
! [K: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K)),bit0(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L_1)) ) ).
tff(fact_70_rel__simps_I19_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),pls)) ).
tff(fact_71_less__eq__int__code_I13_J,axiom,
! [K1: int,K2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K1)),bit0(K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).
tff(fact_72_rel__simps_I31_J,axiom,
! [K: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K)),bit0(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L_1)) ) ).
tff(fact_73_less__number__of__int__code,axiom,
! [K: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(K)),number_number_of_int(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L_1)) ) ).
tff(fact_74_less__eq__number__of__int__code,axiom,
! [K: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(K)),number_number_of_int(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L_1)) ) ).
tff(fact_75_zadd__strict__right__mono,axiom,
! [K_1: int,I_1: int,J_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),J_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(I_1),K_1)),hAPP_int_int(plus_plus_int(J_1),K_1))) ) ).
tff(fact_76_zadd__left__mono,axiom,
! [K_1: int,I_1: int,J_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),J_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(K_1),I_1)),hAPP_int_int(plus_plus_int(K_1),J_1))) ) ).
tff(fact_77_add__nat__number__of,axiom,
! [V_6: int,V: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),number_number_of_nat(V_6)) = number_number_of_nat(V_6) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_6),pls))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),number_number_of_nat(V_6)) = number_number_of_nat(V) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_6),pls))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),number_number_of_nat(V_6)) = number_number_of_nat(hAPP_int_int(plus_plus_int(V),V_6)) ) ) ) ) ) ).
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: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),pls)) ) ).
tff(fact_81_rel__simps_I5_J,axiom,
! [K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),bit1(K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K)) ) ).
tff(fact_82_less__eq__int__code_I15_J,axiom,
! [K1: int,K2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K1)),bit0(K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).
tff(fact_83_rel__simps_I33_J,axiom,
! [K: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K)),bit0(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L_1)) ) ).
tff(fact_84_less__int__code_I14_J,axiom,
! [K1: int,K2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K1)),bit1(K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).
tff(fact_85_rel__simps_I15_J,axiom,
! [K: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K)),bit1(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L_1)) ) ).
tff(fact_86_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(W),one_one_int)),Z)) ) ).
tff(fact_87_add1__zle__eq,axiom,
! [W_1: int,Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(W_1),one_one_int)),Z_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_1)) ) ).
tff(fact_88_zle__add1__eq__le,axiom,
! [W_1: int,Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_int_int(plus_plus_int(Z_1),one_one_int)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_1)) ) ).
tff(fact_89_zprime__2,axiom,
hBOOL(hAPP_int_bool(zprime,number_number_of_int(bit0(bit1(pls))))) ).
tff(fact_90_is__mult__sum2sq,axiom,
! [Y_1: int,X_1: int] :
( hBOOL(hAPP_int_bool(twoSqu820444569sum2sq,X_1))
=> ( hBOOL(hAPP_int_bool(twoSqu820444569sum2sq,Y_1))
=> hBOOL(hAPP_int_bool(twoSqu820444569sum2sq,hAPP_int_int(times_times_int(X_1),Y_1))) ) ) ).
tff(fact_91_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: int,Ly_4: int,Rx_6: int,Ry_4: int] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_6),Ly_4)),hAPP_int_int(times_times_int(Rx_6),Ry_4)) = hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_6),Rx_6)),hAPP_int_int(times_times_int(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] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_6),Ly_4)),hAPP_nat_nat(times_times_nat(Rx_6),Ry_4)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_6),Rx_6)),hAPP_nat_nat(times_times_nat(Ly_4),Ry_4)) ).
tff(fact_93_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: real,Ly_4: real,Rx_6: real,Ry_4: real] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_6),Ly_4)),hAPP_real_real(times_times_real(Rx_6),Ry_4)) = hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_6),Rx_6)),hAPP_real_real(times_times_real(Ly_4),Ry_4)) ).
tff(fact_94_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: int,Ly_3: int,Rx_5: int,Ry_3: int] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_5),Ly_3)),hAPP_int_int(times_times_int(Rx_5),Ry_3)) = hAPP_int_int(times_times_int(Rx_5),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(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] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_5),Ly_3)),hAPP_nat_nat(times_times_nat(Rx_5),Ry_3)) = hAPP_nat_nat(times_times_nat(Rx_5),hAPP_nat_nat(times_times_nat(hAPP_nat_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: real,Ly_3: real,Rx_5: real,Ry_3: real] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_5),Ly_3)),hAPP_real_real(times_times_real(Rx_5),Ry_3)) = hAPP_real_real(times_times_real(Rx_5),hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_5),Ly_3)),Ry_3)) ).
tff(fact_97_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: int,Ly_2: int,Rx_4: int,Ry_2: int] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_4),Ly_2)),hAPP_int_int(times_times_int(Rx_4),Ry_2)) = hAPP_int_int(times_times_int(Lx_4),hAPP_int_int(times_times_int(Ly_2),hAPP_int_int(times_times_int(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] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_4),Ly_2)),hAPP_nat_nat(times_times_nat(Rx_4),Ry_2)) = hAPP_nat_nat(times_times_nat(Lx_4),hAPP_nat_nat(times_times_nat(Ly_2),hAPP_nat_nat(times_times_nat(Rx_4),Ry_2))) ).
tff(fact_99_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: real,Ly_2: real,Rx_4: real,Ry_2: real] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_4),Ly_2)),hAPP_real_real(times_times_real(Rx_4),Ry_2)) = hAPP_real_real(times_times_real(Lx_4),hAPP_real_real(times_times_real(Ly_2),hAPP_real_real(times_times_real(Rx_4),Ry_2))) ).
tff(fact_100_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: int,Ly_1: int,Rx_3: int] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_3),Ly_1)),Rx_3) = hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(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] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_3),Ly_1)),Rx_3) = hAPP_nat_nat(times_times_nat(hAPP_nat_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: real,Ly_1: real,Rx_3: real] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_3),Ly_1)),Rx_3) = hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_3),Rx_3)),Ly_1) ).
tff(fact_103_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: int,Ly: int,Rx_2: int] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_2),Ly)),Rx_2) = hAPP_int_int(times_times_int(Lx_2),hAPP_int_int(times_times_int(Ly),Rx_2)) ).
tff(fact_104_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: nat,Ly: nat,Rx_2: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_2),Ly)),Rx_2) = hAPP_nat_nat(times_times_nat(Lx_2),hAPP_nat_nat(times_times_nat(Ly),Rx_2)) ).
tff(fact_105_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: real,Ly: real,Rx_2: real] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_2),Ly)),Rx_2) = hAPP_real_real(times_times_real(Lx_2),hAPP_real_real(times_times_real(Ly),Rx_2)) ).
tff(fact_106_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: int,Rx_1: int,Ry_1: int] : hAPP_int_int(times_times_int(Lx_1),hAPP_int_int(times_times_int(Rx_1),Ry_1)) = hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(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] : hAPP_nat_nat(times_times_nat(Lx_1),hAPP_nat_nat(times_times_nat(Rx_1),Ry_1)) = hAPP_nat_nat(times_times_nat(hAPP_nat_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: real,Rx_1: real,Ry_1: real] : hAPP_real_real(times_times_real(Lx_1),hAPP_real_real(times_times_real(Rx_1),Ry_1)) = hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_1),Rx_1)),Ry_1) ).
tff(fact_109_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: int,Rx: int,Ry: int] : hAPP_int_int(times_times_int(Lx),hAPP_int_int(times_times_int(Rx),Ry)) = hAPP_int_int(times_times_int(Rx),hAPP_int_int(times_times_int(Lx),Ry)) ).
tff(fact_110_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: nat,Rx: nat,Ry: nat] : hAPP_nat_nat(times_times_nat(Lx),hAPP_nat_nat(times_times_nat(Rx),Ry)) = hAPP_nat_nat(times_times_nat(Rx),hAPP_nat_nat(times_times_nat(Lx),Ry)) ).
tff(fact_111_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: real,Rx: real,Ry: real] : hAPP_real_real(times_times_real(Lx),hAPP_real_real(times_times_real(Rx),Ry)) = hAPP_real_real(times_times_real(Rx),hAPP_real_real(times_times_real(Lx),Ry)) ).
tff(fact_112_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_111: int,B_63: int] : hAPP_int_int(times_times_int(A_111),B_63) = hAPP_int_int(times_times_int(B_63),A_111) ).
tff(fact_113_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_111: nat,B_63: nat] : hAPP_nat_nat(times_times_nat(A_111),B_63) = hAPP_nat_nat(times_times_nat(B_63),A_111) ).
tff(fact_114_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_111: real,B_63: real] : hAPP_real_real(times_times_real(A_111),B_63) = hAPP_real_real(times_times_real(B_63),A_111) ).
tff(fact_115_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_110: int,B_62: int,C_35: int,D_11: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_110),B_62)),hAPP_int_int(plus_plus_int(C_35),D_11)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_110),C_35)),hAPP_int_int(plus_plus_int(B_62),D_11)) ).
tff(fact_116_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_110: nat,B_62: nat,C_35: nat,D_11: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_110),B_62)),hAPP_nat_nat(plus_plus_nat(C_35),D_11)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_110),C_35)),hAPP_nat_nat(plus_plus_nat(B_62),D_11)) ).
tff(fact_117_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_110: real,B_62: real,C_35: real,D_11: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_110),B_62)),hAPP_real_real(plus_plus_real(C_35),D_11)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_110),C_35)),hAPP_real_real(plus_plus_real(B_62),D_11)) ).
tff(fact_118_mem__def,axiom,
! [X_2: int,A_109: fun_int_bool] :
( hBOOL(member_int(X_2,A_109))
<=> hBOOL(hAPP_int_bool(A_109,X_2)) ) ).
tff(fact_119_Collect__def,axiom,
! [P_2: fun_int_bool] : collect_int(P_2) = P_2 ).
tff(fact_120_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_108: int,B_61: int,C_34: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_108),B_61)),C_34) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_108),C_34)),B_61) ).
tff(fact_121_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_108: nat,B_61: nat,C_34: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_108),B_61)),C_34) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_108),C_34)),B_61) ).
tff(fact_122_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_108: real,B_61: real,C_34: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_108),B_61)),C_34) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_108),C_34)),B_61) ).
tff(fact_123_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_107: int,B_60: int,C_33: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_107),B_60)),C_33) = hAPP_int_int(plus_plus_int(A_107),hAPP_int_int(plus_plus_int(B_60),C_33)) ).
tff(fact_124_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_107: nat,B_60: nat,C_33: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_107),B_60)),C_33) = hAPP_nat_nat(plus_plus_nat(A_107),hAPP_nat_nat(plus_plus_nat(B_60),C_33)) ).
tff(fact_125_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_107: real,B_60: real,C_33: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_107),B_60)),C_33) = hAPP_real_real(plus_plus_real(A_107),hAPP_real_real(plus_plus_real(B_60),C_33)) ).
tff(fact_126_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_106: int,C_32: int,D_10: int] : hAPP_int_int(plus_plus_int(A_106),hAPP_int_int(plus_plus_int(C_32),D_10)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_106),C_32)),D_10) ).
tff(fact_127_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_106: nat,C_32: nat,D_10: nat] : hAPP_nat_nat(plus_plus_nat(A_106),hAPP_nat_nat(plus_plus_nat(C_32),D_10)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_106),C_32)),D_10) ).
tff(fact_128_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_106: real,C_32: real,D_10: real] : hAPP_real_real(plus_plus_real(A_106),hAPP_real_real(plus_plus_real(C_32),D_10)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_106),C_32)),D_10) ).
tff(fact_129_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_105: int,C_31: int,D_9: int] : hAPP_int_int(plus_plus_int(A_105),hAPP_int_int(plus_plus_int(C_31),D_9)) = hAPP_int_int(plus_plus_int(C_31),hAPP_int_int(plus_plus_int(A_105),D_9)) ).
tff(fact_130_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_105: nat,C_31: nat,D_9: nat] : hAPP_nat_nat(plus_plus_nat(A_105),hAPP_nat_nat(plus_plus_nat(C_31),D_9)) = hAPP_nat_nat(plus_plus_nat(C_31),hAPP_nat_nat(plus_plus_nat(A_105),D_9)) ).
tff(fact_131_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_105: real,C_31: real,D_9: real] : hAPP_real_real(plus_plus_real(A_105),hAPP_real_real(plus_plus_real(C_31),D_9)) = hAPP_real_real(plus_plus_real(C_31),hAPP_real_real(plus_plus_real(A_105),D_9)) ).
tff(fact_132_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_104: int,C_30: int] : hAPP_int_int(plus_plus_int(A_104),C_30) = hAPP_int_int(plus_plus_int(C_30),A_104) ).
tff(fact_133_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_104: nat,C_30: nat] : hAPP_nat_nat(plus_plus_nat(A_104),C_30) = hAPP_nat_nat(plus_plus_nat(C_30),A_104) ).
tff(fact_134_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_104: real,C_30: real] : hAPP_real_real(plus_plus_real(A_104),C_30) = hAPP_real_real(plus_plus_real(C_30),A_104) ).
tff(fact_135_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_136_eq__number__of,axiom,
! [X_2: int,Y_2: int] :
( ( number267125858f_real(X_2) = number267125858f_real(Y_2) )
<=> ( X_2 = Y_2 ) ) ).
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_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_139_number__of__reorient,axiom,
! [W_1: int,X_2: real] :
( ( number267125858f_real(W_1) = X_2 )
<=> ( X_2 = number267125858f_real(W_1) ) ) ).
tff(fact_140_rel__simps_I51_J,axiom,
! [K: int,L_1: int] :
( ( bit1(K) = bit1(L_1) )
<=> ( K = L_1 ) ) ).
tff(fact_141_rel__simps_I48_J,axiom,
! [K: int,L_1: int] :
( ( bit0(K) = bit0(L_1) )
<=> ( K = L_1 ) ) ).
tff(fact_142_zmult__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Z1),Z2)),Z3) = hAPP_int_int(times_times_int(Z1),hAPP_int_int(times_times_int(Z2),Z3)) ).
tff(fact_143_zmult__commute,axiom,
! [Z: int,W: int] : hAPP_int_int(times_times_int(Z),W) = hAPP_int_int(times_times_int(W),Z) ).
tff(fact_144_number__of__is__id,axiom,
! [K_1: int] : number_number_of_int(K_1) = K_1 ).
tff(fact_145_zadd__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(Z1),Z2)),Z3) = hAPP_int_int(plus_plus_int(Z1),hAPP_int_int(plus_plus_int(Z2),Z3)) ).
tff(fact_146_zadd__left__commute,axiom,
! [X_1: int,Y_1: int,Z: int] : hAPP_int_int(plus_plus_int(X_1),hAPP_int_int(plus_plus_int(Y_1),Z)) = hAPP_int_int(plus_plus_int(Y_1),hAPP_int_int(plus_plus_int(X_1),Z)) ).
tff(fact_147_zadd__commute,axiom,
! [Z: int,W: int] : hAPP_int_int(plus_plus_int(Z),W) = hAPP_int_int(plus_plus_int(W),Z) ).
tff(fact_148_rel__simps_I12_J,axiom,
! [K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),pls)) ) ).
tff(fact_149_less__int__code_I15_J,axiom,
! [K1: int,K2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K1)),bit0(K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).
tff(fact_150_rel__simps_I16_J,axiom,
! [K: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K)),bit0(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L_1)) ) ).
tff(fact_151_rel__simps_I10_J,axiom,
! [K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),pls)) ) ).
tff(fact_152_rel__simps_I4_J,axiom,
! [K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),bit0(K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),K)) ) ).
tff(fact_153_rel__simps_I22_J,axiom,
! [K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),bit1(K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K)) ) ).
tff(fact_154_less__eq__int__code_I14_J,axiom,
! [K1: int,K2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K1)),bit1(K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).
tff(fact_155_rel__simps_I32_J,axiom,
! [K: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K)),bit1(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L_1)) ) ).
tff(fact_156_rel__simps_I27_J,axiom,
! [K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),pls)) ) ).
tff(fact_157_rel__simps_I21_J,axiom,
! [K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),bit0(K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K)) ) ).
tff(fact_158_zless__add1__eq,axiom,
! [W_1: int,Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_int_int(plus_plus_int(Z_1),one_one_int)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_1))
| ( W_1 = Z_1 ) ) ) ).
tff(fact_159_power__even__eq,axiom,
! [A_103: int,N_39: nat] : hAPP_nat_int(power_power_int(A_103),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_39)) = hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(A_103),N_39)),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_160_power__even__eq,axiom,
! [A_103: real,N_39: nat] : hAPP_nat_real(power_power_real(A_103),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_39)) = hAPP_nat_real(power_power_real(hAPP_nat_real(power_power_real(A_103),N_39)),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_161_power__even__eq,axiom,
! [A_103: nat,N_39: nat] : hAPP_nat_nat(power_power_nat(A_103),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_39)) = hAPP_nat_nat(power_power_nat(hAPP_nat_nat(power_power_nat(A_103),N_39)),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_162_less__special_I4_J,axiom,
! [X_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(X_2)),one_one_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),bit1(pls))) ) ).
tff(fact_163_less__special_I4_J,axiom,
! [X_2: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(X_2)),one_one_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),bit1(pls))) ) ).
tff(fact_164_less__special_I2_J,axiom,
! [Y_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),number_number_of_int(Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(pls)),Y_2)) ) ).
tff(fact_165_less__special_I2_J,axiom,
! [Y_2: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),number267125858f_real(Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(pls)),Y_2)) ) ).
tff(fact_166_le__special_I4_J,axiom,
! [X_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(X_2)),one_one_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),bit1(pls))) ) ).
tff(fact_167_le__special_I4_J,axiom,
! [X_2: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(X_2)),one_one_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),bit1(pls))) ) ).
tff(fact_168_le__special_I2_J,axiom,
! [Y_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),number_number_of_int(Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(pls)),Y_2)) ) ).
tff(fact_169_le__special_I2_J,axiom,
! [Y_2: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),number267125858f_real(Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(pls)),Y_2)) ) ).
tff(fact_170_crossproduct__eq,axiom,
! [W_1: int,Y_2: int,X_2: int,Z_1: int] :
( ( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(W_1),Y_2)),hAPP_int_int(times_times_int(X_2),Z_1)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(W_1),Z_1)),hAPP_int_int(times_times_int(X_2),Y_2)) )
<=> ( ( W_1 = X_2 )
| ( Y_2 = Z_1 ) ) ) ).
tff(fact_171_crossproduct__eq,axiom,
! [W_1: nat,Y_2: nat,X_2: nat,Z_1: nat] :
( ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(W_1),Y_2)),hAPP_nat_nat(times_times_nat(X_2),Z_1)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(W_1),Z_1)),hAPP_nat_nat(times_times_nat(X_2),Y_2)) )
<=> ( ( W_1 = X_2 )
| ( Y_2 = Z_1 ) ) ) ).
tff(fact_172_crossproduct__eq,axiom,
! [W_1: real,Y_2: real,X_2: real,Z_1: real] :
( ( hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(W_1),Y_2)),hAPP_real_real(times_times_real(X_2),Z_1)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(W_1),Z_1)),hAPP_real_real(times_times_real(X_2),Y_2)) )
<=> ( ( W_1 = X_2 )
| ( Y_2 = Z_1 ) ) ) ).
tff(fact_173_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_102: int,M_13: int,B_59: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_102),M_13)),hAPP_int_int(times_times_int(B_59),M_13)) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_102),B_59)),M_13) ).
tff(fact_174_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_102: nat,M_13: nat,B_59: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_102),M_13)),hAPP_nat_nat(times_times_nat(B_59),M_13)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_102),B_59)),M_13) ).
tff(fact_175_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_102: real,M_13: real,B_59: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_102),M_13)),hAPP_real_real(times_times_real(B_59),M_13)) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_102),B_59)),M_13) ).
tff(fact_176_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_101: int,B_58: int,C_29: int] : hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_101),B_58)),C_29) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_101),C_29)),hAPP_int_int(times_times_int(B_58),C_29)) ).
tff(fact_177_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_101: nat,B_58: nat,C_29: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_101),B_58)),C_29) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_101),C_29)),hAPP_nat_nat(times_times_nat(B_58),C_29)) ).
tff(fact_178_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_101: real,B_58: real,C_29: real] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_101),B_58)),C_29) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_101),C_29)),hAPP_real_real(times_times_real(B_58),C_29)) ).
tff(fact_179_crossproduct__noteq,axiom,
! [C_2: int,D: int,A_2: int,B_2: int] :
( ( ( A_2 != B_2 )
& ( C_2 != D ) )
<=> ( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_2),C_2)),hAPP_int_int(times_times_int(B_2),D)) != hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_2),D)),hAPP_int_int(times_times_int(B_2),C_2)) ) ) ).
tff(fact_180_crossproduct__noteq,axiom,
! [C_2: nat,D: nat,A_2: nat,B_2: nat] :
( ( ( A_2 != B_2 )
& ( C_2 != D ) )
<=> ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_2),C_2)),hAPP_nat_nat(times_times_nat(B_2),D)) != hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_2),D)),hAPP_nat_nat(times_times_nat(B_2),C_2)) ) ) ).
tff(fact_181_crossproduct__noteq,axiom,
! [C_2: real,D: real,A_2: real,B_2: real] :
( ( ( A_2 != B_2 )
& ( C_2 != D ) )
<=> ( hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_2),C_2)),hAPP_real_real(times_times_real(B_2),D)) != hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_2),D)),hAPP_real_real(times_times_real(B_2),C_2)) ) ) ).
tff(fact_182_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_21: int,Y_18: int,Z_9: int] : hAPP_int_int(times_times_int(X_21),hAPP_int_int(plus_plus_int(Y_18),Z_9)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(X_21),Y_18)),hAPP_int_int(times_times_int(X_21),Z_9)) ).
tff(fact_183_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_21: nat,Y_18: nat,Z_9: nat] : hAPP_nat_nat(times_times_nat(X_21),hAPP_nat_nat(plus_plus_nat(Y_18),Z_9)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(X_21),Y_18)),hAPP_nat_nat(times_times_nat(X_21),Z_9)) ).
tff(fact_184_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_21: real,Y_18: real,Z_9: real] : hAPP_real_real(times_times_real(X_21),hAPP_real_real(plus_plus_real(Y_18),Z_9)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_21),Y_18)),hAPP_real_real(times_times_real(X_21),Z_9)) ).
tff(fact_185_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_100: int] : hAPP_int_int(times_times_int(A_100),one_one_int) = A_100 ).
tff(fact_186_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_100: nat] : hAPP_nat_nat(times_times_nat(A_100),one_one_nat) = A_100 ).
tff(fact_187_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_100: real] : hAPP_real_real(times_times_real(A_100),one_one_real) = A_100 ).
tff(fact_188_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_99: int] : hAPP_int_int(times_times_int(one_one_int),A_99) = A_99 ).
tff(fact_189_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_99: nat] : hAPP_nat_nat(times_times_nat(one_one_nat),A_99) = A_99 ).
tff(fact_190_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_99: real] : hAPP_real_real(times_times_real(one_one_real),A_99) = A_99 ).
tff(fact_191_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_20: int,Y_17: int,Q_5: nat] : hAPP_nat_int(power_power_int(hAPP_int_int(times_times_int(X_20),Y_17)),Q_5) = hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(X_20),Q_5)),hAPP_nat_int(power_power_int(Y_17),Q_5)) ).
tff(fact_192_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_20: real,Y_17: real,Q_5: nat] : hAPP_nat_real(power_power_real(hAPP_real_real(times_times_real(X_20),Y_17)),Q_5) = hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(X_20),Q_5)),hAPP_nat_real(power_power_real(Y_17),Q_5)) ).
tff(fact_193_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_20: nat,Y_17: nat,Q_5: nat] : hAPP_nat_nat(power_power_nat(hAPP_nat_nat(times_times_nat(X_20),Y_17)),Q_5) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(X_20),Q_5)),hAPP_nat_nat(power_power_nat(Y_17),Q_5)) ).
tff(fact_194_rel__simps_I46_J,axiom,
! [K_1: int] : bit1(K_1) != pls ).
tff(fact_195_rel__simps_I39_J,axiom,
! [L: int] : pls != bit1(L) ).
tff(fact_196_rel__simps_I50_J,axiom,
! [K_1: int,L: int] : bit1(K_1) != bit0(L) ).
tff(fact_197_rel__simps_I49_J,axiom,
! [K_1: int,L: int] : bit0(K_1) != bit1(L) ).
tff(fact_198_rel__simps_I44_J,axiom,
! [K: int] :
( ( bit0(K) = pls )
<=> ( K = pls ) ) ).
tff(fact_199_rel__simps_I38_J,axiom,
! [L_1: int] :
( ( pls = bit0(L_1) )
<=> ( pls = L_1 ) ) ).
tff(fact_200_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_201_mult__Pls,axiom,
! [W: int] : hAPP_int_int(times_times_int(pls),W) = pls ).
tff(fact_202_mult__Bit0,axiom,
! [K_1: int,L: int] : hAPP_int_int(times_times_int(bit0(K_1)),L) = bit0(hAPP_int_int(times_times_int(K_1),L)) ).
tff(fact_203_add__Pls__right,axiom,
! [K_1: int] : hAPP_int_int(plus_plus_int(K_1),pls) = K_1 ).
tff(fact_204_add__Pls,axiom,
! [K_1: int] : hAPP_int_int(plus_plus_int(pls),K_1) = K_1 ).
tff(fact_205_add__Bit0__Bit0,axiom,
! [K_1: int,L: int] : hAPP_int_int(plus_plus_int(bit0(K_1)),bit0(L)) = bit0(hAPP_int_int(plus_plus_int(K_1),L)) ).
tff(fact_206_Bit0__def,axiom,
! [K_1: int] : bit0(K_1) = hAPP_int_int(plus_plus_int(K_1),K_1) ).
tff(fact_207_zmult__1__right,axiom,
! [Z: int] : hAPP_int_int(times_times_int(Z),one_one_int) = Z ).
tff(fact_208_zmult__1,axiom,
! [Z: int] : hAPP_int_int(times_times_int(one_one_int),Z) = Z ).
tff(fact_209_times__numeral__code_I5_J,axiom,
! [V: int,W: int] : hAPP_int_int(times_times_int(number_number_of_int(V)),number_number_of_int(W)) = number_number_of_int(hAPP_int_int(times_times_int(V),W)) ).
tff(fact_210_zadd__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] : hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(Z1),Z2)),W) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Z1),W)),hAPP_int_int(times_times_int(Z2),W)) ).
tff(fact_211_zadd__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] : hAPP_int_int(times_times_int(W),hAPP_int_int(plus_plus_int(Z1),Z2)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(W),Z1)),hAPP_int_int(times_times_int(W),Z2)) ).
tff(fact_212_plus__numeral__code_I9_J,axiom,
! [V: int,W: int] : hAPP_int_int(plus_plus_int(number_number_of_int(V)),number_number_of_int(W)) = number_number_of_int(hAPP_int_int(plus_plus_int(V),W)) ).
tff(fact_213_semiring__mult__number__of,axiom,
! [V_20: int,V_19: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20))
=> ( hAPP_int_int(times_times_int(number_number_of_int(V_19)),number_number_of_int(V_20)) = number_number_of_int(hAPP_int_int(times_times_int(V_19),V_20)) ) ) ) ).
tff(fact_214_semiring__mult__number__of,axiom,
! [V_20: int,V_19: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20))
=> ( hAPP_nat_nat(times_times_nat(number_number_of_nat(V_19)),number_number_of_nat(V_20)) = number_number_of_nat(hAPP_int_int(times_times_int(V_19),V_20)) ) ) ) ).
tff(fact_215_semiring__mult__number__of,axiom,
! [V_20: int,V_19: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20))
=> ( hAPP_real_real(times_times_real(number267125858f_real(V_19)),number267125858f_real(V_20)) = number267125858f_real(hAPP_int_int(times_times_int(V_19),V_20)) ) ) ) ).
tff(fact_216_semiring__add__number__of,axiom,
! [V_18: int,V_17: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_17))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_18))
=> ( hAPP_int_int(plus_plus_int(number_number_of_int(V_17)),number_number_of_int(V_18)) = number_number_of_int(hAPP_int_int(plus_plus_int(V_17),V_18)) ) ) ) ).
tff(fact_217_semiring__add__number__of,axiom,
! [V_18: int,V_17: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_17))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_18))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_17)),number_number_of_nat(V_18)) = number_number_of_nat(hAPP_int_int(plus_plus_int(V_17),V_18)) ) ) ) ).
tff(fact_218_semiring__add__number__of,axiom,
! [V_18: int,V_17: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_17))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_18))
=> ( hAPP_real_real(plus_plus_real(number267125858f_real(V_17)),number267125858f_real(V_18)) = number267125858f_real(hAPP_int_int(plus_plus_int(V_17),V_18)) ) ) ) ).
tff(fact_219_power2__ge__self,axiom,
! [X_1: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),hAPP_nat_int(power_power_int(X_1),number_number_of_nat(bit0(bit1(pls)))))) ).
tff(fact_220_left__distrib__number__of,axiom,
! [A_98: int,B_57: int,V_16: int] : hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_98),B_57)),number_number_of_int(V_16)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_98),number_number_of_int(V_16))),hAPP_int_int(times_times_int(B_57),number_number_of_int(V_16))) ).
tff(fact_221_left__distrib__number__of,axiom,
! [A_98: nat,B_57: nat,V_16: int] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_98),B_57)),number_number_of_nat(V_16)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_98),number_number_of_nat(V_16))),hAPP_nat_nat(times_times_nat(B_57),number_number_of_nat(V_16))) ).
tff(fact_222_left__distrib__number__of,axiom,
! [A_98: real,B_57: real,V_16: int] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_98),B_57)),number267125858f_real(V_16)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_98),number267125858f_real(V_16))),hAPP_real_real(times_times_real(B_57),number267125858f_real(V_16))) ).
tff(fact_223_right__distrib__number__of,axiom,
! [V_15: int,B_56: int,C_28: int] : hAPP_int_int(times_times_int(number_number_of_int(V_15)),hAPP_int_int(plus_plus_int(B_56),C_28)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(V_15)),B_56)),hAPP_int_int(times_times_int(number_number_of_int(V_15)),C_28)) ).
tff(fact_224_right__distrib__number__of,axiom,
! [V_15: int,B_56: nat,C_28: nat] : hAPP_nat_nat(times_times_nat(number_number_of_nat(V_15)),hAPP_nat_nat(plus_plus_nat(B_56),C_28)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(number_number_of_nat(V_15)),B_56)),hAPP_nat_nat(times_times_nat(number_number_of_nat(V_15)),C_28)) ).
tff(fact_225_right__distrib__number__of,axiom,
! [V_15: int,B_56: real,C_28: real] : hAPP_real_real(times_times_real(number267125858f_real(V_15)),hAPP_real_real(plus_plus_real(B_56),C_28)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(number267125858f_real(V_15)),B_56)),hAPP_real_real(times_times_real(number267125858f_real(V_15)),C_28)) ).
tff(fact_226_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_97: int,M_12: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_97),M_12)),M_12) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_97),one_one_int)),M_12) ).
tff(fact_227_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_97: nat,M_12: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_97),M_12)),M_12) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_97),one_one_nat)),M_12) ).
tff(fact_228_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_97: real,M_12: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_97),M_12)),M_12) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_97),one_one_real)),M_12) ).
tff(fact_229_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_11: int,A_96: int] : hAPP_int_int(plus_plus_int(M_11),hAPP_int_int(times_times_int(A_96),M_11)) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_96),one_one_int)),M_11) ).
tff(fact_230_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_11: nat,A_96: nat] : hAPP_nat_nat(plus_plus_nat(M_11),hAPP_nat_nat(times_times_nat(A_96),M_11)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_96),one_one_nat)),M_11) ).
tff(fact_231_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_11: real,A_96: real] : hAPP_real_real(plus_plus_real(M_11),hAPP_real_real(times_times_real(A_96),M_11)) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_96),one_one_real)),M_11) ).
tff(fact_232_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_10: int] : hAPP_int_int(plus_plus_int(M_10),M_10) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(one_one_int),one_one_int)),M_10) ).
tff(fact_233_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_10: nat] : hAPP_nat_nat(plus_plus_nat(M_10),M_10) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(one_one_nat),one_one_nat)),M_10) ).
tff(fact_234_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_10: real] : hAPP_real_real(plus_plus_real(M_10),M_10) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(one_one_real),one_one_real)),M_10) ).
tff(fact_235_add__numeral__0,axiom,
! [A_95: int] : hAPP_int_int(plus_plus_int(number_number_of_int(pls)),A_95) = A_95 ).
tff(fact_236_add__numeral__0,axiom,
! [A_95: real] : hAPP_real_real(plus_plus_real(number267125858f_real(pls)),A_95) = A_95 ).
tff(fact_237_add__numeral__0__right,axiom,
! [A_94: int] : hAPP_int_int(plus_plus_int(A_94),number_number_of_int(pls)) = A_94 ).
tff(fact_238_add__numeral__0__right,axiom,
! [A_94: real] : hAPP_real_real(plus_plus_real(A_94),number267125858f_real(pls)) = A_94 ).
tff(fact_239_mult__number__of__left,axiom,
! [V_14: int,W_13: int,Z_8: int] : hAPP_int_int(times_times_int(number_number_of_int(V_14)),hAPP_int_int(times_times_int(number_number_of_int(W_13)),Z_8)) = hAPP_int_int(times_times_int(number_number_of_int(hAPP_int_int(times_times_int(V_14),W_13))),Z_8) ).
tff(fact_240_mult__number__of__left,axiom,
! [V_14: int,W_13: int,Z_8: real] : hAPP_real_real(times_times_real(number267125858f_real(V_14)),hAPP_real_real(times_times_real(number267125858f_real(W_13)),Z_8)) = hAPP_real_real(times_times_real(number267125858f_real(hAPP_int_int(times_times_int(V_14),W_13))),Z_8) ).
tff(fact_241_arith__simps_I32_J,axiom,
! [V_13: int,W_12: int] : hAPP_int_int(times_times_int(number_number_of_int(V_13)),number_number_of_int(W_12)) = number_number_of_int(hAPP_int_int(times_times_int(V_13),W_12)) ).
tff(fact_242_arith__simps_I32_J,axiom,
! [V_13: int,W_12: int] : hAPP_real_real(times_times_real(number267125858f_real(V_13)),number267125858f_real(W_12)) = number267125858f_real(hAPP_int_int(times_times_int(V_13),W_12)) ).
tff(fact_243_number__of__mult,axiom,
! [V_12: int,W_11: int] : number_number_of_int(hAPP_int_int(times_times_int(V_12),W_11)) = hAPP_int_int(times_times_int(number_number_of_int(V_12)),number_number_of_int(W_11)) ).
tff(fact_244_number__of__mult,axiom,
! [V_12: int,W_11: int] : number267125858f_real(hAPP_int_int(times_times_int(V_12),W_11)) = hAPP_real_real(times_times_real(number267125858f_real(V_12)),number267125858f_real(W_11)) ).
tff(fact_245_add__number__of__left,axiom,
! [V_11: int,W_10: int,Z_7: int] : hAPP_int_int(plus_plus_int(number_number_of_int(V_11)),hAPP_int_int(plus_plus_int(number_number_of_int(W_10)),Z_7)) = hAPP_int_int(plus_plus_int(number_number_of_int(hAPP_int_int(plus_plus_int(V_11),W_10))),Z_7) ).
tff(fact_246_add__number__of__left,axiom,
! [V_11: int,W_10: int,Z_7: real] : hAPP_real_real(plus_plus_real(number267125858f_real(V_11)),hAPP_real_real(plus_plus_real(number267125858f_real(W_10)),Z_7)) = hAPP_real_real(plus_plus_real(number267125858f_real(hAPP_int_int(plus_plus_int(V_11),W_10))),Z_7) ).
tff(fact_247_add__number__of__eq,axiom,
! [V_10: int,W_9: int] : hAPP_int_int(plus_plus_int(number_number_of_int(V_10)),number_number_of_int(W_9)) = number_number_of_int(hAPP_int_int(plus_plus_int(V_10),W_9)) ).
tff(fact_248_add__number__of__eq,axiom,
! [V_10: int,W_9: int] : hAPP_real_real(plus_plus_real(number267125858f_real(V_10)),number267125858f_real(W_9)) = number267125858f_real(hAPP_int_int(plus_plus_int(V_10),W_9)) ).
tff(fact_249_number__of__add,axiom,
! [V_9: int,W_8: int] : number_number_of_int(hAPP_int_int(plus_plus_int(V_9),W_8)) = hAPP_int_int(plus_plus_int(number_number_of_int(V_9)),number_number_of_int(W_8)) ).
tff(fact_250_number__of__add,axiom,
! [V_9: int,W_8: int] : number267125858f_real(hAPP_int_int(plus_plus_int(V_9),W_8)) = hAPP_real_real(plus_plus_real(number267125858f_real(V_9)),number267125858f_real(W_8)) ).
tff(fact_251_add__Bit1__Bit0,axiom,
! [K_1: int,L: int] : hAPP_int_int(plus_plus_int(bit1(K_1)),bit0(L)) = bit1(hAPP_int_int(plus_plus_int(K_1),L)) ).
tff(fact_252_add__Bit0__Bit1,axiom,
! [K_1: int,L: int] : hAPP_int_int(plus_plus_int(bit0(K_1)),bit1(L)) = bit1(hAPP_int_int(plus_plus_int(K_1),L)) ).
tff(fact_253_Bit1__def,axiom,
! [K_1: int] : bit1(K_1) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),K_1)),K_1) ).
tff(fact_254_number__of__Bit1,axiom,
! [W_7: int] : number_number_of_int(bit1(W_7)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),number_number_of_int(W_7))),number_number_of_int(W_7)) ).
tff(fact_255_number__of__Bit1,axiom,
! [W_7: int] : number267125858f_real(bit1(W_7)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(one_one_real),number267125858f_real(W_7))),number267125858f_real(W_7)) ).
tff(fact_256_mult__numeral__1,axiom,
! [A_93: int] : hAPP_int_int(times_times_int(number_number_of_int(bit1(pls))),A_93) = A_93 ).
tff(fact_257_mult__numeral__1,axiom,
! [A_93: real] : hAPP_real_real(times_times_real(number267125858f_real(bit1(pls))),A_93) = A_93 ).
tff(fact_258_mult__numeral__1__right,axiom,
! [A_92: int] : hAPP_int_int(times_times_int(A_92),number_number_of_int(bit1(pls))) = A_92 ).
tff(fact_259_mult__numeral__1__right,axiom,
! [A_92: real] : hAPP_real_real(times_times_real(A_92),number267125858f_real(bit1(pls))) = A_92 ).
tff(fact_260_semiring__numeral__1__eq__1,axiom,
number_number_of_int(bit1(pls)) = one_one_int ).
tff(fact_261_semiring__numeral__1__eq__1,axiom,
number_number_of_nat(bit1(pls)) = one_one_nat ).
tff(fact_262_semiring__numeral__1__eq__1,axiom,
number267125858f_real(bit1(pls)) = one_one_real ).
tff(fact_263_numeral__1__eq__1,axiom,
number_number_of_int(bit1(pls)) = one_one_int ).
tff(fact_264_numeral__1__eq__1,axiom,
number267125858f_real(bit1(pls)) = one_one_real ).
tff(fact_265_semiring__norm_I110_J,axiom,
one_one_int = number_number_of_int(bit1(pls)) ).
tff(fact_266_semiring__norm_I110_J,axiom,
one_one_real = number267125858f_real(bit1(pls)) ).
tff(fact_267_one__is__num__one,axiom,
one_one_int = number_number_of_int(bit1(pls)) ).
tff(fact_268_mult__Bit1,axiom,
! [K_1: int,L: int] : hAPP_int_int(times_times_int(bit1(K_1)),L) = hAPP_int_int(plus_plus_int(bit0(hAPP_int_int(times_times_int(K_1),L))),L) ).
tff(fact_269_double__number__of__Bit0,axiom,
! [W_6: int] : hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(one_one_int),one_one_int)),number_number_of_int(W_6)) = number_number_of_int(bit0(W_6)) ).
tff(fact_270_double__number__of__Bit0,axiom,
! [W_6: int] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(one_one_real),one_one_real)),number267125858f_real(W_6)) = number267125858f_real(bit0(W_6)) ).
tff(fact_271_power3__eq__cube,axiom,
! [A_91: int] : hAPP_nat_int(power_power_int(A_91),number_number_of_nat(bit1(bit1(pls)))) = hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(A_91),A_91)),A_91) ).
tff(fact_272_power3__eq__cube,axiom,
! [A_91: real] : hAPP_nat_real(power_power_real(A_91),number_number_of_nat(bit1(bit1(pls)))) = hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(A_91),A_91)),A_91) ).
tff(fact_273_power3__eq__cube,axiom,
! [A_91: nat] : hAPP_nat_nat(power_power_nat(A_91),number_number_of_nat(bit1(bit1(pls)))) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(A_91),A_91)),A_91) ).
tff(fact_274_quartic__square__square,axiom,
! [X_1: int] : hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(X_1),number_number_of_nat(bit0(bit1(pls))))),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_int(power_power_int(X_1),number_number_of_nat(bit0(bit0(bit1(pls))))) ).
tff(fact_275_semiring__mult__2,axiom,
! [Z_6: int] : hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),Z_6) = hAPP_int_int(plus_plus_int(Z_6),Z_6) ).
tff(fact_276_semiring__mult__2,axiom,
! [Z_6: nat] : hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),Z_6) = hAPP_nat_nat(plus_plus_nat(Z_6),Z_6) ).
tff(fact_277_semiring__mult__2,axiom,
! [Z_6: real] : hAPP_real_real(times_times_real(number267125858f_real(bit0(bit1(pls)))),Z_6) = hAPP_real_real(plus_plus_real(Z_6),Z_6) ).
tff(fact_278_mult__2,axiom,
! [Z_5: int] : hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),Z_5) = hAPP_int_int(plus_plus_int(Z_5),Z_5) ).
tff(fact_279_mult__2,axiom,
! [Z_5: real] : hAPP_real_real(times_times_real(number267125858f_real(bit0(bit1(pls)))),Z_5) = hAPP_real_real(plus_plus_real(Z_5),Z_5) ).
tff(fact_280_semiring__mult__2__right,axiom,
! [Z_4: int] : hAPP_int_int(times_times_int(Z_4),number_number_of_int(bit0(bit1(pls)))) = hAPP_int_int(plus_plus_int(Z_4),Z_4) ).
tff(fact_281_semiring__mult__2__right,axiom,
! [Z_4: nat] : hAPP_nat_nat(times_times_nat(Z_4),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_nat(plus_plus_nat(Z_4),Z_4) ).
tff(fact_282_semiring__mult__2__right,axiom,
! [Z_4: real] : hAPP_real_real(times_times_real(Z_4),number267125858f_real(bit0(bit1(pls)))) = hAPP_real_real(plus_plus_real(Z_4),Z_4) ).
tff(fact_283_mult__2__right,axiom,
! [Z_3: int] : hAPP_int_int(times_times_int(Z_3),number_number_of_int(bit0(bit1(pls)))) = hAPP_int_int(plus_plus_int(Z_3),Z_3) ).
tff(fact_284_mult__2__right,axiom,
! [Z_3: real] : hAPP_real_real(times_times_real(Z_3),number267125858f_real(bit0(bit1(pls)))) = hAPP_real_real(plus_plus_real(Z_3),Z_3) ).
tff(fact_285_semiring__one__add__one__is__two,axiom,
hAPP_int_int(plus_plus_int(one_one_int),one_one_int) = number_number_of_int(bit0(bit1(pls))) ).
tff(fact_286_semiring__one__add__one__is__two,axiom,
hAPP_nat_nat(plus_plus_nat(one_one_nat),one_one_nat) = number_number_of_nat(bit0(bit1(pls))) ).
tff(fact_287_semiring__one__add__one__is__two,axiom,
hAPP_real_real(plus_plus_real(one_one_real),one_one_real) = number267125858f_real(bit0(bit1(pls))) ).
tff(fact_288_p0,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int))) ).
tff(fact_289__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_L_A1_096,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int)),hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls))))),one_one_int))) ).
tff(fact_290_prime__g__5,axiom,
! [P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( ( P_1 != number_number_of_int(bit0(bit1(pls))) )
=> ( ( P_1 != number_number_of_int(bit1(bit1(pls))) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(bit1(bit0(bit1(pls))))),P_1)) ) ) ) ).
tff(fact_291__096sum2sq_A_Is_M_A1_J_A_061_A_I4_A_K_Am_A_L_A1_J_A_K_At_096,axiom,
twoSqu949963151sum2sq(product_Pair_int_int(s,one_one_int)) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int)),t) ).
tff(fact_292_real__sum__squared__expand,axiom,
! [X_1: real,Y_1: real] : hAPP_nat_real(power_power_real(hAPP_real_real(plus_plus_real(X_1),Y_1)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_1),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_1),number_number_of_nat(bit0(bit1(pls)))))),hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(number267125858f_real(bit0(bit1(pls)))),X_1)),Y_1)) ).
tff(fact_293_four__x__squared,axiom,
! [X_1: real] : hAPP_real_real(times_times_real(number267125858f_real(bit0(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(X_1),number_number_of_nat(bit0(bit1(pls))))) = hAPP_nat_real(power_power_real(hAPP_real_real(times_times_real(number267125858f_real(bit0(bit1(pls)))),X_1)),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_294_power__less__power__Suc,axiom,
! [N_38: nat,A_90: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_90))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_90),N_38)),hAPP_int_int(times_times_int(A_90),hAPP_nat_int(power_power_int(A_90),N_38)))) ) ).
tff(fact_295_power__less__power__Suc,axiom,
! [N_38: nat,A_90: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_90))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_90),N_38)),hAPP_nat_nat(times_times_nat(A_90),hAPP_nat_nat(power_power_nat(A_90),N_38)))) ) ).
tff(fact_296_power__less__power__Suc,axiom,
! [N_38: nat,A_90: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_90))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_90),N_38)),hAPP_real_real(times_times_real(A_90),hAPP_nat_real(power_power_real(A_90),N_38)))) ) ).
tff(fact_297_power__gt1__lemma,axiom,
! [N_37: nat,A_89: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_89))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(times_times_int(A_89),hAPP_nat_int(power_power_int(A_89),N_37)))) ) ).
tff(fact_298_power__gt1__lemma,axiom,
! [N_37: nat,A_89: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_89))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(times_times_nat(A_89),hAPP_nat_nat(power_power_nat(A_89),N_37)))) ) ).
tff(fact_299_power__gt1__lemma,axiom,
! [N_37: nat,A_89: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_89))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(times_times_real(A_89),hAPP_nat_real(power_power_real(A_89),N_37)))) ) ).
tff(fact_300_power__le__imp__le__exp,axiom,
! [M_9: nat,N_36: nat,A_88: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_88))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_88),M_9)),hAPP_nat_int(power_power_int(A_88),N_36)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_9),N_36)) ) ) ).
tff(fact_301_power__le__imp__le__exp,axiom,
! [M_9: nat,N_36: nat,A_88: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_88))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_88),M_9)),hAPP_nat_nat(power_power_nat(A_88),N_36)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_9),N_36)) ) ) ).
tff(fact_302_power__le__imp__le__exp,axiom,
! [M_9: nat,N_36: nat,A_88: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_88))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_88),M_9)),hAPP_nat_real(power_power_real(A_88),N_36)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_9),N_36)) ) ) ).
tff(fact_303_power__increasing__iff,axiom,
! [X_2: nat,Y_2: nat,B_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(B_2),X_2)),hAPP_nat_int(power_power_int(B_2),Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2)) ) ) ).
tff(fact_304_power__increasing__iff,axiom,
! [X_2: nat,Y_2: nat,B_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(B_2),X_2)),hAPP_nat_nat(power_power_nat(B_2),Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2)) ) ) ).
tff(fact_305_power__increasing__iff,axiom,
! [X_2: nat,Y_2: nat,B_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(B_2),X_2)),hAPP_nat_real(power_power_real(B_2),Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2)) ) ) ).
tff(fact_306__096_091s_A_094_A2_A_061_As1_A_094_A2_093_A_Imod_A4_A_K_Am_A_L_A1_J_096,axiom,
hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(s1),number_number_of_nat(bit0(bit1(pls))))),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int))) ).
tff(fact_307_s0p,axiom,
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),s))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,s),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int)))
& hBOOL(hAPP_int_bool(zcong(s1,s),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int))) ) ).
tff(fact_308__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] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int)))
& hBOOL(hAPP_int_bool(zcong(s1,X),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int)))
& ! [Y: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int)))
& hBOOL(hAPP_int_bool(zcong(s1,Y),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int))) )
=> ( Y = X ) ) ) ).
tff(fact_309__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_1: int] :
~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),S_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,S_1),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int)))
& hBOOL(hAPP_int_bool(zcong(s1,S_1),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int))) ) ).
tff(fact_310_s1,axiom,
hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(s1),number_number_of_nat(bit0(bit1(pls)))),number_number_of_int(min)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int))) ).
tff(fact_311_power__eq__0__iff,axiom,
! [A_2: int,N: nat] :
( ( hAPP_nat_int(power_power_int(A_2),N) = zero_zero_int )
<=> ( ( A_2 = zero_zero_int )
& ( N != zero_zero_nat ) ) ) ).
tff(fact_312_power__eq__0__iff,axiom,
! [A_2: nat,N: nat] :
( ( hAPP_nat_nat(power_power_nat(A_2),N) = zero_zero_nat )
<=> ( ( A_2 = zero_zero_nat )
& ( N != zero_zero_nat ) ) ) ).
tff(fact_313_power__eq__0__iff,axiom,
! [A_2: real,N: nat] :
( ( hAPP_nat_real(power_power_real(A_2),N) = zero_zero_real )
<=> ( ( A_2 = zero_zero_real )
& ( N != zero_zero_nat ) ) ) ).
tff(fact_314_le__imp__power__dvd,axiom,
! [A_87: real,M_8: nat,N_35: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_8),N_35))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(power_power_real(A_87),M_8)),hAPP_nat_real(power_power_real(A_87),N_35))) ) ).
tff(fact_315_le__imp__power__dvd,axiom,
! [A_87: int,M_8: nat,N_35: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_8),N_35))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(A_87),M_8)),hAPP_nat_int(power_power_int(A_87),N_35))) ) ).
tff(fact_316_le__imp__power__dvd,axiom,
! [A_87: nat,M_8: nat,N_35: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_8),N_35))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(A_87),M_8)),hAPP_nat_nat(power_power_nat(A_87),N_35))) ) ).
tff(fact_317_dvd__power__le,axiom,
! [N_34: nat,M_7: nat,X_19: real,Y_16: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_19),Y_16))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_34),M_7))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(power_power_real(X_19),N_34)),hAPP_nat_real(power_power_real(Y_16),M_7))) ) ) ).
tff(fact_318_dvd__power__le,axiom,
! [N_34: nat,M_7: nat,X_19: int,Y_16: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_19),Y_16))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_34),M_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(X_19),N_34)),hAPP_nat_int(power_power_int(Y_16),M_7))) ) ) ).
tff(fact_319_dvd__power__le,axiom,
! [N_34: nat,M_7: nat,X_19: nat,Y_16: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_19),Y_16))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_34),M_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(X_19),N_34)),hAPP_nat_nat(power_power_nat(Y_16),M_7))) ) ) ).
tff(fact_320_power__le__dvd,axiom,
! [M_6: nat,A_86: real,N_33: nat,B_55: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(power_power_real(A_86),N_33)),B_55))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_6),N_33))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(power_power_real(A_86),M_6)),B_55)) ) ) ).
tff(fact_321_power__le__dvd,axiom,
! [M_6: nat,A_86: int,N_33: nat,B_55: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(A_86),N_33)),B_55))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_6),N_33))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(A_86),M_6)),B_55)) ) ) ).
tff(fact_322_power__le__dvd,axiom,
! [M_6: nat,A_86: nat,N_33: nat,B_55: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(A_86),N_33)),B_55))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_6),N_33))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(A_86),M_6)),B_55)) ) ) ).
tff(fact_323_power__eq__imp__eq__base,axiom,
! [A_85: int,N_32: nat,B_54: int] :
( ( hAPP_nat_int(power_power_int(A_85),N_32) = hAPP_nat_int(power_power_int(B_54),N_32) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_85))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_54))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32))
=> ( A_85 = B_54 ) ) ) ) ) ).
tff(fact_324_power__eq__imp__eq__base,axiom,
! [A_85: nat,N_32: nat,B_54: nat] :
( ( hAPP_nat_nat(power_power_nat(A_85),N_32) = hAPP_nat_nat(power_power_nat(B_54),N_32) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_85))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_54))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32))
=> ( A_85 = B_54 ) ) ) ) ) ).
tff(fact_325_power__eq__imp__eq__base,axiom,
! [A_85: real,N_32: nat,B_54: real] :
( ( hAPP_nat_real(power_power_real(A_85),N_32) = hAPP_nat_real(power_power_real(B_54),N_32) )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_85))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_54))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32))
=> ( A_85 = B_54 ) ) ) ) ) ).
tff(fact_326_zdvd__not__zless,axiom,
! [N_1: int,M: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M),N_1))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),M)) ) ) ).
tff(fact_327_zdvd__antisym__nonneg,axiom,
! [N_1: int,M: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),M))
=> ( M = N_1 ) ) ) ) ) ).
tff(fact_328_zdvd__mult__cancel,axiom,
! [K_1: int,M: int,N_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(times_times_int(K_1),M)),hAPP_int_int(times_times_int(K_1),N_1)))
=> ( ( K_1 != zero_zero_int )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N_1)) ) ) ).
tff(fact_329_dvd__power__same,axiom,
! [N_31: nat,X_18: real,Y_15: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_18),Y_15))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(power_power_real(X_18),N_31)),hAPP_nat_real(power_power_real(Y_15),N_31))) ) ).
tff(fact_330_dvd__power__same,axiom,
! [N_31: nat,X_18: int,Y_15: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_18),Y_15))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(X_18),N_31)),hAPP_nat_int(power_power_int(Y_15),N_31))) ) ).
tff(fact_331_dvd__power__same,axiom,
! [N_31: nat,X_18: nat,Y_15: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_18),Y_15))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(X_18),N_31)),hAPP_nat_nat(power_power_nat(Y_15),N_31))) ) ).
tff(fact_332_field__power__not__zero,axiom,
! [N_30: nat,A_84: int] :
( ( A_84 != zero_zero_int )
=> ( hAPP_nat_int(power_power_int(A_84),N_30) != zero_zero_int ) ) ).
tff(fact_333_field__power__not__zero,axiom,
! [N_30: nat,A_84: real] :
( ( A_84 != zero_zero_real )
=> ( hAPP_nat_real(power_power_real(A_84),N_30) != zero_zero_real ) ) ).
tff(fact_334_power__0__left,axiom,
! [N_29: nat] :
( ( ( N_29 = zero_zero_nat )
=> ( hAPP_nat_int(power_power_int(zero_zero_int),N_29) = one_one_int ) )
& ( ( N_29 != zero_zero_nat )
=> ( hAPP_nat_int(power_power_int(zero_zero_int),N_29) = zero_zero_int ) ) ) ).
tff(fact_335_power__0__left,axiom,
! [N_29: nat] :
( ( ( N_29 = zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(zero_zero_nat),N_29) = one_one_nat ) )
& ( ( N_29 != zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(zero_zero_nat),N_29) = zero_zero_nat ) ) ) ).
tff(fact_336_power__0__left,axiom,
! [N_29: nat] :
( ( ( N_29 = zero_zero_nat )
=> ( hAPP_nat_real(power_power_real(zero_zero_real),N_29) = one_one_real ) )
& ( ( N_29 != zero_zero_nat )
=> ( hAPP_nat_real(power_power_real(zero_zero_real),N_29) = zero_zero_real ) ) ) ).
tff(fact_337_zdvd__imp__le,axiom,
! [Z: int,N_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z),N_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),N_1)) ) ) ).
tff(fact_338_power__strict__mono,axiom,
! [N_28: nat,A_83: int,B_53: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_83),B_53))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_83))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_28))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_83),N_28)),hAPP_nat_int(power_power_int(B_53),N_28))) ) ) ) ).
tff(fact_339_power__strict__mono,axiom,
! [N_28: nat,A_83: nat,B_53: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_83),B_53))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_83))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_28))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_83),N_28)),hAPP_nat_nat(power_power_nat(B_53),N_28))) ) ) ) ).
tff(fact_340_power__strict__mono,axiom,
! [N_28: nat,A_83: real,B_53: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_83),B_53))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_83))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_28))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_83),N_28)),hAPP_nat_real(power_power_real(B_53),N_28))) ) ) ) ).
tff(fact_341_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_82: int] : hAPP_int_int(times_times_int(zero_zero_int),A_82) = zero_zero_int ).
tff(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_82: nat] : hAPP_nat_nat(times_times_nat(zero_zero_nat),A_82) = zero_zero_nat ).
tff(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_82: real] : hAPP_real_real(times_times_real(zero_zero_real),A_82) = zero_zero_real ).
tff(fact_344_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_81: int] : hAPP_int_int(times_times_int(A_81),zero_zero_int) = zero_zero_int ).
tff(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_81: nat] : hAPP_nat_nat(times_times_nat(A_81),zero_zero_nat) = zero_zero_nat ).
tff(fact_346_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_81: real] : hAPP_real_real(times_times_real(A_81),zero_zero_real) = zero_zero_real ).
tff(fact_347_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_80: int] : hAPP_int_int(plus_plus_int(zero_zero_int),A_80) = A_80 ).
tff(fact_348_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_80: nat] : hAPP_nat_nat(plus_plus_nat(zero_zero_nat),A_80) = A_80 ).
tff(fact_349_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_80: real] : hAPP_real_real(plus_plus_real(zero_zero_real),A_80) = A_80 ).
tff(fact_350_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_79: int] : hAPP_int_int(plus_plus_int(A_79),zero_zero_int) = A_79 ).
tff(fact_351_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_79: nat] : hAPP_nat_nat(plus_plus_nat(A_79),zero_zero_nat) = A_79 ).
tff(fact_352_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_79: real] : hAPP_real_real(plus_plus_real(A_79),zero_zero_real) = A_79 ).
tff(fact_353_add__0__iff,axiom,
! [B_2: int,A_2: int] :
( ( B_2 = hAPP_int_int(plus_plus_int(B_2),A_2) )
<=> ( A_2 = zero_zero_int ) ) ).
tff(fact_354_add__0__iff,axiom,
! [B_2: nat,A_2: nat] :
( ( B_2 = hAPP_nat_nat(plus_plus_nat(B_2),A_2) )
<=> ( A_2 = zero_zero_nat ) ) ).
tff(fact_355_add__0__iff,axiom,
! [B_2: real,A_2: real] :
( ( B_2 = hAPP_real_real(plus_plus_real(B_2),A_2) )
<=> ( A_2 = zero_zero_real ) ) ).
tff(fact_356_double__eq__0__iff,axiom,
! [A_2: int] :
( ( hAPP_int_int(plus_plus_int(A_2),A_2) = zero_zero_int )
<=> ( A_2 = zero_zero_int ) ) ).
tff(fact_357_double__eq__0__iff,axiom,
! [A_2: real] :
( ( hAPP_real_real(plus_plus_real(A_2),A_2) = zero_zero_real )
<=> ( A_2 = zero_zero_real ) ) ).
tff(fact_358_Pls__def,axiom,
pls = zero_zero_int ).
tff(fact_359_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
tff(fact_360_zadd__0,axiom,
! [Z: int] : hAPP_int_int(plus_plus_int(zero_zero_int),Z) = Z ).
tff(fact_361_zadd__0__right,axiom,
! [Z: int] : hAPP_int_int(plus_plus_int(Z),zero_zero_int) = Z ).
tff(fact_362_zero__le__power,axiom,
! [N_27: nat,A_78: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_78))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(A_78),N_27))) ) ).
tff(fact_363_zero__le__power,axiom,
! [N_27: nat,A_78: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_78))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(A_78),N_27))) ) ).
tff(fact_364_zero__le__power,axiom,
! [N_27: nat,A_78: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_78))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(power_power_real(A_78),N_27))) ) ).
tff(fact_365_power__mono,axiom,
! [N_26: nat,A_77: int,B_52: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_77),B_52))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_77))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_77),N_26)),hAPP_nat_int(power_power_int(B_52),N_26))) ) ) ).
tff(fact_366_power__mono,axiom,
! [N_26: nat,A_77: nat,B_52: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_77),B_52))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_77))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_77),N_26)),hAPP_nat_nat(power_power_nat(B_52),N_26))) ) ) ).
tff(fact_367_power__mono,axiom,
! [N_26: nat,A_77: real,B_52: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_77),B_52))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_77))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_77),N_26)),hAPP_nat_real(power_power_real(B_52),N_26))) ) ) ).
tff(fact_368_zero__less__power,axiom,
! [N_25: nat,A_76: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_76))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(power_power_int(A_76),N_25))) ) ).
tff(fact_369_zero__less__power,axiom,
! [N_25: nat,A_76: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_76))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(A_76),N_25))) ) ).
tff(fact_370_zero__less__power,axiom,
! [N_25: nat,A_76: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_76))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(power_power_real(A_76),N_25))) ) ).
tff(fact_371_zcong__zpower__zmult,axiom,
! [Z: nat,X_1: int,Y_1: nat,P_1: int] :
( hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X_1),Y_1),one_one_int),P_1))
=> hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X_1),hAPP_nat_nat(times_times_nat(Y_1),Z)),one_one_int),P_1)) ) ).
tff(fact_372_zdvd__reduce,axiom,
! [K: int,N: int,Ma: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),hAPP_int_int(plus_plus_int(N),hAPP_int_int(times_times_int(K),Ma))))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),N)) ) ).
tff(fact_373_zdvd__period,axiom,
! [C_2: int,X_2: int,Ta: int,A_2: int,D: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_2),D))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_2),hAPP_int_int(plus_plus_int(X_2),Ta)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_2),hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(X_2),hAPP_int_int(times_times_int(C_2),D))),Ta))) ) ) ).
tff(fact_374_power__less__imp__less__base,axiom,
! [A_75: int,N_24: nat,B_51: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_75),N_24)),hAPP_nat_int(power_power_int(B_51),N_24)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_51))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_75),B_51)) ) ) ).
tff(fact_375_power__less__imp__less__base,axiom,
! [A_75: nat,N_24: nat,B_51: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_75),N_24)),hAPP_nat_nat(power_power_nat(B_51),N_24)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_51))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_75),B_51)) ) ) ).
tff(fact_376_power__less__imp__less__base,axiom,
! [A_75: real,N_24: nat,B_51: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_75),N_24)),hAPP_nat_real(power_power_real(B_51),N_24)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_51))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_75),B_51)) ) ) ).
tff(fact_377_power__decreasing,axiom,
! [A_74: int,N_23: nat,N_22: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_23),N_22))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_74))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_74),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_74),N_22)),hAPP_nat_int(power_power_int(A_74),N_23))) ) ) ) ).
tff(fact_378_power__decreasing,axiom,
! [A_74: nat,N_23: nat,N_22: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_23),N_22))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_74))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_74),one_one_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_74),N_22)),hAPP_nat_nat(power_power_nat(A_74),N_23))) ) ) ) ).
tff(fact_379_power__decreasing,axiom,
! [A_74: real,N_23: nat,N_22: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_23),N_22))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_74))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_74),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_74),N_22)),hAPP_nat_real(power_power_real(A_74),N_23))) ) ) ) ).
tff(fact_380_power__strict__decreasing,axiom,
! [A_73: int,N_21: nat,N_20: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_21),N_20))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_73))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_73),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_73),N_20)),hAPP_nat_int(power_power_int(A_73),N_21))) ) ) ) ).
tff(fact_381_power__strict__decreasing,axiom,
! [A_73: nat,N_21: nat,N_20: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_21),N_20))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_73))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_73),one_one_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_73),N_20)),hAPP_nat_nat(power_power_nat(A_73),N_21))) ) ) ) ).
tff(fact_382_power__strict__decreasing,axiom,
! [A_73: real,N_21: nat,N_20: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_21),N_20))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_73))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_73),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_73),N_20)),hAPP_nat_real(power_power_real(A_73),N_21))) ) ) ) ).
tff(fact_383_even__less__0__iff,axiom,
! [A_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_2),A_2)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),zero_zero_int)) ) ).
tff(fact_384_even__less__0__iff,axiom,
! [A_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_2),A_2)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_2),zero_zero_real)) ) ).
tff(fact_385_sum__squares__eq__zero__iff,axiom,
! [X_2: int,Y_2: int] :
( ( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(X_2),X_2)),hAPP_int_int(times_times_int(Y_2),Y_2)) = zero_zero_int )
<=> ( ( X_2 = zero_zero_int )
& ( Y_2 = zero_zero_int ) ) ) ).
tff(fact_386_sum__squares__eq__zero__iff,axiom,
! [X_2: real,Y_2: real] :
( ( hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_2),X_2)),hAPP_real_real(times_times_real(Y_2),Y_2)) = zero_zero_real )
<=> ( ( X_2 = zero_zero_real )
& ( Y_2 = zero_zero_real ) ) ) ).
tff(fact_387_add__scale__eq__noteq,axiom,
! [C_27: int,D_8: int,A_72: int,B_50: int,R_4: int] :
( ( R_4 != zero_zero_int )
=> ( ( ( A_72 = B_50 )
& ( C_27 != D_8 ) )
=> ( hAPP_int_int(plus_plus_int(A_72),hAPP_int_int(times_times_int(R_4),C_27)) != hAPP_int_int(plus_plus_int(B_50),hAPP_int_int(times_times_int(R_4),D_8)) ) ) ) ).
tff(fact_388_add__scale__eq__noteq,axiom,
! [C_27: nat,D_8: nat,A_72: nat,B_50: nat,R_4: nat] :
( ( R_4 != zero_zero_nat )
=> ( ( ( A_72 = B_50 )
& ( C_27 != D_8 ) )
=> ( hAPP_nat_nat(plus_plus_nat(A_72),hAPP_nat_nat(times_times_nat(R_4),C_27)) != hAPP_nat_nat(plus_plus_nat(B_50),hAPP_nat_nat(times_times_nat(R_4),D_8)) ) ) ) ).
tff(fact_389_add__scale__eq__noteq,axiom,
! [C_27: real,D_8: real,A_72: real,B_50: real,R_4: real] :
( ( R_4 != zero_zero_real )
=> ( ( ( A_72 = B_50 )
& ( C_27 != D_8 ) )
=> ( hAPP_real_real(plus_plus_real(A_72),hAPP_real_real(times_times_real(R_4),C_27)) != hAPP_real_real(plus_plus_real(B_50),hAPP_real_real(times_times_real(R_4),D_8)) ) ) ) ).
tff(fact_390_zprime__zdvd__power,axiom,
! [A: int,N_1: nat,P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),hAPP_nat_int(power_power_int(A),N_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),A)) ) ) ).
tff(fact_391_semiring__norm_I112_J,axiom,
zero_zero_int = number_number_of_int(pls) ).
tff(fact_392_semiring__norm_I112_J,axiom,
zero_zero_real = number267125858f_real(pls) ).
tff(fact_393_number__of__Pls,axiom,
number_number_of_int(pls) = zero_zero_int ).
tff(fact_394_number__of__Pls,axiom,
number267125858f_real(pls) = zero_zero_real ).
tff(fact_395_semiring__numeral__0__eq__0,axiom,
number_number_of_int(pls) = zero_zero_int ).
tff(fact_396_semiring__numeral__0__eq__0,axiom,
number_number_of_nat(pls) = zero_zero_nat ).
tff(fact_397_semiring__numeral__0__eq__0,axiom,
number267125858f_real(pls) = zero_zero_real ).
tff(fact_398_bin__less__0__simps_I4_J,axiom,
! [W_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(W_1)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),zero_zero_int)) ) ).
tff(fact_399_bin__less__0__simps_I1_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),zero_zero_int)) ).
tff(fact_400_bin__less__0__simps_I3_J,axiom,
! [W_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(W_1)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),zero_zero_int)) ) ).
tff(fact_401_zero__is__num__zero,axiom,
zero_zero_int = number_number_of_int(pls) ).
tff(fact_402_int__0__less__1,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
tff(fact_403_pos__zmult__pos,axiom,
! [B: int,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(A),B)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) ) ) ).
tff(fact_404_zmult__zless__mono2,axiom,
! [K_1: int,I_1: int,J_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),J_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(K_1),I_1)),hAPP_int_int(times_times_int(K_1),J_1))) ) ) ).
tff(fact_405_odd__nonzero,axiom,
! [Z: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),Z)),Z) != zero_zero_int ).
tff(fact_406_power__Suc__less,axiom,
! [N_19: nat,A_71: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_71))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_71),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_71),hAPP_nat_int(power_power_int(A_71),N_19))),hAPP_nat_int(power_power_int(A_71),N_19))) ) ) ).
tff(fact_407_power__Suc__less,axiom,
! [N_19: nat,A_71: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_71))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_71),one_one_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_71),hAPP_nat_nat(power_power_nat(A_71),N_19))),hAPP_nat_nat(power_power_nat(A_71),N_19))) ) ) ).
tff(fact_408_power__Suc__less,axiom,
! [N_19: nat,A_71: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_71))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_71),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_71),hAPP_nat_real(power_power_real(A_71),N_19))),hAPP_nat_real(power_power_real(A_71),N_19))) ) ) ).
tff(fact_409_zprime__power__zdvd__cancel__left,axiom,
! [N_1: nat,B: int,A: int,P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_1),N_1)),hAPP_int_int(times_times_int(A),B)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_1),N_1)),B)) ) ) ) ).
tff(fact_410_zprime__power__zdvd__cancel__right,axiom,
! [N_1: nat,A: int,B: int,P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_1),N_1)),hAPP_int_int(times_times_int(A),B)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_1),N_1)),A)) ) ) ) ).
tff(fact_411_sum__squares__ge__zero,axiom,
! [X_17: int,Y_14: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(X_17),X_17)),hAPP_int_int(times_times_int(Y_14),Y_14)))) ).
tff(fact_412_sum__squares__ge__zero,axiom,
! [X_17: real,Y_14: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_17),X_17)),hAPP_real_real(times_times_real(Y_14),Y_14)))) ).
tff(fact_413_sum__squares__le__zero__iff,axiom,
! [X_2: int,Y_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(X_2),X_2)),hAPP_int_int(times_times_int(Y_2),Y_2))),zero_zero_int))
<=> ( ( X_2 = zero_zero_int )
& ( Y_2 = zero_zero_int ) ) ) ).
tff(fact_414_sum__squares__le__zero__iff,axiom,
! [X_2: real,Y_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_2),X_2)),hAPP_real_real(times_times_real(Y_2),Y_2))),zero_zero_real))
<=> ( ( X_2 = zero_zero_real )
& ( Y_2 = zero_zero_real ) ) ) ).
tff(fact_415_less__nat__number__of,axiom,
! [V_7: int,V_8: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,number_number_of_nat(V_7)),number_number_of_nat(V_8)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_7),V_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_8)) )
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_7),V_8)) ) ) ).
tff(fact_416_not__sum__squares__lt__zero,axiom,
! [X_16: int,Y_13: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(X_16),X_16)),hAPP_int_int(times_times_int(Y_13),Y_13))),zero_zero_int)) ).
tff(fact_417_not__sum__squares__lt__zero,axiom,
! [X_16: real,Y_13: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_16),X_16)),hAPP_real_real(times_times_real(Y_13),Y_13))),zero_zero_real)) ).
tff(fact_418_sum__squares__gt__zero__iff,axiom,
! [X_2: int,Y_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(X_2),X_2)),hAPP_int_int(times_times_int(Y_2),Y_2))))
<=> ( ( X_2 != zero_zero_int )
| ( Y_2 != zero_zero_int ) ) ) ).
tff(fact_419_sum__squares__gt__zero__iff,axiom,
! [X_2: real,Y_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_2),X_2)),hAPP_real_real(times_times_real(Y_2),Y_2))))
<=> ( ( X_2 != zero_zero_real )
| ( Y_2 != zero_zero_real ) ) ) ).
tff(fact_420_le__nat__number__of,axiom,
! [V_7: int,V_8: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,number_number_of_nat(V_7)),number_number_of_nat(V_8)))
<=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_7),V_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_7),pls)) ) ) ).
tff(fact_421_number__of__Bit0,axiom,
! [W_5: int] : number_number_of_int(bit0(W_5)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(zero_zero_int),number_number_of_int(W_5))),number_number_of_int(W_5)) ).
tff(fact_422_number__of__Bit0,axiom,
! [W_5: int] : number267125858f_real(bit0(W_5)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(zero_zero_real),number267125858f_real(W_5))),number267125858f_real(W_5)) ).
tff(fact_423_power__one__right,axiom,
! [A_70: int] : hAPP_nat_int(power_power_int(A_70),one_one_nat) = A_70 ).
tff(fact_424_power__one__right,axiom,
! [A_70: real] : hAPP_nat_real(power_power_real(A_70),one_one_nat) = A_70 ).
tff(fact_425_power__one__right,axiom,
! [A_70: nat] : hAPP_nat_nat(power_power_nat(A_70),one_one_nat) = A_70 ).
tff(fact_426_int__one__le__iff__zero__less,axiom,
! [Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Z_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1)) ) ).
tff(fact_427_pos__zmult__eq__1__iff,axiom,
! [N: int,Ma: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( ( hAPP_int_int(times_times_int(Ma),N) = one_one_int )
<=> ( ( Ma = one_one_int )
& ( N = one_one_int ) ) ) ) ).
tff(fact_428_odd__less__0,axiom,
! [Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),Z_1)),Z_1)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),zero_zero_int)) ) ).
tff(fact_429_less__special_I1_J,axiom,
! [Y_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),number_number_of_int(Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Y_2)) ) ).
tff(fact_430_less__special_I1_J,axiom,
! [Y_2: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),number267125858f_real(Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Y_2)) ) ).
tff(fact_431_less__special_I3_J,axiom,
! [X_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(X_2)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),pls)) ) ).
tff(fact_432_less__special_I3_J,axiom,
! [X_2: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(X_2)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),pls)) ) ).
tff(fact_433_le__special_I1_J,axiom,
! [Y_2: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),number267125858f_real(Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Y_2)) ) ).
tff(fact_434_le__special_I1_J,axiom,
! [Y_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Y_2)) ) ).
tff(fact_435_le__special_I3_J,axiom,
! [X_2: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(X_2)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),pls)) ) ).
tff(fact_436_le__special_I3_J,axiom,
! [X_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(X_2)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),pls)) ) ).
tff(fact_437_le__imp__0__less,axiom,
! [Z: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(one_one_int),Z))) ) ).
tff(fact_438_zero__power2,axiom,
hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_real ).
tff(fact_439_zero__power2,axiom,
hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_nat ).
tff(fact_440_zero__power2,axiom,
hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_int ).
tff(fact_441_zero__eq__power2,axiom,
! [A_2: real] :
( ( hAPP_nat_real(power_power_real(A_2),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_real )
<=> ( A_2 = zero_zero_real ) ) ).
tff(fact_442_zero__eq__power2,axiom,
! [A_2: int] :
( ( hAPP_nat_int(power_power_int(A_2),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_int )
<=> ( A_2 = zero_zero_int ) ) ).
tff(fact_443_zero__le__power2,axiom,
! [A_69: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(power_power_real(A_69),number_number_of_nat(bit0(bit1(pls)))))) ).
tff(fact_444_zero__le__power2,axiom,
! [A_69: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(A_69),number_number_of_nat(bit0(bit1(pls)))))) ).
tff(fact_445_power2__le__imp__le,axiom,
! [X_15: real,Y_12: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(X_15),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_12),number_number_of_nat(bit0(bit1(pls))))))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_12))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_15),Y_12)) ) ) ).
tff(fact_446_power2__le__imp__le,axiom,
! [X_15: nat,Y_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(X_15),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_nat(power_power_nat(Y_12),number_number_of_nat(bit0(bit1(pls))))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_15),Y_12)) ) ) ).
tff(fact_447_power2__le__imp__le,axiom,
! [X_15: int,Y_12: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(X_15),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_12),number_number_of_nat(bit0(bit1(pls))))))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_15),Y_12)) ) ) ).
tff(fact_448_power2__eq__imp__eq,axiom,
! [X_14: real,Y_11: real] :
( ( hAPP_nat_real(power_power_real(X_14),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_real(power_power_real(Y_11),number_number_of_nat(bit0(bit1(pls)))) )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_14))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_11))
=> ( X_14 = Y_11 ) ) ) ) ).
tff(fact_449_power2__eq__imp__eq,axiom,
! [X_14: nat,Y_11: nat] :
( ( hAPP_nat_nat(power_power_nat(X_14),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_nat(power_power_nat(Y_11),number_number_of_nat(bit0(bit1(pls)))) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),X_14))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_11))
=> ( X_14 = Y_11 ) ) ) ) ).
tff(fact_450_power2__eq__imp__eq,axiom,
! [X_14: int,Y_11: int] :
( ( hAPP_nat_int(power_power_int(X_14),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_int(power_power_int(Y_11),number_number_of_nat(bit0(bit1(pls)))) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_14))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_11))
=> ( X_14 = Y_11 ) ) ) ) ).
tff(fact_451_power2__less__0,axiom,
! [A_68: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_68),number_number_of_nat(bit0(bit1(pls))))),zero_zero_real)) ).
tff(fact_452_power2__less__0,axiom,
! [A_68: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_68),number_number_of_nat(bit0(bit1(pls))))),zero_zero_int)) ).
tff(fact_453_zero__less__power2,axiom,
! [A_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(power_power_real(A_2),number_number_of_nat(bit0(bit1(pls))))))
<=> ( A_2 != zero_zero_real ) ) ).
tff(fact_454_zero__less__power2,axiom,
! [A_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(power_power_int(A_2),number_number_of_nat(bit0(bit1(pls))))))
<=> ( A_2 != zero_zero_int ) ) ).
tff(fact_455_sum__power2__eq__zero__iff,axiom,
! [X_2: real,Y_2: real] :
( ( hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(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_456_sum__power2__eq__zero__iff,axiom,
! [X_2: int,Y_2: int] :
( ( hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(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_457_power__commutes,axiom,
! [A_67: nat,N_18: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(A_67),N_18)),A_67) = hAPP_nat_nat(times_times_nat(A_67),hAPP_nat_nat(power_power_nat(A_67),N_18)) ).
tff(fact_458_power__commutes,axiom,
! [A_67: real,N_18: nat] : hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(A_67),N_18)),A_67) = hAPP_real_real(times_times_real(A_67),hAPP_nat_real(power_power_real(A_67),N_18)) ).
tff(fact_459_power__commutes,axiom,
! [A_67: int,N_18: nat] : hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(A_67),N_18)),A_67) = hAPP_int_int(times_times_int(A_67),hAPP_nat_int(power_power_int(A_67),N_18)) ).
tff(fact_460_power__mult__distrib,axiom,
! [A_66: nat,B_49: nat,N_17: nat] : hAPP_nat_nat(power_power_nat(hAPP_nat_nat(times_times_nat(A_66),B_49)),N_17) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(A_66),N_17)),hAPP_nat_nat(power_power_nat(B_49),N_17)) ).
tff(fact_461_power__mult__distrib,axiom,
! [A_66: real,B_49: real,N_17: nat] : hAPP_nat_real(power_power_real(hAPP_real_real(times_times_real(A_66),B_49)),N_17) = hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(A_66),N_17)),hAPP_nat_real(power_power_real(B_49),N_17)) ).
tff(fact_462_power__mult__distrib,axiom,
! [A_66: int,B_49: int,N_17: nat] : hAPP_nat_int(power_power_int(hAPP_int_int(times_times_int(A_66),B_49)),N_17) = hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(A_66),N_17)),hAPP_nat_int(power_power_int(B_49),N_17)) ).
tff(fact_463_power__add,axiom,
! [A_65: nat,M_5: nat,N_16: nat] : hAPP_nat_nat(power_power_nat(A_65),hAPP_nat_nat(plus_plus_nat(M_5),N_16)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(A_65),M_5)),hAPP_nat_nat(power_power_nat(A_65),N_16)) ).
tff(fact_464_power__add,axiom,
! [A_65: real,M_5: nat,N_16: nat] : hAPP_nat_real(power_power_real(A_65),hAPP_nat_nat(plus_plus_nat(M_5),N_16)) = hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(A_65),M_5)),hAPP_nat_real(power_power_real(A_65),N_16)) ).
tff(fact_465_power__add,axiom,
! [A_65: int,M_5: nat,N_16: nat] : hAPP_nat_int(power_power_int(A_65),hAPP_nat_nat(plus_plus_nat(M_5),N_16)) = hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(A_65),M_5)),hAPP_nat_int(power_power_int(A_65),N_16)) ).
tff(fact_466_power__one,axiom,
! [N_15: nat] : hAPP_nat_real(power_power_real(one_one_real),N_15) = one_one_real ).
tff(fact_467_power__one,axiom,
! [N_15: nat] : hAPP_nat_nat(power_power_nat(one_one_nat),N_15) = one_one_nat ).
tff(fact_468_power__one,axiom,
! [N_15: nat] : hAPP_nat_int(power_power_int(one_one_int),N_15) = one_one_int ).
tff(fact_469_power__mult,axiom,
! [A_64: nat,M_4: nat,N_14: nat] : hAPP_nat_nat(power_power_nat(A_64),hAPP_nat_nat(times_times_nat(M_4),N_14)) = hAPP_nat_nat(power_power_nat(hAPP_nat_nat(power_power_nat(A_64),M_4)),N_14) ).
tff(fact_470_power__mult,axiom,
! [A_64: real,M_4: nat,N_14: nat] : hAPP_nat_real(power_power_real(A_64),hAPP_nat_nat(times_times_nat(M_4),N_14)) = hAPP_nat_real(power_power_real(hAPP_nat_real(power_power_real(A_64),M_4)),N_14) ).
tff(fact_471_power__mult,axiom,
! [A_64: int,M_4: nat,N_14: nat] : hAPP_nat_int(power_power_int(A_64),hAPP_nat_nat(times_times_nat(M_4),N_14)) = hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(A_64),M_4)),N_14) ).
tff(fact_472_power2__less__imp__less,axiom,
! [X_13: real,Y_10: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(X_13),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_10),number_number_of_nat(bit0(bit1(pls))))))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_10))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_13),Y_10)) ) ) ).
tff(fact_473_power2__less__imp__less,axiom,
! [X_13: nat,Y_10: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(X_13),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_nat(power_power_nat(Y_10),number_number_of_nat(bit0(bit1(pls))))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_10))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_13),Y_10)) ) ) ).
tff(fact_474_power2__less__imp__less,axiom,
! [X_13: int,Y_10: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(X_13),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_10),number_number_of_nat(bit0(bit1(pls))))))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_10))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_13),Y_10)) ) ) ).
tff(fact_475_sum__power2__ge__zero,axiom,
! [X_12: real,Y_9: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_12),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_9),number_number_of_nat(bit0(bit1(pls))))))) ).
tff(fact_476_sum__power2__ge__zero,axiom,
! [X_12: int,Y_9: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X_12),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_9),number_number_of_nat(bit0(bit1(pls))))))) ).
tff(fact_477_sum__power2__le__zero__iff,axiom,
! [X_2: real,Y_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(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_478_sum__power2__le__zero__iff,axiom,
! [X_2: int,Y_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(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_479_not__sum__power2__lt__zero,axiom,
! [X_11: real,Y_8: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_11),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_8),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_real)) ).
tff(fact_480_not__sum__power2__lt__zero,axiom,
! [X_11: int,Y_8: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X_11),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_8),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_int)) ).
tff(fact_481_sum__power2__gt__zero__iff,axiom,
! [X_2: real,Y_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_2),number_number_of_nat(bit0(bit1(pls)))))))
<=> ( ( X_2 != zero_zero_real )
| ( Y_2 != zero_zero_real ) ) ) ).
tff(fact_482_sum__power2__gt__zero__iff,axiom,
! [X_2: int,Y_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_2),number_number_of_nat(bit0(bit1(pls)))))))
<=> ( ( X_2 != zero_zero_int )
| ( Y_2 != zero_zero_int ) ) ) ).
tff(fact_483_zero__le__even__power_H,axiom,
! [A_63: real,N_13: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(power_power_real(A_63),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_13)))) ).
tff(fact_484_zero__le__even__power_H,axiom,
! [A_63: int,N_13: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(A_63),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_13)))) ).
tff(fact_485_one__le__power,axiom,
! [N_12: nat,A_62: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_62))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(power_power_real(A_62),N_12))) ) ).
tff(fact_486_one__le__power,axiom,
! [N_12: nat,A_62: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_62))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(power_power_nat(A_62),N_12))) ) ).
tff(fact_487_one__le__power,axiom,
! [N_12: nat,A_62: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_62))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_nat_int(power_power_int(A_62),N_12))) ) ).
tff(fact_488_power__increasing,axiom,
! [A_61: real,N_11: nat,N_10: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_11),N_10))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_61))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_61),N_11)),hAPP_nat_real(power_power_real(A_61),N_10))) ) ) ).
tff(fact_489_power__increasing,axiom,
! [A_61: nat,N_11: nat,N_10: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_11),N_10))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_61))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_61),N_11)),hAPP_nat_nat(power_power_nat(A_61),N_10))) ) ) ).
tff(fact_490_power__increasing,axiom,
! [A_61: int,N_11: nat,N_10: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_11),N_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_61))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_61),N_11)),hAPP_nat_int(power_power_int(A_61),N_10))) ) ) ).
tff(fact_491_power__inject__exp,axiom,
! [Ma: nat,N: nat,A_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_2))
=> ( ( hAPP_nat_real(power_power_real(A_2),Ma) = hAPP_nat_real(power_power_real(A_2),N) )
<=> ( Ma = N ) ) ) ).
tff(fact_492_power__inject__exp,axiom,
! [Ma: nat,N: nat,A_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_2))
=> ( ( hAPP_nat_nat(power_power_nat(A_2),Ma) = hAPP_nat_nat(power_power_nat(A_2),N) )
<=> ( Ma = N ) ) ) ).
tff(fact_493_power__inject__exp,axiom,
! [Ma: nat,N: nat,A_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_2))
=> ( ( hAPP_nat_int(power_power_int(A_2),Ma) = hAPP_nat_int(power_power_int(A_2),N) )
<=> ( Ma = N ) ) ) ).
tff(fact_494_power__strict__increasing__iff,axiom,
! [X_2: nat,Y_2: nat,B_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(B_2),X_2)),hAPP_nat_real(power_power_real(B_2),Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) ) ) ).
tff(fact_495_power__strict__increasing__iff,axiom,
! [X_2: nat,Y_2: nat,B_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(B_2),X_2)),hAPP_nat_nat(power_power_nat(B_2),Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) ) ) ).
tff(fact_496_power__strict__increasing__iff,axiom,
! [X_2: nat,Y_2: nat,B_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(B_2),X_2)),hAPP_nat_int(power_power_int(B_2),Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) ) ) ).
tff(fact_497_power__less__imp__less__exp,axiom,
! [M_3: nat,N_9: nat,A_60: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_60))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_60),M_3)),hAPP_nat_real(power_power_real(A_60),N_9)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_3),N_9)) ) ) ).
tff(fact_498_power__less__imp__less__exp,axiom,
! [M_3: nat,N_9: nat,A_60: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_60))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_60),M_3)),hAPP_nat_nat(power_power_nat(A_60),N_9)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_3),N_9)) ) ) ).
tff(fact_499_power__less__imp__less__exp,axiom,
! [M_3: nat,N_9: nat,A_60: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_60))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_60),M_3)),hAPP_nat_int(power_power_int(A_60),N_9)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_3),N_9)) ) ) ).
tff(fact_500_power__strict__increasing,axiom,
! [A_59: real,N_8: nat,N_7: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_8),N_7))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_59))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_59),N_8)),hAPP_nat_real(power_power_real(A_59),N_7))) ) ) ).
tff(fact_501_power__strict__increasing,axiom,
! [A_59: nat,N_8: nat,N_7: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_8),N_7))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_59))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_59),N_8)),hAPP_nat_nat(power_power_nat(A_59),N_7))) ) ) ).
tff(fact_502_power__strict__increasing,axiom,
! [A_59: int,N_8: nat,N_7: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_8),N_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_59))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_59),N_8)),hAPP_nat_int(power_power_int(A_59),N_7))) ) ) ).
tff(fact_503_s,axiom,
hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls)))),number_number_of_int(min)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int))) ).
tff(fact_504_Euler_Oaux____1,axiom,
! [Y_1: int,X_1: int,P_1: int] :
( ~ hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_1))
=> ( hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(Y_1),number_number_of_nat(bit0(bit1(pls)))),X_1),P_1))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),Y_1)) ) ) ).
tff(fact_505_int__pos__lt__two__imp__zero__or__one,axiom,
! [X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),number_number_of_int(bit0(bit1(pls)))))
=> ( ( X_1 = zero_zero_int )
| ( X_1 = one_one_int ) ) ) ) ).
tff(fact_506_even__power__le__0__imp__0,axiom,
! [A_58: real,K_4: nat] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_58),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),K_4))),zero_zero_real))
=> ( A_58 = zero_zero_real ) ) ).
tff(fact_507_even__power__le__0__imp__0,axiom,
! [A_58: int,K_4: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_58),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),K_4))),zero_zero_int))
=> ( A_58 = zero_zero_int ) ) ).
tff(fact_508_zprime__def,axiom,
! [P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),P_3))
& ! [M_2: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M_2))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M_2),P_3)) )
=> ( ( M_2 = one_one_int )
| ( M_2 = P_3 ) ) ) ) ) ).
tff(fact_509_self__quotient__aux2,axiom,
! [R_1: int,Q_1: int,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( ( A = hAPP_int_int(plus_plus_int(R_1),hAPP_int_int(times_times_int(A),Q_1)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),one_one_int)) ) ) ) ).
tff(fact_510_self__quotient__aux1,axiom,
! [R_1: int,Q_1: int,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( ( A = hAPP_int_int(plus_plus_int(R_1),hAPP_int_int(times_times_int(A),Q_1)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),A))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Q_1)) ) ) ) ).
tff(fact_511_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit0(bit1(pls))))) ).
tff(fact_512__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] : ~ hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(S1),number_number_of_nat(bit0(bit1(pls)))),number_number_of_int(min)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int))) ).
tff(fact_513__096Legendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_061_A1_096,axiom,
legendre(number_number_of_int(min),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int)) = one_one_int ).
tff(fact_514_nat__zero__less__power__iff,axiom,
! [X_2: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(X_2),N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_2))
| ( N = zero_zero_nat ) ) ) ).
tff(fact_515_zero__less__power__nat__eq,axiom,
! [X_2: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(X_2),N)))
<=> ( ( N = zero_zero_nat )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_2)) ) ) ).
tff(fact_516_zero__less__power__nat__eq__number__of,axiom,
! [X_2: nat,W_1: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(X_2),number_number_of_nat(W_1))))
<=> ( ( number_number_of_nat(W_1) = zero_zero_nat )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_2)) ) ) ).
tff(fact_517_nat__power__less__imp__less,axiom,
! [M: nat,N_1: nat,I_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),I_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(I_1),M)),hAPP_nat_nat(power_power_nat(I_1),N_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ) ).
tff(fact_518_rel__simps_I47_J,axiom,
! [K: int] :
( ( bit1(K) = min )
<=> ( K = min ) ) ).
tff(fact_519_rel__simps_I43_J,axiom,
! [L_1: int] :
( ( min = bit1(L_1) )
<=> ( min = L_1 ) ) ).
tff(fact_520_Bit1__Min,axiom,
bit1(min) = min ).
tff(fact_521_rel__simps_I37_J,axiom,
pls != min ).
tff(fact_522_rel__simps_I40_J,axiom,
min != pls ).
tff(fact_523_rel__simps_I45_J,axiom,
! [K_1: int] : bit0(K_1) != min ).
tff(fact_524_rel__simps_I42_J,axiom,
! [L: int] : min != bit0(L) ).
tff(fact_525_rel__simps_I7_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),min)) ).
tff(fact_526_rel__simps_I24_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),min)) ).
tff(fact_527_not__real__square__gt__zero,axiom,
! [X_2: real] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(X_2),X_2)))
<=> ( X_2 = zero_zero_real ) ) ).
tff(fact_528_rel__simps_I13_J,axiom,
! [K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K)),min))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),min)) ) ).
tff(fact_529_rel__simps_I9_J,axiom,
! [K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),bit1(K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K)) ) ).
tff(fact_530_rel__simps_I3_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),min)) ).
tff(fact_531_rel__simps_I6_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),pls)) ).
tff(fact_532_rel__simps_I8_J,axiom,
! [K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),bit0(K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K)) ) ).
tff(fact_533_bin__less__0__simps_I2_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),zero_zero_int)) ).
tff(fact_534_rel__simps_I30_J,axiom,
! [K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K)),min))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),min)) ) ).
tff(fact_535_rel__simps_I26_J,axiom,
! [K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),bit1(K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),K)) ) ).
tff(fact_536_rel__simps_I20_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),min)) ).
tff(fact_537_rel__simps_I23_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),pls)) ).
tff(fact_538_rel__simps_I28_J,axiom,
! [K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K)),min))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),min)) ) ).
tff(fact_539_eq__number__of__Pls__Min,axiom,
number_number_of_int(pls) != number_number_of_int(min) ).
tff(fact_540_power__dvd__imp__le,axiom,
! [I_1: nat,M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(I_1),M)),hAPP_nat_nat(power_power_nat(I_1),N_1)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),I_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ) ).
tff(fact_541_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_10: real] : hAPP_nat_real(power_power_real(X_10),zero_zero_nat) = one_one_real ).
tff(fact_542_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_10: nat] : hAPP_nat_nat(power_power_nat(X_10),zero_zero_nat) = one_one_nat ).
tff(fact_543_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_10: int] : hAPP_nat_int(power_power_int(X_10),zero_zero_nat) = one_one_int ).
tff(fact_544_power__0,axiom,
! [A_57: real] : hAPP_nat_real(power_power_real(A_57),zero_zero_nat) = one_one_real ).
tff(fact_545_power__0,axiom,
! [A_57: nat] : hAPP_nat_nat(power_power_nat(A_57),zero_zero_nat) = one_one_nat ).
tff(fact_546_power__0,axiom,
! [A_57: int] : hAPP_nat_int(power_power_int(A_57),zero_zero_nat) = one_one_int ).
tff(fact_547_nat__number__of__Pls,axiom,
number_number_of_nat(pls) = zero_zero_nat ).
tff(fact_548_semiring__norm_I113_J,axiom,
zero_zero_nat = number_number_of_nat(pls) ).
tff(fact_549_rel__simps_I25_J,axiom,
! [K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),bit0(K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K)) ) ).
tff(fact_550_rel__simps_I11_J,axiom,
! [K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K)),min))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),min)) ) ).
tff(fact_551_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N_1: int] :
( ( hAPP_int_int(times_times_int(M),N_1) = one_one_int )
=> ( ( M = one_one_int )
| ( M = number_number_of_int(min) ) ) ) ).
tff(fact_552_zmult__eq__1__iff,axiom,
! [Ma: int,N: int] :
( ( hAPP_int_int(times_times_int(Ma),N) = one_one_int )
<=> ( ( ( Ma = one_one_int )
& ( N = one_one_int ) )
| ( ( Ma = number_number_of_int(min) )
& ( N = number_number_of_int(min) ) ) ) ) ).
tff(fact_553_one__less__power,axiom,
! [N_6: nat,A_56: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_56))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_6))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_nat_real(power_power_real(A_56),N_6))) ) ) ).
tff(fact_554_one__less__power,axiom,
! [N_6: nat,A_56: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_56))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(power_power_nat(A_56),N_6))) ) ) ).
tff(fact_555_one__less__power,axiom,
! [N_6: nat,A_56: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_56))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_6))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_nat_int(power_power_int(A_56),N_6))) ) ) ).
tff(fact_556_dvd__power,axiom,
! [X_9: real,N_5: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_5))
| ( X_9 = one_one_real ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_9),hAPP_nat_real(power_power_real(X_9),N_5))) ) ).
tff(fact_557_dvd__power,axiom,
! [X_9: nat,N_5: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_5))
| ( X_9 = one_one_nat ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_9),hAPP_nat_nat(power_power_nat(X_9),N_5))) ) ).
tff(fact_558_dvd__power,axiom,
! [X_9: int,N_5: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_5))
| ( X_9 = one_one_int ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_9),hAPP_nat_int(power_power_int(X_9),N_5))) ) ).
tff(fact_559_less__0__number__of,axiom,
! [V_7: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),number_number_of_nat(V_7)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_7)) ) ).
tff(fact_560_eq__number__of__0,axiom,
! [V_7: int] :
( ( number_number_of_nat(V_7) = zero_zero_nat )
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_7),pls)) ) ).
tff(fact_561_eq__0__number__of,axiom,
! [V_7: int] :
( ( zero_zero_nat = number_number_of_nat(V_7) )
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_7),pls)) ) ).
tff(fact_562_zcong__sym,axiom,
! [A_2: int,B_2: int,Ma: int] :
( hBOOL(hAPP_int_bool(zcong(A_2,B_2),Ma))
<=> hBOOL(hAPP_int_bool(zcong(B_2,A_2),Ma)) ) ).
tff(fact_563_zcong__refl,axiom,
! [K_1: int,M: int] : hBOOL(hAPP_int_bool(zcong(K_1,K_1),M)) ).
tff(fact_564_zcong__trans,axiom,
! [C: int,A: int,B: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(A,B),M))
=> ( hBOOL(hAPP_int_bool(zcong(B,C),M))
=> hBOOL(hAPP_int_bool(zcong(A,C),M)) ) ) ).
tff(fact_565_pos2,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_566_nat__number__of__mult__left,axiom,
! [V_6: int,K_1: nat,V: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> ( hAPP_nat_nat(times_times_nat(number_number_of_nat(V)),hAPP_nat_nat(times_times_nat(number_number_of_nat(V_6)),K_1)) = zero_zero_nat ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> ( hAPP_nat_nat(times_times_nat(number_number_of_nat(V)),hAPP_nat_nat(times_times_nat(number_number_of_nat(V_6)),K_1)) = hAPP_nat_nat(times_times_nat(number_number_of_nat(hAPP_int_int(times_times_int(V),V_6))),K_1) ) ) ) ).
tff(fact_567_mult__nat__number__of,axiom,
! [V_6: int,V: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> ( hAPP_nat_nat(times_times_nat(number_number_of_nat(V)),number_number_of_nat(V_6)) = zero_zero_nat ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> ( hAPP_nat_nat(times_times_nat(number_number_of_nat(V)),number_number_of_nat(V_6)) = number_number_of_nat(hAPP_int_int(times_times_int(V),V_6)) ) ) ) ).
tff(fact_568_order__le__neq__implies__less,axiom,
! [X_8: real,Y_7: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_8),Y_7))
=> ( ( X_8 != Y_7 )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_8),Y_7)) ) ) ).
tff(fact_569_order__le__neq__implies__less,axiom,
! [X_8: nat,Y_7: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_8),Y_7))
=> ( ( X_8 != Y_7 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_8),Y_7)) ) ) ).
tff(fact_570_order__le__neq__implies__less,axiom,
! [X_8: int,Y_7: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_8),Y_7))
=> ( ( X_8 != Y_7 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_8),Y_7)) ) ) ).
tff(fact_571_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).
tff(fact_572_Euler_Oaux2,axiom,
! [B: int,A: int,C: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),C))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),C))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) ) ) ) ).
tff(fact_573_IntPrimes_Ozcong__zero,axiom,
! [A_2: int,B_2: int] :
( hBOOL(hAPP_int_bool(zcong(A_2,B_2),zero_zero_int))
<=> ( A_2 = B_2 ) ) ).
tff(fact_574_zcong__1,axiom,
! [A: int,B: int] : hBOOL(hAPP_int_bool(zcong(A,B),one_one_int)) ).
tff(fact_575_zcong__zmult,axiom,
! [C: int,D_5: int,A: int,B: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(A,B),M))
=> ( hBOOL(hAPP_int_bool(zcong(C,D_5),M))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),C),hAPP_int_int(times_times_int(B),D_5)),M)) ) ) ).
tff(fact_576_zcong__scalar2,axiom,
! [K_1: int,A: int,B: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(A,B),M))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(K_1),A),hAPP_int_int(times_times_int(K_1),B)),M)) ) ).
tff(fact_577_zcong__scalar,axiom,
! [K_1: int,A: int,B: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(A,B),M))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),K_1),hAPP_int_int(times_times_int(B),K_1)),M)) ) ).
tff(fact_578_zcong__zmult__self,axiom,
! [A: int,M: int,B: int] : hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),M),hAPP_int_int(times_times_int(B),M)),M)) ).
tff(fact_579_zcong__zadd,axiom,
! [C: int,D_5: int,A: int,B: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(A,B),M))
=> ( hBOOL(hAPP_int_bool(zcong(C,D_5),M))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(plus_plus_int(A),C),hAPP_int_int(plus_plus_int(B),D_5)),M)) ) ) ).
tff(fact_580_power__m1__even,axiom,
! [N_4: nat] : hAPP_nat_real(power_power_real(number267125858f_real(min)),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_4)) = one_one_real ).
tff(fact_581_power__m1__even,axiom,
! [N_4: nat] : hAPP_nat_int(power_power_int(number_number_of_int(min)),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_4)) = one_one_int ).
tff(fact_582_power__eq__0__iff__number__of,axiom,
! [A_2: real,W_1: int] :
( ( hAPP_nat_real(power_power_real(A_2),number_number_of_nat(W_1)) = zero_zero_real )
<=> ( ( A_2 = zero_zero_real )
& ( number_number_of_nat(W_1) != zero_zero_nat ) ) ) ).
tff(fact_583_power__eq__0__iff__number__of,axiom,
! [A_2: nat,W_1: int] :
( ( hAPP_nat_nat(power_power_nat(A_2),number_number_of_nat(W_1)) = zero_zero_nat )
<=> ( ( A_2 = zero_zero_nat )
& ( number_number_of_nat(W_1) != zero_zero_nat ) ) ) ).
tff(fact_584_power__eq__0__iff__number__of,axiom,
! [A_2: int,W_1: int] :
( ( hAPP_nat_int(power_power_int(A_2),number_number_of_nat(W_1)) = zero_zero_int )
<=> ( ( A_2 = zero_zero_int )
& ( number_number_of_nat(W_1) != zero_zero_nat ) ) ) ).
tff(fact_585_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).
tff(fact_586_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y_1: int,X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(X_1),Y_1))) ) ) ).
tff(fact_587_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y_1: int,X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(X_1),Y_1))) ) ) ).
tff(fact_588_zcong__not,axiom,
! [B: int,M: int,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A))
=> ~ hBOOL(hAPP_int_bool(zcong(A,B),M)) ) ) ) ) ).
tff(fact_589_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [N_1: nat,X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(X_1),N_1))) ) ).
tff(fact_590_zcong__iff__lin,axiom,
! [A_2: int,B_2: int,Ma: int] :
( hBOOL(hAPP_int_bool(zcong(A_2,B_2),Ma))
<=> ? [K_2: int] : B_2 = hAPP_int_int(plus_plus_int(A_2),hAPP_int_int(times_times_int(Ma),K_2)) ) ).
tff(fact_591_power__0__left__number__of,axiom,
! [W_4: int] :
( ( ( number_number_of_nat(W_4) = zero_zero_nat )
=> ( hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(W_4)) = one_one_real ) )
& ( ( number_number_of_nat(W_4) != zero_zero_nat )
=> ( hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(W_4)) = zero_zero_real ) ) ) ).
tff(fact_592_power__0__left__number__of,axiom,
! [W_4: int] :
( ( ( number_number_of_nat(W_4) = zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(W_4)) = one_one_nat ) )
& ( ( number_number_of_nat(W_4) != zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(W_4)) = zero_zero_nat ) ) ) ).
tff(fact_593_power__0__left__number__of,axiom,
! [W_4: int] :
( ( ( number_number_of_nat(W_4) = zero_zero_nat )
=> ( hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(W_4)) = one_one_int ) )
& ( ( number_number_of_nat(W_4) != zero_zero_nat )
=> ( hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(W_4)) = zero_zero_int ) ) ) ).
tff(fact_594_zcong__zless__imp__eq,axiom,
! [B: int,M: int,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),M))
=> ( hBOOL(hAPP_int_bool(zcong(A,B),M))
=> ( A = B ) ) ) ) ) ) ).
tff(fact_595_zcong__zless__0,axiom,
! [M: int,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),M))
=> ( hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),M))
=> ( A = zero_zero_int ) ) ) ) ).
tff(fact_596_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit1(bit1(pls))))) ).
tff(fact_597_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q_1: int,R_1: int,B_48: int,Q_4: int,R_3: int] :
( ( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q_1)),R_1) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_48),Q_4)),R_3) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_48),Q_4)),R_3)),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_48))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_48),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_4),Q_1)) ) ) ) ) ) ) ).
tff(fact_598_unique__quotient__lemma__neg,axiom,
! [B: int,Q_4: int,R_3: int,Q_1: int,R_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q_4)),R_3)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q_1)),R_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_1),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),R_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),R_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q_4)) ) ) ) ) ).
tff(fact_599_zdiv__mono2__lemma,axiom,
! [B: int,Q_1: int,R_1: int,B_48: int,Q_4: int,R_3: int] :
( ( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q_1)),R_1) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_48),Q_4)),R_3) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_48),Q_4)),R_3)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_48))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_48))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_48),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q_4)) ) ) ) ) ) ) ).
tff(fact_600_unique__quotient__lemma,axiom,
! [B: int,Q_4: int,R_3: int,Q_1: int,R_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q_4)),R_3)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q_1)),R_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_4),Q_1)) ) ) ) ) ).
tff(fact_601_q__neg__lemma,axiom,
! [B_48: int,Q_4: int,R_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_48),Q_4)),R_3)),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_48))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_4),zero_zero_int)) ) ) ) ).
tff(fact_602_q__pos__lemma,axiom,
! [B_48: int,Q_4: int,R_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_48),Q_4)),R_3)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_48))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_48))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Q_4)) ) ) ) ).
tff(fact_603_zprime__zdvd__zmult,axiom,
! [N_1: int,P_1: int,M: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M))
=> ( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),hAPP_int_int(times_times_int(M),N_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),M))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),N_1)) ) ) ) ) ).
tff(fact_604__096QuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_096,axiom,
hBOOL(hAPP_int_bool(quadRes(hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int)),number_number_of_int(min))) ).
tff(fact_605__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_N_A_N1_096,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int)),hAPP_int_int(minus_minus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls))))),number_number_of_int(min)))) ).
tff(fact_606_neg__one__power__eq__mod__m,axiom,
! [J_1: nat,K_1: nat,M: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),M))
=> ( hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(number_number_of_int(min)),J_1),hAPP_nat_int(power_power_int(number_number_of_int(min)),K_1)),M))
=> ( hAPP_nat_int(power_power_int(number_number_of_int(min)),J_1) = hAPP_nat_int(power_power_int(number_number_of_int(min)),K_1) ) ) ) ).
tff(fact_607_zcong__neg__1__impl__ne__1,axiom,
! [X_1: int,P_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1))
=> ( hBOOL(hAPP_int_bool(zcong(X_1,number_number_of_int(min)),P_1))
=> ~ hBOOL(hAPP_int_bool(zcong(X_1,one_one_int),P_1)) ) ) ).
tff(fact_608_one__not__neg__one__mod__m,axiom,
! [M: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),M))
=> ~ hBOOL(hAPP_int_bool(zcong(one_one_int,number_number_of_int(min)),M)) ) ).
tff(fact_609__096s_A_094_A2_A_N_A_N1_A_061_As_A_094_A2_A_L_A1_096,axiom,
hAPP_int_int(minus_minus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls))))),number_number_of_int(min)) = hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls))))),one_one_int) ).
tff(fact_610__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_N,axiom,
( ~ hBOOL(hAPP_int_bool(quadRes(hAPP_int_int(plus_plus_int(hAPP_int_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),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int)) != one_one_int ) ) ).
tff(fact_611_Int2_Oaux1,axiom,
! [A: int,B: int,C: int] :
( ( hAPP_int_int(minus_minus_int(A),B) = C )
=> ( A = hAPP_int_int(plus_plus_int(C),B) ) ) ).
tff(fact_612_number__of__diff,axiom,
! [V_5: int,W_3: int] : number267125858f_real(hAPP_int_int(minus_minus_int(V_5),W_3)) = hAPP_real_real(minus_minus_real(number267125858f_real(V_5)),number267125858f_real(W_3)) ).
tff(fact_613_number__of__diff,axiom,
! [V_5: int,W_3: int] : number_number_of_int(hAPP_int_int(minus_minus_int(V_5),W_3)) = hAPP_int_int(minus_minus_int(number_number_of_int(V_5)),number_number_of_int(W_3)) ).
tff(fact_614_diff__bin__simps_I1_J,axiom,
! [K_1: int] : hAPP_int_int(minus_minus_int(K_1),pls) = K_1 ).
tff(fact_615_diff__bin__simps_I7_J,axiom,
! [K_1: int,L: int] : hAPP_int_int(minus_minus_int(bit0(K_1)),bit0(L)) = bit0(hAPP_int_int(minus_minus_int(K_1),L)) ).
tff(fact_616_zdiff__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] : hAPP_int_int(times_times_int(W),hAPP_int_int(minus_minus_int(Z1),Z2)) = hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(W),Z1)),hAPP_int_int(times_times_int(W),Z2)) ).
tff(fact_617_zdiff__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] : hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(Z1),Z2)),W) = hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(Z1),W)),hAPP_int_int(times_times_int(Z2),W)) ).
tff(fact_618_zcong__zdiff,axiom,
! [C: int,D_5: int,A: int,B: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(A,B),M))
=> ( hBOOL(hAPP_int_bool(zcong(C,D_5),M))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(minus_minus_int(A),C),hAPP_int_int(minus_minus_int(B),D_5)),M)) ) ) ).
tff(fact_619_zdvd__zdiffD,axiom,
! [K_1: int,M: int,N_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(minus_minus_int(M),N_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),M)) ) ) ).
tff(fact_620_right__diff__distrib__number__of,axiom,
! [V_4: int,B_47: real,C_26: real] : hAPP_real_real(times_times_real(number267125858f_real(V_4)),hAPP_real_real(minus_minus_real(B_47),C_26)) = hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(number267125858f_real(V_4)),B_47)),hAPP_real_real(times_times_real(number267125858f_real(V_4)),C_26)) ).
tff(fact_621_right__diff__distrib__number__of,axiom,
! [V_4: int,B_47: int,C_26: int] : hAPP_int_int(times_times_int(number_number_of_int(V_4)),hAPP_int_int(minus_minus_int(B_47),C_26)) = hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(number_number_of_int(V_4)),B_47)),hAPP_int_int(times_times_int(number_number_of_int(V_4)),C_26)) ).
tff(fact_622_left__diff__distrib__number__of,axiom,
! [A_55: real,B_46: real,V_3: int] : hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(A_55),B_46)),number267125858f_real(V_3)) = hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(A_55),number267125858f_real(V_3))),hAPP_real_real(times_times_real(B_46),number267125858f_real(V_3))) ).
tff(fact_623_left__diff__distrib__number__of,axiom,
! [A_55: int,B_46: int,V_3: int] : hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(A_55),B_46)),number_number_of_int(V_3)) = hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(A_55),number_number_of_int(V_3))),hAPP_int_int(times_times_int(B_46),number_number_of_int(V_3))) ).
tff(fact_624_diff__bin__simps_I9_J,axiom,
! [K_1: int,L: int] : hAPP_int_int(minus_minus_int(bit1(K_1)),bit0(L)) = bit1(hAPP_int_int(minus_minus_int(K_1),L)) ).
tff(fact_625_diff__bin__simps_I10_J,axiom,
! [K_1: int,L: int] : hAPP_int_int(minus_minus_int(bit1(K_1)),bit1(L)) = bit0(hAPP_int_int(minus_minus_int(K_1),L)) ).
tff(fact_626_diff__bin__simps_I3_J,axiom,
! [L: int] : hAPP_int_int(minus_minus_int(pls),bit0(L)) = bit0(hAPP_int_int(minus_minus_int(pls),L)) ).
tff(fact_627_less__bin__lemma,axiom,
! [K: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(minus_minus_int(K),L_1)),zero_zero_int)) ) ).
tff(fact_628_xzgcda__linear__aux1,axiom,
! [A: int,R_1: int,B: int,M: int,C: int,D_5: int,N_1: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(A),hAPP_int_int(times_times_int(R_1),B))),M)),hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(C),hAPP_int_int(times_times_int(R_1),D_5))),N_1)) = hAPP_int_int(minus_minus_int(hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A),M)),hAPP_int_int(times_times_int(C),N_1))),hAPP_int_int(times_times_int(R_1),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),M)),hAPP_int_int(times_times_int(D_5),N_1)))) ).
tff(fact_629_zcong__def,axiom,
! [A_2: int,B_2: int,Ma: int] :
( hBOOL(hAPP_int_bool(zcong(A_2,B_2),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),hAPP_int_int(minus_minus_int(A_2),B_2))) ) ).
tff(fact_630_add__number__of__diff1,axiom,
! [V_2: int,W_2: int,C_25: real] : hAPP_real_real(plus_plus_real(number267125858f_real(V_2)),hAPP_real_real(minus_minus_real(number267125858f_real(W_2)),C_25)) = hAPP_real_real(minus_minus_real(number267125858f_real(hAPP_int_int(plus_plus_int(V_2),W_2))),C_25) ).
tff(fact_631_add__number__of__diff1,axiom,
! [V_2: int,W_2: int,C_25: int] : hAPP_int_int(plus_plus_int(number_number_of_int(V_2)),hAPP_int_int(minus_minus_int(number_number_of_int(W_2)),C_25)) = hAPP_int_int(minus_minus_int(number_number_of_int(hAPP_int_int(plus_plus_int(V_2),W_2))),C_25) ).
tff(fact_632_Euler_Oaux1,axiom,
! [A: int,X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),A))
=> ( ( X_1 != hAPP_int_int(minus_minus_int(A),one_one_int) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),hAPP_int_int(minus_minus_int(A),one_one_int))) ) ) ) ).
tff(fact_633_zle__diff1__eq,axiom,
! [W_1: int,Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),hAPP_int_int(minus_minus_int(Z_1),one_one_int)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_1)) ) ).
tff(fact_634_diff__bin__simps_I4_J,axiom,
! [L: int] : hAPP_int_int(minus_minus_int(pls),bit1(L)) = bit1(hAPP_int_int(minus_minus_int(min),L)) ).
tff(fact_635_diff__bin__simps_I5_J,axiom,
! [L: int] : hAPP_int_int(minus_minus_int(min),bit0(L)) = bit1(hAPP_int_int(minus_minus_int(min),L)) ).
tff(fact_636_diff__bin__simps_I6_J,axiom,
! [L: int] : hAPP_int_int(minus_minus_int(min),bit1(L)) = bit0(hAPP_int_int(minus_minus_int(min),L)) ).
tff(fact_637_inv__not__p__minus__1__aux,axiom,
! [A_2: int,P_3: int] :
( hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A_2),hAPP_int_int(minus_minus_int(P_3),one_one_int)),one_one_int),P_3))
<=> hBOOL(hAPP_int_bool(zcong(A_2,hAPP_int_int(minus_minus_int(P_3),one_one_int)),P_3)) ) ).
tff(fact_638_zcong__eq__trans,axiom,
! [D_5: int,C: int,A: int,B: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(A,B),M))
=> ( ( B = C )
=> ( hBOOL(hAPP_int_bool(zcong(C,D_5),M))
=> hBOOL(hAPP_int_bool(zcong(A,D_5),M)) ) ) ) ).
tff(fact_639_mult__sum2sq,axiom,
! [A: int,B: int,P_1: int,Q_1: int] : hAPP_int_int(times_times_int(twoSqu949963151sum2sq(product_Pair_int_int(A,B))),twoSqu949963151sum2sq(product_Pair_int_int(P_1,Q_1))) = twoSqu949963151sum2sq(product_Pair_int_int(hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A),P_1)),hAPP_int_int(times_times_int(B),Q_1)),hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(A),Q_1)),hAPP_int_int(times_times_int(B),P_1)))) ).
tff(fact_640_zcong__square,axiom,
! [A: int,P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),A),one_one_int),P_1))
=> ( hBOOL(hAPP_int_bool(zcong(A,one_one_int),P_1))
| hBOOL(hAPP_int_bool(zcong(A,hAPP_int_int(minus_minus_int(P_1),one_one_int)),P_1)) ) ) ) ) ).
tff(fact_641_zcong__square__zless,axiom,
! [A: int,P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),P_1))
=> ( hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),A),one_one_int),P_1))
=> ( ( A = one_one_int )
| ( A = hAPP_int_int(minus_minus_int(P_1),one_one_int) ) ) ) ) ) ) ).
tff(fact_642_zspecial__product,axiom,
! [A: int,B: int] : hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A),B)),hAPP_int_int(minus_minus_int(A),B)) = hAPP_int_int(minus_minus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(B),number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_643_zcong__id,axiom,
! [M: int] : hBOOL(hAPP_int_bool(zcong(M,zero_zero_int),M)) ).
tff(fact_644_zcong__zmult__prop1,axiom,
! [C_2: int,D: int,A_2: int,B_2: int,Ma: int] :
( hBOOL(hAPP_int_bool(zcong(A_2,B_2),Ma))
=> ( hBOOL(hAPP_int_bool(zcong(C_2,hAPP_int_int(times_times_int(A_2),D)),Ma))
<=> hBOOL(hAPP_int_bool(zcong(C_2,hAPP_int_int(times_times_int(B_2),D)),Ma)) ) ) ).
tff(fact_645_zcong__zmult__prop2,axiom,
! [C_2: int,D: int,A_2: int,B_2: int,Ma: int] :
( hBOOL(hAPP_int_bool(zcong(A_2,B_2),Ma))
=> ( hBOOL(hAPP_int_bool(zcong(C_2,hAPP_int_int(times_times_int(D),A_2)),Ma))
<=> hBOOL(hAPP_int_bool(zcong(C_2,hAPP_int_int(times_times_int(D),B_2)),Ma)) ) ) ).
tff(fact_646_zcong__shift,axiom,
! [C: int,A: int,B: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(A,B),M))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(plus_plus_int(A),C),hAPP_int_int(plus_plus_int(B),C)),M)) ) ).
tff(fact_647_zcong__zpower,axiom,
! [Z: nat,X_1: int,Y_1: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(X_1,Y_1),M))
=> hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X_1),Z),hAPP_nat_int(power_power_int(Y_1),Z)),M)) ) ).
tff(fact_648_power2__diff,axiom,
! [X_7: real,Y_6: real] : hAPP_nat_real(power_power_real(hAPP_real_real(minus_minus_real(X_7),Y_6)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_real_real(minus_minus_real(hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_7),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_6),number_number_of_nat(bit0(bit1(pls)))))),hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(number267125858f_real(bit0(bit1(pls)))),X_7)),Y_6)) ).
tff(fact_649_power2__diff,axiom,
! [X_7: int,Y_6: int] : hAPP_nat_int(power_power_int(hAPP_int_int(minus_minus_int(X_7),Y_6)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_int_int(minus_minus_int(hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X_7),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_6),number_number_of_nat(bit0(bit1(pls)))))),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),X_7)),Y_6)) ).
tff(fact_650_zdiff__power2,axiom,
! [A: int,B: int] : hAPP_nat_int(power_power_int(hAPP_int_int(minus_minus_int(A),B)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_int_int(plus_plus_int(hAPP_int_int(minus_minus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls))))),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),A)),B))),hAPP_nat_int(power_power_int(B),number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_651_zdiff__power3,axiom,
! [A: int,B: int] : hAPP_nat_int(power_power_int(hAPP_int_int(minus_minus_int(A),B)),number_number_of_nat(bit1(bit1(pls)))) = hAPP_int_int(minus_minus_int(hAPP_int_int(plus_plus_int(hAPP_int_int(minus_minus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit1(bit1(pls))))),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(number_number_of_int(bit1(bit1(pls)))),hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))))),B))),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(number_number_of_int(bit1(bit1(pls)))),A)),hAPP_nat_int(power_power_int(B),number_number_of_nat(bit0(bit1(pls))))))),hAPP_nat_int(power_power_int(B),number_number_of_nat(bit1(bit1(pls))))) ).
tff(fact_652_zcong__less__eq,axiom,
! [M: int,Y_1: int,X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
=> ( hBOOL(hAPP_int_bool(zcong(X_1,Y_1),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),M))
=> ( X_1 = Y_1 ) ) ) ) ) ) ) ).
tff(fact_653_zcong__not__zero,axiom,
! [M: int,X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),M))
=> ~ hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),M)) ) ) ).
tff(fact_654_zdvd__bounds,axiom,
! [N_1: int,M: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),zero_zero_int))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),M)) ) ) ).
tff(fact_655_zcong__eq__zdvd__prop,axiom,
! [X_2: int,P_3: int] :
( hBOOL(hAPP_int_bool(zcong(X_2,zero_zero_int),P_3))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),X_2)) ) ).
tff(fact_656_zcong__zero__equiv__div,axiom,
! [A_2: int,Ma: int] :
( hBOOL(hAPP_int_bool(zcong(A_2,zero_zero_int),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),A_2)) ) ).
tff(fact_657_zprime__zdvd__zmult__better,axiom,
! [M: int,N_1: int,P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),hAPP_int_int(times_times_int(M),N_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),M))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),N_1)) ) ) ) ).
tff(fact_658_Int2_Ozcong__zero,axiom,
! [M: int,X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),M))
=> ( hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),M))
=> ( X_1 = zero_zero_int ) ) ) ) ).
tff(fact_659_zpower__zdvd__prop1,axiom,
! [P_1: int,Y_1: int,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),Y_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),hAPP_nat_int(power_power_int(Y_1),N_1))) ) ) ).
tff(fact_660_neg__one__power,axiom,
! [N_1: nat] :
( ( hAPP_nat_int(power_power_int(number_number_of_int(min)),N_1) = one_one_int )
| ( hAPP_nat_int(power_power_int(number_number_of_int(min)),N_1) = number_number_of_int(min) ) ) ).
tff(fact_661_zcong__zmult__prop3,axiom,
! [Y_1: int,X_1: int,P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( ~ hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_1))
=> ( ~ hBOOL(hAPP_int_bool(zcong(Y_1,zero_zero_int),P_1))
=> ~ hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(X_1),Y_1),zero_zero_int),P_1)) ) ) ) ).
tff(fact_662_zcong__zprime__prod__zero,axiom,
! [B: int,A: int,P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),B),zero_zero_int),P_1))
=> ( hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_1))
| hBOOL(hAPP_int_bool(zcong(B,zero_zero_int),P_1)) ) ) ) ) ).
tff(fact_663_zcong__zprime__prod__zero__contra,axiom,
! [B: int,A: int,P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( ( ~ hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_1))
& ~ hBOOL(hAPP_int_bool(zcong(B,zero_zero_int),P_1)) )
=> ~ hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),B),zero_zero_int),P_1)) ) ) ) ).
tff(fact_664_zpower__zdvd__prop2,axiom,
! [Y_1: int,N_1: nat,P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),hAPP_nat_int(power_power_int(Y_1),N_1)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),Y_1)) ) ) ) ).
tff(fact_665_Legendre__1mod4,axiom,
! [M: int] :
( hBOOL(hAPP_int_bool(zprime,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),M)),one_one_int)))
=> ( legendre(number_number_of_int(min),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),M)),one_one_int)) = one_one_int ) ) ).
tff(fact_666_Legendre__def,axiom,
! [A: int,P_1: int] :
( ( hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_1))
=> ( legendre(A,P_1) = zero_zero_int ) )
& ( ~ hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_1))
=> ( ( hBOOL(hAPP_int_bool(quadRes(P_1),A))
=> ( legendre(A,P_1) = one_one_int ) )
& ( ~ hBOOL(hAPP_int_bool(quadRes(P_1),A))
=> ( legendre(A,P_1) = number_number_of_int(min) ) ) ) ) ) ).
tff(fact_667_divides__cases,axiom,
! [N_1: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),M))
=> ( ( M = zero_zero_nat )
| ( M = N_1 )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_1)),M)) ) ) ).
tff(fact_668_QuadRes__def,axiom,
! [Ma: int,X_2: int] :
( hBOOL(hAPP_int_bool(quadRes(Ma),X_2))
<=> ? [Y: int] : hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(Y),number_number_of_nat(bit0(bit1(pls)))),X_2),Ma)) ) ).
tff(fact_669_realpow__two__sum__zero__iff,axiom,
! [X_2: real,Y_2: real] :
( ( hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(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_670_convex__bound__lt,axiom,
! [V_1: real,U_2: real,Y_5: real,X_6: real,A_54: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_6),A_54))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_5),A_54))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),U_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),V_1))
=> ( ( hAPP_real_real(plus_plus_real(U_2),V_1) = one_one_real )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(U_2),X_6)),hAPP_real_real(times_times_real(V_1),Y_5))),A_54)) ) ) ) ) ) ).
tff(fact_671_convex__bound__lt,axiom,
! [V_1: int,U_2: int,Y_5: int,X_6: int,A_54: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_6),A_54))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_5),A_54))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V_1))
=> ( ( hAPP_int_int(plus_plus_int(U_2),V_1) = one_one_int )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(U_2),X_6)),hAPP_int_int(times_times_int(V_1),Y_5))),A_54)) ) ) ) ) ) ).
tff(fact_672_dvd__0__right,axiom,
! [A_53: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_53),zero_zero_real)) ).
tff(fact_673_dvd__0__right,axiom,
! [A_53: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_53),zero_zero_nat)) ).
tff(fact_674_dvd__0__right,axiom,
! [A_53: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_53),zero_zero_int)) ).
tff(fact_675_real__le__eq__diff,axiom,
! [X_2: real,Y_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(minus_minus_real(X_2),Y_2)),zero_zero_real)) ) ).
tff(fact_676_linorder__neqE__linordered__idom,axiom,
! [X_5: real,Y_4: real] :
( ( X_5 != Y_4 )
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_5),Y_4))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_4),X_5)) ) ) ).
tff(fact_677_linorder__neqE__linordered__idom,axiom,
! [X_5: int,Y_4: int] :
( ( X_5 != Y_4 )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_5),Y_4))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_4),X_5)) ) ) ).
tff(fact_678_dvd__refl,axiom,
! [A_52: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_52),A_52)) ).
tff(fact_679_dvd__refl,axiom,
! [A_52: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_52),A_52)) ).
tff(fact_680_dvd__refl,axiom,
! [A_52: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_52),A_52)) ).
tff(fact_681_dvd__trans,axiom,
! [C_24: real,A_51: real,B_45: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_51),B_45))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_45),C_24))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_51),C_24)) ) ) ).
tff(fact_682_dvd__trans,axiom,
! [C_24: nat,A_51: nat,B_45: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_51),B_45))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_45),C_24))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_51),C_24)) ) ) ).
tff(fact_683_dvd__trans,axiom,
! [C_24: int,A_51: int,B_45: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_51),B_45))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_45),C_24))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_51),C_24)) ) ) ).
tff(fact_684_real__zero__not__eq__one,axiom,
zero_zero_real != one_one_real ).
tff(fact_685_less__eq__real__def,axiom,
! [X_2: real,Y_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2))
| ( X_2 = Y_2 ) ) ) ).
tff(fact_686_real__less__def,axiom,
! [X_2: real,Y_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2))
& ( X_2 != Y_2 ) ) ) ).
tff(fact_687_divides__antisym,axiom,
! [X_2: nat,Y_2: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) )
<=> ( X_2 = Y_2 ) ) ).
tff(fact_688_real__mult__assoc,axiom,
! [Z1: real,Z2: real,Z3: real] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Z1),Z2)),Z3) = hAPP_real_real(times_times_real(Z1),hAPP_real_real(times_times_real(Z2),Z3)) ).
tff(fact_689_real__mult__commute,axiom,
! [Z: real,W: real] : hAPP_real_real(times_times_real(Z),W) = hAPP_real_real(times_times_real(W),Z) ).
tff(fact_690_real__mult__1,axiom,
! [Z: real] : hAPP_real_real(times_times_real(one_one_real),Z) = Z ).
tff(fact_691_real__add__left__mono,axiom,
! [Z: real,X_1: real,Y_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(Z),X_1)),hAPP_real_real(plus_plus_real(Z),Y_1))) ) ).
tff(fact_692_realpow__minus__mult,axiom,
! [X_4: nat,N_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_3))
=> ( hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(X_4),hAPP_nat_nat(minus_minus_nat(N_3),one_one_nat))),X_4) = hAPP_nat_nat(power_power_nat(X_4),N_3) ) ) ).
tff(fact_693_realpow__minus__mult,axiom,
! [X_4: real,N_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_3))
=> ( hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(X_4),hAPP_nat_nat(minus_minus_nat(N_3),one_one_nat))),X_4) = hAPP_nat_real(power_power_real(X_4),N_3) ) ) ).
tff(fact_694_realpow__minus__mult,axiom,
! [X_4: int,N_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_3))
=> ( hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(X_4),hAPP_nat_nat(minus_minus_nat(N_3),one_one_nat))),X_4) = hAPP_nat_int(power_power_int(X_4),N_3) ) ) ).
tff(fact_695_mult__eq__if,axiom,
! [N_1: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( hAPP_nat_nat(times_times_nat(M),N_1) = zero_zero_nat ) )
& ( ( M != zero_zero_nat )
=> ( hAPP_nat_nat(times_times_nat(M),N_1) = hAPP_nat_nat(plus_plus_nat(N_1),hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(M),one_one_nat)),N_1)) ) ) ) ).
tff(fact_696_power__eq__if,axiom,
! [P_1: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(P_1),M) = one_one_nat ) )
& ( ( M != zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(P_1),M) = hAPP_nat_nat(times_times_nat(P_1),hAPP_nat_nat(power_power_nat(P_1),hAPP_nat_nat(minus_minus_nat(M),one_one_nat))) ) ) ) ).
tff(fact_697_diff__square,axiom,
! [X_1: nat,Y_1: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(power_power_nat(X_1),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_nat(power_power_nat(Y_1),number_number_of_nat(bit0(bit1(pls))))) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(X_1),Y_1)),hAPP_nat_nat(minus_minus_nat(X_1),Y_1)) ).
tff(fact_698_divisors__zero,axiom,
! [A_50: real,B_44: real] :
( ( hAPP_real_real(times_times_real(A_50),B_44) = zero_zero_real )
=> ( ( A_50 = zero_zero_real )
| ( B_44 = zero_zero_real ) ) ) ).
tff(fact_699_divisors__zero,axiom,
! [A_50: nat,B_44: nat] :
( ( hAPP_nat_nat(times_times_nat(A_50),B_44) = zero_zero_nat )
=> ( ( A_50 = zero_zero_nat )
| ( B_44 = zero_zero_nat ) ) ) ).
tff(fact_700_divisors__zero,axiom,
! [A_50: int,B_44: int] :
( ( hAPP_int_int(times_times_int(A_50),B_44) = zero_zero_int )
=> ( ( A_50 = zero_zero_int )
| ( B_44 = zero_zero_int ) ) ) ).
tff(fact_701_no__zero__divisors,axiom,
! [B_43: real,A_49: real] :
( ( A_49 != zero_zero_real )
=> ( ( B_43 != zero_zero_real )
=> ( hAPP_real_real(times_times_real(A_49),B_43) != zero_zero_real ) ) ) ).
tff(fact_702_no__zero__divisors,axiom,
! [B_43: nat,A_49: nat] :
( ( A_49 != zero_zero_nat )
=> ( ( B_43 != zero_zero_nat )
=> ( hAPP_nat_nat(times_times_nat(A_49),B_43) != zero_zero_nat ) ) ) ).
tff(fact_703_no__zero__divisors,axiom,
! [B_43: int,A_49: int] :
( ( A_49 != zero_zero_int )
=> ( ( B_43 != zero_zero_int )
=> ( hAPP_int_int(times_times_int(A_49),B_43) != zero_zero_int ) ) ) ).
tff(fact_704_mult__eq__0__iff,axiom,
! [A_2: real,B_2: real] :
( ( hAPP_real_real(times_times_real(A_2),B_2) = zero_zero_real )
<=> ( ( A_2 = zero_zero_real )
| ( B_2 = zero_zero_real ) ) ) ).
tff(fact_705_mult__eq__0__iff,axiom,
! [A_2: int,B_2: int] :
( ( hAPP_int_int(times_times_int(A_2),B_2) = zero_zero_int )
<=> ( ( A_2 = zero_zero_int )
| ( B_2 = zero_zero_int ) ) ) ).
tff(fact_706_mult__zero__right,axiom,
! [A_48: real] : hAPP_real_real(times_times_real(A_48),zero_zero_real) = zero_zero_real ).
tff(fact_707_mult__zero__right,axiom,
! [A_48: nat] : hAPP_nat_nat(times_times_nat(A_48),zero_zero_nat) = zero_zero_nat ).
tff(fact_708_mult__zero__right,axiom,
! [A_48: int] : hAPP_int_int(times_times_int(A_48),zero_zero_int) = zero_zero_int ).
tff(fact_709_mult__zero__left,axiom,
! [A_47: real] : hAPP_real_real(times_times_real(zero_zero_real),A_47) = zero_zero_real ).
tff(fact_710_mult__zero__left,axiom,
! [A_47: nat] : hAPP_nat_nat(times_times_nat(zero_zero_nat),A_47) = zero_zero_nat ).
tff(fact_711_mult__zero__left,axiom,
! [A_47: int] : hAPP_int_int(times_times_int(zero_zero_int),A_47) = zero_zero_int ).
tff(fact_712_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
tff(fact_713_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
tff(fact_714_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
tff(fact_715_one__neq__zero,axiom,
one_one_real != zero_zero_real ).
tff(fact_716_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
tff(fact_717_one__neq__zero,axiom,
one_one_int != zero_zero_int ).
tff(fact_718_combine__common__factor,axiom,
! [A_46: real,E: real,B_42: real,C_23: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_46),E)),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(B_42),E)),C_23)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_46),B_42)),E)),C_23) ).
tff(fact_719_combine__common__factor,axiom,
! [A_46: nat,E: nat,B_42: nat,C_23: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_46),E)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(B_42),E)),C_23)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_46),B_42)),E)),C_23) ).
tff(fact_720_combine__common__factor,axiom,
! [A_46: int,E: int,B_42: int,C_23: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_46),E)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_42),E)),C_23)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_46),B_42)),E)),C_23) ).
tff(fact_721_comm__semiring__class_Odistrib,axiom,
! [A_45: real,B_41: real,C_22: real] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_45),B_41)),C_22) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_45),C_22)),hAPP_real_real(times_times_real(B_41),C_22)) ).
tff(fact_722_comm__semiring__class_Odistrib,axiom,
! [A_45: nat,B_41: nat,C_22: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_45),B_41)),C_22) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_45),C_22)),hAPP_nat_nat(times_times_nat(B_41),C_22)) ).
tff(fact_723_comm__semiring__class_Odistrib,axiom,
! [A_45: int,B_41: int,C_22: int] : hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_45),B_41)),C_22) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_45),C_22)),hAPP_int_int(times_times_int(B_41),C_22)) ).
tff(fact_724_dvd__0__left,axiom,
! [A_44: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,zero_zero_real),A_44))
=> ( A_44 = zero_zero_real ) ) ).
tff(fact_725_dvd__0__left,axiom,
! [A_44: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_44))
=> ( A_44 = zero_zero_nat ) ) ).
tff(fact_726_dvd__0__left,axiom,
! [A_44: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,zero_zero_int),A_44))
=> ( A_44 = zero_zero_int ) ) ).
tff(fact_727_add__diff__add,axiom,
! [A_43: real,C_21: real,B_40: real,D_7: real] : hAPP_real_real(minus_minus_real(hAPP_real_real(plus_plus_real(A_43),C_21)),hAPP_real_real(plus_plus_real(B_40),D_7)) = hAPP_real_real(plus_plus_real(hAPP_real_real(minus_minus_real(A_43),B_40)),hAPP_real_real(minus_minus_real(C_21),D_7)) ).
tff(fact_728_add__diff__add,axiom,
! [A_43: int,C_21: int,B_40: int,D_7: int] : hAPP_int_int(minus_minus_int(hAPP_int_int(plus_plus_int(A_43),C_21)),hAPP_int_int(plus_plus_int(B_40),D_7)) = hAPP_int_int(plus_plus_int(hAPP_int_int(minus_minus_int(A_43),B_40)),hAPP_int_int(minus_minus_int(C_21),D_7)) ).
tff(fact_729_dvd__mult__right,axiom,
! [A_42: real,B_39: real,C_20: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(times_times_real(A_42),B_39)),C_20))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_39),C_20)) ) ).
tff(fact_730_dvd__mult__right,axiom,
! [A_42: nat,B_39: nat,C_20: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(A_42),B_39)),C_20))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_39),C_20)) ) ).
tff(fact_731_dvd__mult__right,axiom,
! [A_42: int,B_39: int,C_20: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(times_times_int(A_42),B_39)),C_20))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_39),C_20)) ) ).
tff(fact_732_dvd__mult__left,axiom,
! [A_41: real,B_38: real,C_19: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(times_times_real(A_41),B_38)),C_19))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_41),C_19)) ) ).
tff(fact_733_dvd__mult__left,axiom,
! [A_41: nat,B_38: nat,C_19: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(A_41),B_38)),C_19))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_41),C_19)) ) ).
tff(fact_734_dvd__mult__left,axiom,
! [A_41: int,B_38: int,C_19: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(times_times_int(A_41),B_38)),C_19))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_41),C_19)) ) ).
tff(fact_735_dvdI,axiom,
! [A_40: real,B_37: real,K_3: real] :
( ( A_40 = hAPP_real_real(times_times_real(B_37),K_3) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_37),A_40)) ) ).
tff(fact_736_dvdI,axiom,
! [A_40: nat,B_37: nat,K_3: nat] :
( ( A_40 = hAPP_nat_nat(times_times_nat(B_37),K_3) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_37),A_40)) ) ).
tff(fact_737_dvdI,axiom,
! [A_40: int,B_37: int,K_3: int] :
( ( A_40 = hAPP_int_int(times_times_int(B_37),K_3) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_37),A_40)) ) ).
tff(fact_738_mult__dvd__mono,axiom,
! [C_18: real,D_6: real,A_39: real,B_36: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_39),B_36))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,C_18),D_6))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(times_times_real(A_39),C_18)),hAPP_real_real(times_times_real(B_36),D_6))) ) ) ).
tff(fact_739_mult__dvd__mono,axiom,
! [C_18: nat,D_6: nat,A_39: nat,B_36: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_39),B_36))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_18),D_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(A_39),C_18)),hAPP_nat_nat(times_times_nat(B_36),D_6))) ) ) ).
tff(fact_740_mult__dvd__mono,axiom,
! [C_18: int,D_6: int,A_39: int,B_36: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_39),B_36))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C_18),D_6))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(times_times_int(A_39),C_18)),hAPP_int_int(times_times_int(B_36),D_6))) ) ) ).
tff(fact_741_dvd__mult,axiom,
! [B_35: real,A_38: real,C_17: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_38),C_17))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_38),hAPP_real_real(times_times_real(B_35),C_17))) ) ).
tff(fact_742_dvd__mult,axiom,
! [B_35: nat,A_38: nat,C_17: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_38),C_17))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_38),hAPP_nat_nat(times_times_nat(B_35),C_17))) ) ).
tff(fact_743_dvd__mult,axiom,
! [B_35: int,A_38: int,C_17: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_38),C_17))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_38),hAPP_int_int(times_times_int(B_35),C_17))) ) ).
tff(fact_744_dvd__mult2,axiom,
! [C_16: real,A_37: real,B_34: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_37),B_34))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_37),hAPP_real_real(times_times_real(B_34),C_16))) ) ).
tff(fact_745_dvd__mult2,axiom,
! [C_16: nat,A_37: nat,B_34: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_37),B_34))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_37),hAPP_nat_nat(times_times_nat(B_34),C_16))) ) ).
tff(fact_746_dvd__mult2,axiom,
! [C_16: int,A_37: int,B_34: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_37),B_34))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_37),hAPP_int_int(times_times_int(B_34),C_16))) ) ).
tff(fact_747_dvd__triv__right,axiom,
! [A_36: real,B_33: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_36),hAPP_real_real(times_times_real(B_33),A_36))) ).
tff(fact_748_dvd__triv__right,axiom,
! [A_36: nat,B_33: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_36),hAPP_nat_nat(times_times_nat(B_33),A_36))) ).
tff(fact_749_dvd__triv__right,axiom,
! [A_36: int,B_33: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_36),hAPP_int_int(times_times_int(B_33),A_36))) ).
tff(fact_750_dvd__triv__left,axiom,
! [A_35: real,B_32: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_35),hAPP_real_real(times_times_real(A_35),B_32))) ).
tff(fact_751_dvd__triv__left,axiom,
! [A_35: nat,B_32: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_35),hAPP_nat_nat(times_times_nat(A_35),B_32))) ).
tff(fact_752_dvd__triv__left,axiom,
! [A_35: int,B_32: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_35),hAPP_int_int(times_times_int(A_35),B_32))) ).
tff(fact_753_dvd__add,axiom,
! [C_15: real,A_34: real,B_31: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_34),B_31))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_34),C_15))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_34),hAPP_real_real(plus_plus_real(B_31),C_15))) ) ) ).
tff(fact_754_dvd__add,axiom,
! [C_15: nat,A_34: nat,B_31: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_34),B_31))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_34),C_15))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_34),hAPP_nat_nat(plus_plus_nat(B_31),C_15))) ) ) ).
tff(fact_755_dvd__add,axiom,
! [C_15: int,A_34: int,B_31: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_34),B_31))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_34),C_15))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_34),hAPP_int_int(plus_plus_int(B_31),C_15))) ) ) ).
tff(fact_756_one__dvd,axiom,
! [A_33: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,one_one_real),A_33)) ).
tff(fact_757_one__dvd,axiom,
! [A_33: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_33)) ).
tff(fact_758_one__dvd,axiom,
! [A_33: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,one_one_int),A_33)) ).
tff(fact_759_dvd__diff,axiom,
! [Z_2: real,X_3: real,Y_3: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_3),Y_3))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_3),Z_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_3),hAPP_real_real(minus_minus_real(Y_3),Z_2))) ) ) ).
tff(fact_760_dvd__diff,axiom,
! [Z_2: int,X_3: int,Y_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_3),Y_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_3),Z_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_3),hAPP_int_int(minus_minus_int(Y_3),Z_2))) ) ) ).
tff(fact_761_real__mult__right__cancel,axiom,
! [A_2: real,B_2: real,C_2: real] :
( ( C_2 != zero_zero_real )
=> ( ( hAPP_real_real(times_times_real(A_2),C_2) = hAPP_real_real(times_times_real(B_2),C_2) )
<=> ( A_2 = B_2 ) ) ) ).
tff(fact_762_real__mult__left__cancel,axiom,
! [A_2: real,B_2: real,C_2: real] :
( ( C_2 != zero_zero_real )
=> ( ( hAPP_real_real(times_times_real(C_2),A_2) = hAPP_real_real(times_times_real(C_2),B_2) )
<=> ( A_2 = B_2 ) ) ) ).
tff(fact_763_divides__add__revr,axiom,
! [B: nat,D_5: nat,A: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_5),A))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_5),hAPP_nat_nat(plus_plus_nat(A),B)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_5),B)) ) ) ).
tff(fact_764_divides__mul__l,axiom,
! [C: nat,A: nat,B: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(C),A)),hAPP_nat_nat(times_times_nat(C),B))) ) ).
tff(fact_765_divides__mul__r,axiom,
! [C: nat,A: nat,B: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(A),C)),hAPP_nat_nat(times_times_nat(B),C))) ) ).
tff(fact_766_nat__mult__eq__one,axiom,
! [N: nat,Ma: nat] :
( ( hAPP_nat_nat(times_times_nat(N),Ma) = one_one_nat )
<=> ( ( N = one_one_nat )
& ( Ma = one_one_nat ) ) ) ).
tff(fact_767_real__add__mult__distrib,axiom,
! [Z1: real,Z2: real,W: real] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(Z1),Z2)),W) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Z1),W)),hAPP_real_real(times_times_real(Z2),W)) ).
tff(fact_768_nat__power__eq__0__iff,axiom,
! [Ma: nat,N: nat] :
( ( hAPP_nat_nat(power_power_nat(Ma),N) = zero_zero_nat )
<=> ( ( N != zero_zero_nat )
& ( Ma = zero_zero_nat ) ) ) ).
tff(fact_769_divides__exp,axiom,
! [N_1: nat,X_1: nat,Y_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(X_1),N_1)),hAPP_nat_nat(power_power_nat(Y_1),N_1))) ) ).
tff(fact_770_split__mult__neg__le,axiom,
! [B_30: real,A_32: real] :
( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_32))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_30),zero_zero_real)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_32),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_30)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_32),B_30)),zero_zero_real)) ) ).
tff(fact_771_split__mult__neg__le,axiom,
! [B_30: nat,A_32: nat] :
( ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_32))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_30),zero_zero_nat)) )
| ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_32),zero_zero_nat))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_30)) ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_32),B_30)),zero_zero_nat)) ) ).
tff(fact_772_split__mult__neg__le,axiom,
! [B_30: int,A_32: int] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_32))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_30),zero_zero_int)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_32),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_30)) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_32),B_30)),zero_zero_int)) ) ).
tff(fact_773_split__mult__pos__le,axiom,
! [B_29: real,A_31: real] :
( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_31))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_29)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_31),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_29),zero_zero_real)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_31),B_29))) ) ).
tff(fact_774_split__mult__pos__le,axiom,
! [B_29: int,A_31: int] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_31))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_29)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_31),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_29),zero_zero_int)) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_31),B_29))) ) ).
tff(fact_775_mult__mono,axiom,
! [C_14: real,D_4: real,A_30: real,B_28: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_30),B_28))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_14),D_4))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_28))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_14))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_30),C_14)),hAPP_real_real(times_times_real(B_28),D_4))) ) ) ) ) ).
tff(fact_776_mult__mono,axiom,
! [C_14: nat,D_4: nat,A_30: nat,B_28: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_30),B_28))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_14),D_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_28))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_14))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_30),C_14)),hAPP_nat_nat(times_times_nat(B_28),D_4))) ) ) ) ) ).
tff(fact_777_mult__mono,axiom,
! [C_14: int,D_4: int,A_30: int,B_28: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_30),B_28))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_14),D_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_28))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_14))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_30),C_14)),hAPP_int_int(times_times_int(B_28),D_4))) ) ) ) ) ).
tff(fact_778_mult__mono_H,axiom,
! [C_13: real,D_3: real,A_29: real,B_27: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_29),B_27))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_13),D_3))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_29))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_13))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_29),C_13)),hAPP_real_real(times_times_real(B_27),D_3))) ) ) ) ) ).
tff(fact_779_mult__mono_H,axiom,
! [C_13: nat,D_3: nat,A_29: nat,B_27: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_29),B_27))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_13),D_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_29))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_13))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_29),C_13)),hAPP_nat_nat(times_times_nat(B_27),D_3))) ) ) ) ) ).
tff(fact_780_mult__mono_H,axiom,
! [C_13: int,D_3: int,A_29: int,B_27: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_29),B_27))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_13),D_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_29))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_29),C_13)),hAPP_int_int(times_times_int(B_27),D_3))) ) ) ) ) ).
tff(fact_781_mult__left__mono__neg,axiom,
! [C_12: real,B_26: real,A_28: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_26),A_28))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_12),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(C_12),A_28)),hAPP_real_real(times_times_real(C_12),B_26))) ) ) ).
tff(fact_782_mult__left__mono__neg,axiom,
! [C_12: int,B_26: int,A_28: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_26),A_28))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_12),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(C_12),A_28)),hAPP_int_int(times_times_int(C_12),B_26))) ) ) ).
tff(fact_783_mult__right__mono__neg,axiom,
! [C_11: real,B_25: real,A_27: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_25),A_27))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_11),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_27),C_11)),hAPP_real_real(times_times_real(B_25),C_11))) ) ) ).
tff(fact_784_mult__right__mono__neg,axiom,
! [C_11: int,B_25: int,A_27: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_25),A_27))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_11),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_27),C_11)),hAPP_int_int(times_times_int(B_25),C_11))) ) ) ).
tff(fact_785_comm__mult__left__mono,axiom,
! [C_10: real,A_26: real,B_24: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_26),B_24))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_10))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(C_10),A_26)),hAPP_real_real(times_times_real(C_10),B_24))) ) ) ).
tff(fact_786_comm__mult__left__mono,axiom,
! [C_10: nat,A_26: nat,B_24: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_26),B_24))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_10))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(C_10),A_26)),hAPP_nat_nat(times_times_nat(C_10),B_24))) ) ) ).
tff(fact_787_comm__mult__left__mono,axiom,
! [C_10: int,A_26: int,B_24: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_26),B_24))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_10))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(C_10),A_26)),hAPP_int_int(times_times_int(C_10),B_24))) ) ) ).
tff(fact_788_mult__left__mono,axiom,
! [C_9: real,A_25: real,B_23: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_25),B_23))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_9))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(C_9),A_25)),hAPP_real_real(times_times_real(C_9),B_23))) ) ) ).
tff(fact_789_mult__left__mono,axiom,
! [C_9: nat,A_25: nat,B_23: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_25),B_23))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(C_9),A_25)),hAPP_nat_nat(times_times_nat(C_9),B_23))) ) ) ).
tff(fact_790_mult__left__mono,axiom,
! [C_9: int,A_25: int,B_23: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_25),B_23))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(C_9),A_25)),hAPP_int_int(times_times_int(C_9),B_23))) ) ) ).
tff(fact_791_mult__right__mono,axiom,
! [C_8: real,A_24: real,B_22: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_24),B_22))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_8))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_24),C_8)),hAPP_real_real(times_times_real(B_22),C_8))) ) ) ).
tff(fact_792_mult__right__mono,axiom,
! [C_8: nat,A_24: nat,B_22: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_24),B_22))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_24),C_8)),hAPP_nat_nat(times_times_nat(B_22),C_8))) ) ) ).
tff(fact_793_mult__right__mono,axiom,
! [C_8: int,A_24: int,B_22: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_24),B_22))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_24),C_8)),hAPP_int_int(times_times_int(B_22),C_8))) ) ) ).
tff(fact_794_mult__nonpos__nonpos,axiom,
! [B_21: real,A_23: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_23),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_21),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_23),B_21))) ) ) ).
tff(fact_795_mult__nonpos__nonpos,axiom,
! [B_21: int,A_23: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_23),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_21),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_23),B_21))) ) ) ).
tff(fact_796_mult__nonpos__nonneg,axiom,
! [B_20: real,A_22: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_22),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_20))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_22),B_20)),zero_zero_real)) ) ) ).
tff(fact_797_mult__nonpos__nonneg,axiom,
! [B_20: nat,A_22: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_22),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_20))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_22),B_20)),zero_zero_nat)) ) ) ).
tff(fact_798_mult__nonpos__nonneg,axiom,
! [B_20: int,A_22: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_22),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_20))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_22),B_20)),zero_zero_int)) ) ) ).
tff(fact_799_mult__nonneg__nonpos2,axiom,
! [B_19: real,A_21: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_21))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_19),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(B_19),A_21)),zero_zero_real)) ) ) ).
tff(fact_800_mult__nonneg__nonpos2,axiom,
! [B_19: nat,A_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_21))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_19),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(B_19),A_21)),zero_zero_nat)) ) ) ).
tff(fact_801_mult__nonneg__nonpos2,axiom,
! [B_19: int,A_21: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_19),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(B_19),A_21)),zero_zero_int)) ) ) ).
tff(fact_802_mult__nonneg__nonpos,axiom,
! [B_18: real,A_20: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_20))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_18),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_20),B_18)),zero_zero_real)) ) ) ).
tff(fact_803_mult__nonneg__nonpos,axiom,
! [B_18: nat,A_20: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_20))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_18),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_20),B_18)),zero_zero_nat)) ) ) ).
tff(fact_804_mult__nonneg__nonpos,axiom,
! [B_18: int,A_20: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_20))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_18),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_20),B_18)),zero_zero_int)) ) ) ).
tff(fact_805_mult__nonneg__nonneg,axiom,
! [B_17: real,A_19: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_19))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_17))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_19),B_17))) ) ) ).
tff(fact_806_mult__nonneg__nonneg,axiom,
! [B_17: nat,A_19: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_19))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_17))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(A_19),B_17))) ) ) ).
tff(fact_807_mult__nonneg__nonneg,axiom,
! [B_17: int,A_19: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_19))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_17))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_19),B_17))) ) ) ).
tff(fact_808_mult__le__0__iff,axiom,
! [A_2: real,B_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_2),B_2)),zero_zero_real))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_2))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_2),zero_zero_real)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_2)) ) ) ) ).
tff(fact_809_mult__le__0__iff,axiom,
! [A_2: int,B_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_2),B_2)),zero_zero_int))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_2))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2)) ) ) ) ).
tff(fact_810_zero__le__mult__iff,axiom,
! [A_2: real,B_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_2),B_2)))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_2))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_2)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_2),zero_zero_real)) ) ) ) ).
tff(fact_811_zero__le__mult__iff,axiom,
! [A_2: int,B_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_2),B_2)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_2))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) ) ) ) ).
tff(fact_812_zero__le__square,axiom,
! [A_18: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_18),A_18))) ).
tff(fact_813_zero__le__square,axiom,
! [A_18: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_18),A_18))) ).
tff(fact_814_mult__strict__left__mono__neg,axiom,
! [C_7: real,B_16: real,A_17: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_16),A_17))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_7),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_7),A_17)),hAPP_real_real(times_times_real(C_7),B_16))) ) ) ).
tff(fact_815_mult__strict__left__mono__neg,axiom,
! [C_7: int,B_16: int,A_17: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_16),A_17))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_7),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_7),A_17)),hAPP_int_int(times_times_int(C_7),B_16))) ) ) ).
tff(fact_816_mult__strict__right__mono__neg,axiom,
! [C_6: real,B_15: real,A_16: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_15),A_16))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_6),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_16),C_6)),hAPP_real_real(times_times_real(B_15),C_6))) ) ) ).
tff(fact_817_mult__strict__right__mono__neg,axiom,
! [C_6: int,B_15: int,A_16: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_15),A_16))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_6),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_16),C_6)),hAPP_int_int(times_times_int(B_15),C_6))) ) ) ).
tff(fact_818_comm__mult__strict__left__mono,axiom,
! [C_5: real,A_15: real,B_14: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_15),B_14))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_5))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_5),A_15)),hAPP_real_real(times_times_real(C_5),B_14))) ) ) ).
tff(fact_819_comm__mult__strict__left__mono,axiom,
! [C_5: nat,A_15: nat,B_14: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_15),B_14))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(C_5),A_15)),hAPP_nat_nat(times_times_nat(C_5),B_14))) ) ) ).
tff(fact_820_comm__mult__strict__left__mono,axiom,
! [C_5: int,A_15: int,B_14: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_15),B_14))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_5),A_15)),hAPP_int_int(times_times_int(C_5),B_14))) ) ) ).
tff(fact_821_mult__strict__left__mono,axiom,
! [C_4: real,A_14: real,B_13: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_14),B_13))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_4))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_4),A_14)),hAPP_real_real(times_times_real(C_4),B_13))) ) ) ).
tff(fact_822_mult__strict__left__mono,axiom,
! [C_4: nat,A_14: nat,B_13: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_14),B_13))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(C_4),A_14)),hAPP_nat_nat(times_times_nat(C_4),B_13))) ) ) ).
tff(fact_823_mult__strict__left__mono,axiom,
! [C_4: int,A_14: int,B_13: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_14),B_13))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_4))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_4),A_14)),hAPP_int_int(times_times_int(C_4),B_13))) ) ) ).
tff(fact_824_mult__strict__right__mono,axiom,
! [C_3: real,A_13: real,B_12: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_13),B_12))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_3))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_13),C_3)),hAPP_real_real(times_times_real(B_12),C_3))) ) ) ).
tff(fact_825_mult__strict__right__mono,axiom,
! [C_3: nat,A_13: nat,B_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_13),B_12))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_13),C_3)),hAPP_nat_nat(times_times_nat(B_12),C_3))) ) ) ).
tff(fact_826_mult__strict__right__mono,axiom,
! [C_3: int,A_13: int,B_12: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_13),B_12))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_13),C_3)),hAPP_int_int(times_times_int(B_12),C_3))) ) ) ).
tff(fact_827_mult__neg__neg,axiom,
! [B_11: real,A_12: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_12),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_11),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(A_12),B_11))) ) ) ).
tff(fact_828_mult__neg__neg,axiom,
! [B_11: int,A_12: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_12),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_11),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(A_12),B_11))) ) ) ).
tff(fact_829_mult__neg__pos,axiom,
! [B_10: real,A_11: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_11),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_10))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_11),B_10)),zero_zero_real)) ) ) ).
tff(fact_830_mult__neg__pos,axiom,
! [B_10: nat,A_11: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_11),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_10))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_11),B_10)),zero_zero_nat)) ) ) ).
tff(fact_831_mult__neg__pos,axiom,
! [B_10: int,A_11: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_11),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_11),B_10)),zero_zero_int)) ) ) ).
tff(fact_832_mult__less__cancel__left__neg,axiom,
! [A_2: real,B_2: real,C_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_2),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_2),A_2)),hAPP_real_real(times_times_real(C_2),B_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_2),A_2)) ) ) ).
tff(fact_833_mult__less__cancel__left__neg,axiom,
! [A_2: int,B_2: int,C_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_2),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_2),A_2)),hAPP_int_int(times_times_int(C_2),B_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_2)) ) ) ).
tff(fact_834_zero__less__mult__pos2,axiom,
! [B_9: real,A_10: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(B_9),A_10)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_10))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_9)) ) ) ).
tff(fact_835_zero__less__mult__pos2,axiom,
! [B_9: nat,A_10: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(B_9),A_10)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_10))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_9)) ) ) ).
tff(fact_836_zero__less__mult__pos2,axiom,
! [B_9: int,A_10: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(B_9),A_10)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_10))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_9)) ) ) ).
tff(fact_837_zero__less__mult__pos,axiom,
! [A_9: real,B_8: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(A_9),B_8)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_9))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_8)) ) ) ).
tff(fact_838_zero__less__mult__pos,axiom,
! [A_9: nat,B_8: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(A_9),B_8)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_8)) ) ) ).
tff(fact_839_zero__less__mult__pos,axiom,
! [A_9: int,B_8: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(A_9),B_8)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_8)) ) ) ).
tff(fact_840_mult__pos__neg2,axiom,
! [B_7: real,A_8: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_8))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_7),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(B_7),A_8)),zero_zero_real)) ) ) ).
tff(fact_841_mult__pos__neg2,axiom,
! [B_7: nat,A_8: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_8))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_7),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(B_7),A_8)),zero_zero_nat)) ) ) ).
tff(fact_842_mult__pos__neg2,axiom,
! [B_7: int,A_8: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_8))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_7),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(B_7),A_8)),zero_zero_int)) ) ) ).
tff(fact_843_mult__pos__neg,axiom,
! [B_6: real,A_7: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_7))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_6),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_7),B_6)),zero_zero_real)) ) ) ).
tff(fact_844_mult__pos__neg,axiom,
! [B_6: nat,A_7: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_7))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_6),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_7),B_6)),zero_zero_nat)) ) ) ).
tff(fact_845_mult__pos__neg,axiom,
! [B_6: int,A_7: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_6),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_7),B_6)),zero_zero_int)) ) ) ).
tff(fact_846_mult__pos__pos,axiom,
! [B_5: real,A_6: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_6))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_5))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(A_6),B_5))) ) ) ).
tff(fact_847_mult__pos__pos,axiom,
! [B_5: nat,A_6: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(A_6),B_5))) ) ) ).
tff(fact_848_mult__pos__pos,axiom,
! [B_5: int,A_6: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(A_6),B_5))) ) ) ).
tff(fact_849_mult__less__cancel__left__pos,axiom,
! [A_2: real,B_2: real,C_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_2),A_2)),hAPP_real_real(times_times_real(C_2),B_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_2),B_2)) ) ) ).
tff(fact_850_mult__less__cancel__left__pos,axiom,
! [A_2: int,B_2: int,C_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_2),A_2)),hAPP_int_int(times_times_int(C_2),B_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_2)) ) ) ).
tff(fact_851_mult__less__cancel__left__disj,axiom,
! [C_2: real,A_2: real,B_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_2),A_2)),hAPP_real_real(times_times_real(C_2),B_2)))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_2))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_2),B_2)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_2),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_2),A_2)) ) ) ) ).
tff(fact_852_mult__less__cancel__left__disj,axiom,
! [C_2: int,A_2: int,B_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_2),A_2)),hAPP_int_int(times_times_int(C_2),B_2)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_2))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_2)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_2),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_2)) ) ) ) ).
tff(fact_853_mult__less__cancel__right__disj,axiom,
! [A_2: real,C_2: real,B_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_2),C_2)),hAPP_real_real(times_times_real(B_2),C_2)))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_2))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_2),B_2)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_2),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_2),A_2)) ) ) ) ).
tff(fact_854_mult__less__cancel__right__disj,axiom,
! [A_2: int,C_2: int,B_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_2),C_2)),hAPP_int_int(times_times_int(B_2),C_2)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_2))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_2)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_2),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_2)) ) ) ) ).
tff(fact_855_not__square__less__zero,axiom,
! [A_5: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_5),A_5)),zero_zero_real)) ).
tff(fact_856_not__square__less__zero,axiom,
! [A_5: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_5),A_5)),zero_zero_int)) ).
tff(fact_857_pos__add__strict,axiom,
! [B_4: nat,C_1: nat,A_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_4),C_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_4),hAPP_nat_nat(plus_plus_nat(A_4),C_1))) ) ) ).
tff(fact_858_pos__add__strict,axiom,
! [B_4: int,C_1: int,A_4: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_4),C_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_4),hAPP_int_int(plus_plus_int(A_4),C_1))) ) ) ).
tff(fact_859_divides__ge,axiom,
! [A: nat,B: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))
=> ( ( B = zero_zero_nat )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B)) ) ) ).
tff(fact_860_nat__mult__dvd__cancel__disj_H,axiom,
! [Ma: nat,K: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(Ma),K)),hAPP_nat_nat(times_times_nat(N),K)))
<=> ( ( K = zero_zero_nat )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),N)) ) ) ).
tff(fact_861_real__mult__less__iff1,axiom,
! [X_2: real,Y_2: real,Z_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(X_2),Z_1)),hAPP_real_real(times_times_real(Y_2),Z_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) ) ) ).
tff(fact_862_real__mult__le__cancel__iff1,axiom,
! [X_2: real,Y_2: real,Z_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(X_2),Z_1)),hAPP_real_real(times_times_real(Y_2),Z_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)) ) ) ).
tff(fact_863_real__mult__le__cancel__iff2,axiom,
! [X_2: real,Y_2: real,Z_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(Z_1),X_2)),hAPP_real_real(times_times_real(Z_1),Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)) ) ) ).
tff(fact_864_real__mult__order,axiom,
! [Y_1: real,X_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(X_1),Y_1))) ) ) ).
tff(fact_865_real__mult__less__mono2,axiom,
! [X_1: real,Y_1: real,Z: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(Z),X_1)),hAPP_real_real(times_times_real(Z),Y_1))) ) ) ).
tff(fact_866_real__two__squares__add__zero__iff,axiom,
! [X_2: real,Y_2: real] :
( ( hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_2),X_2)),hAPP_real_real(times_times_real(Y_2),Y_2)) = zero_zero_real )
<=> ( ( X_2 = zero_zero_real )
& ( Y_2 = zero_zero_real ) ) ) ).
tff(fact_867_divides__exp2,axiom,
! [X_1: nat,Y_1: nat,N_1: nat] :
( ( N_1 != zero_zero_nat )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(X_1),N_1)),Y_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) ) ) ).
tff(fact_868_divides__rev,axiom,
! [A: nat,N_1: nat,B: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(A),N_1)),hAPP_nat_nat(power_power_nat(B),N_1)))
=> ( ( N_1 != zero_zero_nat )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) ) ) ).
tff(fact_869_exp__eq__1,axiom,
! [X_2: nat,N: nat] :
( ( hAPP_nat_nat(power_power_nat(X_2),N) = one_one_nat )
<=> ( ( X_2 = one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
tff(fact_870_divides__div__not,axiom,
! [X_1: nat,Q_1: nat,N_1: nat,R_1: nat] :
( ( X_1 = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(Q_1),N_1)),R_1) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),R_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_1),N_1))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),X_1)) ) ) ) ).
tff(fact_871_two__realpow__ge__one,axiom,
! [N_1: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(power_power_real(number267125858f_real(bit0(bit1(pls)))),N_1))) ).
tff(fact_872_realpow__pos__nth,axiom,
! [A: real,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A))
=> ? [R_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_2))
& ( hAPP_nat_real(power_power_real(R_2),N_1) = A ) ) ) ) ).
tff(fact_873_realpow__pos__nth__unique,axiom,
! [A: real,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A))
=> ? [X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
& ( hAPP_nat_real(power_power_real(X),N_1) = A )
& ! [Y: real] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
& ( hAPP_nat_real(power_power_real(Y),N_1) = A ) )
=> ( Y = X ) ) ) ) ) ).
tff(fact_874_dvd__mult__cancel2,axiom,
! [N: nat,Ma: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(N),Ma)),Ma))
<=> ( N = one_one_nat ) ) ) ).
tff(fact_875_dvd__mult__cancel1,axiom,
! [N: nat,Ma: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(Ma),N)),Ma))
<=> ( N = one_one_nat ) ) ) ).
tff(fact_876_le0,axiom,
! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1)) ).
tff(fact_877_dvd_Oorder__refl,axiom,
! [X_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),X_1)) ).
tff(fact_878_less__zeroE,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).
tff(fact_879_diff__commute,axiom,
! [I_1: nat,J_1: nat,K_1: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I_1),J_1)),K_1) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I_1),K_1)),J_1) ).
tff(fact_880_real__le__refl,axiom,
! [W: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),W)) ).
tff(fact_881_real__le__linear,axiom,
! [Z: real,W: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z),W))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z)) ) ).
tff(fact_882_real__le__trans,axiom,
! [K_1: real,I_1: real,J_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I_1),J_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,J_1),K_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I_1),K_1)) ) ) ).
tff(fact_883_real__le__antisym,axiom,
! [Z: real,W: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z),W))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z))
=> ( Z = W ) ) ) ).
tff(fact_884_diff__0__eq__0,axiom,
! [N_1: nat] : hAPP_nat_nat(minus_minus_nat(zero_zero_nat),N_1) = zero_zero_nat ).
tff(fact_885_minus__nat_Odiff__0,axiom,
! [M: nat] : hAPP_nat_nat(minus_minus_nat(M),zero_zero_nat) = M ).
tff(fact_886_diff__self__eq__0,axiom,
! [M: nat] : hAPP_nat_nat(minus_minus_nat(M),M) = zero_zero_nat ).
tff(fact_887_diffs0__imp__equal,axiom,
! [M: nat,N_1: nat] :
( ( hAPP_nat_nat(minus_minus_nat(M),N_1) = zero_zero_nat )
=> ( ( hAPP_nat_nat(minus_minus_nat(N_1),M) = zero_zero_nat )
=> ( M = N_1 ) ) ) ).
tff(fact_888_nat__less__cases,axiom,
! [P_2: fun_nat_fun_nat_bool,Ma: nat,N: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_2,N),Ma)) )
=> ( ( ( Ma = N )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_2,N),Ma)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),Ma))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_2,N),Ma)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_2,N),Ma)) ) ) ) ).
tff(fact_889_less__not__refl3,axiom,
! [S: nat,T: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S),T))
=> ( S != T ) ) ).
tff(fact_890_less__not__refl2,axiom,
! [N_1: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M))
=> ( M != N_1 ) ) ).
tff(fact_891_less__irrefl__nat,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1)) ).
tff(fact_892_linorder__neqE__nat,axiom,
! [X_1: nat,Y_1: nat] :
( ( X_1 != Y_1 )
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_1),X_1)) ) ) ).
tff(fact_893_nat__neq__iff,axiom,
! [Ma: nat,N: nat] :
( ( Ma != N )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),Ma)) ) ) ).
tff(fact_894_less__not__refl,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1)) ).
tff(fact_895_diff__less__mono2,axiom,
! [L: nat,M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(L),N_1)),hAPP_nat_nat(minus_minus_nat(L),M))) ) ) ).
tff(fact_896_less__imp__diff__less,axiom,
! [N_1: nat,J_1: nat,K_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(J_1),N_1)),K_1)) ) ).
tff(fact_897_dvd_Oless__asym,axiom,
! [X_1: nat,Y_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) ) ) ).
tff(fact_898_dvd_Oless__trans,axiom,
! [Z: nat,X_1: nat,Y_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),Z))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),Y_1)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),X_1)) ) ) ) ).
tff(fact_899_dvd_Oless__asym_H,axiom,
! [A: nat,B: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B),A)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B),A))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) ) ) ).
tff(fact_900_dvd_Oless__le__trans,axiom,
! [Z: nat,X_1: nat,Y_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),Z))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),X_1)) ) ) ) ).
tff(fact_901_dvd_Oord__less__eq__trans,axiom,
! [C: nat,A: nat,B: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B),A)) )
=> ( ( B = C )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C),A)) ) ) ) ).
tff(fact_902_dvd_Oless__imp__triv,axiom,
! [P_2: bool,X_2: nat,Y_2: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) )
=> hBOOL(P_2) ) ) ).
tff(fact_903_dvd_Oless__imp__not__eq2,axiom,
! [X_1: nat,Y_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
=> ( Y_1 != X_1 ) ) ).
tff(fact_904_dvd_Oless__imp__not__eq,axiom,
! [X_1: nat,Y_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
=> ( X_1 != Y_1 ) ) ).
tff(fact_905_dvd_Oless__imp__not__less,axiom,
! [X_1: nat,Y_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) ) ) ).
tff(fact_906_dvd_Oless__imp__le,axiom,
! [X_1: nat,Y_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) ) ).
tff(fact_907_dvd_Oless__not__sym,axiom,
! [X_1: nat,Y_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) ) ) ).
tff(fact_908_dvd_Oless__imp__neq,axiom,
! [X_1: nat,Y_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
=> ( X_1 != Y_1 ) ) ).
tff(fact_909_dvd_Ole__less__trans,axiom,
! [Z: nat,X_1: nat,Y_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),Z))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),Y_1)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),X_1)) ) ) ) ).
tff(fact_910_dvd_Oord__eq__less__trans,axiom,
! [C: nat,A: nat,B: nat] :
( ( A = B )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B),C))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C),B)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C),A)) ) ) ) ).
tff(fact_911_dvd_Oorder__trans,axiom,
! [Z: nat,X_1: nat,Y_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),Z))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z)) ) ) ).
tff(fact_912_dvd_Oantisym,axiom,
! [X_1: nat,Y_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1))
=> ( X_1 = Y_1 ) ) ) ).
tff(fact_913_dvd__antisym,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),M))
=> ( M = N_1 ) ) ) ).
tff(fact_914_dvd_Oord__le__eq__trans,axiom,
! [C: nat,A: nat,B: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))
=> ( ( B = C )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C)) ) ) ).
tff(fact_915_dvd_Oord__eq__le__trans,axiom,
! [C: nat,A: nat,B: nat] :
( ( A = B )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B),C))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C)) ) ) ).
tff(fact_916_dvd_Ole__neq__trans,axiom,
! [A: nat,B: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))
=> ( ( A != B )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B),A)) ) ) ) ).
tff(fact_917_dvd_Ole__imp__less__or__eq,axiom,
! [X_1: nat,Y_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
| ( X_1 = Y_1 ) ) ) ).
tff(fact_918_dvd_Oantisym__conv,axiom,
! [Y_2: nat,X_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
<=> ( X_2 = Y_2 ) ) ) ).
tff(fact_919_dvd_Oeq__refl,axiom,
! [X_1: nat,Y_1: nat] :
( ( X_1 = Y_1 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) ) ).
tff(fact_920_dvd_Oneq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B),A)) ) ) ) ).
tff(fact_921_dvd_Oless__le__not__le,axiom,
! [X_2: nat,Y_2: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) ) ) ).
tff(fact_922_dvd_Oless__le,axiom,
! [X_2: nat,Y_2: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
& ( X_2 != Y_2 ) ) ) ).
tff(fact_923_dvd_Ole__less,axiom,
! [X_2: nat,Y_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) )
| ( X_2 = Y_2 ) ) ) ).
tff(fact_924_dvd_Oeq__iff,axiom,
! [X_2: nat,Y_2: nat] :
( ( X_2 = Y_2 )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) ) ) ).
tff(fact_925_dvd_Oless__irrefl,axiom,
! [X_1: nat] :
~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),X_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),X_1)) ) ).
tff(fact_926_dvd__diff__nat,axiom,
! [N_1: nat,K_1: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(minus_minus_nat(M),N_1))) ) ) ).
tff(fact_927_nat__add__commute,axiom,
! [M: nat,N_1: nat] : hAPP_nat_nat(plus_plus_nat(M),N_1) = hAPP_nat_nat(plus_plus_nat(N_1),M) ).
tff(fact_928_nat__add__left__commute,axiom,
! [X_1: nat,Y_1: nat,Z: nat] : hAPP_nat_nat(plus_plus_nat(X_1),hAPP_nat_nat(plus_plus_nat(Y_1),Z)) = hAPP_nat_nat(plus_plus_nat(Y_1),hAPP_nat_nat(plus_plus_nat(X_1),Z)) ).
tff(fact_929_nat__add__assoc,axiom,
! [M: nat,N_1: nat,K_1: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(M),N_1)),K_1) = hAPP_nat_nat(plus_plus_nat(M),hAPP_nat_nat(plus_plus_nat(N_1),K_1)) ).
tff(fact_930_nat__add__left__cancel,axiom,
! [K: nat,Ma: nat,N: nat] :
( ( hAPP_nat_nat(plus_plus_nat(K),Ma) = hAPP_nat_nat(plus_plus_nat(K),N) )
<=> ( Ma = N ) ) ).
tff(fact_931_nat__add__right__cancel,axiom,
! [Ma: nat,K: nat,N: nat] :
( ( hAPP_nat_nat(plus_plus_nat(Ma),K) = hAPP_nat_nat(plus_plus_nat(N),K) )
<=> ( Ma = N ) ) ).
tff(fact_932_diff__add__inverse2,axiom,
! [M: nat,N_1: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(M),N_1)),N_1) = M ).
tff(fact_933_diff__add__inverse,axiom,
! [N_1: nat,M: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(N_1),M)),N_1) = M ).
tff(fact_934_diff__diff__left,axiom,
! [I_1: nat,J_1: nat,K_1: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I_1),J_1)),K_1) = hAPP_nat_nat(minus_minus_nat(I_1),hAPP_nat_nat(plus_plus_nat(J_1),K_1)) ).
tff(fact_935_Nat_Odiff__cancel,axiom,
! [K_1: nat,M: nat,N_1: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(K_1),M)),hAPP_nat_nat(plus_plus_nat(K_1),N_1)) = hAPP_nat_nat(minus_minus_nat(M),N_1) ).
tff(fact_936_diff__cancel2,axiom,
! [M: nat,K_1: nat,N_1: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(M),K_1)),hAPP_nat_nat(plus_plus_nat(N_1),K_1)) = hAPP_nat_nat(minus_minus_nat(M),N_1) ).
tff(fact_937_le__refl,axiom,
! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),N_1)) ).
tff(fact_938_nat__le__linear,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) ) ).
tff(fact_939_eq__imp__le,axiom,
! [M: nat,N_1: nat] :
( ( M = N_1 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).
tff(fact_940_le__trans,axiom,
! [K_1: nat,I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),K_1)) ) ) ).
tff(fact_941_le__antisym,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> ( M = N_1 ) ) ) ).
tff(fact_942_Nat_Odiff__le__self,axiom,
! [M: nat,N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(M),N_1)),M)) ).
tff(fact_943_diff__le__mono2,axiom,
! [L: nat,M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(L),N_1)),hAPP_nat_nat(minus_minus_nat(L),M))) ) ).
tff(fact_944_diff__le__mono,axiom,
! [L: nat,M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(M),L)),hAPP_nat_nat(minus_minus_nat(N_1),L))) ) ).
tff(fact_945_diff__diff__cancel,axiom,
! [I_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),N_1))
=> ( hAPP_nat_nat(minus_minus_nat(N_1),hAPP_nat_nat(minus_minus_nat(N_1),I_1)) = I_1 ) ) ).
tff(fact_946_eq__diff__iff,axiom,
! [N: nat,K: nat,Ma: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N))
=> ( ( hAPP_nat_nat(minus_minus_nat(Ma),K) = hAPP_nat_nat(minus_minus_nat(N),K) )
<=> ( Ma = N ) ) ) ) ).
tff(fact_947_Nat_Odiff__diff__eq,axiom,
! [N_1: nat,K_1: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(M),K_1)),hAPP_nat_nat(minus_minus_nat(N_1),K_1)) = hAPP_nat_nat(minus_minus_nat(M),N_1) ) ) ) ).
tff(fact_948_le__diff__iff,axiom,
! [N: nat,K: nat,Ma: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(Ma),K)),hAPP_nat_nat(minus_minus_nat(N),K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ) ) ).
tff(fact_949_nat__mult__assoc,axiom,
! [M: nat,N_1: nat,K_1: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(M),N_1)),K_1) = hAPP_nat_nat(times_times_nat(M),hAPP_nat_nat(times_times_nat(N_1),K_1)) ).
tff(fact_950_nat__mult__commute,axiom,
! [M: nat,N_1: nat] : hAPP_nat_nat(times_times_nat(M),N_1) = hAPP_nat_nat(times_times_nat(N_1),M) ).
tff(fact_951_diff__mult__distrib,axiom,
! [M: nat,N_1: nat,K_1: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(M),N_1)),K_1) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(times_times_nat(M),K_1)),hAPP_nat_nat(times_times_nat(N_1),K_1)) ).
tff(fact_952_diff__mult__distrib2,axiom,
! [K_1: nat,M: nat,N_1: nat] : hAPP_nat_nat(times_times_nat(K_1),hAPP_nat_nat(minus_minus_nat(M),N_1)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(times_times_nat(K_1),M)),hAPP_nat_nat(times_times_nat(K_1),N_1)) ).
tff(fact_953_gr0I,axiom,
! [N_1: nat] :
( ( N_1 != zero_zero_nat )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).
tff(fact_954_gr__implies__not0,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> ( N_1 != zero_zero_nat ) ) ).
tff(fact_955_less__nat__zero__code,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).
tff(fact_956_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
tff(fact_957_not__less0,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).
tff(fact_958_diff__less,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(M),N_1)),M)) ) ) ).
tff(fact_959_zero__less__diff,axiom,
! [N: nat,Ma: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(minus_minus_nat(N),Ma)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ).
tff(fact_960_add__eq__self__zero,axiom,
! [M: nat,N_1: nat] :
( ( hAPP_nat_nat(plus_plus_nat(M),N_1) = M )
=> ( N_1 = zero_zero_nat ) ) ).
tff(fact_961_add__is__0,axiom,
! [Ma: nat,N: nat] :
( ( hAPP_nat_nat(plus_plus_nat(Ma),N) = zero_zero_nat )
<=> ( ( Ma = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
tff(fact_962_Nat_Oadd__0__right,axiom,
! [M: nat] : hAPP_nat_nat(plus_plus_nat(M),zero_zero_nat) = M ).
tff(fact_963_plus__nat_Oadd__0,axiom,
! [N_1: nat] : hAPP_nat_nat(plus_plus_nat(zero_zero_nat),N_1) = N_1 ).
tff(fact_964_diff__add__0,axiom,
! [N_1: nat,M: nat] : hAPP_nat_nat(minus_minus_nat(N_1),hAPP_nat_nat(plus_plus_nat(N_1),M)) = zero_zero_nat ).
tff(fact_965_le__0__eq,axiom,
! [N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),zero_zero_nat))
<=> ( N = zero_zero_nat ) ) ).
tff(fact_966_less__eq__nat_Osimps_I1_J,axiom,
! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1)) ).
tff(fact_967_diff__is__0__eq_H,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> ( hAPP_nat_nat(minus_minus_nat(M),N_1) = zero_zero_nat ) ) ).
tff(fact_968_diff__is__0__eq,axiom,
! [Ma: nat,N: nat] :
( ( hAPP_nat_nat(minus_minus_nat(Ma),N) = zero_zero_nat )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ).
tff(fact_969_mult__0,axiom,
! [N_1: nat] : hAPP_nat_nat(times_times_nat(zero_zero_nat),N_1) = zero_zero_nat ).
tff(fact_970_mult__0__right,axiom,
! [M: nat] : hAPP_nat_nat(times_times_nat(M),zero_zero_nat) = zero_zero_nat ).
tff(fact_971_mult__is__0,axiom,
! [Ma: nat,N: nat] :
( ( hAPP_nat_nat(times_times_nat(Ma),N) = zero_zero_nat )
<=> ( ( Ma = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
tff(fact_972_mult__cancel1,axiom,
! [K: nat,Ma: nat,N: nat] :
( ( hAPP_nat_nat(times_times_nat(K),Ma) = hAPP_nat_nat(times_times_nat(K),N) )
<=> ( ( Ma = N )
| ( K = zero_zero_nat ) ) ) ).
tff(fact_973_mult__cancel2,axiom,
! [Ma: nat,K: nat,N: nat] :
( ( hAPP_nat_nat(times_times_nat(Ma),K) = hAPP_nat_nat(times_times_nat(N),K) )
<=> ( ( Ma = N )
| ( K = zero_zero_nat ) ) ) ).
tff(fact_974_not__add__less1,axiom,
! [I_1: nat,J_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_1),J_1)),I_1)) ).
tff(fact_975_not__add__less2,axiom,
! [J_1: nat,I_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(J_1),I_1)),I_1)) ).
tff(fact_976_nat__add__left__cancel__less,axiom,
! [K: nat,Ma: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(K),Ma)),hAPP_nat_nat(plus_plus_nat(K),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ).
tff(fact_977_trans__less__add1,axiom,
! [M: nat,I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(plus_plus_nat(J_1),M))) ) ).
tff(fact_978_trans__less__add2,axiom,
! [M: nat,I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(plus_plus_nat(M),J_1))) ) ).
tff(fact_979_add__less__mono1,axiom,
! [K_1: nat,I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_1),K_1)),hAPP_nat_nat(plus_plus_nat(J_1),K_1))) ) ).
tff(fact_980_add__less__mono,axiom,
! [K_1: nat,L: nat,I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_1),K_1)),hAPP_nat_nat(plus_plus_nat(J_1),L))) ) ) ).
tff(fact_981_less__add__eq__less,axiom,
! [M: nat,N_1: nat,K_1: nat,L: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),L))
=> ( ( hAPP_nat_nat(plus_plus_nat(M),L) = hAPP_nat_nat(plus_plus_nat(K_1),N_1) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ) ).
tff(fact_982_add__lessD1,axiom,
! [I_1: nat,J_1: nat,K_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_1),J_1)),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),K_1)) ) ).
tff(fact_983_add__diff__inverse,axiom,
! [M: nat,N_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> ( hAPP_nat_nat(plus_plus_nat(N_1),hAPP_nat_nat(minus_minus_nat(M),N_1)) = M ) ) ).
tff(fact_984_less__diff__conv,axiom,
! [I_2: nat,J_2: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),hAPP_nat_nat(minus_minus_nat(J_2),K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_2),K)),J_2)) ) ).
tff(fact_985_nat__less__le,axiom,
! [Ma: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N))
& ( Ma != N ) ) ) ).
tff(fact_986_le__eq__less__or__eq,axiom,
! [Ma: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N))
| ( Ma = N ) ) ) ).
tff(fact_987_less__imp__le__nat,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).
tff(fact_988_le__neq__implies__less,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> ( ( M != N_1 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ) ).
tff(fact_989_less__or__eq__imp__le,axiom,
! [M: nat,N_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
| ( M = N_1 ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).
tff(fact_990_less__diff__iff,axiom,
! [N: nat,K: nat,Ma: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(Ma),K)),hAPP_nat_nat(minus_minus_nat(N),K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ) ) ).
tff(fact_991_diff__less__mono,axiom,
! [C: nat,A: nat,B: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C),A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(A),C)),hAPP_nat_nat(minus_minus_nat(B),C))) ) ) ).
tff(fact_992_dvd__reduce,axiom,
! [K: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),hAPP_nat_nat(plus_plus_nat(N),K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),N)) ) ).
tff(fact_993_dvd__diffD1,axiom,
! [K_1: nat,M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(minus_minus_nat(M),N_1)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N_1)) ) ) ) ).
tff(fact_994_dvd__diffD,axiom,
! [K_1: nat,M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(minus_minus_nat(M),N_1)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),M)) ) ) ) ).
tff(fact_995_le__add2,axiom,
! [N_1: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(plus_plus_nat(M),N_1))) ).
tff(fact_996_le__add1,axiom,
! [N_1: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(plus_plus_nat(N_1),M))) ).
tff(fact_997_le__iff__add,axiom,
! [Ma: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N))
<=> ? [K_2: nat] : N = hAPP_nat_nat(plus_plus_nat(Ma),K_2) ) ).
tff(fact_998_nat__add__left__cancel__le,axiom,
! [K: nat,Ma: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(K),Ma)),hAPP_nat_nat(plus_plus_nat(K),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ).
tff(fact_999_trans__le__add1,axiom,
! [M: nat,I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(plus_plus_nat(J_1),M))) ) ).
tff(fact_1000_trans__le__add2,axiom,
! [M: nat,I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(plus_plus_nat(M),J_1))) ) ).
tff(fact_1001_add__le__mono1,axiom,
! [K_1: nat,I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(I_1),K_1)),hAPP_nat_nat(plus_plus_nat(J_1),K_1))) ) ).
tff(fact_1002_add__le__mono,axiom,
! [K_1: nat,L: nat,I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(I_1),K_1)),hAPP_nat_nat(plus_plus_nat(J_1),L))) ) ) ).
tff(fact_1003_add__leD2,axiom,
! [M: nat,K_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M),K_1)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1)) ) ).
tff(fact_1004_add__leD1,axiom,
! [M: nat,K_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M),K_1)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).
tff(fact_1005_add__leE,axiom,
! [M: nat,K_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M),K_1)),N_1))
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1)) ) ) ).
tff(fact_1006_diff__diff__right,axiom,
! [I_1: nat,K_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1))
=> ( hAPP_nat_nat(minus_minus_nat(I_1),hAPP_nat_nat(minus_minus_nat(J_1),K_1)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(I_1),K_1)),J_1) ) ) ).
tff(fact_1007_le__diff__conv,axiom,
! [J_2: nat,K: nat,I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(J_2),K)),I_2))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),hAPP_nat_nat(plus_plus_nat(I_2),K))) ) ).
tff(fact_1008_le__add__diff,axiom,
! [M: nat,K_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(N_1),M)),K_1))) ) ).
tff(fact_1009_le__add__diff__inverse,axiom,
! [N_1: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> ( hAPP_nat_nat(plus_plus_nat(N_1),hAPP_nat_nat(minus_minus_nat(M),N_1)) = M ) ) ).
tff(fact_1010_add__diff__assoc,axiom,
! [I_1: nat,K_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1))
=> ( hAPP_nat_nat(plus_plus_nat(I_1),hAPP_nat_nat(minus_minus_nat(J_1),K_1)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(I_1),J_1)),K_1) ) ) ).
tff(fact_1011_le__diff__conv2,axiom,
! [I_2: nat,K: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),hAPP_nat_nat(minus_minus_nat(J_2),K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(I_2),K)),J_2)) ) ) ).
tff(fact_1012_le__add__diff__inverse2,axiom,
! [N_1: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(minus_minus_nat(M),N_1)),N_1) = M ) ) ).
tff(fact_1013_le__imp__diff__is__add,axiom,
! [K: nat,I_2: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2))
=> ( ( hAPP_nat_nat(minus_minus_nat(J_2),I_2) = K )
<=> ( J_2 = hAPP_nat_nat(plus_plus_nat(K),I_2) ) ) ) ).
tff(fact_1014_diff__add__assoc,axiom,
! [I_1: nat,K_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(I_1),J_1)),K_1) = hAPP_nat_nat(plus_plus_nat(I_1),hAPP_nat_nat(minus_minus_nat(J_1),K_1)) ) ) ).
tff(fact_1015_add__diff__assoc2,axiom,
! [I_1: nat,K_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1))
=> ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(minus_minus_nat(J_1),K_1)),I_1) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(J_1),I_1)),K_1) ) ) ).
tff(fact_1016_diff__add__assoc2,axiom,
! [I_1: nat,K_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(J_1),I_1)),K_1) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(minus_minus_nat(J_1),K_1)),I_1) ) ) ).
tff(fact_1017_add__mult__distrib,axiom,
! [M: nat,N_1: nat,K_1: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(M),N_1)),K_1) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(M),K_1)),hAPP_nat_nat(times_times_nat(N_1),K_1)) ).
tff(fact_1018_add__mult__distrib2,axiom,
! [K_1: nat,M: nat,N_1: nat] : hAPP_nat_nat(times_times_nat(K_1),hAPP_nat_nat(plus_plus_nat(M),N_1)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(K_1),M)),hAPP_nat_nat(times_times_nat(K_1),N_1)) ).
tff(fact_1019_nat__dvd__1__iff__1,axiom,
! [Ma: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),one_one_nat))
<=> ( Ma = one_one_nat ) ) ).
tff(fact_1020_mult__le__mono,axiom,
! [K_1: nat,L: nat,I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(I_1),K_1)),hAPP_nat_nat(times_times_nat(J_1),L))) ) ) ).
tff(fact_1021_mult__le__mono2,axiom,
! [K_1: nat,I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(K_1),I_1)),hAPP_nat_nat(times_times_nat(K_1),J_1))) ) ).
tff(fact_1022_mult__le__mono1,axiom,
! [K_1: nat,I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(I_1),K_1)),hAPP_nat_nat(times_times_nat(J_1),K_1))) ) ).
tff(fact_1023_le__cube,axiom,
! [M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(times_times_nat(M),hAPP_nat_nat(times_times_nat(M),M)))) ).
tff(fact_1024_le__square,axiom,
! [M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(times_times_nat(M),M))) ).
tff(fact_1025_nat__mult__eq__1__iff,axiom,
! [Ma: nat,N: nat] :
( ( hAPP_nat_nat(times_times_nat(Ma),N) = one_one_nat )
<=> ( ( Ma = one_one_nat )
& ( N = one_one_nat ) ) ) ).
tff(fact_1026_nat__mult__1__right,axiom,
! [N_1: nat] : hAPP_nat_nat(times_times_nat(N_1),one_one_nat) = N_1 ).
tff(fact_1027_nat__1__eq__mult__iff,axiom,
! [Ma: nat,N: nat] :
( ( one_one_nat = hAPP_nat_nat(times_times_nat(Ma),N) )
<=> ( ( Ma = one_one_nat )
& ( N = one_one_nat ) ) ) ).
tff(fact_1028_nat__mult__1,axiom,
! [N_1: nat] : hAPP_nat_nat(times_times_nat(one_one_nat),N_1) = N_1 ).
tff(fact_1029_nat__dvd__not__less,axiom,
! [N_1: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),M)) ) ) ).
tff(fact_1030_add__gr__0,axiom,
! [Ma: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(plus_plus_nat(Ma),N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
tff(fact_1031_nat__diff__split,axiom,
! [P_2: fun_nat_bool,A_2: nat,B_2: nat] :
( hBOOL(hAPP_nat_bool(P_2,hAPP_nat_nat(minus_minus_nat(A_2),B_2)))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_2),B_2))
=> hBOOL(hAPP_nat_bool(P_2,zero_zero_nat)) )
& ! [D_2: nat] :
( ( A_2 = hAPP_nat_nat(plus_plus_nat(B_2),D_2) )
=> hBOOL(hAPP_nat_bool(P_2,D_2)) ) ) ) ).
tff(fact_1032_nat__diff__split__asm,axiom,
! [P_2: fun_nat_bool,A_2: nat,B_2: nat] :
( hBOOL(hAPP_nat_bool(P_2,hAPP_nat_nat(minus_minus_nat(A_2),B_2)))
<=> ~ ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_2),B_2))
& ~ hBOOL(hAPP_nat_bool(P_2,zero_zero_nat)) )
| ? [D_2: nat] :
( ( A_2 = hAPP_nat_nat(plus_plus_nat(B_2),D_2) )
& ~ hBOOL(hAPP_nat_bool(P_2,D_2)) ) ) ) ).
tff(fact_1033_mult__less__mono2,axiom,
! [K_1: nat,I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(K_1),I_1)),hAPP_nat_nat(times_times_nat(K_1),J_1))) ) ) ).
tff(fact_1034_mult__less__mono1,axiom,
! [K_1: nat,I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(I_1),K_1)),hAPP_nat_nat(times_times_nat(J_1),K_1))) ) ) ).
tff(fact_1035_mult__less__cancel2,axiom,
! [Ma: nat,K: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(Ma),K)),hAPP_nat_nat(times_times_nat(N),K)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ) ).
tff(fact_1036_mult__less__cancel1,axiom,
! [K: nat,Ma: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(K),Ma)),hAPP_nat_nat(times_times_nat(K),N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ) ).
tff(fact_1037_nat__0__less__mult__iff,axiom,
! [Ma: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(Ma),N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
tff(fact_1038_mult__eq__self__implies__10,axiom,
! [M: nat,N_1: nat] :
( ( M = hAPP_nat_nat(times_times_nat(M),N_1) )
=> ( ( N_1 = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
tff(fact_1039_dvd__imp__le,axiom,
! [K_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1)) ) ) ).
tff(fact_1040_dvd__mult__cancel,axiom,
! [K_1: nat,M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(K_1),M)),hAPP_nat_nat(times_times_nat(K_1),N_1)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N_1)) ) ) ).
tff(fact_1041_mult__le__cancel1,axiom,
! [K: nat,Ma: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(K),Ma)),hAPP_nat_nat(times_times_nat(K),N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ) ).
tff(fact_1042_mult__le__cancel2,axiom,
! [Ma: nat,K: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(Ma),K)),hAPP_nat_nat(times_times_nat(N),K)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ) ).
tff(fact_1043_ex__least__nat__less,axiom,
! [N: nat,P_2: fun_nat_bool] :
( ~ hBOOL(hAPP_nat_bool(P_2,zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(P_2,N))
=> ? [K_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),N))
& ! [I: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),K_2))
=> ~ hBOOL(hAPP_nat_bool(P_2,I)) )
& hBOOL(hAPP_nat_bool(P_2,hAPP_nat_nat(plus_plus_nat(K_2),one_one_nat))) ) ) ) ).
tff(fact_1044_nat__less__add__iff2,axiom,
! [U: nat,Ma: nat,N: nat,I_2: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_2),U)),Ma)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(J_2),I_2)),U)),N))) ) ) ).
tff(fact_1045_nat__mult__eq__cancel__disj,axiom,
! [K: nat,Ma: nat,N: nat] :
( ( hAPP_nat_nat(times_times_nat(K),Ma) = hAPP_nat_nat(times_times_nat(K),N) )
<=> ( ( K = zero_zero_nat )
| ( Ma = N ) ) ) ).
tff(fact_1046_left__add__mult__distrib,axiom,
! [I_1: nat,U_1: nat,J_1: nat,K_1: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_1),U_1)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_1),U_1)),K_1)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(I_1),J_1)),U_1)),K_1) ).
tff(fact_1047_nat__mult__eq__cancel1,axiom,
! [Ma: nat,N: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( ( hAPP_nat_nat(times_times_nat(K),Ma) = hAPP_nat_nat(times_times_nat(K),N) )
<=> ( Ma = N ) ) ) ).
tff(fact_1048_nat__mult__less__cancel1,axiom,
! [Ma: nat,N: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(K),Ma)),hAPP_nat_nat(times_times_nat(K),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ) ).
tff(fact_1049_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,Ma: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(K),Ma)),hAPP_nat_nat(times_times_nat(K),N)))
<=> ( ( K = zero_zero_nat )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),N)) ) ) ).
tff(fact_1050_nat__mult__dvd__cancel1,axiom,
! [Ma: nat,N: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(K),Ma)),hAPP_nat_nat(times_times_nat(K),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),N)) ) ) ).
tff(fact_1051_nat__mult__le__cancel1,axiom,
! [Ma: nat,N: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(K),Ma)),hAPP_nat_nat(times_times_nat(K),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ) ).
tff(fact_1052_nat__le__add__iff1,axiom,
! [U: nat,Ma: nat,N: nat,J_2: nat,I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_2),U)),Ma)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(I_2),J_2)),U)),Ma)),N)) ) ) ).
tff(fact_1053_nat__diff__add__eq1,axiom,
! [U_1: nat,M: nat,N_1: nat,J_1: nat,I_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),I_1))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_1),U_1)),M)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_1),U_1)),N_1)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(I_1),J_1)),U_1)),M)),N_1) ) ) ).
tff(fact_1054_nat__eq__add__iff1,axiom,
! [U: nat,Ma: nat,N: nat,J_2: nat,I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I_2))
=> ( ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_2),U)),Ma) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N) )
<=> ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(I_2),J_2)),U)),Ma) = N ) ) ) ).
tff(fact_1055_nat__le__add__iff2,axiom,
! [U: nat,Ma: nat,N: nat,I_2: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_2),U)),Ma)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(J_2),I_2)),U)),N))) ) ) ).
tff(fact_1056_nat__diff__add__eq2,axiom,
! [U_1: nat,M: nat,N_1: nat,I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_1),U_1)),M)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_1),U_1)),N_1)) = hAPP_nat_nat(minus_minus_nat(M),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(J_1),I_1)),U_1)),N_1)) ) ) ).
tff(fact_1057_nat__eq__add__iff2,axiom,
! [U: nat,Ma: nat,N: nat,I_2: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2))
=> ( ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_2),U)),Ma) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N) )
<=> ( Ma = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(J_2),I_2)),U)),N) ) ) ) ).
tff(fact_1058_nat__less__add__iff1,axiom,
! [U: nat,Ma: nat,N: nat,J_2: nat,I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_2),U)),Ma)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(I_2),J_2)),U)),Ma)),N)) ) ) ).
tff(fact_1059_number__of1,axiom,
! [N_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(N_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit0(N_1))))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit1(N_1)))) ) ) ).
tff(fact_1060_is__sum2sq__def,axiom,
! [X_2: int] :
( hBOOL(hAPP_int_bool(twoSqu820444569sum2sq,X_2))
<=> ? [A_3: int,B_3: int] : twoSqu949963151sum2sq(product_Pair_int_int(A_3,B_3)) = X_2 ) ).
tff(fact_1061_imp__le__cong,axiom,
! [P_5: bool,P_2: bool,X_2: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> ( hBOOL(P_2)
<=> hBOOL(P_5) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> hBOOL(P_2) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> hBOOL(P_5) ) ) ) ).
tff(fact_1062_conj__le__cong,axiom,
! [P_5: bool,P_2: bool,X_2: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> ( hBOOL(P_2)
<=> hBOOL(P_5) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
& hBOOL(P_2) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
& hBOOL(P_5) ) ) ) ).
tff(fact_1063_zdvd__mono,axiom,
! [Ma: int,Ta: int,K: int] :
( ( K != zero_zero_int )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),Ta))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(times_times_int(K),Ma)),hAPP_int_int(times_times_int(K),Ta))) ) ) ).
tff(fact_1064_number__of2,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(pls))) ).
tff(fact_1065_decr__mult__lemma,axiom,
! [K: int,P_2: fun_int_bool,D: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X: int] :
( hBOOL(hAPP_int_bool(P_2,X))
=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(minus_minus_int(X),D))) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
=> ! [X: int] :
( hBOOL(hAPP_int_bool(P_2,X))
=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(minus_minus_int(X),hAPP_int_int(times_times_int(K),D)))) ) ) ) ) ).
tff(fact_1066_incr__mult__lemma,axiom,
! [K: int,P_2: fun_int_bool,D: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X: int] :
( hBOOL(hAPP_int_bool(P_2,X))
=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(plus_plus_int(X),D))) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
=> ! [X: int] :
( hBOOL(hAPP_int_bool(P_2,X))
=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(plus_plus_int(X),hAPP_int_int(times_times_int(K),D)))) ) ) ) ) ).
tff(fact_1067_zprime__factor__exists,axiom,
! [A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A))
=> ? [P_4: int] :
( hBOOL(hAPP_int_bool(zprime,P_4))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),A)) ) ) ).
tff(fact_1068_zcong__zless__unique,axiom,
! [A: int,M: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
=> ? [X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M))
& hBOOL(hAPP_int_bool(zcong(A,X),M))
& ! [Y: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),M))
& hBOOL(hAPP_int_bool(zcong(A,Y),M)) )
=> ( Y = X ) ) ) ) ).
tff(fact_1069_norR__mem__unique__aux,axiom,
! [A: int,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),hAPP_int_int(minus_minus_int(B),one_one_int)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) ) ).
tff(fact_1070_Wilson__Russ,axiom,
! [P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> hBOOL(hAPP_int_bool(zcong(zfact(hAPP_int_int(minus_minus_int(P_1),one_one_int)),number_number_of_int(min)),P_1)) ) ).
tff(fact_1071_zfact_Osimps,axiom,
! [N_1: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),zero_zero_int))
=> ( zfact(N_1) = one_one_int ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),zero_zero_int))
=> ( zfact(N_1) = hAPP_int_int(times_times_int(N_1),zfact(hAPP_int_int(minus_minus_int(N_1),one_one_int))) ) ) ) ).
tff(fact_1072_inv__inv,axiom,
! [A: int,P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(bit1(bit0(bit1(pls))))),P_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),P_1))
=> ( inv(P_1,inv(P_1,A)) = A ) ) ) ) ) ).
tff(fact_1073_MultInvPair__distinct,axiom,
! [J_1: int,A: int,P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1))
=> ( ~ hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_1))
=> ( ~ hBOOL(hAPP_int_bool(zcong(J_1,zero_zero_int),P_1))
=> ( ~ hBOOL(hAPP_int_bool(quadRes(P_1),A))
=> ~ hBOOL(hAPP_int_bool(zcong(J_1,hAPP_int_int(times_times_int(A),multInv(P_1,J_1))),P_1)) ) ) ) ) ) ).
tff(fact_1074_aux______3,axiom,
! [J_1: int,K_1: int,A: int,P_1: int] :
( hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(J_1),K_1),A),P_1))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(multInv(P_1,J_1)),J_1)),K_1),hAPP_int_int(times_times_int(multInv(P_1,J_1)),A)),P_1)) ) ).
tff(fact_1075_aux______1,axiom,
! [J_1: int,A: int,P_1: int,K_1: int] :
( hBOOL(hAPP_int_bool(zcong(J_1,hAPP_int_int(times_times_int(A),multInv(P_1,K_1))),P_1))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(J_1),K_1),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(A),multInv(P_1,K_1))),K_1)),P_1)) ) ).
tff(fact_1076_inv__distinct,axiom,
! [A: int,P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(minus_minus_int(P_1),one_one_int)))
=> ( A != inv(P_1,A) ) ) ) ) ).
tff(fact_1077_inv__not__1,axiom,
! [A: int,P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(minus_minus_int(P_1),one_one_int)))
=> ( inv(P_1,A) != one_one_int ) ) ) ) ).
tff(fact_1078_inv__not__p__minus__1,axiom,
! [A: int,P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(minus_minus_int(P_1),one_one_int)))
=> ( inv(P_1,A) != hAPP_int_int(minus_minus_int(P_1),one_one_int) ) ) ) ) ).
tff(fact_1079_inv__g__1,axiom,
! [A: int,P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(minus_minus_int(P_1),one_one_int)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),inv(P_1,A))) ) ) ) ).
tff(fact_1080_inv__less__p__minus__1,axiom,
! [A: int,P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(minus_minus_int(P_1),one_one_int)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,inv(P_1,A)),hAPP_int_int(minus_minus_int(P_1),one_one_int))) ) ) ) ).
tff(fact_1081_MultInv__prop1,axiom,
! [X_1: int,Y_1: int,P_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1))
=> ( hBOOL(hAPP_int_bool(zcong(X_1,Y_1),P_1))
=> hBOOL(hAPP_int_bool(zcong(multInv(P_1,X_1),multInv(P_1,Y_1)),P_1)) ) ) ).
tff(fact_1082_inv__not__0,axiom,
! [A: int,P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(minus_minus_int(P_1),one_one_int)))
=> ( inv(P_1,A) != zero_zero_int ) ) ) ) ).
tff(fact_1083_MultInv__zcong__prop1,axiom,
! [A: int,J_1: int,K_1: int,P_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1))
=> ( hBOOL(hAPP_int_bool(zcong(J_1,K_1),P_1))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),multInv(P_1,J_1)),hAPP_int_int(times_times_int(A),multInv(P_1,K_1))),P_1)) ) ) ).
tff(fact_1084_MultInv__prop5,axiom,
! [Y_1: int,X_1: int,P_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1))
=> ( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( ~ hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_1))
=> ( ~ hBOOL(hAPP_int_bool(zcong(Y_1,zero_zero_int),P_1))
=> ( hBOOL(hAPP_int_bool(zcong(multInv(P_1,X_1),multInv(P_1,Y_1)),P_1))
=> hBOOL(hAPP_int_bool(zcong(X_1,Y_1),P_1)) ) ) ) ) ) ).
tff(fact_1085_MultInv__prop4,axiom,
! [X_1: int,P_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1))
=> ( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( ~ hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_1))
=> hBOOL(hAPP_int_bool(zcong(multInv(P_1,multInv(P_1,X_1)),X_1),P_1)) ) ) ) ).
tff(fact_1086_MultInv__prop3,axiom,
! [X_1: int,P_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1))
=> ( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( ~ hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_1))
=> ~ hBOOL(hAPP_int_bool(zcong(multInv(P_1,X_1),zero_zero_int),P_1)) ) ) ) ).
tff(fact_1087_inv__is__inv,axiom,
! [A: int,P_1: int] :
( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),P_1))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),inv(P_1,A)),one_one_int),P_1)) ) ) ) ).
tff(fact_1088_aux______4,axiom,
! [K_1: int,A: int,J_1: int,P_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1))
=> ( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( ~ hBOOL(hAPP_int_bool(zcong(J_1,zero_zero_int),P_1))
=> ( hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(multInv(P_1,J_1)),J_1)),K_1),hAPP_int_int(times_times_int(multInv(P_1,J_1)),A)),P_1))
=> hBOOL(hAPP_int_bool(zcong(K_1,hAPP_int_int(times_times_int(A),multInv(P_1,J_1))),P_1)) ) ) ) ) ).
tff(fact_1089_aux______2,axiom,
! [J_1: int,A: int,K_1: int,P_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1))
=> ( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( ~ hBOOL(hAPP_int_bool(zcong(K_1,zero_zero_int),P_1))
=> ( hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(J_1),K_1),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(A),multInv(P_1,K_1))),K_1)),P_1))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(J_1),K_1),A),P_1)) ) ) ) ) ).
tff(fact_1090_MultInv__zcong__prop2,axiom,
! [A: int,J_1: int,K_1: int,P_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1))
=> ( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( ~ hBOOL(hAPP_int_bool(zcong(K_1,zero_zero_int),P_1))
=> ( ~ hBOOL(hAPP_int_bool(zcong(J_1,zero_zero_int),P_1))
=> ( hBOOL(hAPP_int_bool(zcong(J_1,hAPP_int_int(times_times_int(A),multInv(P_1,K_1))),P_1))
=> hBOOL(hAPP_int_bool(zcong(K_1,hAPP_int_int(times_times_int(A),multInv(P_1,J_1))),P_1)) ) ) ) ) ) ).
tff(fact_1091_MultInv__zcong__prop3,axiom,
! [J_1: int,K_1: int,A: int,P_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1))
=> ( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( ~ hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_1))
=> ( ~ hBOOL(hAPP_int_bool(zcong(K_1,zero_zero_int),P_1))
=> ( ~ hBOOL(hAPP_int_bool(zcong(J_1,zero_zero_int),P_1))
=> ( hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),multInv(P_1,J_1)),hAPP_int_int(times_times_int(A),multInv(P_1,K_1))),P_1))
=> hBOOL(hAPP_int_bool(zcong(J_1,K_1),P_1)) ) ) ) ) ) ) ).
tff(fact_1092_Int2_Oaux____2,axiom,
! [X_1: int,P_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1))
=> ( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( ~ hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_1))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(X_1),multInv(P_1,X_1))),multInv(P_1,multInv(P_1,X_1))),X_1),P_1)) ) ) ) ).
tff(fact_1093_Int2_Oaux____1,axiom,
! [X_1: int,P_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1))
=> ( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( ~ hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_1))
=> hBOOL(hAPP_int_bool(zcong(multInv(P_1,multInv(P_1,X_1)),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(X_1),multInv(P_1,X_1))),multInv(P_1,multInv(P_1,X_1)))),P_1)) ) ) ) ).
tff(fact_1094_MultInv__prop2a,axiom,
! [X_1: int,P_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1))
=> ( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( ~ hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_1))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(multInv(P_1,X_1)),X_1),one_one_int),P_1)) ) ) ) ).
tff(fact_1095_MultInv__prop2,axiom,
! [X_1: int,P_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1))
=> ( hBOOL(hAPP_int_bool(zprime,P_1))
=> ( ~ hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_1))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(X_1),multInv(P_1,X_1)),one_one_int),P_1)) ) ) ) ).
tff(fact_1096_wset__mem__inv__mem,axiom,
! [B_2: int,A_2: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(bit1(bit0(bit1(pls))))),P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(minus_minus_int(P_3),one_one_int)))
=> ( hBOOL(member_int(B_2,wset(A_2,P_3)))
=> hBOOL(member_int(inv(P_3,B_2),wset(A_2,P_3))) ) ) ) ) ).
tff(fact_1097_wset__inv__mem__mem,axiom,
! [B_2: int,A_2: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(bit1(bit0(bit1(pls))))),P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(minus_minus_int(P_3),one_one_int)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),hAPP_int_int(minus_minus_int(P_3),one_one_int)))
=> ( hBOOL(member_int(inv(P_3,B_2),wset(A_2,P_3)))
=> hBOOL(member_int(B_2,wset(A_2,P_3))) ) ) ) ) ) ) ).
tff(fact_1098_wset__mem__mem,axiom,
! [P_3: int,A_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_2))
=> hBOOL(member_int(A_2,wset(A_2,P_3))) ) ).
tff(fact_1099_wset__subset,axiom,
! [B_2: int,P_3: int,A_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_2))
=> ( hBOOL(member_int(B_2,wset(hAPP_int_int(minus_minus_int(A_2),one_one_int),P_3)))
=> hBOOL(member_int(B_2,wset(A_2,P_3))) ) ) ).
tff(fact_1100_wset__less,axiom,
! [B_2: int,A_2: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(minus_minus_int(P_3),one_one_int)))
=> ( hBOOL(member_int(B_2,wset(A_2,P_3)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),hAPP_int_int(minus_minus_int(P_3),one_one_int))) ) ) ) ).
tff(fact_1101_wset__g__1,axiom,
! [B_2: int,A_2: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(minus_minus_int(P_3),one_one_int)))
=> ( hBOOL(member_int(B_2,wset(A_2,P_3)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_2)) ) ) ) ).
tff(fact_1102_wset__mem__imp__or,axiom,
! [B_2: int,P_3: int,A_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_2))
=> ( ~ hBOOL(member_int(B_2,wset(hAPP_int_int(minus_minus_int(A_2),one_one_int),P_3)))
=> ( hBOOL(member_int(B_2,wset(A_2,P_3)))
=> ( ( B_2 = A_2 )
| ( B_2 = inv(P_3,A_2) ) ) ) ) ) ).
tff(fact_1103_wset__mem,axiom,
! [B_2: int,A_2: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(minus_minus_int(P_3),one_one_int)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),A_2))
=> hBOOL(member_int(B_2,wset(A_2,P_3))) ) ) ) ) ).
tff(fact_1104_int__le__induct,axiom,
! [P_2: fun_int_bool,I_2: int,K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K))
=> ( hBOOL(hAPP_int_bool(P_2,K))
=> ( ! [I: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),K))
=> ( hBOOL(hAPP_int_bool(P_2,I))
=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(minus_minus_int(I),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_2,I_2)) ) ) ) ).
tff(fact_1105_int__less__induct,axiom,
! [P_2: fun_int_bool,I_2: int,K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),K))
=> ( hBOOL(hAPP_int_bool(P_2,hAPP_int_int(minus_minus_int(K),one_one_int)))
=> ( ! [I: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),K))
=> ( hBOOL(hAPP_int_bool(P_2,I))
=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(minus_minus_int(I),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_2,I_2)) ) ) ) ).
tff(fact_1106_d22set__induct__old,axiom,
! [X_2: int,P_2: fun_int_bool] :
( ! [A_3: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3))
=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(minus_minus_int(A_3),one_one_int))) )
=> hBOOL(hAPP_int_bool(P_2,A_3)) )
=> hBOOL(hAPP_int_bool(P_2,X_2)) ) ).
tff(fact_1107_int__ge__induct,axiom,
! [P_2: fun_int_bool,K: int,I_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),I_2))
=> ( hBOOL(hAPP_int_bool(P_2,K))
=> ( ! [I: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),I))
=> ( hBOOL(hAPP_int_bool(P_2,I))
=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(plus_plus_int(I),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_2,I_2)) ) ) ) ).
tff(fact_1108_int__gr__induct,axiom,
! [P_2: fun_int_bool,K: int,I_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),I_2))
=> ( hBOOL(hAPP_int_bool(P_2,hAPP_int_int(plus_plus_int(K),one_one_int)))
=> ( ! [I: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),I))
=> ( hBOOL(hAPP_int_bool(P_2,I))
=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(plus_plus_int(I),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_2,I_2)) ) ) ) ).
tff(fact_1109_mono__nat__linear__lb,axiom,
! [Ma: nat,K: nat,F_1: fun_nat_nat] :
( ! [M_2: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F_1,M_2)),hAPP_nat_nat(F_1,N_2))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(F_1,Ma)),K)),hAPP_nat_nat(F_1,hAPP_nat_nat(plus_plus_nat(Ma),K)))) ) ).
tff(fact_1110_d22set__eq__wset,axiom,
! [P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( d22set(hAPP_int_int(minus_minus_int(P_3),number_number_of_int(bit0(bit1(pls))))) = wset(hAPP_int_int(minus_minus_int(P_3),number_number_of_int(bit0(bit1(pls)))),P_3) ) ) ).
tff(fact_1111_ex__least__nat__le,axiom,
! [N: nat,P_2: fun_nat_bool] :
( ~ hBOOL(hAPP_nat_bool(P_2,zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(P_2,N))
=> ? [K_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N))
& ! [I: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K_2))
=> ~ hBOOL(hAPP_nat_bool(P_2,I)) )
& hBOOL(hAPP_nat_bool(P_2,K_2)) ) ) ) ).
tff(fact_1112_d22set__le,axiom,
! [B_2: int,A_2: int] :
( hBOOL(member_int(B_2,d22set(A_2)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),A_2)) ) ).
tff(fact_1113_d22set__le__swap,axiom,
! [A_2: int,B_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_2))
=> ~ hBOOL(member_int(B_2,d22set(A_2))) ) ).
tff(fact_1114_d22set__g__1,axiom,
! [B_2: int,A_2: int] :
( hBOOL(member_int(B_2,d22set(A_2)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_2)) ) ).
tff(fact_1115_d22set__mem,axiom,
! [A_2: int,B_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),A_2))
=> hBOOL(member_int(B_2,d22set(A_2))) ) ) ).
tff(fact_1116_less__imp__add__positive,axiom,
! [I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_1))
=> ? [K_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
& ( hAPP_nat_nat(plus_plus_nat(I_1),K_2) = J_1 ) ) ) ).
tff(fact_1117_pow__divides__eq__int,axiom,
! [A_2: int,B_2: int,N: nat] :
( ( N != zero_zero_nat )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(A_2),N)),hAPP_nat_int(power_power_int(B_2),N)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_2),B_2)) ) ) ).
tff(fact_1118_gcd__lcm__complete__lattice__nat_Otop__le,axiom,
! [A: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A))
=> ( A = zero_zero_nat ) ) ).
tff(fact_1119_gcd__lcm__complete__lattice__nat_Otop__unique,axiom,
! [A_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_2))
<=> ( A_2 = zero_zero_nat ) ) ).
tff(fact_1120_gcd__lcm__complete__lattice__nat_Oless__top,axiom,
! [A_2: nat] :
( ( A_2 != zero_zero_nat )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),zero_zero_nat))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_2)) ) ) ).
tff(fact_1121_gcd__lcm__complete__lattice__nat_Otop__greatest,axiom,
! [A: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),zero_zero_nat)) ).
tff(fact_1122_gcd__lcm__complete__lattice__nat_Onot__top__less,axiom,
! [A: nat] :
~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),zero_zero_nat)) ) ).
tff(fact_1123_gcd__lcm__complete__lattice__nat_Onot__less__bot,axiom,
! [A: nat] :
~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),one_one_nat))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A)) ) ).
tff(fact_1124_gcd__lcm__complete__lattice__nat_Obot__least,axiom,
! [A: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A)) ).
tff(fact_1125_gcd__lcm__complete__lattice__nat_Obot__less,axiom,
! [A_2: nat] :
( ( A_2 != one_one_nat )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_2))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),one_one_nat)) ) ) ).
tff(fact_1126_gcd__lcm__complete__lattice__nat_Obot__unique,axiom,
! [A_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),one_one_nat))
<=> ( A_2 = one_one_nat ) ) ).
tff(fact_1127_gcd__lcm__complete__lattice__nat_Ole__bot,axiom,
! [A: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),one_one_nat))
=> ( A = one_one_nat ) ) ).
tff(fact_1128_dvd__pos__nat,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) ) ) ).
tff(fact_1129_pow__divides__eq__nat,axiom,
! [A_2: nat,B_2: nat,N: nat] :
( ( N != zero_zero_nat )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(A_2),N)),hAPP_nat_nat(power_power_nat(B_2),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),B_2)) ) ) ).
tff(fact_1130_pow__divides__pow__int,axiom,
! [A: int,N_1: nat,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(A),N_1)),hAPP_nat_int(power_power_int(B),N_1)))
=> ( ( N_1 != zero_zero_nat )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B)) ) ) ).
tff(fact_1131_divides__le,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
| ( N_1 = zero_zero_nat ) ) ) ).
tff(fact_1132_mult__left__cancel,axiom,
! [N_1: nat,M: nat,K_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> ( ( hAPP_nat_nat(times_times_nat(K_1),N_1) = hAPP_nat_nat(times_times_nat(K_1),M) )
=> ( N_1 = M ) ) ) ).
tff(fact_1133_SR__def,axiom,
! [P_3: int] : sr(P_3) = collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int)),cOMBC_int_int_bool(ord_less_int,P_3))) ).
tff(fact_1134_neg__zmod__mult__2,axiom,
! [B: int,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int))
=> ( hAPP_int_int(div_mod_int(hAPP_int_int(plus_plus_int(one_one_int),hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),B))),hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),A)) = hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(plus_plus_int(B),one_one_int)),A))),one_one_int) ) ) ).
tff(fact_1135_zmod__number__of__Bit1,axiom,
! [V: int,W: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(W)))
=> ( hAPP_int_int(div_mod_int(number_number_of_int(bit1(V))),number_number_of_int(bit0(W))) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),hAPP_int_int(div_mod_int(number_number_of_int(V)),number_number_of_int(W)))),one_one_int) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(W)))
=> ( hAPP_int_int(div_mod_int(number_number_of_int(bit1(V))),number_number_of_int(bit0(W))) = hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(plus_plus_int(number_number_of_int(V)),one_one_int)),number_number_of_int(W)))),one_one_int) ) ) ) ).
tff(fact_1136_zdvd__iff__zmod__eq__0__number__of,axiom,
! [X_2: int,Y_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,number_number_of_int(X_2)),number_number_of_int(Y_2)))
<=> ( hAPP_int_int(div_mod_int(number_number_of_int(Y_2)),number_number_of_int(X_2)) = zero_zero_int ) ) ).
tff(fact_1137_neg__mod__bound,axiom,
! [A: int,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),hAPP_int_int(div_mod_int(A),B))) ) ).
tff(fact_1138_pos__mod__bound,axiom,
! [A: int,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(div_mod_int(A),B)),B)) ) ).
tff(fact_1139_Divides_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y_1: int,X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(X_1),Y_1))) ) ) ).
tff(fact_1140_zmod__le__nonneg__dividend,axiom,
! [K_1: int,M: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(M),K_1)),M)) ) ).
tff(fact_1141_zmod__eq__0__iff,axiom,
! [Ma: int,D: int] :
( ( hAPP_int_int(div_mod_int(Ma),D) = zero_zero_int )
<=> ? [Q_2: int] : Ma = hAPP_int_int(times_times_int(D),Q_2) ) ).
tff(fact_1142_zmod__eq__dvd__iff,axiom,
! [X_2: int,N: int,Y_2: int] :
( ( hAPP_int_int(div_mod_int(X_2),N) = hAPP_int_int(div_mod_int(Y_2),N) )
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),hAPP_int_int(minus_minus_int(X_2),Y_2))) ) ).
tff(fact_1143_Residues_Oaux,axiom,
! [X_1: int,M: int,Y_1: int] :
( ( hAPP_int_int(div_mod_int(X_1),M) = hAPP_int_int(div_mod_int(Y_1),M) )
=> hBOOL(hAPP_int_bool(zcong(X_1,Y_1),M)) ) ).
tff(fact_1144_mod__mod__is__mod,axiom,
! [X_1: int,M: int] : hBOOL(hAPP_int_bool(zcong(X_1,hAPP_int_int(div_mod_int(X_1),M)),M)) ).
tff(fact_1145_zcong__zmod,axiom,
! [A_2: int,B_2: int,Ma: int] :
( hBOOL(hAPP_int_bool(zcong(A_2,B_2),Ma))
<=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(div_mod_int(A_2),Ma),hAPP_int_int(div_mod_int(B_2),Ma)),Ma)) ) ).
tff(fact_1146_zdvd__zmod__imp__zdvd,axiom,
! [K_1: int,M: int,N_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(div_mod_int(M),N_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),M)) ) ) ).
tff(fact_1147_zdvd__zmod,axiom,
! [N_1: int,F: int,M: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F),hAPP_int_int(div_mod_int(M),N_1))) ) ) ).
tff(fact_1148_zpower__zmod,axiom,
! [X_1: int,M: int,Y_1: nat] : hAPP_int_int(div_mod_int(hAPP_nat_int(power_power_int(hAPP_int_int(div_mod_int(X_1),M)),Y_1)),M) = hAPP_int_int(div_mod_int(hAPP_nat_int(power_power_int(X_1),Y_1)),M) ).
tff(fact_1149_zmod__zmult1__eq,axiom,
! [A: int,B: int,C: int] : hAPP_int_int(div_mod_int(hAPP_int_int(times_times_int(A),B)),C) = hAPP_int_int(div_mod_int(hAPP_int_int(times_times_int(A),hAPP_int_int(div_mod_int(B),C))),C) ).
tff(fact_1150_zmod__simps_I3_J,axiom,
! [A: int,B: int,C: int] : hAPP_int_int(div_mod_int(hAPP_int_int(times_times_int(A),hAPP_int_int(div_mod_int(B),C))),C) = hAPP_int_int(div_mod_int(hAPP_int_int(times_times_int(A),B)),C) ).
tff(fact_1151_zmod__self,axiom,
! [A: int] : hAPP_int_int(div_mod_int(A),A) = zero_zero_int ).
tff(fact_1152_zmod__zero,axiom,
! [B: int] : hAPP_int_int(div_mod_int(zero_zero_int),B) = zero_zero_int ).
tff(fact_1153_zdiff__zmod__right,axiom,
! [X_1: int,Y_1: int,M: int] : hAPP_int_int(div_mod_int(hAPP_int_int(minus_minus_int(X_1),hAPP_int_int(div_mod_int(Y_1),M))),M) = hAPP_int_int(div_mod_int(hAPP_int_int(minus_minus_int(X_1),Y_1)),M) ).
tff(fact_1154_zdiff__zmod__left,axiom,
! [X_1: int,M: int,Y_1: int] : hAPP_int_int(div_mod_int(hAPP_int_int(minus_minus_int(hAPP_int_int(div_mod_int(X_1),M)),Y_1)),M) = hAPP_int_int(div_mod_int(hAPP_int_int(minus_minus_int(X_1),Y_1)),M) ).
tff(fact_1155_zmod__minus1__right,axiom,
! [A: int] : hAPP_int_int(div_mod_int(A),number_number_of_int(min)) = zero_zero_int ).
tff(fact_1156_zmod__zdvd__zmod,axiom,
! [A: int,B: int,M: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),B))
=> ( hAPP_int_int(div_mod_int(hAPP_int_int(div_mod_int(A),B)),M) = hAPP_int_int(div_mod_int(A),M) ) ) ) ).
tff(fact_1157_zcong__zmod__eq,axiom,
! [A_2: int,B_2: int,Ma: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( hBOOL(hAPP_int_bool(zcong(A_2,B_2),Ma))
<=> ( hAPP_int_int(div_mod_int(A_2),Ma) = hAPP_int_int(div_mod_int(B_2),Ma) ) ) ) ).
tff(fact_1158_pos__mod__sign,axiom,
! [A: int,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(A),B))) ) ).
tff(fact_1159_pos__mod__conj,axiom,
! [A: int,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(A),B)))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(div_mod_int(A),B)),B)) ) ) ).
tff(fact_1160_mod__pos__pos__trivial,axiom,
! [B: int,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B))
=> ( hAPP_int_int(div_mod_int(A),B) = A ) ) ) ).
tff(fact_1161_neg__mod__sign,axiom,
! [A: int,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(A),B)),zero_zero_int)) ) ).
tff(fact_1162_neg__mod__conj,axiom,
! [A: int,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(A),B)),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),hAPP_int_int(div_mod_int(A),B))) ) ) ).
tff(fact_1163_mod__neg__neg__trivial,axiom,
! [B: int,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A))
=> ( hAPP_int_int(div_mod_int(A),B) = A ) ) ) ).
tff(fact_1164_mod__pos__neg__trivial,axiom,
! [B: int,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A),B)),zero_zero_int))
=> ( hAPP_int_int(div_mod_int(A),B) = hAPP_int_int(plus_plus_int(A),B) ) ) ) ).
tff(fact_1165_zmod__number__of__Bit0,axiom,
! [V: int,W: int] : hAPP_int_int(div_mod_int(number_number_of_int(bit0(V))),number_number_of_int(bit0(W))) = hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),hAPP_int_int(div_mod_int(number_number_of_int(V)),number_number_of_int(W))) ).
tff(fact_1166_split__zmod,axiom,
! [P_2: fun_int_bool,N: int,K: int] :
( hBOOL(hAPP_int_bool(P_2,hAPP_int_int(div_mod_int(N),K)))
<=> ( ( ( K = zero_zero_int )
=> hBOOL(hAPP_int_bool(P_2,N)) )
& ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K))
=> ! [I: int,J: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),J))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J),K))
& ( N = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(K),I)),J) ) )
=> hBOOL(hAPP_int_bool(P_2,J)) ) )
& ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),zero_zero_int))
=> ! [I: int,J: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),J))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J),zero_zero_int))
& ( N = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(K),I)),J) ) )
=> hBOOL(hAPP_int_bool(P_2,J)) ) ) ) ) ).
tff(fact_1167_zmult2__lemma__aux3,axiom,
! [Q_1: int,B: int,R_1: int,C: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),hAPP_int_int(div_mod_int(Q_1),C))),R_1))) ) ) ) ).
tff(fact_1168_zmult2__lemma__aux4,axiom,
! [Q_1: int,B: int,R_1: int,C: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),hAPP_int_int(div_mod_int(Q_1),C))),R_1)),hAPP_int_int(times_times_int(B),C))) ) ) ) ).
tff(fact_1169_zmult2__lemma__aux1,axiom,
! [Q_1: int,B: int,R_1: int,C: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),R_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_1),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(B),C)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),hAPP_int_int(div_mod_int(Q_1),C))),R_1))) ) ) ) ).
tff(fact_1170_zmult2__lemma__aux2,axiom,
! [Q_1: int,B: int,R_1: int,C: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),R_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_1),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),hAPP_int_int(div_mod_int(Q_1),C))),R_1)),zero_zero_int)) ) ) ) ).
tff(fact_1171_divmod__int__rel__mod__eq,axiom,
! [A_1: int,B_1: int,Q_3: int,Y_1: int] :
( ( A_1 = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_1),Q_3)),Y_1) )
=> ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),B_1)) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),Y_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),zero_zero_int)) ) ) )
=> ( ( B_1 != zero_zero_int )
=> ( hAPP_int_int(div_mod_int(A_1),B_1) = Y_1 ) ) ) ) ).
tff(fact_1172_neq__one__mod__two,axiom,
! [X_2: int] :
( ( hAPP_int_int(div_mod_int(X_2),number_number_of_int(bit0(bit1(pls)))) != zero_zero_int )
<=> ( hAPP_int_int(div_mod_int(X_2),number_number_of_int(bit0(bit1(pls)))) = one_one_int ) ) ).
tff(fact_1173_zmod__minus1,axiom,
! [B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hAPP_int_int(div_mod_int(number_number_of_int(min)),B) = hAPP_int_int(minus_minus_int(B),one_one_int) ) ) ).
tff(fact_1174_pos__zmod__mult__2,axiom,
! [B: int,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A))
=> ( hAPP_int_int(div_mod_int(hAPP_int_int(plus_plus_int(one_one_int),hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),B))),hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),A)) = hAPP_int_int(plus_plus_int(one_one_int),hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),hAPP_int_int(div_mod_int(B),A))) ) ) ).
tff(fact_1175_zmod__eq__0D,axiom,
! [M_1: int,D_1: int] :
( ( hAPP_int_int(div_mod_int(M_1),D_1) = zero_zero_int )
=> ? [Q_2: int] : M_1 = hAPP_int_int(times_times_int(D_1),Q_2) ) ).
tff(fact_1176_StandardRes__prop4,axiom,
! [X_1: int,Y_1: int,M: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),M))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(standardRes(M,X_1)),standardRes(M,Y_1)),hAPP_int_int(times_times_int(X_1),Y_1)),M)) ) ).
tff(fact_1177_mod__le__divisor,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(div_mod_nat(M),N_1)),N_1)) ) ).
tff(fact_1178_mod__less__divisor,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(div_mod_nat(M),N_1)),N_1)) ) ).
tff(fact_1179_mod__eq__0__iff,axiom,
! [Ma: nat,D: nat] :
( ( hAPP_nat_nat(div_mod_nat(Ma),D) = zero_zero_nat )
<=> ? [Q_2: nat] : Ma = hAPP_nat_nat(times_times_nat(D),Q_2) ) ).
tff(fact_1180_mod__geq,axiom,
! [M: nat,N_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> ( hAPP_nat_nat(div_mod_nat(M),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(minus_minus_nat(M),N_1)),N_1) ) ) ).
tff(fact_1181_mod__if,axiom,
! [M: nat,N_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> ( hAPP_nat_nat(div_mod_nat(M),N_1) = M ) )
& ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> ( hAPP_nat_nat(div_mod_nat(M),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(minus_minus_nat(M),N_1)),N_1) ) ) ) ).
tff(fact_1182_mod__mult__self3,axiom,
! [K_1: nat,N_1: nat,M: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(K_1),N_1)),M)),N_1) = hAPP_nat_nat(div_mod_nat(M),N_1) ).
tff(fact_1183_le__mod__geq,axiom,
! [N_1: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> ( hAPP_nat_nat(div_mod_nat(M),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(minus_minus_nat(M),N_1)),N_1) ) ) ).
tff(fact_1184_StandardRes__def,axiom,
! [M: int,X_1: int] : standardRes(M,X_1) = hAPP_int_int(div_mod_int(X_1),M) ).
tff(fact_1185_mod__mult__distrib,axiom,
! [M: nat,N_1: nat,K_1: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(div_mod_nat(M),N_1)),K_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(times_times_nat(M),K_1)),hAPP_nat_nat(times_times_nat(N_1),K_1)) ).
tff(fact_1186_mod__mult__distrib2,axiom,
! [K_1: nat,M: nat,N_1: nat] : hAPP_nat_nat(times_times_nat(K_1),hAPP_nat_nat(div_mod_nat(M),N_1)) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(times_times_nat(K_1),M)),hAPP_nat_nat(times_times_nat(K_1),N_1)) ).
tff(fact_1187_mod__less,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> ( hAPP_nat_nat(div_mod_nat(M),N_1) = M ) ) ).
tff(fact_1188_mod__less__eq__dividend,axiom,
! [M: nat,N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(div_mod_nat(M),N_1)),M)) ).
tff(fact_1189_StandardRes__eq__zcong,axiom,
! [Ma: int,X_2: int] :
( ( standardRes(Ma,X_2) = zero_zero_int )
<=> hBOOL(hAPP_int_bool(zcong(X_2,zero_zero_int),Ma)) ) ).
tff(fact_1190_StandardRes__prop3,axiom,
! [X_2: int,P_3: int] :
( ~ hBOOL(hAPP_int_bool(zcong(X_2,zero_zero_int),P_3))
<=> ( standardRes(P_3,X_2) != zero_zero_int ) ) ).
tff(fact_1191_StandardRes__prop1,axiom,
! [X_1: int,M: int] : hBOOL(hAPP_int_bool(zcong(X_1,standardRes(M,X_1)),M)) ).
tff(fact_1192_StandardRes__ubound,axiom,
! [X_1: int,P_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,standardRes(P_1,X_1)),P_1)) ) ).
tff(fact_1193_StandardRes__SR__prop,axiom,
! [X_2: int,P_3: int] :
( hBOOL(member_int(X_2,sr(P_3)))
=> ( standardRes(P_3,X_2) = X_2 ) ) ).
tff(fact_1194_split__mod,axiom,
! [P_2: fun_nat_bool,N: nat,K: nat] :
( hBOOL(hAPP_nat_bool(P_2,hAPP_nat_nat(div_mod_nat(N),K)))
<=> ( ( ( K = zero_zero_nat )
=> hBOOL(hAPP_nat_bool(P_2,N)) )
& ( ( K != zero_zero_nat )
=> ! [I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),K))
=> ( ( N = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(K),I)),J) )
=> hBOOL(hAPP_nat_bool(P_2,J)) ) ) ) ) ) ).
tff(fact_1195_mod__lemma,axiom,
! [Q_1: nat,R_1: nat,B: nat,C: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_1),B))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(B),hAPP_nat_nat(div_mod_nat(Q_1),C))),R_1)),hAPP_nat_nat(times_times_nat(B),C))) ) ) ).
tff(fact_1196_StandardRes__lbound,axiom,
! [X_1: int,P_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),standardRes(P_1,X_1))) ) ).
tff(fact_1197_StandardRes__prop2,axiom,
! [X1: int,X2: int,Ma: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( ( standardRes(Ma,X1) = standardRes(Ma,X2) )
<=> hBOOL(hAPP_int_bool(zcong(X1,X2),Ma)) ) ) ).
%----Helper facts (6)
tff(help_fconj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ hBOOL(P)
| ~ hBOOL(Q)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) ) ).
tff(help_fconj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
| hBOOL(P) ) ).
tff(help_fconj_3_1_U,axiom,
! [P: bool,Q: bool] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
| hBOOL(Q) ) ).
tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__HOL__Obool_U,axiom,
! [P: fun_int_fun_int_bool,Q: int,R: int] : hAPP_int_bool(cOMBC_int_int_bool(P,Q),R) = hAPP_int_bool(hAPP_i1948725293t_bool(P,R),Q) ).
tff(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P: fun_in531499254l_bool,Q: fun_int_bool,R: int] : hAPP_int_bool(cOMBS_int_bool_bool(P,Q),R) = hAPP_bool_bool(hAPP_i68813070l_bool(P,R),hAPP_int_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_int_bool,R: int] : hAPP_i68813070l_bool(cOMBB_1652995168ol_int(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_int_bool(Q,R)) ).
%----Conjectures (1)
tff(conj_0,conjecture,
? [X: int,Y: int] : hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y),number_number_of_nat(bit0(bit1(pls))))) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int) ).
%------------------------------------------------------------------------------