TPTP Problem File: NUM924_3.p
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%------------------------------------------------------------------------------
% File : NUM924_3 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 102, 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_l102 [Bla11]
% Status : Theorem
% Rating : 0.00 v8.1.0, 0.09 v7.5.0, 0.12 v7.4.0, 0.20 v7.3.0, 0.17 v7.1.0, 0.00 v5.5.0, 0.25 v5.4.0, 0.33 v5.3.0
% Syntax : Number of formulae : 1299 ( 464 unt; 94 typ; 0 def)
% Number of atoms : 2741 ( 810 equ)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 1803 ( 267 ~; 75 |; 162 &)
% ( 255 <=>;1044 =>; 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 : 2576 (2559 !; 17 ?;2576 :)
% SPC : TF0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:39:52
%------------------------------------------------------------------------------
%----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__096t_A_060_A0_096,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,t),zero_zero_int)) ).
tff(fact_1_calculation_I1_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,t),one_one_int)) ).
tff(fact_2__096_I4_A_K_Am_A_L_A1_J_A_K_At_A_060_A_I4_A_K_Am_A_L_A1_J_A_K_A0_096,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_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)),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)),zero_zero_int))) ).
tff(fact_3_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_4_calculation_I2_J,axiom,
( ( t = zero_zero_int )
=> ( hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls))))),one_one_int) = zero_zero_int ) ) ).
tff(fact_5__096_126_A1_A_060_061_At_096,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),t)) ).
tff(fact_6_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_7_not__sum__power2__lt__zero,axiom,
! [X_1: real,Y: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_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),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_real)) ).
tff(fact_8_not__sum__power2__lt__zero,axiom,
! [X_1: int,Y: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X_1),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_int)) ).
tff(fact_9_sum__power2__gt__zero__iff,axiom,
! [X: real,Y_1: 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),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_1),number_number_of_nat(bit0(bit1(pls)))))))
<=> ( ( X != zero_zero_real )
| ( Y_1 != zero_zero_real ) ) ) ).
tff(fact_10_sum__power2__gt__zero__iff,axiom,
! [X: int,Y_1: 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),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_1),number_number_of_nat(bit0(bit1(pls)))))))
<=> ( ( X != zero_zero_int )
| ( Y_1 != zero_zero_int ) ) ) ).
tff(fact_11_sum__power2__eq__zero__iff,axiom,
! [X: real,Y_1: real] :
( ( hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_1),number_number_of_nat(bit0(bit1(pls))))) = zero_zero_real )
<=> ( ( X = zero_zero_real )
& ( Y_1 = zero_zero_real ) ) ) ).
tff(fact_12_sum__power2__eq__zero__iff,axiom,
! [X: int,Y_1: 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_1),number_number_of_nat(bit0(bit1(pls))))) = zero_zero_int )
<=> ( ( X = zero_zero_int )
& ( Y_1 = zero_zero_int ) ) ) ).
tff(fact_13_power2__less__0,axiom,
! [A_110: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_110),number_number_of_nat(bit0(bit1(pls))))),zero_zero_real)) ).
tff(fact_14_power2__less__0,axiom,
! [A_110: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_110),number_number_of_nat(bit0(bit1(pls))))),zero_zero_int)) ).
tff(fact_15_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_16_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_17_one__power2,axiom,
hAPP_nat_real(power_power_real(one_one_real),number_number_of_nat(bit0(bit1(pls)))) = one_one_real ).
tff(fact_18_one__power2,axiom,
hAPP_nat_nat(power_power_nat(one_one_nat),number_number_of_nat(bit0(bit1(pls)))) = one_one_nat ).
tff(fact_19_one__power2,axiom,
hAPP_nat_int(power_power_int(one_one_int),number_number_of_nat(bit0(bit1(pls)))) = one_one_int ).
tff(fact_20_zero__power2,axiom,
hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_real ).
tff(fact_21_zero__power2,axiom,
hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_nat ).
tff(fact_22_zero__power2,axiom,
hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_int ).
tff(fact_23_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_24_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_25_add__special_I2_J,axiom,
! [W_14: int] : hAPP_real_real(plus_plus_real(one_one_real),number267125858f_real(W_14)) = number267125858f_real(hAPP_int_int(plus_plus_int(bit1(pls)),W_14)) ).
tff(fact_26_add__special_I2_J,axiom,
! [W_14: int] : hAPP_int_int(plus_plus_int(one_one_int),number_number_of_int(W_14)) = number_number_of_int(hAPP_int_int(plus_plus_int(bit1(pls)),W_14)) ).
tff(fact_27_add__special_I3_J,axiom,
! [V_17: int] : hAPP_real_real(plus_plus_real(number267125858f_real(V_17)),one_one_real) = number267125858f_real(hAPP_int_int(plus_plus_int(V_17),bit1(pls))) ).
tff(fact_28_add__special_I3_J,axiom,
! [V_17: int] : hAPP_int_int(plus_plus_int(number_number_of_int(V_17)),one_one_int) = number_number_of_int(hAPP_int_int(plus_plus_int(V_17),bit1(pls))) ).
tff(fact_29_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_30__096_B_Bthesis_O_A_I_B_Bt_O_As_A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_L_A1_,axiom,
~ ! [T_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_31_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_32_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_33_zle__refl,axiom,
! [W: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),W)) ).
tff(fact_34_number__of__is__id,axiom,
! [K_1: int] : number_number_of_int(K_1) = K_1 ).
tff(fact_35_zmult__commute,axiom,
! [Z: int,W: int] : hAPP_int_int(times_times_int(Z),W) = hAPP_int_int(times_times_int(W),Z) ).
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_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_38_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_39_le__number__of,axiom,
! [X: int,Y_1: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(X)),number267125858f_real(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),Y_1)) ) ).
tff(fact_40_le__number__of,axiom,
! [X: int,Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(X)),number_number_of_int(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),Y_1)) ) ).
tff(fact_41_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_42_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_43_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_44_zpower__zadd__distrib,axiom,
! [X_1: int,Y: nat,Z: nat] : hAPP_nat_int(power_power_int(X_1),hAPP_nat_nat(plus_plus_nat(Y),Z)) = hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(X_1),Y)),hAPP_nat_int(power_power_int(X_1),Z)) ).
tff(fact_45_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_46_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_47_rel__simps_I19_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),pls)) ).
tff(fact_48_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_49_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_50_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_51_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_52_eq__number__of__0,axiom,
! [V_4: int] :
( ( number_number_of_nat(V_4) = zero_zero_nat )
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_4),pls)) ) ).
tff(fact_53_eq__0__number__of,axiom,
! [V_4: int] :
( ( zero_zero_nat = number_number_of_nat(V_4) )
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_4),pls)) ) ).
tff(fact_54_semiring__mult__number__of,axiom,
! [V_16: int,V_15: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_15))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_16))
=> ( hAPP_real_real(times_times_real(number267125858f_real(V_15)),number267125858f_real(V_16)) = number267125858f_real(hAPP_int_int(times_times_int(V_15),V_16)) ) ) ) ).
tff(fact_55_semiring__mult__number__of,axiom,
! [V_16: int,V_15: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_15))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_16))
=> ( hAPP_nat_nat(times_times_nat(number_number_of_nat(V_15)),number_number_of_nat(V_16)) = number_number_of_nat(hAPP_int_int(times_times_int(V_15),V_16)) ) ) ) ).
tff(fact_56_semiring__mult__number__of,axiom,
! [V_16: int,V_15: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_15))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_16))
=> ( hAPP_int_int(times_times_int(number_number_of_int(V_15)),number_number_of_int(V_16)) = number_number_of_int(hAPP_int_int(times_times_int(V_15),V_16)) ) ) ) ).
tff(fact_57_mult__number__of__left,axiom,
! [V_14: int,W_13: int,Z_9: real] : hAPP_real_real(times_times_real(number267125858f_real(V_14)),hAPP_real_real(times_times_real(number267125858f_real(W_13)),Z_9)) = hAPP_real_real(times_times_real(number267125858f_real(hAPP_int_int(times_times_int(V_14),W_13))),Z_9) ).
tff(fact_58_mult__number__of__left,axiom,
! [V_14: int,W_13: int,Z_9: 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_9)) = hAPP_int_int(times_times_int(number_number_of_int(hAPP_int_int(times_times_int(V_14),W_13))),Z_9) ).
tff(fact_59_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_60_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_61_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_62_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_63_sum__squares__le__zero__iff,axiom,
! [X: real,Y_1: 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),X)),hAPP_real_real(times_times_real(Y_1),Y_1))),zero_zero_real))
<=> ( ( X = zero_zero_real )
& ( Y_1 = zero_zero_real ) ) ) ).
tff(fact_64_sum__squares__le__zero__iff,axiom,
! [X: int,Y_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(X),X)),hAPP_int_int(times_times_int(Y_1),Y_1))),zero_zero_int))
<=> ( ( X = zero_zero_int )
& ( Y_1 = zero_zero_int ) ) ) ).
tff(fact_65_sum__squares__ge__zero,axiom,
! [X_24: real,Y_18: 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_24),X_24)),hAPP_real_real(times_times_real(Y_18),Y_18)))) ).
tff(fact_66_sum__squares__ge__zero,axiom,
! [X_24: int,Y_18: 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_24),X_24)),hAPP_int_int(times_times_int(Y_18),Y_18)))) ).
tff(fact_67_le__special_I3_J,axiom,
! [X: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(X)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),pls)) ) ).
tff(fact_68_le__special_I3_J,axiom,
! [X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(X)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),pls)) ) ).
tff(fact_69_le__special_I1_J,axiom,
! [Y_1: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),number267125858f_real(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Y_1)) ) ).
tff(fact_70_le__special_I1_J,axiom,
! [Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Y_1)) ) ).
tff(fact_71_less__0__number__of,axiom,
! [V_4: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),number_number_of_nat(V_4)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_4)) ) ).
tff(fact_72_le__number__of__eq__not__less,axiom,
! [V_4: int,W_1: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(V_4)),number267125858f_real(W_1)))
<=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(W_1)),number267125858f_real(V_4))) ) ).
tff(fact_73_le__number__of__eq__not__less,axiom,
! [V_4: int,W_1: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,number_number_of_nat(V_4)),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_4))) ) ).
tff(fact_74_le__number__of__eq__not__less,axiom,
! [V_4: int,W_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(V_4)),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_4))) ) ).
tff(fact_75_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_76_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_77_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_78_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_79_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_80_zadd__zless__mono,axiom,
! [Z_8: int,Z: int,W_10: int,W: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_10),W))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_8),Z))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(W_10),Z_8)),hAPP_int_int(plus_plus_int(W),Z))) ) ) ).
tff(fact_81_nat__number__of__Pls,axiom,
number_number_of_nat(pls) = zero_zero_nat ).
tff(fact_82_semiring__norm_I113_J,axiom,
zero_zero_nat = number_number_of_nat(pls) ).
tff(fact_83_le__special_I4_J,axiom,
! [X: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(X)),one_one_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),bit1(pls))) ) ).
tff(fact_84_le__special_I4_J,axiom,
! [X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(X)),one_one_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),bit1(pls))) ) ).
tff(fact_85_le__special_I2_J,axiom,
! [Y_1: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),number267125858f_real(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(pls)),Y_1)) ) ).
tff(fact_86_le__special_I2_J,axiom,
! [Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),number_number_of_int(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(pls)),Y_1)) ) ).
tff(fact_87_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_88_mult__Pls,axiom,
! [W: int] : hAPP_int_int(times_times_int(pls),W) = pls ).
tff(fact_89_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_90_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_91_zmult__1__right,axiom,
! [Z: int] : hAPP_int_int(times_times_int(Z),one_one_int) = Z ).
tff(fact_92_zmult__1,axiom,
! [Z: int] : hAPP_int_int(times_times_int(one_one_int),Z) = Z ).
tff(fact_93_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_94_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_95_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_96_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_97_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_98_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_99_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_100_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_101_mem__def,axiom,
! [X: int,A_109: fun_int_bool] :
( hBOOL(member_int(X,A_109))
<=> hBOOL(hAPP_int_bool(A_109,X)) ) ).
tff(fact_102_Collect__def,axiom,
! [P_1: fun_int_bool] : collect_int(P_1) = P_1 ).
tff(fact_103_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_104_less__nat__number__of,axiom,
! [V_4: int,V_3: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,number_number_of_nat(V_4)),number_number_of_nat(V_3)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_4),V_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_3)) )
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_4),V_3)) ) ) ).
tff(fact_105_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_106_nat__numeral__1__eq__1,axiom,
number_number_of_nat(bit1(pls)) = one_one_nat ).
tff(fact_107_Numeral1__eq1__nat,axiom,
one_one_nat = number_number_of_nat(bit1(pls)) ).
tff(fact_108_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_109_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_110_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_111_semiring__add__number__of,axiom,
! [V_11: int,V_10: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_11))
=> ( hAPP_real_real(plus_plus_real(number267125858f_real(V_10)),number267125858f_real(V_11)) = number267125858f_real(hAPP_int_int(plus_plus_int(V_10),V_11)) ) ) ) ).
tff(fact_112_semiring__add__number__of,axiom,
! [V_11: int,V_10: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_11))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_10)),number_number_of_nat(V_11)) = number_number_of_nat(hAPP_int_int(plus_plus_int(V_10),V_11)) ) ) ) ).
tff(fact_113_semiring__add__number__of,axiom,
! [V_11: int,V_10: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_11))
=> ( hAPP_int_int(plus_plus_int(number_number_of_int(V_10)),number_number_of_int(V_11)) = number_number_of_int(hAPP_int_int(plus_plus_int(V_10),V_11)) ) ) ) ).
tff(fact_114_add__nat__number__of,axiom,
! [V_2: 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_2)) = number_number_of_nat(V_2) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),number_number_of_nat(V_2)) = number_number_of_nat(V) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),number_number_of_nat(V_2)) = number_number_of_nat(hAPP_int_int(plus_plus_int(V),V_2)) ) ) ) ) ) ).
tff(fact_115_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_116_eq__number__of,axiom,
! [X: int,Y_1: int] :
( ( number267125858f_real(X) = number267125858f_real(Y_1) )
<=> ( X = Y_1 ) ) ).
tff(fact_117_eq__number__of,axiom,
! [X: int,Y_1: int] :
( ( number_number_of_int(X) = number_number_of_int(Y_1) )
<=> ( X = Y_1 ) ) ).
tff(fact_118_number__of__reorient,axiom,
! [W_1: int,X: real] :
( ( number267125858f_real(W_1) = X )
<=> ( X = number267125858f_real(W_1) ) ) ).
tff(fact_119_number__of__reorient,axiom,
! [W_1: int,X: int] :
( ( number_number_of_int(W_1) = X )
<=> ( X = number_number_of_int(W_1) ) ) ).
tff(fact_120_number__of__reorient,axiom,
! [W_1: int,X: nat] :
( ( number_number_of_nat(W_1) = X )
<=> ( X = number_number_of_nat(W_1) ) ) ).
tff(fact_121_rel__simps_I51_J,axiom,
! [K: int,L_1: int] :
( ( bit1(K) = bit1(L_1) )
<=> ( K = L_1 ) ) ).
tff(fact_122_rel__simps_I48_J,axiom,
! [K: int,L_1: int] :
( ( bit0(K) = bit0(L_1) )
<=> ( K = L_1 ) ) ).
tff(fact_123_zless__linear,axiom,
! [X_1: int,Y: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y))
| ( X_1 = Y )
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),X_1)) ) ).
tff(fact_124_sum__squares__eq__zero__iff,axiom,
! [X: real,Y_1: real] :
( ( hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X),X)),hAPP_real_real(times_times_real(Y_1),Y_1)) = zero_zero_real )
<=> ( ( X = zero_zero_real )
& ( Y_1 = zero_zero_real ) ) ) ).
tff(fact_125_sum__squares__eq__zero__iff,axiom,
! [X: int,Y_1: int] :
( ( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(X),X)),hAPP_int_int(times_times_int(Y_1),Y_1)) = zero_zero_int )
<=> ( ( X = zero_zero_int )
& ( Y_1 = zero_zero_int ) ) ) ).
tff(fact_126_left__distrib__number__of,axiom,
! [A_108: real,B_68: real,V_9: int] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_108),B_68)),number267125858f_real(V_9)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_108),number267125858f_real(V_9))),hAPP_real_real(times_times_real(B_68),number267125858f_real(V_9))) ).
tff(fact_127_left__distrib__number__of,axiom,
! [A_108: nat,B_68: nat,V_9: int] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_108),B_68)),number_number_of_nat(V_9)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_108),number_number_of_nat(V_9))),hAPP_nat_nat(times_times_nat(B_68),number_number_of_nat(V_9))) ).
tff(fact_128_left__distrib__number__of,axiom,
! [A_108: int,B_68: int,V_9: int] : hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_108),B_68)),number_number_of_int(V_9)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_108),number_number_of_int(V_9))),hAPP_int_int(times_times_int(B_68),number_number_of_int(V_9))) ).
tff(fact_129_right__distrib__number__of,axiom,
! [V_8: int,B_67: real,C_42: real] : hAPP_real_real(times_times_real(number267125858f_real(V_8)),hAPP_real_real(plus_plus_real(B_67),C_42)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(number267125858f_real(V_8)),B_67)),hAPP_real_real(times_times_real(number267125858f_real(V_8)),C_42)) ).
tff(fact_130_right__distrib__number__of,axiom,
! [V_8: int,B_67: nat,C_42: nat] : hAPP_nat_nat(times_times_nat(number_number_of_nat(V_8)),hAPP_nat_nat(plus_plus_nat(B_67),C_42)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(number_number_of_nat(V_8)),B_67)),hAPP_nat_nat(times_times_nat(number_number_of_nat(V_8)),C_42)) ).
tff(fact_131_right__distrib__number__of,axiom,
! [V_8: int,B_67: int,C_42: int] : hAPP_int_int(times_times_int(number_number_of_int(V_8)),hAPP_int_int(plus_plus_int(B_67),C_42)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(V_8)),B_67)),hAPP_int_int(times_times_int(number_number_of_int(V_8)),C_42)) ).
tff(fact_132_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_133_zadd__left__commute,axiom,
! [X_1: int,Y: int,Z: int] : hAPP_int_int(plus_plus_int(X_1),hAPP_int_int(plus_plus_int(Y),Z)) = hAPP_int_int(plus_plus_int(Y),hAPP_int_int(plus_plus_int(X_1),Z)) ).
tff(fact_134_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_135_zero__is__num__zero,axiom,
zero_zero_int = number_number_of_int(pls) ).
tff(fact_136_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_137_power2__eq__imp__eq,axiom,
! [X_23: real,Y_17: real] :
( ( hAPP_nat_real(power_power_real(X_23),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_real(power_power_real(Y_17),number_number_of_nat(bit0(bit1(pls)))) )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_23))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_17))
=> ( X_23 = Y_17 ) ) ) ) ).
tff(fact_138_power2__eq__imp__eq,axiom,
! [X_23: nat,Y_17: nat] :
( ( hAPP_nat_nat(power_power_nat(X_23),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_nat(power_power_nat(Y_17),number_number_of_nat(bit0(bit1(pls)))) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),X_23))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_17))
=> ( X_23 = Y_17 ) ) ) ) ).
tff(fact_139_power2__eq__imp__eq,axiom,
! [X_23: int,Y_17: int] :
( ( hAPP_nat_int(power_power_int(X_23),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_int(power_power_int(Y_17),number_number_of_nat(bit0(bit1(pls)))) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_23))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_17))
=> ( X_23 = Y_17 ) ) ) ) ).
tff(fact_140_power2__le__imp__le,axiom,
! [X_22: real,Y_16: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(X_22),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_16),number_number_of_nat(bit0(bit1(pls))))))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_16))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_22),Y_16)) ) ) ).
tff(fact_141_power2__le__imp__le,axiom,
! [X_22: nat,Y_16: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(X_22),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_nat(power_power_nat(Y_16),number_number_of_nat(bit0(bit1(pls))))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_16))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_22),Y_16)) ) ) ).
tff(fact_142_power2__le__imp__le,axiom,
! [X_22: int,Y_16: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(X_22),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_16),number_number_of_nat(bit0(bit1(pls))))))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_16))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_22),Y_16)) ) ) ).
tff(fact_143_zero__le__power2,axiom,
! [A_107: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(power_power_real(A_107),number_number_of_nat(bit0(bit1(pls)))))) ).
tff(fact_144_zero__le__power2,axiom,
! [A_107: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(A_107),number_number_of_nat(bit0(bit1(pls)))))) ).
tff(fact_145_power2__less__imp__less,axiom,
! [X_21: real,Y_15: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(X_21),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_15),number_number_of_nat(bit0(bit1(pls))))))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_15))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_21),Y_15)) ) ) ).
tff(fact_146_power2__less__imp__less,axiom,
! [X_21: nat,Y_15: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(X_21),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_nat(power_power_nat(Y_15),number_number_of_nat(bit0(bit1(pls))))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_15))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_21),Y_15)) ) ) ).
tff(fact_147_power2__less__imp__less,axiom,
! [X_21: int,Y_15: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(X_21),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_15),number_number_of_nat(bit0(bit1(pls))))))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_15))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_21),Y_15)) ) ) ).
tff(fact_148_sum__power2__le__zero__iff,axiom,
! [X: real,Y_1: 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),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_1),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_real))
<=> ( ( X = zero_zero_real )
& ( Y_1 = zero_zero_real ) ) ) ).
tff(fact_149_sum__power2__le__zero__iff,axiom,
! [X: int,Y_1: 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),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_1),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_int))
<=> ( ( X = zero_zero_int )
& ( Y_1 = zero_zero_int ) ) ) ).
tff(fact_150_sum__power2__ge__zero,axiom,
! [X_20: 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_nat_real(power_power_real(X_20),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_14),number_number_of_nat(bit0(bit1(pls))))))) ).
tff(fact_151_sum__power2__ge__zero,axiom,
! [X_20: 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_nat_int(power_power_int(X_20),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_14),number_number_of_nat(bit0(bit1(pls))))))) ).
tff(fact_152_sum__squares__gt__zero__iff,axiom,
! [X: real,Y_1: 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),X)),hAPP_real_real(times_times_real(Y_1),Y_1))))
<=> ( ( X != zero_zero_real )
| ( Y_1 != zero_zero_real ) ) ) ).
tff(fact_153_sum__squares__gt__zero__iff,axiom,
! [X: int,Y_1: 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),X)),hAPP_int_int(times_times_int(Y_1),Y_1))))
<=> ( ( X != zero_zero_int )
| ( Y_1 != zero_zero_int ) ) ) ).
tff(fact_154_not__sum__squares__lt__zero,axiom,
! [X_19: 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_19),X_19)),hAPP_real_real(times_times_real(Y_13),Y_13))),zero_zero_real)) ).
tff(fact_155_not__sum__squares__lt__zero,axiom,
! [X_19: 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_19),X_19)),hAPP_int_int(times_times_int(Y_13),Y_13))),zero_zero_int)) ).
tff(fact_156_mult__numeral__1,axiom,
! [A_106: real] : hAPP_real_real(times_times_real(number267125858f_real(bit1(pls))),A_106) = A_106 ).
tff(fact_157_mult__numeral__1,axiom,
! [A_106: int] : hAPP_int_int(times_times_int(number_number_of_int(bit1(pls))),A_106) = A_106 ).
tff(fact_158_mult__numeral__1__right,axiom,
! [A_105: real] : hAPP_real_real(times_times_real(A_105),number267125858f_real(bit1(pls))) = A_105 ).
tff(fact_159_mult__numeral__1__right,axiom,
! [A_105: int] : hAPP_int_int(times_times_int(A_105),number_number_of_int(bit1(pls))) = A_105 ).
tff(fact_160_one__is__num__one,axiom,
one_one_int = number_number_of_int(bit1(pls)) ).
tff(fact_161_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_162_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_163_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_164_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_165_rel__simps_I46_J,axiom,
! [K_1: int] : bit1(K_1) != pls ).
tff(fact_166_rel__simps_I39_J,axiom,
! [L: int] : pls != bit1(L) ).
tff(fact_167_rel__simps_I50_J,axiom,
! [K_1: int,L: int] : bit1(K_1) != bit0(L) ).
tff(fact_168_rel__simps_I49_J,axiom,
! [K_1: int,L: int] : bit0(K_1) != bit1(L) ).
tff(fact_169_rel__simps_I44_J,axiom,
! [K: int] :
( ( bit0(K) = pls )
<=> ( K = pls ) ) ).
tff(fact_170_rel__simps_I38_J,axiom,
! [L_1: int] :
( ( pls = bit0(L_1) )
<=> ( pls = L_1 ) ) ).
tff(fact_171_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_172_Pls__def,axiom,
pls = zero_zero_int ).
tff(fact_173_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_174_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_175_rel__simps_I2_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),pls)) ).
tff(fact_176_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_177_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_178_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
tff(fact_179_add__Pls__right,axiom,
! [K_1: int] : hAPP_int_int(plus_plus_int(K_1),pls) = K_1 ).
tff(fact_180_add__Pls,axiom,
! [K_1: int] : hAPP_int_int(plus_plus_int(pls),K_1) = K_1 ).
tff(fact_181_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_182_Bit0__def,axiom,
! [K_1: int] : bit0(K_1) = hAPP_int_int(plus_plus_int(K_1),K_1) ).
tff(fact_183_zadd__0__right,axiom,
! [Z: int] : hAPP_int_int(plus_plus_int(Z),zero_zero_int) = Z ).
tff(fact_184_zadd__0,axiom,
! [Z: int] : hAPP_int_int(plus_plus_int(zero_zero_int),Z) = Z ).
tff(fact_185_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_186_double__number__of__Bit0,axiom,
! [W_9: int] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(one_one_real),one_one_real)),number267125858f_real(W_9)) = number267125858f_real(bit0(W_9)) ).
tff(fact_187_double__number__of__Bit0,axiom,
! [W_9: int] : hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(one_one_int),one_one_int)),number_number_of_int(W_9)) = number_number_of_int(bit0(W_9)) ).
tff(fact_188_power3__eq__cube,axiom,
! [A_104: real] : hAPP_nat_real(power_power_real(A_104),number_number_of_nat(bit1(bit1(pls)))) = hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(A_104),A_104)),A_104) ).
tff(fact_189_power3__eq__cube,axiom,
! [A_104: nat] : hAPP_nat_nat(power_power_nat(A_104),number_number_of_nat(bit1(bit1(pls)))) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(A_104),A_104)),A_104) ).
tff(fact_190_power3__eq__cube,axiom,
! [A_104: int] : hAPP_nat_int(power_power_int(A_104),number_number_of_nat(bit1(bit1(pls)))) = hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(A_104),A_104)),A_104) ).
tff(fact_191_semiring__mult__2,axiom,
! [Z_7: real] : hAPP_real_real(times_times_real(number267125858f_real(bit0(bit1(pls)))),Z_7) = hAPP_real_real(plus_plus_real(Z_7),Z_7) ).
tff(fact_192_semiring__mult__2,axiom,
! [Z_7: nat] : hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),Z_7) = hAPP_nat_nat(plus_plus_nat(Z_7),Z_7) ).
tff(fact_193_semiring__mult__2,axiom,
! [Z_7: int] : hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),Z_7) = hAPP_int_int(plus_plus_int(Z_7),Z_7) ).
tff(fact_194_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_195_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_196_semiring__mult__2__right,axiom,
! [Z_5: real] : hAPP_real_real(times_times_real(Z_5),number267125858f_real(bit0(bit1(pls)))) = hAPP_real_real(plus_plus_real(Z_5),Z_5) ).
tff(fact_197_semiring__mult__2__right,axiom,
! [Z_5: nat] : hAPP_nat_nat(times_times_nat(Z_5),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_nat(plus_plus_nat(Z_5),Z_5) ).
tff(fact_198_semiring__mult__2__right,axiom,
! [Z_5: int] : hAPP_int_int(times_times_int(Z_5),number_number_of_int(bit0(bit1(pls)))) = hAPP_int_int(plus_plus_int(Z_5),Z_5) ).
tff(fact_199_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_200_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_201_power2__eq__square,axiom,
! [A_103: real] : hAPP_nat_real(power_power_real(A_103),number_number_of_nat(bit0(bit1(pls)))) = hAPP_real_real(times_times_real(A_103),A_103) ).
tff(fact_202_power2__eq__square,axiom,
! [A_103: nat] : hAPP_nat_nat(power_power_nat(A_103),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_nat(times_times_nat(A_103),A_103) ).
tff(fact_203_power2__eq__square,axiom,
! [A_103: int] : hAPP_nat_int(power_power_int(A_103),number_number_of_nat(bit0(bit1(pls)))) = hAPP_int_int(times_times_int(A_103),A_103) ).
tff(fact_204_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_205_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_206_semiring__numeral__0__eq__0,axiom,
number267125858f_real(pls) = zero_zero_real ).
tff(fact_207_semiring__numeral__0__eq__0,axiom,
number_number_of_nat(pls) = zero_zero_nat ).
tff(fact_208_semiring__numeral__0__eq__0,axiom,
number_number_of_int(pls) = zero_zero_int ).
tff(fact_209_number__of__Pls,axiom,
number267125858f_real(pls) = zero_zero_real ).
tff(fact_210_number__of__Pls,axiom,
number_number_of_int(pls) = zero_zero_int ).
tff(fact_211_semiring__norm_I112_J,axiom,
zero_zero_real = number267125858f_real(pls) ).
tff(fact_212_semiring__norm_I112_J,axiom,
zero_zero_int = number_number_of_int(pls) ).
tff(fact_213_add__numeral__0,axiom,
! [A_102: real] : hAPP_real_real(plus_plus_real(number267125858f_real(pls)),A_102) = A_102 ).
tff(fact_214_add__numeral__0,axiom,
! [A_102: int] : hAPP_int_int(plus_plus_int(number_number_of_int(pls)),A_102) = A_102 ).
tff(fact_215_add__numeral__0__right,axiom,
! [A_101: real] : hAPP_real_real(plus_plus_real(A_101),number267125858f_real(pls)) = A_101 ).
tff(fact_216_add__numeral__0__right,axiom,
! [A_101: int] : hAPP_int_int(plus_plus_int(A_101),number_number_of_int(pls)) = A_101 ).
tff(fact_217_less__number__of,axiom,
! [X: int,Y_1: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(X)),number267125858f_real(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Y_1)) ) ).
tff(fact_218_less__number__of,axiom,
! [X: int,Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(X)),number_number_of_int(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Y_1)) ) ).
tff(fact_219_add__number__of__left,axiom,
! [V_7: int,W_8: int,Z_3: real] : hAPP_real_real(plus_plus_real(number267125858f_real(V_7)),hAPP_real_real(plus_plus_real(number267125858f_real(W_8)),Z_3)) = hAPP_real_real(plus_plus_real(number267125858f_real(hAPP_int_int(plus_plus_int(V_7),W_8))),Z_3) ).
tff(fact_220_add__number__of__left,axiom,
! [V_7: int,W_8: int,Z_3: int] : hAPP_int_int(plus_plus_int(number_number_of_int(V_7)),hAPP_int_int(plus_plus_int(number_number_of_int(W_8)),Z_3)) = hAPP_int_int(plus_plus_int(number_number_of_int(hAPP_int_int(plus_plus_int(V_7),W_8))),Z_3) ).
tff(fact_221_add__number__of__eq,axiom,
! [V_6: int,W_7: int] : hAPP_real_real(plus_plus_real(number267125858f_real(V_6)),number267125858f_real(W_7)) = number267125858f_real(hAPP_int_int(plus_plus_int(V_6),W_7)) ).
tff(fact_222_add__number__of__eq,axiom,
! [V_6: int,W_7: int] : hAPP_int_int(plus_plus_int(number_number_of_int(V_6)),number_number_of_int(W_7)) = number_number_of_int(hAPP_int_int(plus_plus_int(V_6),W_7)) ).
tff(fact_223_number__of__add,axiom,
! [V_5: int,W_6: int] : number267125858f_real(hAPP_int_int(plus_plus_int(V_5),W_6)) = hAPP_real_real(plus_plus_real(number267125858f_real(V_5)),number267125858f_real(W_6)) ).
tff(fact_224_number__of__add,axiom,
! [V_5: int,W_6: int] : number_number_of_int(hAPP_int_int(plus_plus_int(V_5),W_6)) = hAPP_int_int(plus_plus_int(number_number_of_int(V_5)),number_number_of_int(W_6)) ).
tff(fact_225_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_226_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_227_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_228_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_229_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_230_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_231_bin__less__0__simps_I1_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),zero_zero_int)) ).
tff(fact_232_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_233_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_234_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_235_int__0__less__1,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
tff(fact_236_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_237_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_238_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_239_power2__eq__square__number__of,axiom,
! [W_5: int] : hAPP_nat_real(power_power_real(number267125858f_real(W_5)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_real_real(times_times_real(number267125858f_real(W_5)),number267125858f_real(W_5)) ).
tff(fact_240_power2__eq__square__number__of,axiom,
! [W_5: int] : hAPP_nat_nat(power_power_nat(number_number_of_nat(W_5)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_nat(times_times_nat(number_number_of_nat(W_5)),number_number_of_nat(W_5)) ).
tff(fact_241_power2__eq__square__number__of,axiom,
! [W_5: int] : hAPP_nat_int(power_power_int(number_number_of_int(W_5)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_int_int(times_times_int(number_number_of_int(W_5)),number_number_of_int(W_5)) ).
tff(fact_242_power2__sum,axiom,
! [X_18: real,Y_12: real] : hAPP_nat_real(power_power_real(hAPP_real_real(plus_plus_real(X_18),Y_12)),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_18),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_12),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_18)),Y_12)) ).
tff(fact_243_power2__sum,axiom,
! [X_18: nat,Y_12: nat] : hAPP_nat_nat(power_power_nat(hAPP_nat_nat(plus_plus_nat(X_18),Y_12)),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_18),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_nat(power_power_nat(Y_12),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_18)),Y_12)) ).
tff(fact_244_power2__sum,axiom,
! [X_18: int,Y_12: int] : hAPP_nat_int(power_power_int(hAPP_int_int(plus_plus_int(X_18),Y_12)),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_18),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_12),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_18)),Y_12)) ).
tff(fact_245_number__of__Bit0,axiom,
! [W_4: int] : number267125858f_real(bit0(W_4)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(zero_zero_real),number267125858f_real(W_4))),number267125858f_real(W_4)) ).
tff(fact_246_number__of__Bit0,axiom,
! [W_4: int] : number_number_of_int(bit0(W_4)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(zero_zero_int),number_number_of_int(W_4))),number_number_of_int(W_4)) ).
tff(fact_247_number__of__Bit1,axiom,
! [W_3: int] : number267125858f_real(bit1(W_3)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(one_one_real),number267125858f_real(W_3))),number267125858f_real(W_3)) ).
tff(fact_248_number__of__Bit1,axiom,
! [W_3: int] : number_number_of_int(bit1(W_3)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),number_number_of_int(W_3))),number_number_of_int(W_3)) ).
tff(fact_249_semiring__numeral__1__eq__1,axiom,
number267125858f_real(bit1(pls)) = one_one_real ).
tff(fact_250_semiring__numeral__1__eq__1,axiom,
number_number_of_nat(bit1(pls)) = one_one_nat ).
tff(fact_251_semiring__numeral__1__eq__1,axiom,
number_number_of_int(bit1(pls)) = one_one_int ).
tff(fact_252_numeral__1__eq__1,axiom,
number267125858f_real(bit1(pls)) = one_one_real ).
tff(fact_253_numeral__1__eq__1,axiom,
number_number_of_int(bit1(pls)) = one_one_int ).
tff(fact_254_semiring__norm_I110_J,axiom,
one_one_real = number267125858f_real(bit1(pls)) ).
tff(fact_255_semiring__norm_I110_J,axiom,
one_one_int = number_number_of_int(bit1(pls)) ).
tff(fact_256_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_257_less__special_I3_J,axiom,
! [X: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(X)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),pls)) ) ).
tff(fact_258_less__special_I3_J,axiom,
! [X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(X)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),pls)) ) ).
tff(fact_259_less__special_I1_J,axiom,
! [Y_1: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),number267125858f_real(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Y_1)) ) ).
tff(fact_260_less__special_I1_J,axiom,
! [Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),number_number_of_int(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Y_1)) ) ).
tff(fact_261_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_262_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_263_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_264_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_265_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_266_less__special_I4_J,axiom,
! [X: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(X)),one_one_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),bit1(pls))) ) ).
tff(fact_267_less__special_I4_J,axiom,
! [X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(X)),one_one_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),bit1(pls))) ) ).
tff(fact_268_less__special_I2_J,axiom,
! [Y_1: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),number267125858f_real(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(pls)),Y_1)) ) ).
tff(fact_269_less__special_I2_J,axiom,
! [Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),number_number_of_int(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(pls)),Y_1)) ) ).
tff(fact_270__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_271_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_272_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_273_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_274_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_275_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_276_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_277_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_278_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_279_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_280_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_17: real,N_24: nat] : hAPP_nat_real(power_power_real(X_17),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_24)) = hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(X_17),N_24)),hAPP_nat_real(power_power_real(X_17),N_24)) ).
tff(fact_281_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_17: nat,N_24: nat] : hAPP_nat_nat(power_power_nat(X_17),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_24)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(X_17),N_24)),hAPP_nat_nat(power_power_nat(X_17),N_24)) ).
tff(fact_282_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_17: int,N_24: nat] : hAPP_nat_int(power_power_int(X_17),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_24)) = hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(X_17),N_24)),hAPP_nat_int(power_power_int(X_17),N_24)) ).
tff(fact_283_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_16: real] : hAPP_real_real(times_times_real(X_16),X_16) = hAPP_nat_real(power_power_real(X_16),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_284_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_16: nat] : hAPP_nat_nat(times_times_nat(X_16),X_16) = hAPP_nat_nat(power_power_nat(X_16),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_285_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_16: int] : hAPP_int_int(times_times_int(X_16),X_16) = hAPP_nat_int(power_power_int(X_16),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_286__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_287__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_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),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_2),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_2: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_2),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_2),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_2 = X_2 ) ) ) ).
tff(fact_288__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_289_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_290_zero__less__power__nat__eq,axiom,
! [X: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(X),N)))
<=> ( ( N = zero_zero_nat )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X)) ) ) ).
tff(fact_291_zprime__zdvd__power,axiom,
! [A: int,N_1: nat,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(power_power_int(A),N_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),A)) ) ) ).
tff(fact_292_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_15: real,P_7: nat,Q_7: nat] : hAPP_nat_real(power_power_real(hAPP_nat_real(power_power_real(X_15),P_7)),Q_7) = hAPP_nat_real(power_power_real(X_15),hAPP_nat_nat(times_times_nat(P_7),Q_7)) ).
tff(fact_293_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_15: nat,P_7: nat,Q_7: nat] : hAPP_nat_nat(power_power_nat(hAPP_nat_nat(power_power_nat(X_15),P_7)),Q_7) = hAPP_nat_nat(power_power_nat(X_15),hAPP_nat_nat(times_times_nat(P_7),Q_7)) ).
tff(fact_294_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_15: int,P_7: nat,Q_7: nat] : hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(X_15),P_7)),Q_7) = hAPP_nat_int(power_power_int(X_15),hAPP_nat_nat(times_times_nat(P_7),Q_7)) ).
tff(fact_295_zprime__power__zdvd__cancel__right,axiom,
! [N_1: nat,A: int,B: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_3),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_3),N_1)),A)) ) ) ) ).
tff(fact_296_zprime__power__zdvd__cancel__left,axiom,
! [N_1: nat,B: int,A: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_3),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_3),N_1)),B)) ) ) ) ).
tff(fact_297_zpower__zpower,axiom,
! [X_1: int,Y: nat,Z: nat] : hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(X_1),Y)),Z) = hAPP_nat_int(power_power_int(X_1),hAPP_nat_nat(times_times_nat(Y),Z)) ).
tff(fact_298_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_299_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_300_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_301_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_302_zdvd__period,axiom,
! [C_1: int,X: 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),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),hAPP_int_int(times_times_int(C_1),D))),Ta))) ) ) ).
tff(fact_303_zprime__2,axiom,
hBOOL(hAPP_int_bool(zprime,number_number_of_int(bit0(bit1(pls))))) ).
tff(fact_304_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_305_is__mult__sum2sq,axiom,
! [Y: int,X_1: int] :
( hBOOL(hAPP_int_bool(twoSqu820444569sum2sq,X_1))
=> ( hBOOL(hAPP_int_bool(twoSqu820444569sum2sq,Y))
=> hBOOL(hAPP_int_bool(twoSqu820444569sum2sq,hAPP_int_int(times_times_int(X_1),Y))) ) ) ).
tff(fact_306_le__nat__number__of,axiom,
! [V_4: int,V_3: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,number_number_of_nat(V_4)),number_number_of_nat(V_3)))
<=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_4),V_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_4),pls)) ) ) ).
tff(fact_307_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_308_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_309_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_310_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_311_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_312_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_313_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_314_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_315_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_316_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_317_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_318_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_319_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_320_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_321_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_322_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_323_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_324_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_325_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_326_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_327_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_328_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_100: real,B_66: real] : hAPP_real_real(times_times_real(A_100),B_66) = hAPP_real_real(times_times_real(B_66),A_100) ).
tff(fact_329_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_100: nat,B_66: nat] : hAPP_nat_nat(times_times_nat(A_100),B_66) = hAPP_nat_nat(times_times_nat(B_66),A_100) ).
tff(fact_330_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_100: int,B_66: int] : hAPP_int_int(times_times_int(A_100),B_66) = hAPP_int_int(times_times_int(B_66),A_100) ).
tff(fact_331_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_99: real,C_41: real] : hAPP_real_real(plus_plus_real(A_99),C_41) = hAPP_real_real(plus_plus_real(C_41),A_99) ).
tff(fact_332_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_99: nat,C_41: nat] : hAPP_nat_nat(plus_plus_nat(A_99),C_41) = hAPP_nat_nat(plus_plus_nat(C_41),A_99) ).
tff(fact_333_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_99: int,C_41: int] : hAPP_int_int(plus_plus_int(A_99),C_41) = hAPP_int_int(plus_plus_int(C_41),A_99) ).
tff(fact_334_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_98: real,C_40: real,D_14: real] : hAPP_real_real(plus_plus_real(A_98),hAPP_real_real(plus_plus_real(C_40),D_14)) = hAPP_real_real(plus_plus_real(C_40),hAPP_real_real(plus_plus_real(A_98),D_14)) ).
tff(fact_335_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_98: nat,C_40: nat,D_14: nat] : hAPP_nat_nat(plus_plus_nat(A_98),hAPP_nat_nat(plus_plus_nat(C_40),D_14)) = hAPP_nat_nat(plus_plus_nat(C_40),hAPP_nat_nat(plus_plus_nat(A_98),D_14)) ).
tff(fact_336_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_98: int,C_40: int,D_14: int] : hAPP_int_int(plus_plus_int(A_98),hAPP_int_int(plus_plus_int(C_40),D_14)) = hAPP_int_int(plus_plus_int(C_40),hAPP_int_int(plus_plus_int(A_98),D_14)) ).
tff(fact_337_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_97: real,C_39: real,D_13: real] : hAPP_real_real(plus_plus_real(A_97),hAPP_real_real(plus_plus_real(C_39),D_13)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_97),C_39)),D_13) ).
tff(fact_338_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_97: nat,C_39: nat,D_13: nat] : hAPP_nat_nat(plus_plus_nat(A_97),hAPP_nat_nat(plus_plus_nat(C_39),D_13)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_97),C_39)),D_13) ).
tff(fact_339_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_97: int,C_39: int,D_13: int] : hAPP_int_int(plus_plus_int(A_97),hAPP_int_int(plus_plus_int(C_39),D_13)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_97),C_39)),D_13) ).
tff(fact_340_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_96: real,B_65: real,C_38: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_96),B_65)),C_38) = hAPP_real_real(plus_plus_real(A_96),hAPP_real_real(plus_plus_real(B_65),C_38)) ).
tff(fact_341_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_96: nat,B_65: nat,C_38: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_96),B_65)),C_38) = hAPP_nat_nat(plus_plus_nat(A_96),hAPP_nat_nat(plus_plus_nat(B_65),C_38)) ).
tff(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_96: int,B_65: int,C_38: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_96),B_65)),C_38) = hAPP_int_int(plus_plus_int(A_96),hAPP_int_int(plus_plus_int(B_65),C_38)) ).
tff(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_95: real,B_64: real,C_37: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_95),B_64)),C_37) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_95),C_37)),B_64) ).
tff(fact_344_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_95: nat,B_64: nat,C_37: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_95),B_64)),C_37) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_95),C_37)),B_64) ).
tff(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_95: int,B_64: int,C_37: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_95),B_64)),C_37) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_95),C_37)),B_64) ).
tff(fact_346_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_94: real,B_63: real,C_36: real,D_12: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_94),B_63)),hAPP_real_real(plus_plus_real(C_36),D_12)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_94),C_36)),hAPP_real_real(plus_plus_real(B_63),D_12)) ).
tff(fact_347_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_94: nat,B_63: nat,C_36: nat,D_12: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_94),B_63)),hAPP_nat_nat(plus_plus_nat(C_36),D_12)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_94),C_36)),hAPP_nat_nat(plus_plus_nat(B_63),D_12)) ).
tff(fact_348_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_94: int,B_63: int,C_36: int,D_12: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_94),B_63)),hAPP_int_int(plus_plus_int(C_36),D_12)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_94),C_36)),hAPP_int_int(plus_plus_int(B_63),D_12)) ).
tff(fact_349_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_14: real] : hAPP_nat_real(power_power_real(X_14),one_one_nat) = X_14 ).
tff(fact_350_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_14: nat] : hAPP_nat_nat(power_power_nat(X_14),one_one_nat) = X_14 ).
tff(fact_351_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_14: int] : hAPP_nat_int(power_power_int(X_14),one_one_nat) = X_14 ).
tff(fact_352_zero__less__power__nat__eq__number__of,axiom,
! [X: nat,W_1: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(X),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)) ) ) ).
tff(fact_353_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_354_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_355_mult__nat__number__of,axiom,
! [V_2: 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_2)) = 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_2)) = number_number_of_nat(hAPP_int_int(times_times_int(V),V_2)) ) ) ) ).
tff(fact_356_nat__number__of__mult__left,axiom,
! [V_2: 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_2)),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_2)),K_1)) = hAPP_nat_nat(times_times_nat(number_number_of_nat(hAPP_int_int(times_times_int(V),V_2))),K_1) ) ) ) ).
tff(fact_357_power__even__eq,axiom,
! [A_93: real,N_23: nat] : hAPP_nat_real(power_power_real(A_93),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_23)) = hAPP_nat_real(power_power_real(hAPP_nat_real(power_power_real(A_93),N_23)),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_358_power__even__eq,axiom,
! [A_93: nat,N_23: nat] : hAPP_nat_nat(power_power_nat(A_93),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_23)) = hAPP_nat_nat(power_power_nat(hAPP_nat_nat(power_power_nat(A_93),N_23)),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_359_power__even__eq,axiom,
! [A_93: int,N_23: nat] : hAPP_nat_int(power_power_int(A_93),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_23)) = hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(A_93),N_23)),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_360_even__power__le__0__imp__0,axiom,
! [A_92: real,K_4: nat] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_92),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),K_4))),zero_zero_real))
=> ( A_92 = zero_zero_real ) ) ).
tff(fact_361_even__power__le__0__imp__0,axiom,
! [A_92: int,K_4: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_92),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),K_4))),zero_zero_int))
=> ( A_92 = zero_zero_int ) ) ).
tff(fact_362_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_91: real] : hAPP_real_real(times_times_real(zero_zero_real),A_91) = zero_zero_real ).
tff(fact_363_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_91: nat] : hAPP_nat_nat(times_times_nat(zero_zero_nat),A_91) = zero_zero_nat ).
tff(fact_364_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_91: int] : hAPP_int_int(times_times_int(zero_zero_int),A_91) = zero_zero_int ).
tff(fact_365_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_90: real] : hAPP_real_real(times_times_real(A_90),zero_zero_real) = zero_zero_real ).
tff(fact_366_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_90: nat] : hAPP_nat_nat(times_times_nat(A_90),zero_zero_nat) = zero_zero_nat ).
tff(fact_367_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_90: int] : hAPP_int_int(times_times_int(A_90),zero_zero_int) = zero_zero_int ).
tff(fact_368_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_89: real] : hAPP_real_real(plus_plus_real(zero_zero_real),A_89) = A_89 ).
tff(fact_369_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_89: nat] : hAPP_nat_nat(plus_plus_nat(zero_zero_nat),A_89) = A_89 ).
tff(fact_370_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_89: int] : hAPP_int_int(plus_plus_int(zero_zero_int),A_89) = A_89 ).
tff(fact_371_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_88: real] : hAPP_real_real(plus_plus_real(A_88),zero_zero_real) = A_88 ).
tff(fact_372_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_88: nat] : hAPP_nat_nat(plus_plus_nat(A_88),zero_zero_nat) = A_88 ).
tff(fact_373_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_88: int] : hAPP_int_int(plus_plus_int(A_88),zero_zero_int) = A_88 ).
tff(fact_374_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_375_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_376_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_377_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_13: real,Y_11: real,Z_2: real] : hAPP_real_real(times_times_real(X_13),hAPP_real_real(plus_plus_real(Y_11),Z_2)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_13),Y_11)),hAPP_real_real(times_times_real(X_13),Z_2)) ).
tff(fact_378_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_13: nat,Y_11: nat,Z_2: nat] : hAPP_nat_nat(times_times_nat(X_13),hAPP_nat_nat(plus_plus_nat(Y_11),Z_2)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(X_13),Y_11)),hAPP_nat_nat(times_times_nat(X_13),Z_2)) ).
tff(fact_379_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_13: int,Y_11: int,Z_2: int] : hAPP_int_int(times_times_int(X_13),hAPP_int_int(plus_plus_int(Y_11),Z_2)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(X_13),Y_11)),hAPP_int_int(times_times_int(X_13),Z_2)) ).
tff(fact_380_crossproduct__noteq,axiom,
! [C_1: real,D: real,A_2: real,B_2: real] :
( ( ( A_2 != B_2 )
& ( C_1 != D ) )
<=> ( hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_2),C_1)),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_1)) ) ) ).
tff(fact_381_crossproduct__noteq,axiom,
! [C_1: nat,D: nat,A_2: nat,B_2: nat] :
( ( ( A_2 != B_2 )
& ( C_1 != D ) )
<=> ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_2),C_1)),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_1)) ) ) ).
tff(fact_382_crossproduct__noteq,axiom,
! [C_1: int,D: int,A_2: int,B_2: int] :
( ( ( A_2 != B_2 )
& ( C_1 != D ) )
<=> ( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_2),C_1)),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_1)) ) ) ).
tff(fact_383_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_87: real,B_62: real,C_35: real] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_87),B_62)),C_35) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_87),C_35)),hAPP_real_real(times_times_real(B_62),C_35)) ).
tff(fact_384_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_87: nat,B_62: nat,C_35: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_87),B_62)),C_35) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_87),C_35)),hAPP_nat_nat(times_times_nat(B_62),C_35)) ).
tff(fact_385_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_87: int,B_62: int,C_35: int] : hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_87),B_62)),C_35) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_87),C_35)),hAPP_int_int(times_times_int(B_62),C_35)) ).
tff(fact_386_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_86: real,M_12: real,B_61: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_86),M_12)),hAPP_real_real(times_times_real(B_61),M_12)) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_86),B_61)),M_12) ).
tff(fact_387_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_86: nat,M_12: nat,B_61: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_86),M_12)),hAPP_nat_nat(times_times_nat(B_61),M_12)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_86),B_61)),M_12) ).
tff(fact_388_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_86: int,M_12: int,B_61: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_86),M_12)),hAPP_int_int(times_times_int(B_61),M_12)) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_86),B_61)),M_12) ).
tff(fact_389_crossproduct__eq,axiom,
! [W_1: real,Y_1: real,X: real,Z_1: real] :
( ( hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(W_1),Y_1)),hAPP_real_real(times_times_real(X),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),Y_1)) )
<=> ( ( W_1 = X )
| ( Y_1 = Z_1 ) ) ) ).
tff(fact_390_crossproduct__eq,axiom,
! [W_1: nat,Y_1: nat,X: nat,Z_1: nat] :
( ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(W_1),Y_1)),hAPP_nat_nat(times_times_nat(X),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),Y_1)) )
<=> ( ( W_1 = X )
| ( Y_1 = Z_1 ) ) ) ).
tff(fact_391_crossproduct__eq,axiom,
! [W_1: int,Y_1: int,X: int,Z_1: int] :
( ( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(W_1),Y_1)),hAPP_int_int(times_times_int(X),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),Y_1)) )
<=> ( ( W_1 = X )
| ( Y_1 = Z_1 ) ) ) ).
tff(fact_392_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_85: real] : hAPP_real_real(times_times_real(one_one_real),A_85) = A_85 ).
tff(fact_393_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_85: nat] : hAPP_nat_nat(times_times_nat(one_one_nat),A_85) = A_85 ).
tff(fact_394_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_85: int] : hAPP_int_int(times_times_int(one_one_int),A_85) = A_85 ).
tff(fact_395_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_84: real] : hAPP_real_real(times_times_real(A_84),one_one_real) = A_84 ).
tff(fact_396_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_84: nat] : hAPP_nat_nat(times_times_nat(A_84),one_one_nat) = A_84 ).
tff(fact_397_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_84: int] : hAPP_int_int(times_times_int(A_84),one_one_int) = A_84 ).
tff(fact_398_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_12: real,Y_10: real,Q_6: nat] : hAPP_nat_real(power_power_real(hAPP_real_real(times_times_real(X_12),Y_10)),Q_6) = hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(X_12),Q_6)),hAPP_nat_real(power_power_real(Y_10),Q_6)) ).
tff(fact_399_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_12: nat,Y_10: nat,Q_6: nat] : hAPP_nat_nat(power_power_nat(hAPP_nat_nat(times_times_nat(X_12),Y_10)),Q_6) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(X_12),Q_6)),hAPP_nat_nat(power_power_nat(Y_10),Q_6)) ).
tff(fact_400_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_12: int,Y_10: int,Q_6: nat] : hAPP_nat_int(power_power_int(hAPP_int_int(times_times_int(X_12),Y_10)),Q_6) = hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(X_12),Q_6)),hAPP_nat_int(power_power_int(Y_10),Q_6)) ).
tff(fact_401_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_11: real,P_6: nat,Q_5: nat] : hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(X_11),P_6)),hAPP_nat_real(power_power_real(X_11),Q_5)) = hAPP_nat_real(power_power_real(X_11),hAPP_nat_nat(plus_plus_nat(P_6),Q_5)) ).
tff(fact_402_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_11: nat,P_6: nat,Q_5: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(X_11),P_6)),hAPP_nat_nat(power_power_nat(X_11),Q_5)) = hAPP_nat_nat(power_power_nat(X_11),hAPP_nat_nat(plus_plus_nat(P_6),Q_5)) ).
tff(fact_403_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_11: int,P_6: nat,Q_5: nat] : hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(X_11),P_6)),hAPP_nat_int(power_power_int(X_11),Q_5)) = hAPP_nat_int(power_power_int(X_11),hAPP_nat_nat(plus_plus_nat(P_6),Q_5)) ).
tff(fact_404_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_405_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_406_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_407_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_408_zero__le__even__power_H,axiom,
! [A_83: real,N_22: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(power_power_real(A_83),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_22)))) ).
tff(fact_409_zero__le__even__power_H,axiom,
! [A_83: int,N_22: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(A_83),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_22)))) ).
tff(fact_410_add__scale__eq__noteq,axiom,
! [C_34: real,D_11: real,A_82: real,B_60: real,R_4: real] :
( ( R_4 != zero_zero_real )
=> ( ( ( A_82 = B_60 )
& ( C_34 != D_11 ) )
=> ( hAPP_real_real(plus_plus_real(A_82),hAPP_real_real(times_times_real(R_4),C_34)) != hAPP_real_real(plus_plus_real(B_60),hAPP_real_real(times_times_real(R_4),D_11)) ) ) ) ).
tff(fact_411_add__scale__eq__noteq,axiom,
! [C_34: nat,D_11: nat,A_82: nat,B_60: nat,R_4: nat] :
( ( R_4 != zero_zero_nat )
=> ( ( ( A_82 = B_60 )
& ( C_34 != D_11 ) )
=> ( hAPP_nat_nat(plus_plus_nat(A_82),hAPP_nat_nat(times_times_nat(R_4),C_34)) != hAPP_nat_nat(plus_plus_nat(B_60),hAPP_nat_nat(times_times_nat(R_4),D_11)) ) ) ) ).
tff(fact_412_add__scale__eq__noteq,axiom,
! [C_34: int,D_11: int,A_82: int,B_60: int,R_4: int] :
( ( R_4 != zero_zero_int )
=> ( ( ( A_82 = B_60 )
& ( C_34 != D_11 ) )
=> ( hAPP_int_int(plus_plus_int(A_82),hAPP_int_int(times_times_int(R_4),C_34)) != hAPP_int_int(plus_plus_int(B_60),hAPP_int_int(times_times_int(R_4),D_11)) ) ) ) ).
tff(fact_413_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_11: real] : hAPP_real_real(plus_plus_real(M_11),M_11) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(one_one_real),one_one_real)),M_11) ).
tff(fact_414_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_11: nat] : hAPP_nat_nat(plus_plus_nat(M_11),M_11) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(one_one_nat),one_one_nat)),M_11) ).
tff(fact_415_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_11: int] : hAPP_int_int(plus_plus_int(M_11),M_11) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(one_one_int),one_one_int)),M_11) ).
tff(fact_416_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_10: real,A_81: real] : hAPP_real_real(plus_plus_real(M_10),hAPP_real_real(times_times_real(A_81),M_10)) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_81),one_one_real)),M_10) ).
tff(fact_417_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_10: nat,A_81: nat] : hAPP_nat_nat(plus_plus_nat(M_10),hAPP_nat_nat(times_times_nat(A_81),M_10)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_81),one_one_nat)),M_10) ).
tff(fact_418_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_10: int,A_81: int] : hAPP_int_int(plus_plus_int(M_10),hAPP_int_int(times_times_int(A_81),M_10)) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_81),one_one_int)),M_10) ).
tff(fact_419_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_80: real,M_9: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_80),M_9)),M_9) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_80),one_one_real)),M_9) ).
tff(fact_420_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_80: nat,M_9: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_80),M_9)),M_9) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_80),one_one_nat)),M_9) ).
tff(fact_421_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_80: int,M_9: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_80),M_9)),M_9) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_80),one_one_int)),M_9) ).
tff(fact_422_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_423_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_424_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_425_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_426_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_427_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y: 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))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(X_1),Y))) ) ) ).
tff(fact_428_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y: 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))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(X_1),Y))) ) ) ).
tff(fact_429_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_430_power__0__left__number__of,axiom,
! [W_2: int] :
( ( ( number_number_of_nat(W_2) = zero_zero_nat )
=> ( hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(W_2)) = one_one_real ) )
& ( ( number_number_of_nat(W_2) != zero_zero_nat )
=> ( hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(W_2)) = zero_zero_real ) ) ) ).
tff(fact_431_power__0__left__number__of,axiom,
! [W_2: int] :
( ( ( number_number_of_nat(W_2) = zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(W_2)) = one_one_nat ) )
& ( ( number_number_of_nat(W_2) != zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(W_2)) = zero_zero_nat ) ) ) ).
tff(fact_432_power__0__left__number__of,axiom,
! [W_2: int] :
( ( ( number_number_of_nat(W_2) = zero_zero_nat )
=> ( hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(W_2)) = one_one_int ) )
& ( ( number_number_of_nat(W_2) != zero_zero_nat )
=> ( hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(W_2)) = zero_zero_int ) ) ) ).
tff(fact_433_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_434_q__pos__lemma,axiom,
! [B_59: 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_59),Q_4)),R_3)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_59))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_59))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Q_4)) ) ) ) ).
tff(fact_435_q__neg__lemma,axiom,
! [B_59: 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_59),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_59))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_4),zero_zero_int)) ) ) ) ).
tff(fact_436_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_437_zdiv__mono2__lemma,axiom,
! [B: int,Q_1: int,R_1: int,B_59: 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_59),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_59),Q_4)),R_3)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_59))
=> ( 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_59))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_59),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q_4)) ) ) ) ) ) ) ).
tff(fact_438_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_439_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q_1: int,R_1: int,B_59: 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_59),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_59),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_59))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_59),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_4),Q_1)) ) ) ) ) ) ) ).
tff(fact_440_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_441_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_442_Euler_Oaux____1,axiom,
! [Y: int,X_1: int,P_3: int] :
( ~ hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_3))
=> ( hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(Y),number_number_of_nat(bit0(bit1(pls)))),X_1),P_3))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y)) ) ) ).
tff(fact_443_zprime__def,axiom,
! [P_2: int] :
( hBOOL(hAPP_int_bool(zprime,P_2))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),P_2))
& ! [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_2)) )
=> ( ( M_2 = one_one_int )
| ( M_2 = P_2 ) ) ) ) ) ).
tff(fact_444_prime__g__5,axiom,
! [P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ( P_3 != number_number_of_int(bit0(bit1(pls))) )
=> ( ( P_3 != 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_3)) ) ) ) ).
tff(fact_445_pos2,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_446_power__strict__mono,axiom,
! [N_21: nat,A_79: real,B_58: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_79),B_58))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_79))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_21))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_79),N_21)),hAPP_nat_real(power_power_real(B_58),N_21))) ) ) ) ).
tff(fact_447_power__strict__mono,axiom,
! [N_21: nat,A_79: nat,B_58: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_79),B_58))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_79))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_21))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_79),N_21)),hAPP_nat_nat(power_power_nat(B_58),N_21))) ) ) ) ).
tff(fact_448_power__strict__mono,axiom,
! [N_21: nat,A_79: int,B_58: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_79),B_58))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_79))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_21))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_79),N_21)),hAPP_nat_int(power_power_int(B_58),N_21))) ) ) ) ).
tff(fact_449_convex__bound__lt,axiom,
! [V_1: real,U_2: real,Y_9: real,X_9: real,A_78: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_9),A_78))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_9),A_78))
=> ( 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_9)),hAPP_real_real(times_times_real(V_1),Y_9))),A_78)) ) ) ) ) ) ).
tff(fact_450_convex__bound__lt,axiom,
! [V_1: int,U_2: int,Y_9: int,X_9: int,A_78: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_9),A_78))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_9),A_78))
=> ( 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_9)),hAPP_int_int(times_times_int(V_1),Y_9))),A_78)) ) ) ) ) ) ).
tff(fact_451_zpower__zdvd__prop2,axiom,
! [Y: int,N_1: nat,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(power_power_int(Y),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_3),Y)) ) ) ) ).
tff(fact_452_dvd__0__right,axiom,
! [A_77: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_77),zero_zero_nat)) ).
tff(fact_453_dvd__0__right,axiom,
! [A_77: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_77),zero_zero_int)) ).
tff(fact_454_dvd__0__right,axiom,
! [A_77: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_77),zero_zero_real)) ).
tff(fact_455__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_456__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_457_rel__simps_I47_J,axiom,
! [K: int] :
( ( bit1(K) = min )
<=> ( K = min ) ) ).
tff(fact_458_rel__simps_I43_J,axiom,
! [L_1: int] :
( ( min = bit1(L_1) )
<=> ( min = L_1 ) ) ).
tff(fact_459_Bit1__Min,axiom,
bit1(min) = min ).
tff(fact_460_rel__simps_I37_J,axiom,
pls != min ).
tff(fact_461_rel__simps_I40_J,axiom,
min != pls ).
tff(fact_462_rel__simps_I45_J,axiom,
! [K_1: int] : bit0(K_1) != min ).
tff(fact_463_rel__simps_I42_J,axiom,
! [L: int] : min != bit0(L) ).
tff(fact_464_rel__simps_I7_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),min)) ).
tff(fact_465_rel__simps_I24_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),min)) ).
tff(fact_466_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_467_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_468_rel__simps_I3_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),min)) ).
tff(fact_469_rel__simps_I6_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),pls)) ).
tff(fact_470_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_471_bin__less__0__simps_I2_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),zero_zero_int)) ).
tff(fact_472_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_473_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_474_rel__simps_I23_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),pls)) ).
tff(fact_475_rel__simps_I20_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),min)) ).
tff(fact_476_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_477_eq__number__of__Pls__Min,axiom,
number_number_of_int(pls) != number_number_of_int(min) ).
tff(fact_478_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_479_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_480_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_481_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_482_linorder__neqE__linordered__idom,axiom,
! [X_8: real,Y_8: real] :
( ( X_8 != Y_8 )
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_8),Y_8))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_8),X_8)) ) ) ).
tff(fact_483_linorder__neqE__linordered__idom,axiom,
! [X_8: int,Y_8: int] :
( ( X_8 != Y_8 )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_8),Y_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_8),X_8)) ) ) ).
tff(fact_484_dvd__refl,axiom,
! [A_76: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_76),A_76)) ).
tff(fact_485_dvd__refl,axiom,
! [A_76: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_76),A_76)) ).
tff(fact_486_dvd__trans,axiom,
! [C_33: nat,A_75: nat,B_57: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_75),B_57))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_57),C_33))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_75),C_33)) ) ) ).
tff(fact_487_dvd__trans,axiom,
! [C_33: int,A_75: int,B_57: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_75),B_57))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_57),C_33))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_75),C_33)) ) ) ).
tff(fact_488_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_489_zcong__refl,axiom,
! [K_1: int,M: int] : hBOOL(hAPP_int_bool(zcong(K_1,K_1),M)) ).
tff(fact_490_zcong__eq__trans,axiom,
! [D_3: 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_3),M))
=> hBOOL(hAPP_int_bool(zcong(A,D_3),M)) ) ) ) ).
tff(fact_491_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_492_zcong__neg__1__impl__ne__1,axiom,
! [X_1: int,P_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zcong(X_1,number_number_of_int(min)),P_3))
=> ~ hBOOL(hAPP_int_bool(zcong(X_1,one_one_int),P_3)) ) ) ).
tff(fact_493_order__le__neq__implies__less,axiom,
! [X_7: real,Y_7: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_7),Y_7))
=> ( ( X_7 != Y_7 )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_7),Y_7)) ) ) ).
tff(fact_494_order__le__neq__implies__less,axiom,
! [X_7: nat,Y_7: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_7),Y_7))
=> ( ( X_7 != Y_7 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_7),Y_7)) ) ) ).
tff(fact_495_order__le__neq__implies__less,axiom,
! [X_7: int,Y_7: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_7),Y_7))
=> ( ( X_7 != Y_7 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_7),Y_7)) ) ) ).
tff(fact_496_divisors__zero,axiom,
! [A_74: real,B_56: real] :
( ( hAPP_real_real(times_times_real(A_74),B_56) = zero_zero_real )
=> ( ( A_74 = zero_zero_real )
| ( B_56 = zero_zero_real ) ) ) ).
tff(fact_497_divisors__zero,axiom,
! [A_74: nat,B_56: nat] :
( ( hAPP_nat_nat(times_times_nat(A_74),B_56) = zero_zero_nat )
=> ( ( A_74 = zero_zero_nat )
| ( B_56 = zero_zero_nat ) ) ) ).
tff(fact_498_divisors__zero,axiom,
! [A_74: int,B_56: int] :
( ( hAPP_int_int(times_times_int(A_74),B_56) = zero_zero_int )
=> ( ( A_74 = zero_zero_int )
| ( B_56 = zero_zero_int ) ) ) ).
tff(fact_499_no__zero__divisors,axiom,
! [B_55: real,A_73: real] :
( ( A_73 != zero_zero_real )
=> ( ( B_55 != zero_zero_real )
=> ( hAPP_real_real(times_times_real(A_73),B_55) != zero_zero_real ) ) ) ).
tff(fact_500_no__zero__divisors,axiom,
! [B_55: nat,A_73: nat] :
( ( A_73 != zero_zero_nat )
=> ( ( B_55 != zero_zero_nat )
=> ( hAPP_nat_nat(times_times_nat(A_73),B_55) != zero_zero_nat ) ) ) ).
tff(fact_501_no__zero__divisors,axiom,
! [B_55: int,A_73: int] :
( ( A_73 != zero_zero_int )
=> ( ( B_55 != zero_zero_int )
=> ( hAPP_int_int(times_times_int(A_73),B_55) != zero_zero_int ) ) ) ).
tff(fact_502_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_503_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_504_mult__zero__right,axiom,
! [A_72: real] : hAPP_real_real(times_times_real(A_72),zero_zero_real) = zero_zero_real ).
tff(fact_505_mult__zero__right,axiom,
! [A_72: nat] : hAPP_nat_nat(times_times_nat(A_72),zero_zero_nat) = zero_zero_nat ).
tff(fact_506_mult__zero__right,axiom,
! [A_72: int] : hAPP_int_int(times_times_int(A_72),zero_zero_int) = zero_zero_int ).
tff(fact_507_mult__zero__left,axiom,
! [A_71: real] : hAPP_real_real(times_times_real(zero_zero_real),A_71) = zero_zero_real ).
tff(fact_508_mult__zero__left,axiom,
! [A_71: nat] : hAPP_nat_nat(times_times_nat(zero_zero_nat),A_71) = zero_zero_nat ).
tff(fact_509_mult__zero__left,axiom,
! [A_71: int] : hAPP_int_int(times_times_int(zero_zero_int),A_71) = zero_zero_int ).
tff(fact_510_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
tff(fact_511_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
tff(fact_512_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
tff(fact_513_one__neq__zero,axiom,
one_one_real != zero_zero_real ).
tff(fact_514_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
tff(fact_515_one__neq__zero,axiom,
one_one_int != zero_zero_int ).
tff(fact_516_combine__common__factor,axiom,
! [A_70: real,E: real,B_54: real,C_32: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_70),E)),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(B_54),E)),C_32)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_70),B_54)),E)),C_32) ).
tff(fact_517_combine__common__factor,axiom,
! [A_70: nat,E: nat,B_54: nat,C_32: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_70),E)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(B_54),E)),C_32)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_70),B_54)),E)),C_32) ).
tff(fact_518_combine__common__factor,axiom,
! [A_70: int,E: int,B_54: int,C_32: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_70),E)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_54),E)),C_32)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_70),B_54)),E)),C_32) ).
tff(fact_519_comm__semiring__class_Odistrib,axiom,
! [A_69: real,B_53: real,C_31: real] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_69),B_53)),C_31) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_69),C_31)),hAPP_real_real(times_times_real(B_53),C_31)) ).
tff(fact_520_comm__semiring__class_Odistrib,axiom,
! [A_69: nat,B_53: nat,C_31: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_69),B_53)),C_31) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_69),C_31)),hAPP_nat_nat(times_times_nat(B_53),C_31)) ).
tff(fact_521_comm__semiring__class_Odistrib,axiom,
! [A_69: int,B_53: int,C_31: int] : hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_69),B_53)),C_31) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_69),C_31)),hAPP_int_int(times_times_int(B_53),C_31)) ).
tff(fact_522_field__power__not__zero,axiom,
! [N_20: nat,A_68: real] :
( ( A_68 != zero_zero_real )
=> ( hAPP_nat_real(power_power_real(A_68),N_20) != zero_zero_real ) ) ).
tff(fact_523_field__power__not__zero,axiom,
! [N_20: nat,A_68: int] :
( ( A_68 != zero_zero_int )
=> ( hAPP_nat_int(power_power_int(A_68),N_20) != zero_zero_int ) ) ).
tff(fact_524_dvd__0__left,axiom,
! [A_67: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_67))
=> ( A_67 = zero_zero_nat ) ) ).
tff(fact_525_dvd__0__left,axiom,
! [A_67: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,zero_zero_int),A_67))
=> ( A_67 = zero_zero_int ) ) ).
tff(fact_526_dvd__0__left,axiom,
! [A_67: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,zero_zero_real),A_67))
=> ( A_67 = zero_zero_real ) ) ).
tff(fact_527_power__mult__distrib,axiom,
! [A_66: real,B_52: real,N_19: nat] : hAPP_nat_real(power_power_real(hAPP_real_real(times_times_real(A_66),B_52)),N_19) = hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(A_66),N_19)),hAPP_nat_real(power_power_real(B_52),N_19)) ).
tff(fact_528_power__mult__distrib,axiom,
! [A_66: nat,B_52: nat,N_19: nat] : hAPP_nat_nat(power_power_nat(hAPP_nat_nat(times_times_nat(A_66),B_52)),N_19) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(A_66),N_19)),hAPP_nat_nat(power_power_nat(B_52),N_19)) ).
tff(fact_529_power__mult__distrib,axiom,
! [A_66: int,B_52: int,N_19: nat] : hAPP_nat_int(power_power_int(hAPP_int_int(times_times_int(A_66),B_52)),N_19) = hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(A_66),N_19)),hAPP_nat_int(power_power_int(B_52),N_19)) ).
tff(fact_530_power__commutes,axiom,
! [A_65: real,N_18: nat] : hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(A_65),N_18)),A_65) = hAPP_real_real(times_times_real(A_65),hAPP_nat_real(power_power_real(A_65),N_18)) ).
tff(fact_531_power__commutes,axiom,
! [A_65: nat,N_18: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(A_65),N_18)),A_65) = hAPP_nat_nat(times_times_nat(A_65),hAPP_nat_nat(power_power_nat(A_65),N_18)) ).
tff(fact_532_power__commutes,axiom,
! [A_65: int,N_18: nat] : hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(A_65),N_18)),A_65) = hAPP_int_int(times_times_int(A_65),hAPP_nat_int(power_power_int(A_65),N_18)) ).
tff(fact_533_dvd__triv__left,axiom,
! [A_64: nat,B_51: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_64),hAPP_nat_nat(times_times_nat(A_64),B_51))) ).
tff(fact_534_dvd__triv__left,axiom,
! [A_64: int,B_51: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_64),hAPP_int_int(times_times_int(A_64),B_51))) ).
tff(fact_535_dvd__triv__left,axiom,
! [A_64: real,B_51: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_64),hAPP_real_real(times_times_real(A_64),B_51))) ).
tff(fact_536_dvd__triv__right,axiom,
! [A_63: nat,B_50: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_63),hAPP_nat_nat(times_times_nat(B_50),A_63))) ).
tff(fact_537_dvd__triv__right,axiom,
! [A_63: int,B_50: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_63),hAPP_int_int(times_times_int(B_50),A_63))) ).
tff(fact_538_dvd__triv__right,axiom,
! [A_63: real,B_50: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_63),hAPP_real_real(times_times_real(B_50),A_63))) ).
tff(fact_539_dvd__mult2,axiom,
! [C_30: nat,A_62: nat,B_49: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_62),B_49))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_62),hAPP_nat_nat(times_times_nat(B_49),C_30))) ) ).
tff(fact_540_dvd__mult2,axiom,
! [C_30: int,A_62: int,B_49: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_62),B_49))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_62),hAPP_int_int(times_times_int(B_49),C_30))) ) ).
tff(fact_541_dvd__mult2,axiom,
! [C_30: real,A_62: real,B_49: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_62),B_49))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_62),hAPP_real_real(times_times_real(B_49),C_30))) ) ).
tff(fact_542_dvd__mult,axiom,
! [B_48: nat,A_61: nat,C_29: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_61),C_29))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_61),hAPP_nat_nat(times_times_nat(B_48),C_29))) ) ).
tff(fact_543_dvd__mult,axiom,
! [B_48: int,A_61: int,C_29: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_61),C_29))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_61),hAPP_int_int(times_times_int(B_48),C_29))) ) ).
tff(fact_544_dvd__mult,axiom,
! [B_48: real,A_61: real,C_29: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_61),C_29))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_61),hAPP_real_real(times_times_real(B_48),C_29))) ) ).
tff(fact_545_mult__dvd__mono,axiom,
! [C_28: nat,D_10: nat,A_60: nat,B_47: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_60),B_47))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_28),D_10))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(A_60),C_28)),hAPP_nat_nat(times_times_nat(B_47),D_10))) ) ) ).
tff(fact_546_mult__dvd__mono,axiom,
! [C_28: int,D_10: int,A_60: int,B_47: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_60),B_47))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C_28),D_10))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(times_times_int(A_60),C_28)),hAPP_int_int(times_times_int(B_47),D_10))) ) ) ).
tff(fact_547_mult__dvd__mono,axiom,
! [C_28: real,D_10: real,A_60: real,B_47: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_60),B_47))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,C_28),D_10))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(times_times_real(A_60),C_28)),hAPP_real_real(times_times_real(B_47),D_10))) ) ) ).
tff(fact_548_dvdI,axiom,
! [A_59: nat,B_46: nat,K_3: nat] :
( ( A_59 = hAPP_nat_nat(times_times_nat(B_46),K_3) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_46),A_59)) ) ).
tff(fact_549_dvdI,axiom,
! [A_59: int,B_46: int,K_3: int] :
( ( A_59 = hAPP_int_int(times_times_int(B_46),K_3) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_46),A_59)) ) ).
tff(fact_550_dvdI,axiom,
! [A_59: real,B_46: real,K_3: real] :
( ( A_59 = hAPP_real_real(times_times_real(B_46),K_3) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_46),A_59)) ) ).
tff(fact_551_dvd__mult__left,axiom,
! [A_58: nat,B_45: nat,C_27: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(A_58),B_45)),C_27))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_58),C_27)) ) ).
tff(fact_552_dvd__mult__left,axiom,
! [A_58: int,B_45: int,C_27: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(times_times_int(A_58),B_45)),C_27))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_58),C_27)) ) ).
tff(fact_553_dvd__mult__left,axiom,
! [A_58: real,B_45: real,C_27: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(times_times_real(A_58),B_45)),C_27))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_58),C_27)) ) ).
tff(fact_554_dvd__mult__right,axiom,
! [A_57: nat,B_44: nat,C_26: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(A_57),B_44)),C_26))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_44),C_26)) ) ).
tff(fact_555_dvd__mult__right,axiom,
! [A_57: int,B_44: int,C_26: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(times_times_int(A_57),B_44)),C_26))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_44),C_26)) ) ).
tff(fact_556_dvd__mult__right,axiom,
! [A_57: real,B_44: real,C_26: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(times_times_real(A_57),B_44)),C_26))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_44),C_26)) ) ).
tff(fact_557_dvd__add,axiom,
! [C_25: nat,A_56: nat,B_43: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_56),B_43))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_56),C_25))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_56),hAPP_nat_nat(plus_plus_nat(B_43),C_25))) ) ) ).
tff(fact_558_dvd__add,axiom,
! [C_25: int,A_56: int,B_43: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_56),B_43))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_56),C_25))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_56),hAPP_int_int(plus_plus_int(B_43),C_25))) ) ) ).
tff(fact_559_dvd__add,axiom,
! [C_25: real,A_56: real,B_43: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_56),B_43))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_56),C_25))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_56),hAPP_real_real(plus_plus_real(B_43),C_25))) ) ) ).
tff(fact_560_power__one,axiom,
! [N_17: nat] : hAPP_nat_real(power_power_real(one_one_real),N_17) = one_one_real ).
tff(fact_561_power__one,axiom,
! [N_17: nat] : hAPP_nat_nat(power_power_nat(one_one_nat),N_17) = one_one_nat ).
tff(fact_562_power__one,axiom,
! [N_17: nat] : hAPP_nat_int(power_power_int(one_one_int),N_17) = one_one_int ).
tff(fact_563_one__dvd,axiom,
! [A_55: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_55)) ).
tff(fact_564_one__dvd,axiom,
! [A_55: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,one_one_int),A_55)) ).
tff(fact_565_one__dvd,axiom,
! [A_55: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,one_one_real),A_55)) ).
tff(fact_566_dvd__power__same,axiom,
! [N_16: nat,X_6: nat,Y_6: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_6),Y_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(X_6),N_16)),hAPP_nat_nat(power_power_nat(Y_6),N_16))) ) ).
tff(fact_567_dvd__power__same,axiom,
! [N_16: nat,X_6: int,Y_6: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_6),Y_6))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(X_6),N_16)),hAPP_nat_int(power_power_int(Y_6),N_16))) ) ).
tff(fact_568_dvd__power__same,axiom,
! [N_16: nat,X_6: real,Y_6: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_6),Y_6))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(power_power_real(X_6),N_16)),hAPP_nat_real(power_power_real(Y_6),N_16))) ) ).
tff(fact_569_power__le__dvd,axiom,
! [M_8: nat,A_54: nat,N_15: nat,B_42: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(A_54),N_15)),B_42))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_8),N_15))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(A_54),M_8)),B_42)) ) ) ).
tff(fact_570_power__le__dvd,axiom,
! [M_8: nat,A_54: int,N_15: nat,B_42: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(A_54),N_15)),B_42))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_8),N_15))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(A_54),M_8)),B_42)) ) ) ).
tff(fact_571_power__le__dvd,axiom,
! [M_8: nat,A_54: real,N_15: nat,B_42: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(power_power_real(A_54),N_15)),B_42))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_8),N_15))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(power_power_real(A_54),M_8)),B_42)) ) ) ).
tff(fact_572_dvd__power__le,axiom,
! [N_14: nat,M_7: nat,X_5: nat,Y_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_5),Y_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_14),M_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(X_5),N_14)),hAPP_nat_nat(power_power_nat(Y_5),M_7))) ) ) ).
tff(fact_573_dvd__power__le,axiom,
! [N_14: nat,M_7: nat,X_5: int,Y_5: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_5),Y_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_14),M_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(X_5),N_14)),hAPP_nat_int(power_power_int(Y_5),M_7))) ) ) ).
tff(fact_574_dvd__power__le,axiom,
! [N_14: nat,M_7: nat,X_5: real,Y_5: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_5),Y_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_14),M_7))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(power_power_real(X_5),N_14)),hAPP_nat_real(power_power_real(Y_5),M_7))) ) ) ).
tff(fact_575_le__imp__power__dvd,axiom,
! [A_53: nat,M_6: nat,N_13: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_6),N_13))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(A_53),M_6)),hAPP_nat_nat(power_power_nat(A_53),N_13))) ) ).
tff(fact_576_le__imp__power__dvd,axiom,
! [A_53: int,M_6: nat,N_13: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_6),N_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(A_53),M_6)),hAPP_nat_int(power_power_int(A_53),N_13))) ) ).
tff(fact_577_le__imp__power__dvd,axiom,
! [A_53: real,M_6: nat,N_13: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_6),N_13))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(power_power_real(A_53),M_6)),hAPP_nat_real(power_power_real(A_53),N_13))) ) ).
tff(fact_578_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_579_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(X),N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X))
| ( N = zero_zero_nat ) ) ) ).
tff(fact_580_power__mult,axiom,
! [A_52: real,M_5: nat,N_12: nat] : hAPP_nat_real(power_power_real(A_52),hAPP_nat_nat(times_times_nat(M_5),N_12)) = hAPP_nat_real(power_power_real(hAPP_nat_real(power_power_real(A_52),M_5)),N_12) ).
tff(fact_581_power__mult,axiom,
! [A_52: nat,M_5: nat,N_12: nat] : hAPP_nat_nat(power_power_nat(A_52),hAPP_nat_nat(times_times_nat(M_5),N_12)) = hAPP_nat_nat(power_power_nat(hAPP_nat_nat(power_power_nat(A_52),M_5)),N_12) ).
tff(fact_582_power__mult,axiom,
! [A_52: int,M_5: nat,N_12: nat] : hAPP_nat_int(power_power_int(A_52),hAPP_nat_nat(times_times_nat(M_5),N_12)) = hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(A_52),M_5)),N_12) ).
tff(fact_583_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_584_power__one__right,axiom,
! [A_51: real] : hAPP_nat_real(power_power_real(A_51),one_one_nat) = A_51 ).
tff(fact_585_power__one__right,axiom,
! [A_51: nat] : hAPP_nat_nat(power_power_nat(A_51),one_one_nat) = A_51 ).
tff(fact_586_power__one__right,axiom,
! [A_51: int] : hAPP_nat_int(power_power_int(A_51),one_one_nat) = A_51 ).
tff(fact_587_zcong__id,axiom,
! [M: int] : hBOOL(hAPP_int_bool(zcong(M,zero_zero_int),M)) ).
tff(fact_588_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_589_zcong__1,axiom,
! [A: int,B: int] : hBOOL(hAPP_int_bool(zcong(A,B),one_one_int)) ).
tff(fact_590_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_591_zcong__zmult__prop1,axiom,
! [C_1: 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_1,hAPP_int_int(times_times_int(A_2),D)),Ma))
<=> hBOOL(hAPP_int_bool(zcong(C_1,hAPP_int_int(times_times_int(B_2),D)),Ma)) ) ) ).
tff(fact_592_zcong__zmult__prop2,axiom,
! [C_1: 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_1,hAPP_int_int(times_times_int(D),A_2)),Ma))
<=> hBOOL(hAPP_int_bool(zcong(C_1,hAPP_int_int(times_times_int(D),B_2)),Ma)) ) ) ).
tff(fact_593_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_594_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_595_zcong__zmult,axiom,
! [C: int,D_3: int,A: int,B: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(A,B),M))
=> ( hBOOL(hAPP_int_bool(zcong(C,D_3),M))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),C),hAPP_int_int(times_times_int(B),D_3)),M)) ) ) ).
tff(fact_596_zcong__zadd,axiom,
! [C: int,D_3: int,A: int,B: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(A,B),M))
=> ( hBOOL(hAPP_int_bool(zcong(C,D_3),M))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(plus_plus_int(A),C),hAPP_int_int(plus_plus_int(B),D_3)),M)) ) ) ).
tff(fact_597_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_598_zcong__zpower,axiom,
! [Z: nat,X_1: int,Y: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(X_1,Y),M))
=> hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X_1),Z),hAPP_nat_int(power_power_int(Y),Z)),M)) ) ).
tff(fact_599_power__m1__even,axiom,
! [N_11: nat] : hAPP_nat_real(power_power_real(number267125858f_real(min)),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_11)) = one_one_real ).
tff(fact_600_power__m1__even,axiom,
! [N_11: 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_11)) = one_one_int ).
tff(fact_601_split__mult__neg__le,axiom,
! [B_41: real,A_50: real] :
( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_50))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_41),zero_zero_real)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_50),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_41)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_50),B_41)),zero_zero_real)) ) ).
tff(fact_602_split__mult__neg__le,axiom,
! [B_41: nat,A_50: nat] :
( ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_50))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_41),zero_zero_nat)) )
| ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_50),zero_zero_nat))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_41)) ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_50),B_41)),zero_zero_nat)) ) ).
tff(fact_603_split__mult__neg__le,axiom,
! [B_41: int,A_50: int] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_50))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_41),zero_zero_int)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_50),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_41)) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_50),B_41)),zero_zero_int)) ) ).
tff(fact_604_split__mult__pos__le,axiom,
! [B_40: real,A_49: real] :
( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_49))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_40)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_49),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_40),zero_zero_real)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_49),B_40))) ) ).
tff(fact_605_split__mult__pos__le,axiom,
! [B_40: int,A_49: int] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_49))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_40)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_49),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_40),zero_zero_int)) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_49),B_40))) ) ).
tff(fact_606_mult__mono,axiom,
! [C_24: real,D_9: real,A_48: real,B_39: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_48),B_39))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_24),D_9))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_39))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_24))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_48),C_24)),hAPP_real_real(times_times_real(B_39),D_9))) ) ) ) ) ).
tff(fact_607_mult__mono,axiom,
! [C_24: nat,D_9: nat,A_48: nat,B_39: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_48),B_39))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_24),D_9))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_39))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_24))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_48),C_24)),hAPP_nat_nat(times_times_nat(B_39),D_9))) ) ) ) ) ).
tff(fact_608_mult__mono,axiom,
! [C_24: int,D_9: int,A_48: int,B_39: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_48),B_39))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_24),D_9))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_39))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_24))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_48),C_24)),hAPP_int_int(times_times_int(B_39),D_9))) ) ) ) ) ).
tff(fact_609_mult__mono_H,axiom,
! [C_23: real,D_8: real,A_47: real,B_38: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_47),B_38))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_23),D_8))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_47))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_23))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_47),C_23)),hAPP_real_real(times_times_real(B_38),D_8))) ) ) ) ) ).
tff(fact_610_mult__mono_H,axiom,
! [C_23: nat,D_8: nat,A_47: nat,B_38: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_47),B_38))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_23),D_8))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_47))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_23))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_47),C_23)),hAPP_nat_nat(times_times_nat(B_38),D_8))) ) ) ) ) ).
tff(fact_611_mult__mono_H,axiom,
! [C_23: int,D_8: int,A_47: int,B_38: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_47),B_38))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_23),D_8))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_47))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_23))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_47),C_23)),hAPP_int_int(times_times_int(B_38),D_8))) ) ) ) ) ).
tff(fact_612_mult__left__mono__neg,axiom,
! [C_22: real,B_37: real,A_46: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_37),A_46))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_22),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(C_22),A_46)),hAPP_real_real(times_times_real(C_22),B_37))) ) ) ).
tff(fact_613_mult__left__mono__neg,axiom,
! [C_22: int,B_37: int,A_46: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_37),A_46))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_22),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(C_22),A_46)),hAPP_int_int(times_times_int(C_22),B_37))) ) ) ).
tff(fact_614_mult__right__mono__neg,axiom,
! [C_21: real,B_36: real,A_45: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_36),A_45))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_21),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_45),C_21)),hAPP_real_real(times_times_real(B_36),C_21))) ) ) ).
tff(fact_615_mult__right__mono__neg,axiom,
! [C_21: int,B_36: int,A_45: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_36),A_45))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_21),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_45),C_21)),hAPP_int_int(times_times_int(B_36),C_21))) ) ) ).
tff(fact_616_comm__mult__left__mono,axiom,
! [C_20: real,A_44: real,B_35: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_44),B_35))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_20))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(C_20),A_44)),hAPP_real_real(times_times_real(C_20),B_35))) ) ) ).
tff(fact_617_comm__mult__left__mono,axiom,
! [C_20: nat,A_44: nat,B_35: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_44),B_35))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_20))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(C_20),A_44)),hAPP_nat_nat(times_times_nat(C_20),B_35))) ) ) ).
tff(fact_618_comm__mult__left__mono,axiom,
! [C_20: int,A_44: int,B_35: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_44),B_35))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_20))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(C_20),A_44)),hAPP_int_int(times_times_int(C_20),B_35))) ) ) ).
tff(fact_619_mult__left__mono,axiom,
! [C_19: real,A_43: real,B_34: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_43),B_34))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_19))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(C_19),A_43)),hAPP_real_real(times_times_real(C_19),B_34))) ) ) ).
tff(fact_620_mult__left__mono,axiom,
! [C_19: nat,A_43: nat,B_34: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_43),B_34))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_19))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(C_19),A_43)),hAPP_nat_nat(times_times_nat(C_19),B_34))) ) ) ).
tff(fact_621_mult__left__mono,axiom,
! [C_19: int,A_43: int,B_34: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_43),B_34))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_19))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(C_19),A_43)),hAPP_int_int(times_times_int(C_19),B_34))) ) ) ).
tff(fact_622_mult__right__mono,axiom,
! [C_18: real,A_42: real,B_33: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_42),B_33))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_18))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_42),C_18)),hAPP_real_real(times_times_real(B_33),C_18))) ) ) ).
tff(fact_623_mult__right__mono,axiom,
! [C_18: nat,A_42: nat,B_33: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_42),B_33))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_18))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_42),C_18)),hAPP_nat_nat(times_times_nat(B_33),C_18))) ) ) ).
tff(fact_624_mult__right__mono,axiom,
! [C_18: int,A_42: int,B_33: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_42),B_33))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_18))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_42),C_18)),hAPP_int_int(times_times_int(B_33),C_18))) ) ) ).
tff(fact_625_mult__nonpos__nonpos,axiom,
! [B_32: real,A_41: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_41),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_32),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_41),B_32))) ) ) ).
tff(fact_626_mult__nonpos__nonpos,axiom,
! [B_32: int,A_41: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_41),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_32),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_41),B_32))) ) ) ).
tff(fact_627_mult__nonpos__nonneg,axiom,
! [B_31: real,A_40: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_40),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_31))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_40),B_31)),zero_zero_real)) ) ) ).
tff(fact_628_mult__nonpos__nonneg,axiom,
! [B_31: nat,A_40: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_40),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_31))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_40),B_31)),zero_zero_nat)) ) ) ).
tff(fact_629_mult__nonpos__nonneg,axiom,
! [B_31: int,A_40: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_40),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_31))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_40),B_31)),zero_zero_int)) ) ) ).
tff(fact_630_mult__nonneg__nonpos2,axiom,
! [B_30: real,A_39: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_39))
=> ( 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,hAPP_real_real(times_times_real(B_30),A_39)),zero_zero_real)) ) ) ).
tff(fact_631_mult__nonneg__nonpos2,axiom,
! [B_30: nat,A_39: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_39))
=> ( 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,hAPP_nat_nat(times_times_nat(B_30),A_39)),zero_zero_nat)) ) ) ).
tff(fact_632_mult__nonneg__nonpos2,axiom,
! [B_30: int,A_39: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_39))
=> ( 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,hAPP_int_int(times_times_int(B_30),A_39)),zero_zero_int)) ) ) ).
tff(fact_633_mult__nonneg__nonpos,axiom,
! [B_29: real,A_38: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_38))
=> ( 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,hAPP_real_real(times_times_real(A_38),B_29)),zero_zero_real)) ) ) ).
tff(fact_634_mult__nonneg__nonpos,axiom,
! [B_29: nat,A_38: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_38))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_29),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_38),B_29)),zero_zero_nat)) ) ) ).
tff(fact_635_mult__nonneg__nonpos,axiom,
! [B_29: int,A_38: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_38))
=> ( 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,hAPP_int_int(times_times_int(A_38),B_29)),zero_zero_int)) ) ) ).
tff(fact_636_mult__nonneg__nonneg,axiom,
! [B_28: real,A_37: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_37))
=> ( 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),hAPP_real_real(times_times_real(A_37),B_28))) ) ) ).
tff(fact_637_mult__nonneg__nonneg,axiom,
! [B_28: nat,A_37: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_37))
=> ( 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),hAPP_nat_nat(times_times_nat(A_37),B_28))) ) ) ).
tff(fact_638_mult__nonneg__nonneg,axiom,
! [B_28: int,A_37: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_37))
=> ( 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),hAPP_int_int(times_times_int(A_37),B_28))) ) ) ).
tff(fact_639_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_640_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_641_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_642_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_643_zero__le__square,axiom,
! [A_36: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_36),A_36))) ).
tff(fact_644_zero__le__square,axiom,
! [A_36: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_36),A_36))) ).
tff(fact_645_mult__strict__left__mono__neg,axiom,
! [C_17: real,B_27: real,A_35: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_27),A_35))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_17),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_17),A_35)),hAPP_real_real(times_times_real(C_17),B_27))) ) ) ).
tff(fact_646_mult__strict__left__mono__neg,axiom,
! [C_17: int,B_27: int,A_35: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_27),A_35))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_17),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_17),A_35)),hAPP_int_int(times_times_int(C_17),B_27))) ) ) ).
tff(fact_647_mult__strict__right__mono__neg,axiom,
! [C_16: real,B_26: real,A_34: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_26),A_34))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_16),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_34),C_16)),hAPP_real_real(times_times_real(B_26),C_16))) ) ) ).
tff(fact_648_mult__strict__right__mono__neg,axiom,
! [C_16: int,B_26: int,A_34: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_26),A_34))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_16),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_34),C_16)),hAPP_int_int(times_times_int(B_26),C_16))) ) ) ).
tff(fact_649_comm__mult__strict__left__mono,axiom,
! [C_15: real,A_33: real,B_25: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_33),B_25))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_15))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_15),A_33)),hAPP_real_real(times_times_real(C_15),B_25))) ) ) ).
tff(fact_650_comm__mult__strict__left__mono,axiom,
! [C_15: nat,A_33: nat,B_25: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_33),B_25))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_15))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(C_15),A_33)),hAPP_nat_nat(times_times_nat(C_15),B_25))) ) ) ).
tff(fact_651_comm__mult__strict__left__mono,axiom,
! [C_15: int,A_33: int,B_25: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_33),B_25))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_15))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_15),A_33)),hAPP_int_int(times_times_int(C_15),B_25))) ) ) ).
tff(fact_652_mult__strict__left__mono,axiom,
! [C_14: real,A_32: real,B_24: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_32),B_24))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_14))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_14),A_32)),hAPP_real_real(times_times_real(C_14),B_24))) ) ) ).
tff(fact_653_mult__strict__left__mono,axiom,
! [C_14: nat,A_32: nat,B_24: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_32),B_24))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_14))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(C_14),A_32)),hAPP_nat_nat(times_times_nat(C_14),B_24))) ) ) ).
tff(fact_654_mult__strict__left__mono,axiom,
! [C_14: int,A_32: int,B_24: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_32),B_24))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_14))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_14),A_32)),hAPP_int_int(times_times_int(C_14),B_24))) ) ) ).
tff(fact_655_mult__strict__right__mono,axiom,
! [C_13: real,A_31: real,B_23: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_31),B_23))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_13))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_31),C_13)),hAPP_real_real(times_times_real(B_23),C_13))) ) ) ).
tff(fact_656_mult__strict__right__mono,axiom,
! [C_13: nat,A_31: nat,B_23: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_31),B_23))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_13))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_31),C_13)),hAPP_nat_nat(times_times_nat(B_23),C_13))) ) ) ).
tff(fact_657_mult__strict__right__mono,axiom,
! [C_13: int,A_31: int,B_23: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_31),B_23))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_31),C_13)),hAPP_int_int(times_times_int(B_23),C_13))) ) ) ).
tff(fact_658_mult__neg__neg,axiom,
! [B_22: real,A_30: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_30),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_22),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(A_30),B_22))) ) ) ).
tff(fact_659_mult__neg__neg,axiom,
! [B_22: int,A_30: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_30),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_22),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(A_30),B_22))) ) ) ).
tff(fact_660_mult__neg__pos,axiom,
! [B_21: real,A_29: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_29),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_21))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_29),B_21)),zero_zero_real)) ) ) ).
tff(fact_661_mult__neg__pos,axiom,
! [B_21: nat,A_29: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_29),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_21))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_29),B_21)),zero_zero_nat)) ) ) ).
tff(fact_662_mult__neg__pos,axiom,
! [B_21: int,A_29: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_29),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_21))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_29),B_21)),zero_zero_int)) ) ) ).
tff(fact_663_mult__less__cancel__left__neg,axiom,
! [A_2: real,B_2: real,C_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_1),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_1),A_2)),hAPP_real_real(times_times_real(C_1),B_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_2),A_2)) ) ) ).
tff(fact_664_mult__less__cancel__left__neg,axiom,
! [A_2: int,B_2: int,C_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_1),A_2)),hAPP_int_int(times_times_int(C_1),B_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_2)) ) ) ).
tff(fact_665_zero__less__mult__pos2,axiom,
! [B_20: real,A_28: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(B_20),A_28)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_28))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_20)) ) ) ).
tff(fact_666_zero__less__mult__pos2,axiom,
! [B_20: nat,A_28: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(B_20),A_28)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_28))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_20)) ) ) ).
tff(fact_667_zero__less__mult__pos2,axiom,
! [B_20: int,A_28: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(B_20),A_28)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_28))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_20)) ) ) ).
tff(fact_668_zero__less__mult__pos,axiom,
! [A_27: real,B_19: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(A_27),B_19)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_27))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_19)) ) ) ).
tff(fact_669_zero__less__mult__pos,axiom,
! [A_27: nat,B_19: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(A_27),B_19)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_27))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_19)) ) ) ).
tff(fact_670_zero__less__mult__pos,axiom,
! [A_27: int,B_19: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(A_27),B_19)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_27))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_19)) ) ) ).
tff(fact_671_mult__pos__neg2,axiom,
! [B_18: real,A_26: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_26))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_18),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(B_18),A_26)),zero_zero_real)) ) ) ).
tff(fact_672_mult__pos__neg2,axiom,
! [B_18: nat,A_26: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_26))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_18),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(B_18),A_26)),zero_zero_nat)) ) ) ).
tff(fact_673_mult__pos__neg2,axiom,
! [B_18: int,A_26: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_26))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_18),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(B_18),A_26)),zero_zero_int)) ) ) ).
tff(fact_674_mult__pos__neg,axiom,
! [B_17: real,A_25: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_25))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_17),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_25),B_17)),zero_zero_real)) ) ) ).
tff(fact_675_mult__pos__neg,axiom,
! [B_17: nat,A_25: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_25))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_17),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_25),B_17)),zero_zero_nat)) ) ) ).
tff(fact_676_mult__pos__neg,axiom,
! [B_17: int,A_25: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_25))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_17),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_25),B_17)),zero_zero_int)) ) ) ).
tff(fact_677_mult__pos__pos,axiom,
! [B_16: real,A_24: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_24))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_16))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(A_24),B_16))) ) ) ).
tff(fact_678_mult__pos__pos,axiom,
! [B_16: nat,A_24: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_24))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_16))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(A_24),B_16))) ) ) ).
tff(fact_679_mult__pos__pos,axiom,
! [B_16: int,A_24: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_24))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_16))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(A_24),B_16))) ) ) ).
tff(fact_680_mult__less__cancel__left__pos,axiom,
! [A_2: real,B_2: real,C_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_1),A_2)),hAPP_real_real(times_times_real(C_1),B_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_2),B_2)) ) ) ).
tff(fact_681_mult__less__cancel__left__pos,axiom,
! [A_2: int,B_2: int,C_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_1),A_2)),hAPP_int_int(times_times_int(C_1),B_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_2)) ) ) ).
tff(fact_682_mult__less__cancel__left__disj,axiom,
! [C_1: real,A_2: real,B_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_1),A_2)),hAPP_real_real(times_times_real(C_1),B_2)))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_1))
& 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_1),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_2),A_2)) ) ) ) ).
tff(fact_683_mult__less__cancel__left__disj,axiom,
! [C_1: int,A_2: int,B_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_1),A_2)),hAPP_int_int(times_times_int(C_1),B_2)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
& 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_1),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_2)) ) ) ) ).
tff(fact_684_mult__less__cancel__right__disj,axiom,
! [A_2: real,C_1: real,B_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_2),C_1)),hAPP_real_real(times_times_real(B_2),C_1)))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_1))
& 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_1),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_2),A_2)) ) ) ) ).
tff(fact_685_mult__less__cancel__right__disj,axiom,
! [A_2: int,C_1: int,B_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_2),C_1)),hAPP_int_int(times_times_int(B_2),C_1)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
& 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_1),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_2)) ) ) ) ).
tff(fact_686_not__square__less__zero,axiom,
! [A_23: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_23),A_23)),zero_zero_real)) ).
tff(fact_687_not__square__less__zero,axiom,
! [A_23: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_23),A_23)),zero_zero_int)) ).
tff(fact_688_pos__add__strict,axiom,
! [B_15: real,C_12: real,A_22: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_22))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_15),C_12))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_15),hAPP_real_real(plus_plus_real(A_22),C_12))) ) ) ).
tff(fact_689_pos__add__strict,axiom,
! [B_15: nat,C_12: nat,A_22: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_22))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_15),C_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_15),hAPP_nat_nat(plus_plus_nat(A_22),C_12))) ) ) ).
tff(fact_690_pos__add__strict,axiom,
! [B_15: int,C_12: int,A_22: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_22))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_15),C_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_15),hAPP_int_int(plus_plus_int(A_22),C_12))) ) ) ).
tff(fact_691_zero__le__one,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).
tff(fact_692_zero__le__one,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).
tff(fact_693_zero__le__one,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).
tff(fact_694_not__one__le__zero,axiom,
~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),zero_zero_real)) ).
tff(fact_695_not__one__le__zero,axiom,
~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),zero_zero_nat)) ).
tff(fact_696_not__one__le__zero,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),zero_zero_int)) ).
tff(fact_697_zero__less__one,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),one_one_real)) ).
tff(fact_698_zero__less__one,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).
tff(fact_699_zero__less__one,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
tff(fact_700_not__one__less__zero,axiom,
~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),zero_zero_real)) ).
tff(fact_701_not__one__less__zero,axiom,
~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).
tff(fact_702_not__one__less__zero,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),zero_zero_int)) ).
tff(fact_703_power__mono,axiom,
! [N_10: nat,A_21: real,B_14: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_21),B_14))
=> ( 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,hAPP_nat_real(power_power_real(A_21),N_10)),hAPP_nat_real(power_power_real(B_14),N_10))) ) ) ).
tff(fact_704_power__mono,axiom,
! [N_10: nat,A_21: nat,B_14: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_21),B_14))
=> ( 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,hAPP_nat_nat(power_power_nat(A_21),N_10)),hAPP_nat_nat(power_power_nat(B_14),N_10))) ) ) ).
tff(fact_705_power__mono,axiom,
! [N_10: nat,A_21: int,B_14: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_21),B_14))
=> ( 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,hAPP_nat_int(power_power_int(A_21),N_10)),hAPP_nat_int(power_power_int(B_14),N_10))) ) ) ).
tff(fact_706_zero__le__power,axiom,
! [N_9: nat,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,zero_zero_real),hAPP_nat_real(power_power_real(A_20),N_9))) ) ).
tff(fact_707_zero__le__power,axiom,
! [N_9: 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,zero_zero_nat),hAPP_nat_nat(power_power_nat(A_20),N_9))) ) ).
tff(fact_708_zero__le__power,axiom,
! [N_9: nat,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,zero_zero_int),hAPP_nat_int(power_power_int(A_20),N_9))) ) ).
tff(fact_709_less__1__mult,axiom,
! [N_8: real,M_4: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),M_4))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),N_8))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(times_times_real(M_4),N_8))) ) ) ).
tff(fact_710_less__1__mult,axiom,
! [N_8: nat,M_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),M_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(times_times_nat(M_4),N_8))) ) ) ).
tff(fact_711_less__1__mult,axiom,
! [N_8: int,M_4: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),N_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(times_times_int(M_4),N_8))) ) ) ).
tff(fact_712_zero__less__power,axiom,
! [N_7: nat,A_19: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_19))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(power_power_real(A_19),N_7))) ) ).
tff(fact_713_zero__less__power,axiom,
! [N_7: nat,A_19: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_19))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(A_19),N_7))) ) ).
tff(fact_714_zero__less__power,axiom,
! [N_7: nat,A_19: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_19))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(power_power_int(A_19),N_7))) ) ).
tff(fact_715_less__add__one,axiom,
! [A_18: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_18),hAPP_real_real(plus_plus_real(A_18),one_one_real))) ).
tff(fact_716_less__add__one,axiom,
! [A_18: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_18),hAPP_nat_nat(plus_plus_nat(A_18),one_one_nat))) ).
tff(fact_717_less__add__one,axiom,
! [A_18: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_18),hAPP_int_int(plus_plus_int(A_18),one_one_int))) ).
tff(fact_718_dvd__mult__cancel__left,axiom,
! [C_1: int,A_2: int,B_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(times_times_int(C_1),A_2)),hAPP_int_int(times_times_int(C_1),B_2)))
<=> ( ( C_1 = zero_zero_int )
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_2),B_2)) ) ) ).
tff(fact_719_dvd__mult__cancel__left,axiom,
! [C_1: real,A_2: real,B_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(times_times_real(C_1),A_2)),hAPP_real_real(times_times_real(C_1),B_2)))
<=> ( ( C_1 = zero_zero_real )
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_2),B_2)) ) ) ).
tff(fact_720_dvd__mult__cancel__right,axiom,
! [A_2: int,C_1: int,B_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(times_times_int(A_2),C_1)),hAPP_int_int(times_times_int(B_2),C_1)))
<=> ( ( C_1 = zero_zero_int )
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_2),B_2)) ) ) ).
tff(fact_721_dvd__mult__cancel__right,axiom,
! [A_2: real,C_1: real,B_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(times_times_real(A_2),C_1)),hAPP_real_real(times_times_real(B_2),C_1)))
<=> ( ( C_1 = zero_zero_real )
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_2),B_2)) ) ) ).
tff(fact_722_power__increasing,axiom,
! [A_17: real,N_6: nat,N_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_6),N_5))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_17))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_17),N_6)),hAPP_nat_real(power_power_real(A_17),N_5))) ) ) ).
tff(fact_723_power__increasing,axiom,
! [A_17: nat,N_6: nat,N_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_6),N_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_17))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_17),N_6)),hAPP_nat_nat(power_power_nat(A_17),N_5))) ) ) ).
tff(fact_724_power__increasing,axiom,
! [A_17: int,N_6: nat,N_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_6),N_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_17))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_17),N_6)),hAPP_nat_int(power_power_int(A_17),N_5))) ) ) ).
tff(fact_725_one__le__power,axiom,
! [N_4: nat,A_16: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_16))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(power_power_real(A_16),N_4))) ) ).
tff(fact_726_one__le__power,axiom,
! [N_4: nat,A_16: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_16))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(power_power_nat(A_16),N_4))) ) ).
tff(fact_727_one__le__power,axiom,
! [N_4: nat,A_16: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_16))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_nat_int(power_power_int(A_16),N_4))) ) ).
tff(fact_728_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_729_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_730_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_731_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_732_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_733_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_734_power__0,axiom,
! [A_15: real] : hAPP_nat_real(power_power_real(A_15),zero_zero_nat) = one_one_real ).
tff(fact_735_power__0,axiom,
! [A_15: nat] : hAPP_nat_nat(power_power_nat(A_15),zero_zero_nat) = one_one_nat ).
tff(fact_736_power__0,axiom,
! [A_15: int] : hAPP_nat_int(power_power_int(A_15),zero_zero_nat) = one_one_int ).
tff(fact_737_power__add,axiom,
! [A_14: real,M_3: nat,N_3: nat] : hAPP_nat_real(power_power_real(A_14),hAPP_nat_nat(plus_plus_nat(M_3),N_3)) = hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(A_14),M_3)),hAPP_nat_real(power_power_real(A_14),N_3)) ).
tff(fact_738_power__add,axiom,
! [A_14: nat,M_3: nat,N_3: nat] : hAPP_nat_nat(power_power_nat(A_14),hAPP_nat_nat(plus_plus_nat(M_3),N_3)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(A_14),M_3)),hAPP_nat_nat(power_power_nat(A_14),N_3)) ).
tff(fact_739_power__add,axiom,
! [A_14: int,M_3: nat,N_3: nat] : hAPP_nat_int(power_power_int(A_14),hAPP_nat_nat(plus_plus_nat(M_3),N_3)) = hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(A_14),M_3)),hAPP_nat_int(power_power_int(A_14),N_3)) ).
tff(fact_740_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_741_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_742_zcong__less__eq,axiom,
! [M: int,Y: 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))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
=> ( hBOOL(hAPP_int_bool(zcong(X_1,Y),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),M))
=> ( X_1 = Y ) ) ) ) ) ) ) ).
tff(fact_743_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_744_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_745_zcong__eq__zdvd__prop,axiom,
! [X: int,P_2: int] :
( hBOOL(hAPP_int_bool(zcong(X,zero_zero_int),P_2))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),X)) ) ).
tff(fact_746_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_747_zprime__zdvd__zmult__better,axiom,
! [M: int,N_1: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_int_int(times_times_int(M),N_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),M))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),N_1)) ) ) ) ).
tff(fact_748_mult__left__le__imp__le,axiom,
! [C_11: real,A_13: real,B_13: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(C_11),A_13)),hAPP_real_real(times_times_real(C_11),B_13)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_11))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_13),B_13)) ) ) ).
tff(fact_749_mult__left__le__imp__le,axiom,
! [C_11: nat,A_13: nat,B_13: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(C_11),A_13)),hAPP_nat_nat(times_times_nat(C_11),B_13)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_11))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_13),B_13)) ) ) ).
tff(fact_750_mult__left__le__imp__le,axiom,
! [C_11: int,A_13: int,B_13: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(C_11),A_13)),hAPP_int_int(times_times_int(C_11),B_13)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_11))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_13),B_13)) ) ) ).
tff(fact_751_mult__right__le__imp__le,axiom,
! [A_12: real,C_10: real,B_12: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_12),C_10)),hAPP_real_real(times_times_real(B_12),C_10)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_10))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_12),B_12)) ) ) ).
tff(fact_752_mult__right__le__imp__le,axiom,
! [A_12: nat,C_10: nat,B_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_12),C_10)),hAPP_nat_nat(times_times_nat(B_12),C_10)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_10))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_12),B_12)) ) ) ).
tff(fact_753_mult__right__le__imp__le,axiom,
! [A_12: int,C_10: int,B_12: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_12),C_10)),hAPP_int_int(times_times_int(B_12),C_10)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_10))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_12),B_12)) ) ) ).
tff(fact_754_mult__less__imp__less__left,axiom,
! [C_9: real,A_11: real,B_11: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_9),A_11)),hAPP_real_real(times_times_real(C_9),B_11)))
=> ( 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_real,A_11),B_11)) ) ) ).
tff(fact_755_mult__less__imp__less__left,axiom,
! [C_9: nat,A_11: nat,B_11: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(C_9),A_11)),hAPP_nat_nat(times_times_nat(C_9),B_11)))
=> ( 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_nat,A_11),B_11)) ) ) ).
tff(fact_756_mult__less__imp__less__left,axiom,
! [C_9: int,A_11: int,B_11: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_9),A_11)),hAPP_int_int(times_times_int(C_9),B_11)))
=> ( 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_int,A_11),B_11)) ) ) ).
tff(fact_757_mult__left__less__imp__less,axiom,
! [C_8: real,A_10: real,B_10: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_8),A_10)),hAPP_real_real(times_times_real(C_8),B_10)))
=> ( 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_real,A_10),B_10)) ) ) ).
tff(fact_758_mult__left__less__imp__less,axiom,
! [C_8: nat,A_10: nat,B_10: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(C_8),A_10)),hAPP_nat_nat(times_times_nat(C_8),B_10)))
=> ( 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_nat,A_10),B_10)) ) ) ).
tff(fact_759_mult__left__less__imp__less,axiom,
! [C_8: int,A_10: int,B_10: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_8),A_10)),hAPP_int_int(times_times_int(C_8),B_10)))
=> ( 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_int,A_10),B_10)) ) ) ).
tff(fact_760_mult__less__imp__less__right,axiom,
! [A_9: real,C_7: real,B_9: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_9),C_7)),hAPP_real_real(times_times_real(B_9),C_7)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_7))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_9),B_9)) ) ) ).
tff(fact_761_mult__less__imp__less__right,axiom,
! [A_9: nat,C_7: nat,B_9: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_9),C_7)),hAPP_nat_nat(times_times_nat(B_9),C_7)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_9),B_9)) ) ) ).
tff(fact_762_mult__less__imp__less__right,axiom,
! [A_9: int,C_7: int,B_9: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_9),C_7)),hAPP_int_int(times_times_int(B_9),C_7)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_9),B_9)) ) ) ).
tff(fact_763_mult__right__less__imp__less,axiom,
! [A_8: real,C_6: real,B_8: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_8),C_6)),hAPP_real_real(times_times_real(B_8),C_6)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_6))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_8),B_8)) ) ) ).
tff(fact_764_mult__right__less__imp__less,axiom,
! [A_8: nat,C_6: nat,B_8: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_8),C_6)),hAPP_nat_nat(times_times_nat(B_8),C_6)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_8),B_8)) ) ) ).
tff(fact_765_mult__right__less__imp__less,axiom,
! [A_8: int,C_6: int,B_8: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_8),C_6)),hAPP_int_int(times_times_int(B_8),C_6)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_6))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_8),B_8)) ) ) ).
tff(fact_766_mult__le__less__imp__less,axiom,
! [C_5: real,D_7: real,A_7: real,B_7: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_7),B_7))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_5),D_7))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_7))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_5))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_7),C_5)),hAPP_real_real(times_times_real(B_7),D_7))) ) ) ) ) ).
tff(fact_767_mult__le__less__imp__less,axiom,
! [C_5: nat,D_7: nat,A_7: nat,B_7: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_7),B_7))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_5),D_7))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_7))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_7),C_5)),hAPP_nat_nat(times_times_nat(B_7),D_7))) ) ) ) ) ).
tff(fact_768_mult__le__less__imp__less,axiom,
! [C_5: int,D_7: int,A_7: int,B_7: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_7),B_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_5),D_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_7),C_5)),hAPP_int_int(times_times_int(B_7),D_7))) ) ) ) ) ).
tff(fact_769_mult__less__le__imp__less,axiom,
! [C_4: real,D_6: real,A_6: real,B_6: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_6),B_6))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_4),D_6))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_6))
=> ( 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(A_6),C_4)),hAPP_real_real(times_times_real(B_6),D_6))) ) ) ) ) ).
tff(fact_770_mult__less__le__imp__less,axiom,
! [C_4: nat,D_6: nat,A_6: nat,B_6: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_6),B_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_4),D_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_6))
=> ( 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(A_6),C_4)),hAPP_nat_nat(times_times_nat(B_6),D_6))) ) ) ) ) ).
tff(fact_771_mult__less__le__imp__less,axiom,
! [C_4: int,D_6: int,A_6: int,B_6: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_6),B_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_4),D_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_6))
=> ( 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(A_6),C_4)),hAPP_int_int(times_times_int(B_6),D_6))) ) ) ) ) ).
tff(fact_772_mult__strict__mono_H,axiom,
! [C_3: real,D_5: real,A_5: real,B_5: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_5),B_5))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_3),D_5))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_5))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_3))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_5),C_3)),hAPP_real_real(times_times_real(B_5),D_5))) ) ) ) ) ).
tff(fact_773_mult__strict__mono_H,axiom,
! [C_3: nat,D_5: nat,A_5: nat,B_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_5),B_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_3),D_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_5),C_3)),hAPP_nat_nat(times_times_nat(B_5),D_5))) ) ) ) ) ).
tff(fact_774_mult__strict__mono_H,axiom,
! [C_3: int,D_5: int,A_5: int,B_5: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_5),B_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_3),D_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_5),C_3)),hAPP_int_int(times_times_int(B_5),D_5))) ) ) ) ) ).
tff(fact_775_mult__strict__mono,axiom,
! [C_2: real,D_4: real,A_4: real,B_4: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_4),B_4))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_2),D_4))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_4))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_4),C_2)),hAPP_real_real(times_times_real(B_4),D_4))) ) ) ) ) ).
tff(fact_776_mult__strict__mono,axiom,
! [C_2: nat,D_4: nat,A_4: nat,B_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_4),B_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_2),D_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_4),C_2)),hAPP_nat_nat(times_times_nat(B_4),D_4))) ) ) ) ) ).
tff(fact_777_mult__strict__mono,axiom,
! [C_2: int,D_4: int,A_4: int,B_4: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_4),B_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_2),D_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_4),C_2)),hAPP_int_int(times_times_int(B_4),D_4))) ) ) ) ) ).
tff(fact_778_mult__le__cancel__left__neg,axiom,
! [A_2: real,B_2: real,C_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_1),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(C_1),A_2)),hAPP_real_real(times_times_real(C_1),B_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_2),A_2)) ) ) ).
tff(fact_779_mult__le__cancel__left__neg,axiom,
! [A_2: int,B_2: int,C_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(C_1),A_2)),hAPP_int_int(times_times_int(C_1),B_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),A_2)) ) ) ).
tff(fact_780_mult__le__cancel__left__pos,axiom,
! [A_2: real,B_2: real,C_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(C_1),A_2)),hAPP_real_real(times_times_real(C_1),B_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_2)) ) ) ).
tff(fact_781_mult__le__cancel__left__pos,axiom,
! [A_2: int,B_2: int,C_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(C_1),A_2)),hAPP_int_int(times_times_int(C_1),B_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),B_2)) ) ) ).
tff(fact_782_mult__left__le__one__le,axiom,
! [Y_4: real,X_4: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_4))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_4))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_4),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(Y_4),X_4)),X_4)) ) ) ) ).
tff(fact_783_mult__left__le__one__le,axiom,
! [Y_4: int,X_4: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_4),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(Y_4),X_4)),X_4)) ) ) ) ).
tff(fact_784_mult__right__le__one__le,axiom,
! [Y_3: real,X_3: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_3))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_3))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_3),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(X_3),Y_3)),X_3)) ) ) ) ).
tff(fact_785_mult__right__le__one__le,axiom,
! [Y_3: int,X_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_3),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(X_3),Y_3)),X_3)) ) ) ) ).
tff(fact_786_zero__less__two,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(plus_plus_real(one_one_real),one_one_real))) ).
tff(fact_787_zero__less__two,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(plus_plus_nat(one_one_nat),one_one_nat))) ).
tff(fact_788_zero__less__two,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(one_one_int),one_one_int))) ).
tff(fact_789_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_790_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_791_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_792_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_793_zpower__zdvd__prop1,axiom,
! [P_3: int,Y: 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_3),Y))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(power_power_int(Y),N_1))) ) ) ).
tff(fact_794_zcong__zpower__zmult,axiom,
! [Z: nat,X_1: int,Y: nat,P_3: int] :
( hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X_1),Y),one_one_int),P_3))
=> hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X_1),hAPP_nat_nat(times_times_nat(Y),Z)),one_one_int),P_3)) ) ).
tff(fact_795_zcong__zmult__prop3,axiom,
! [Y: int,X_1: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_3))
=> ( ~ hBOOL(hAPP_int_bool(zcong(Y,zero_zero_int),P_3))
=> ~ hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(X_1),Y),zero_zero_int),P_3)) ) ) ) ).
tff(fact_796_zcong__zprime__prod__zero,axiom,
! [B: int,A: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( 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_3))
=> ( hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_3))
| hBOOL(hAPP_int_bool(zcong(B,zero_zero_int),P_3)) ) ) ) ) ).
tff(fact_797_zcong__zprime__prod__zero__contra,axiom,
! [B: int,A: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( ( ~ hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_3))
& ~ hBOOL(hAPP_int_bool(zcong(B,zero_zero_int),P_3)) )
=> ~ hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),B),zero_zero_int),P_3)) ) ) ) ).
tff(fact_798_zprime__zdvd__zmult,axiom,
! [N_1: int,P_3: int,M: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_int_int(times_times_int(M),N_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),M))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),N_1)) ) ) ) ) ).
tff(fact_799__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_800__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_801_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_802_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_803__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_804__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_805_not__real__square__gt__zero,axiom,
! [X: real] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(X),X)))
<=> ( X = zero_zero_real ) ) ).
tff(fact_806_diff__bin__simps_I1_J,axiom,
! [K_1: int] : hAPP_int_int(minus_minus_int(K_1),pls) = K_1 ).
tff(fact_807_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_808_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_809_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_810_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_811_zcong__zdiff,axiom,
! [C: int,D_3: int,A: int,B: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(A,B),M))
=> ( hBOOL(hAPP_int_bool(zcong(C,D_3),M))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(minus_minus_int(A),C),hAPP_int_int(minus_minus_int(B),D_3)),M)) ) ) ).
tff(fact_812_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_813_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_814_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_815_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_816_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_817_xzgcda__linear__aux1,axiom,
! [A: int,R_1: int,B: int,M: int,C: int,D_3: 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_3))),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_3),N_1)))) ).
tff(fact_818_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_819_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_820_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_821_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_822_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_823_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_824_inv__not__p__minus__1__aux,axiom,
! [A_2: int,P_2: int] :
( hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A_2),hAPP_int_int(minus_minus_int(P_2),one_one_int)),one_one_int),P_2))
<=> hBOOL(hAPP_int_bool(zcong(A_2,hAPP_int_int(minus_minus_int(P_2),one_one_int)),P_2)) ) ).
tff(fact_825_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_826_zcong__square__zless,axiom,
! [A: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( 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_3))
=> ( hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),A),one_one_int),P_3))
=> ( ( A = one_one_int )
| ( A = hAPP_int_int(minus_minus_int(P_3),one_one_int) ) ) ) ) ) ) ).
tff(fact_827_zcong__square,axiom,
! [A: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( 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_3))
=> ( hBOOL(hAPP_int_bool(zcong(A,one_one_int),P_3))
| hBOOL(hAPP_int_bool(zcong(A,hAPP_int_int(minus_minus_int(P_3),one_one_int)),P_3)) ) ) ) ) ).
tff(fact_828_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_829_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_830_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_831_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_832_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_833_Legendre__def,axiom,
! [A: int,P_3: int] :
( ( hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_3))
=> ( legendre(A,P_3) = zero_zero_int ) )
& ( ~ hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_3))
=> ( ( hBOOL(hAPP_int_bool(quadRes(P_3),A))
=> ( legendre(A,P_3) = one_one_int ) )
& ( ~ hBOOL(hAPP_int_bool(quadRes(P_3),A))
=> ( legendre(A,P_3) = number_number_of_int(min) ) ) ) ) ) ).
tff(fact_834_QuadRes__def,axiom,
! [Ma: int,X: int] :
( hBOOL(hAPP_int_bool(quadRes(Ma),X))
<=> ? [Y_2: int] : hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(Y_2),number_number_of_nat(bit0(bit1(pls)))),X),Ma)) ) ).
tff(fact_835_real__sum__squared__expand,axiom,
! [X_1: real,Y: real] : hAPP_nat_real(power_power_real(hAPP_real_real(plus_plus_real(X_1),Y)),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),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)) ).
tff(fact_836_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_837_divides__antisym,axiom,
! [X: nat,Y_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X)) )
<=> ( X = Y_1 ) ) ).
tff(fact_838_divides__add__revr,axiom,
! [B: nat,D_3: nat,A: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),A))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),hAPP_nat_nat(plus_plus_nat(A),B)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),B)) ) ) ).
tff(fact_839_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_840_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_841_divides__exp,axiom,
! [N_1: nat,X_1: nat,Y: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y))
=> 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),N_1))) ) ).
tff(fact_842_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_843_power__eq__if,axiom,
! [P_3: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(P_3),M) = one_one_nat ) )
& ( ( M != zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(P_3),M) = hAPP_nat_nat(times_times_nat(P_3),hAPP_nat_nat(power_power_nat(P_3),hAPP_nat_nat(minus_minus_nat(M),one_one_nat))) ) ) ) ).
tff(fact_844_diff__square,axiom,
! [X_1: nat,Y: 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),number_number_of_nat(bit0(bit1(pls))))) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(X_1),Y)),hAPP_nat_nat(minus_minus_nat(X_1),Y)) ).
tff(fact_845_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_846_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_847_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_848_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_849_divides__exp2,axiom,
! [X_1: nat,Y: 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))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y)) ) ) ).
tff(fact_850_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_851_exp__eq__1,axiom,
! [X: nat,N: nat] :
( ( hAPP_nat_nat(power_power_nat(X),N) = one_one_nat )
<=> ( ( X = one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
tff(fact_852_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_853_realpow__two__sum__zero__iff,axiom,
! [X: real,Y_1: real] :
( ( hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_1),number_number_of_nat(bit0(bit1(pls))))) = zero_zero_real )
<=> ( ( X = zero_zero_real )
& ( Y_1 = zero_zero_real ) ) ) ).
tff(fact_854_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_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
& ( hAPP_nat_real(power_power_real(X_2),N_1) = A )
& ! [Y_2: real] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2))
& ( hAPP_nat_real(power_power_real(Y_2),N_1) = A ) )
=> ( Y_2 = X_2 ) ) ) ) ) ).
tff(fact_855_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_856_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_857_dvd_Oorder__refl,axiom,
! [X_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),X_1)) ).
tff(fact_858_less__zeroE,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).
tff(fact_859_le0,axiom,
! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1)) ).
tff(fact_860_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_861_diff__0__eq__0,axiom,
! [N_1: nat] : hAPP_nat_nat(minus_minus_nat(zero_zero_nat),N_1) = zero_zero_nat ).
tff(fact_862_minus__nat_Odiff__0,axiom,
! [M: nat] : hAPP_nat_nat(minus_minus_nat(M),zero_zero_nat) = M ).
tff(fact_863_diff__self__eq__0,axiom,
! [M: nat] : hAPP_nat_nat(minus_minus_nat(M),M) = zero_zero_nat ).
tff(fact_864_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_865_real__zero__not__eq__one,axiom,
zero_zero_real != one_one_real ).
tff(fact_866_real__le__eq__diff,axiom,
! [X: real,Y_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(minus_minus_real(X),Y_1)),zero_zero_real)) ) ).
tff(fact_867_real__less__def,axiom,
! [X: real,Y_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y_1))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y_1))
& ( X != Y_1 ) ) ) ).
tff(fact_868_less__eq__real__def,axiom,
! [X: real,Y_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y_1))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y_1))
| ( X = Y_1 ) ) ) ).
tff(fact_869_less__not__refl,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1)) ).
tff(fact_870_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_871_linorder__neqE__nat,axiom,
! [X_1: nat,Y: nat] :
( ( X_1 != Y )
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y),X_1)) ) ) ).
tff(fact_872_less__irrefl__nat,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1)) ).
tff(fact_873_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_874_less__not__refl3,axiom,
! [S: nat,T: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S),T))
=> ( S != T ) ) ).
tff(fact_875_nat__less__cases,axiom,
! [P_1: 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_1,N),Ma)) )
=> ( ( ( Ma = N )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N),Ma)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),Ma))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N),Ma)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N),Ma)) ) ) ) ).
tff(fact_876_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_877_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_878_dvd_Oless__asym,axiom,
! [X_1: nat,Y: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X_1)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y)) ) ) ).
tff(fact_879_dvd_Oless__trans,axiom,
! [Z: nat,X_1: nat,Y: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X_1)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),Y)) )
=> ( 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_880_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_881_dvd_Oless__le__trans,axiom,
! [Z: nat,X_1: nat,Y: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X_1)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),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_882_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_883_dvd_Oless__imp__triv,axiom,
! [P_1: bool,X: nat,Y_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y_1)) )
=> hBOOL(P_1) ) ) ).
tff(fact_884_dvd_Oless__imp__not__eq2,axiom,
! [X_1: nat,Y: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X_1)) )
=> ( Y != X_1 ) ) ).
tff(fact_885_dvd_Oless__imp__not__eq,axiom,
! [X_1: nat,Y: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X_1)) )
=> ( X_1 != Y ) ) ).
tff(fact_886_dvd_Oless__imp__not__less,axiom,
! [X_1: nat,Y: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X_1)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y)) ) ) ).
tff(fact_887_dvd_Oless__imp__le,axiom,
! [X_1: nat,Y: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X_1)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y)) ) ).
tff(fact_888_dvd_Oless__not__sym,axiom,
! [X_1: nat,Y: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X_1)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y)) ) ) ).
tff(fact_889_dvd_Oless__imp__neq,axiom,
! [X_1: nat,Y: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X_1)) )
=> ( X_1 != Y ) ) ).
tff(fact_890_dvd_Ole__less__trans,axiom,
! [Z: nat,X_1: nat,Y: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y))
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),Y)) )
=> ( 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_891_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_892_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_893_dvd_Oorder__trans,axiom,
! [Z: nat,X_1: nat,Y: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z)) ) ) ).
tff(fact_894_dvd_Oantisym,axiom,
! [X_1: nat,Y: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X_1))
=> ( X_1 = Y ) ) ) ).
tff(fact_895_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_896_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_897_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_898_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_899_dvd_Ole__imp__less__or__eq,axiom,
! [X_1: nat,Y: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y))
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X_1)) )
| ( X_1 = Y ) ) ) ).
tff(fact_900_dvd_Oantisym__conv,axiom,
! [Y_1: nat,X: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y_1))
<=> ( X = Y_1 ) ) ) ).
tff(fact_901_dvd_Oeq__refl,axiom,
! [X_1: nat,Y: nat] :
( ( X_1 = Y )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y)) ) ).
tff(fact_902_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_903_dvd_Oless__le__not__le,axiom,
! [X: nat,Y_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X)) )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X)) ) ) ).
tff(fact_904_dvd_Oless__le,axiom,
! [X: nat,Y_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X)) )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y_1))
& ( X != Y_1 ) ) ) ).
tff(fact_905_dvd_Ole__less,axiom,
! [X: nat,Y_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y_1))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X)) )
| ( X = Y_1 ) ) ) ).
tff(fact_906_dvd_Oeq__iff,axiom,
! [X: nat,Y_1: nat] :
( ( X = Y_1 )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X)) ) ) ).
tff(fact_907_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_908_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_909_nat__add__left__commute,axiom,
! [X_1: nat,Y: nat,Z: nat] : hAPP_nat_nat(plus_plus_nat(X_1),hAPP_nat_nat(plus_plus_nat(Y),Z)) = hAPP_nat_nat(plus_plus_nat(Y),hAPP_nat_nat(plus_plus_nat(X_1),Z)) ).
tff(fact_910_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_911_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_912_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_913_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_914_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_915_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_916_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_917_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_918_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_919_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_920_real__mult__1,axiom,
! [Z: real] : hAPP_real_real(times_times_real(one_one_real),Z) = Z ).
tff(fact_921_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_922_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_923_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_924_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_925_le__refl,axiom,
! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),N_1)) ).
tff(fact_926_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_927_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_928_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_929_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_930_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_931_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_932_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_933_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_934_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_935_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_936_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_937_real__add__left__mono,axiom,
! [Z: real,X_1: real,Y: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y))
=> 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))) ) ).
tff(fact_938_not__less0,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).
tff(fact_939_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_940_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_941_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_942_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_943_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_944_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_945_plus__nat_Oadd__0,axiom,
! [N_1: nat] : hAPP_nat_nat(plus_plus_nat(zero_zero_nat),N_1) = N_1 ).
tff(fact_946_Nat_Oadd__0__right,axiom,
! [M: nat] : hAPP_nat_nat(plus_plus_nat(M),zero_zero_nat) = M ).
tff(fact_947_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_948_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_949_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_950_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_951_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_952_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_953_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_954_mult__0,axiom,
! [N_1: nat] : hAPP_nat_nat(times_times_nat(zero_zero_nat),N_1) = zero_zero_nat ).
tff(fact_955_mult__0__right,axiom,
! [M: nat] : hAPP_nat_nat(times_times_nat(M),zero_zero_nat) = zero_zero_nat ).
tff(fact_956_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_957_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_958_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_959_real__mult__left__cancel,axiom,
! [A_2: real,B_2: real,C_1: real] :
( ( C_1 != zero_zero_real )
=> ( ( hAPP_real_real(times_times_real(C_1),A_2) = hAPP_real_real(times_times_real(C_1),B_2) )
<=> ( A_2 = B_2 ) ) ) ).
tff(fact_960_real__mult__right__cancel,axiom,
! [A_2: real,B_2: real,C_1: real] :
( ( C_1 != zero_zero_real )
=> ( ( hAPP_real_real(times_times_real(A_2),C_1) = hAPP_real_real(times_times_real(B_2),C_1) )
<=> ( A_2 = B_2 ) ) ) ).
tff(fact_961_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_962_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_963_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_964_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_965_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_966_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_967_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_968_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_969_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_970_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_971_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_972_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_973_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_974_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_975_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_976_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_977_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_978_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_979_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_980_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_981_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_982_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_983_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_984_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_985_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_986_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_987_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_988_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_989_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_990_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_991_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_992_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_993_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_994_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_995_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_996_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_997_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_998_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_999_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_1000_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_1001_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_1002_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_1003_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_1004_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_1005_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_1006_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_1007_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_1008_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_1009_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_1010_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_1011_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_1012_nat__mult__1,axiom,
! [N_1: nat] : hAPP_nat_nat(times_times_nat(one_one_nat),N_1) = N_1 ).
tff(fact_1013_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_1014_nat__mult__1__right,axiom,
! [N_1: nat] : hAPP_nat_nat(times_times_nat(N_1),one_one_nat) = N_1 ).
tff(fact_1015_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_1016_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_1017_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_1018_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_1019_nat__diff__split__asm,axiom,
! [P_1: fun_nat_bool,A_2: nat,B_2: nat] :
( hBOOL(hAPP_nat_bool(P_1,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_1,zero_zero_nat)) )
| ? [D_2: nat] :
( ( A_2 = hAPP_nat_nat(plus_plus_nat(B_2),D_2) )
& ~ hBOOL(hAPP_nat_bool(P_1,D_2)) ) ) ) ).
tff(fact_1020_nat__diff__split,axiom,
! [P_1: fun_nat_bool,A_2: nat,B_2: nat] :
( hBOOL(hAPP_nat_bool(P_1,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_1,zero_zero_nat)) )
& ! [D_2: nat] :
( ( A_2 = hAPP_nat_nat(plus_plus_nat(B_2),D_2) )
=> hBOOL(hAPP_nat_bool(P_1,D_2)) ) ) ) ).
tff(fact_1021_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_1022_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_1023_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_1024_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_1025_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_1026_real__mult__le__cancel__iff1,axiom,
! [X: real,Y_1: 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),Z_1)),hAPP_real_real(times_times_real(Y_1),Z_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y_1)) ) ) ).
tff(fact_1027_real__mult__le__cancel__iff2,axiom,
! [X: real,Y_1: 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)),hAPP_real_real(times_times_real(Z_1),Y_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y_1)) ) ) ).
tff(fact_1028_real__mult__less__mono2,axiom,
! [X_1: real,Y: 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))
=> 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))) ) ) ).
tff(fact_1029_real__mult__order,axiom,
! [Y: 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))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(X_1),Y))) ) ) ).
tff(fact_1030_real__mult__less__iff1,axiom,
! [X: real,Y_1: 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),Z_1)),hAPP_real_real(times_times_real(Y_1),Z_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y_1)) ) ) ).
tff(fact_1031_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_1032_real__two__squares__add__zero__iff,axiom,
! [X: real,Y_1: real] :
( ( hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X),X)),hAPP_real_real(times_times_real(Y_1),Y_1)) = zero_zero_real )
<=> ( ( X = zero_zero_real )
& ( Y_1 = zero_zero_real ) ) ) ).
tff(fact_1033_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_1034_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_1035_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_1036_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_1037_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_1038_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_1039_ex__least__nat__less,axiom,
! [N: nat,P_1: fun_nat_bool] :
( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(P_1,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_1,I)) )
& hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(plus_plus_nat(K_2),one_one_nat))) ) ) ) ).
tff(fact_1040_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_1041_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_1042_real__le__refl,axiom,
! [W: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),W)) ).
tff(fact_1043_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_1044_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_1045_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_1046_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_1047_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_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__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_1050_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_1051_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_1052_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_1053_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_1054_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_1055_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_1056_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_1057_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_1058_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_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_zprime__factor__exists,axiom,
! [A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A))
=> ? [P_5: int] :
( hBOOL(hAPP_int_bool(zprime,P_5))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_5),A)) ) ) ).
tff(fact_1061_conj__le__cong,axiom,
! [P_4: bool,P_1: bool,X: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(P_1)
<=> hBOOL(P_4) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
& hBOOL(P_1) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
& hBOOL(P_4) ) ) ) ).
tff(fact_1062_imp__le__cong,axiom,
! [P_4: bool,P_1: bool,X: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(P_1)
<=> hBOOL(P_4) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> hBOOL(P_1) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> hBOOL(P_4) ) ) ) ).
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_1: fun_int_bool,D: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X_2: int] :
( hBOOL(hAPP_int_bool(P_1,X_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(X_2),D))) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
=> ! [X_2: int] :
( hBOOL(hAPP_int_bool(P_1,X_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(X_2),hAPP_int_int(times_times_int(K),D)))) ) ) ) ) ).
tff(fact_1066_incr__mult__lemma,axiom,
! [K: int,P_1: fun_int_bool,D: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X_2: int] :
( hBOOL(hAPP_int_bool(P_1,X_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(X_2),D))) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
=> ! [X_2: int] :
( hBOOL(hAPP_int_bool(P_1,X_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(X_2),hAPP_int_int(times_times_int(K),D)))) ) ) ) ) ).
tff(fact_1067_zcong__zless__unique,axiom,
! [A: int,M: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
=> ? [X_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),M))
& hBOOL(hAPP_int_bool(zcong(A,X_2),M))
& ! [Y_2: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_2),M))
& hBOOL(hAPP_int_bool(zcong(A,Y_2),M)) )
=> ( Y_2 = X_2 ) ) ) ) ).
tff(fact_1068__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_1069_mult__sum2sq,axiom,
! [A: int,B: int,P_3: int,Q_1: int] : hAPP_int_int(times_times_int(twoSqu949963151sum2sq(product_Pair_int_int(A,B))),twoSqu949963151sum2sq(product_Pair_int_int(P_3,Q_1))) = twoSqu949963151sum2sq(product_Pair_int_int(hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A),P_3)),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_3)))) ).
tff(fact_1070_is__sum2sq__def,axiom,
! [X: int] :
( hBOOL(hAPP_int_bool(twoSqu820444569sum2sq,X))
<=> ? [A_3: int,B_3: int] : twoSqu949963151sum2sq(product_Pair_int_int(A_3,B_3)) = X ) ).
tff(fact_1071_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_1072_Wilson__Russ,axiom,
! [P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> hBOOL(hAPP_int_bool(zcong(zfact(hAPP_int_int(minus_minus_int(P_3),one_one_int)),number_number_of_int(min)),P_3)) ) ).
tff(fact_1073_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_1074_inv__inv,axiom,
! [A: 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,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),P_3))
=> ( inv(P_3,inv(P_3,A)) = A ) ) ) ) ) ).
tff(fact_1075_inv__less__p__minus__1,axiom,
! [A: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( 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_3),one_one_int)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,inv(P_3,A)),hAPP_int_int(minus_minus_int(P_3),one_one_int))) ) ) ) ).
tff(fact_1076_inv__g__1,axiom,
! [A: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( 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_3),one_one_int)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),inv(P_3,A))) ) ) ) ).
tff(fact_1077_inv__not__p__minus__1,axiom,
! [A: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( 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_3),one_one_int)))
=> ( inv(P_3,A) != hAPP_int_int(minus_minus_int(P_3),one_one_int) ) ) ) ) ).
tff(fact_1078_inv__not__1,axiom,
! [A: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( 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_3),one_one_int)))
=> ( inv(P_3,A) != one_one_int ) ) ) ) ).
tff(fact_1079_inv__distinct,axiom,
! [A: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( 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_3),one_one_int)))
=> ( A != inv(P_3,A) ) ) ) ) ).
tff(fact_1080_inv__not__0,axiom,
! [A: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( 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_3),one_one_int)))
=> ( inv(P_3,A) != zero_zero_int ) ) ) ) ).
tff(fact_1081_inv__is__inv,axiom,
! [A: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( 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_3))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),inv(P_3,A)),one_one_int),P_3)) ) ) ) ).
tff(fact_1082_wset__mem__inv__mem,axiom,
! [B_2: int,A_2: int,P_2: int] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(bit1(bit0(bit1(pls))))),P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(minus_minus_int(P_2),one_one_int)))
=> ( hBOOL(member_int(B_2,wset(A_2,P_2)))
=> hBOOL(member_int(inv(P_2,B_2),wset(A_2,P_2))) ) ) ) ) ).
tff(fact_1083_wset__inv__mem__mem,axiom,
! [B_2: int,A_2: int,P_2: int] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(bit1(bit0(bit1(pls))))),P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(minus_minus_int(P_2),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_2),one_one_int)))
=> ( hBOOL(member_int(inv(P_2,B_2),wset(A_2,P_2)))
=> hBOOL(member_int(B_2,wset(A_2,P_2))) ) ) ) ) ) ) ).
tff(fact_1084_wset__mem__mem,axiom,
! [P_2: 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_2))) ) ).
tff(fact_1085_wset__subset,axiom,
! [B_2: int,P_2: 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_2)))
=> hBOOL(member_int(B_2,wset(A_2,P_2))) ) ) ).
tff(fact_1086_wset__g__1,axiom,
! [B_2: int,A_2: int,P_2: int] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(minus_minus_int(P_2),one_one_int)))
=> ( hBOOL(member_int(B_2,wset(A_2,P_2)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_2)) ) ) ) ).
tff(fact_1087_wset__less,axiom,
! [B_2: int,A_2: int,P_2: int] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(minus_minus_int(P_2),one_one_int)))
=> ( hBOOL(member_int(B_2,wset(A_2,P_2)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),hAPP_int_int(minus_minus_int(P_2),one_one_int))) ) ) ) ).
tff(fact_1088_wset__mem__imp__or,axiom,
! [B_2: int,P_2: 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_2)))
=> ( hBOOL(member_int(B_2,wset(A_2,P_2)))
=> ( ( B_2 = A_2 )
| ( B_2 = inv(P_2,A_2) ) ) ) ) ) ).
tff(fact_1089_wset__mem,axiom,
! [B_2: int,A_2: int,P_2: int] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(minus_minus_int(P_2),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_2))) ) ) ) ) ).
tff(fact_1090_MultInvPair__distinct,axiom,
! [J_1: int,A: int,P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_3))
=> ( ~ hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_3))
=> ( ~ hBOOL(hAPP_int_bool(zcong(J_1,zero_zero_int),P_3))
=> ( ~ hBOOL(hAPP_int_bool(quadRes(P_3),A))
=> ~ hBOOL(hAPP_int_bool(zcong(J_1,hAPP_int_int(times_times_int(A),multInv(P_3,J_1))),P_3)) ) ) ) ) ) ).
tff(fact_1091_int__le__induct,axiom,
! [P_1: 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_1,K))
=> ( ! [I: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),K))
=> ( hBOOL(hAPP_int_bool(P_1,I))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(I),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_1,I_2)) ) ) ) ).
tff(fact_1092_aux______1,axiom,
! [J_1: int,A: int,P_3: int,K_1: int] :
( hBOOL(hAPP_int_bool(zcong(J_1,hAPP_int_int(times_times_int(A),multInv(P_3,K_1))),P_3))
=> 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_3,K_1))),K_1)),P_3)) ) ).
tff(fact_1093_aux______3,axiom,
! [J_1: int,K_1: int,A: int,P_3: int] :
( hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(J_1),K_1),A),P_3))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(multInv(P_3,J_1)),J_1)),K_1),hAPP_int_int(times_times_int(multInv(P_3,J_1)),A)),P_3)) ) ).
tff(fact_1094_MultInv__prop1,axiom,
! [X_1: int,Y: int,P_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zcong(X_1,Y),P_3))
=> hBOOL(hAPP_int_bool(zcong(multInv(P_3,X_1),multInv(P_3,Y)),P_3)) ) ) ).
tff(fact_1095_MultInv__zcong__prop1,axiom,
! [A: int,J_1: int,K_1: int,P_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zcong(J_1,K_1),P_3))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),multInv(P_3,J_1)),hAPP_int_int(times_times_int(A),multInv(P_3,K_1))),P_3)) ) ) ).
tff(fact_1096_MultInv__prop5,axiom,
! [Y: int,X_1: int,P_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_3))
=> ( ~ hBOOL(hAPP_int_bool(zcong(Y,zero_zero_int),P_3))
=> ( hBOOL(hAPP_int_bool(zcong(multInv(P_3,X_1),multInv(P_3,Y)),P_3))
=> hBOOL(hAPP_int_bool(zcong(X_1,Y),P_3)) ) ) ) ) ) ).
tff(fact_1097_MultInv__prop4,axiom,
! [X_1: int,P_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_3))
=> hBOOL(hAPP_int_bool(zcong(multInv(P_3,multInv(P_3,X_1)),X_1),P_3)) ) ) ) ).
tff(fact_1098_MultInv__prop3,axiom,
! [X_1: int,P_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_3))
=> ~ hBOOL(hAPP_int_bool(zcong(multInv(P_3,X_1),zero_zero_int),P_3)) ) ) ) ).
tff(fact_1099_aux______4,axiom,
! [K_1: int,A: int,J_1: int,P_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(zcong(J_1,zero_zero_int),P_3))
=> ( hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(multInv(P_3,J_1)),J_1)),K_1),hAPP_int_int(times_times_int(multInv(P_3,J_1)),A)),P_3))
=> hBOOL(hAPP_int_bool(zcong(K_1,hAPP_int_int(times_times_int(A),multInv(P_3,J_1))),P_3)) ) ) ) ) ).
tff(fact_1100_aux______2,axiom,
! [J_1: int,A: int,K_1: int,P_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(zcong(K_1,zero_zero_int),P_3))
=> ( 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_3,K_1))),K_1)),P_3))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(J_1),K_1),A),P_3)) ) ) ) ) ).
tff(fact_1101_MultInv__zcong__prop2,axiom,
! [A: int,J_1: int,K_1: int,P_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(zcong(K_1,zero_zero_int),P_3))
=> ( ~ hBOOL(hAPP_int_bool(zcong(J_1,zero_zero_int),P_3))
=> ( hBOOL(hAPP_int_bool(zcong(J_1,hAPP_int_int(times_times_int(A),multInv(P_3,K_1))),P_3))
=> hBOOL(hAPP_int_bool(zcong(K_1,hAPP_int_int(times_times_int(A),multInv(P_3,J_1))),P_3)) ) ) ) ) ) ).
tff(fact_1102_MultInv__zcong__prop3,axiom,
! [J_1: int,K_1: int,A: int,P_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_3))
=> ( ~ hBOOL(hAPP_int_bool(zcong(K_1,zero_zero_int),P_3))
=> ( ~ hBOOL(hAPP_int_bool(zcong(J_1,zero_zero_int),P_3))
=> ( hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),multInv(P_3,J_1)),hAPP_int_int(times_times_int(A),multInv(P_3,K_1))),P_3))
=> hBOOL(hAPP_int_bool(zcong(J_1,K_1),P_3)) ) ) ) ) ) ) ).
tff(fact_1103_Int2_Oaux____2,axiom,
! [X_1: int,P_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_3))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(X_1),multInv(P_3,X_1))),multInv(P_3,multInv(P_3,X_1))),X_1),P_3)) ) ) ) ).
tff(fact_1104_Int2_Oaux____1,axiom,
! [X_1: int,P_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_3))
=> hBOOL(hAPP_int_bool(zcong(multInv(P_3,multInv(P_3,X_1)),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(X_1),multInv(P_3,X_1))),multInv(P_3,multInv(P_3,X_1)))),P_3)) ) ) ) ).
tff(fact_1105_MultInv__prop2a,axiom,
! [X_1: int,P_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_3))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(multInv(P_3,X_1)),X_1),one_one_int),P_3)) ) ) ) ).
tff(fact_1106_MultInv__prop2,axiom,
! [X_1: int,P_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_3))
=> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(X_1),multInv(P_3,X_1)),one_one_int),P_3)) ) ) ) ).
tff(fact_1107_d22set__induct__old,axiom,
! [X: int,P_1: 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_1,hAPP_int_int(minus_minus_int(A_3),one_one_int))) )
=> hBOOL(hAPP_int_bool(P_1,A_3)) )
=> hBOOL(hAPP_int_bool(P_1,X)) ) ).
tff(fact_1108_int__less__induct,axiom,
! [P_1: 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_1,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_1,I))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(I),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_1,I_2)) ) ) ) ).
tff(fact_1109_int__ge__induct,axiom,
! [P_1: 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_1,K))
=> ( ! [I: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),I))
=> ( hBOOL(hAPP_int_bool(P_1,I))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(I),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_1,I_2)) ) ) ) ).
tff(fact_1110_int__gr__induct,axiom,
! [P_1: 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_1,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_1,I))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(I),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_1,I_2)) ) ) ) ).
tff(fact_1111_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_1112_d22set__eq__wset,axiom,
! [P_2: int] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( d22set(hAPP_int_int(minus_minus_int(P_2),number_number_of_int(bit0(bit1(pls))))) = wset(hAPP_int_int(minus_minus_int(P_2),number_number_of_int(bit0(bit1(pls)))),P_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__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_1115_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_1116_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_1117_ex__least__nat__le,axiom,
! [N: nat,P_1: fun_nat_bool] :
( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(P_1,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_1,I)) )
& hBOOL(hAPP_nat_bool(P_1,K_2)) ) ) ) ).
tff(fact_1118_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_1119_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_1120_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_1121_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_1122_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_1123_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_1124_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_1125_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_1126_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_1127_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_1128_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_1129_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_1130_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_1131_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_1132_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_1133_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_1134_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_1135_SR__def,axiom,
! [P_2: int] : sr(P_2) = 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_2))) ).
tff(fact_1136_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_1137_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_1138_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_1139_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_1140_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_1141_zpower__zmod,axiom,
! [X_1: int,M: int,Y: nat] : hAPP_int_int(div_mod_int(hAPP_nat_int(power_power_int(hAPP_int_int(div_mod_int(X_1),M)),Y)),M) = hAPP_int_int(div_mod_int(hAPP_nat_int(power_power_int(X_1),Y)),M) ).
tff(fact_1142_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_1143_Residues_Oaux,axiom,
! [X_1: int,M: int,Y: int] :
( ( hAPP_int_int(div_mod_int(X_1),M) = hAPP_int_int(div_mod_int(Y),M) )
=> hBOOL(hAPP_int_bool(zcong(X_1,Y),M)) ) ).
tff(fact_1144_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_1145_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_1146_zmod__self,axiom,
! [A: int] : hAPP_int_int(div_mod_int(A),A) = zero_zero_int ).
tff(fact_1147_zmod__zero,axiom,
! [B: int] : hAPP_int_int(div_mod_int(zero_zero_int),B) = zero_zero_int ).
tff(fact_1148_zmod__eq__dvd__iff,axiom,
! [X: int,N: int,Y_1: int] :
( ( hAPP_int_int(div_mod_int(X),N) = hAPP_int_int(div_mod_int(Y_1),N) )
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),hAPP_int_int(minus_minus_int(X),Y_1))) ) ).
tff(fact_1149_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_1150_Divides_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y: 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))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(X_1),Y))) ) ) ).
tff(fact_1151_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_1152_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_1153_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_1154_zdiff__zmod__left,axiom,
! [X_1: int,M: int,Y: int] : hAPP_int_int(div_mod_int(hAPP_int_int(minus_minus_int(hAPP_int_int(div_mod_int(X_1),M)),Y)),M) = hAPP_int_int(div_mod_int(hAPP_int_int(minus_minus_int(X_1),Y)),M) ).
tff(fact_1155_zdiff__zmod__right,axiom,
! [X_1: int,Y: int,M: int] : hAPP_int_int(div_mod_int(hAPP_int_int(minus_minus_int(X_1),hAPP_int_int(div_mod_int(Y),M))),M) = hAPP_int_int(div_mod_int(hAPP_int_int(minus_minus_int(X_1),Y)),M) ).
tff(fact_1156_zdvd__iff__zmod__eq__0__number__of,axiom,
! [X: int,Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,number_number_of_int(X)),number_number_of_int(Y_1)))
<=> ( hAPP_int_int(div_mod_int(number_number_of_int(Y_1)),number_number_of_int(X)) = zero_zero_int ) ) ).
tff(fact_1157_zmod__minus1__right,axiom,
! [A: int] : hAPP_int_int(div_mod_int(A),number_number_of_int(min)) = zero_zero_int ).
tff(fact_1158_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_1159_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_1160_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_1161_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_1162_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_1163_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_1164_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_1165_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_1166_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_1167_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_1168_divmod__int__rel__mod__eq,axiom,
! [A_1: int,B_1: int,Q_3: int,Y: int] :
( ( A_1 = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_1),Q_3)),Y) )
=> ( ( ( 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))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),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))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),zero_zero_int)) ) ) )
=> ( ( B_1 != zero_zero_int )
=> ( hAPP_int_int(div_mod_int(A_1),B_1) = Y ) ) ) ) ).
tff(fact_1169_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_1170_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_1171_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_1172_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_1173_split__zmod,axiom,
! [P_1: fun_int_bool,N: int,K: int] :
( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(div_mod_int(N),K)))
<=> ( ( ( K = zero_zero_int )
=> hBOOL(hAPP_int_bool(P_1,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_1,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_1,J)) ) ) ) ) ).
tff(fact_1174_neq__one__mod__two,axiom,
! [X: int] :
( ( hAPP_int_int(div_mod_int(X),number_number_of_int(bit0(bit1(pls)))) != zero_zero_int )
<=> ( hAPP_int_int(div_mod_int(X),number_number_of_int(bit0(bit1(pls)))) = one_one_int ) ) ).
tff(fact_1175_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_1176_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_1177_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_1178_StandardRes__prop4,axiom,
! [X_1: int,Y: 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)),hAPP_int_int(times_times_int(X_1),Y)),M)) ) ).
tff(fact_1179_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_1180_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_1181_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_1182_StandardRes__def,axiom,
! [M: int,X_1: int] : standardRes(M,X_1) = hAPP_int_int(div_mod_int(X_1),M) ).
tff(fact_1183_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_1184_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_1185_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_1186_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_1187_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_1188_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_1189_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_1190_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_1191_StandardRes__eq__zcong,axiom,
! [Ma: int,X: int] :
( ( standardRes(Ma,X) = zero_zero_int )
<=> hBOOL(hAPP_int_bool(zcong(X,zero_zero_int),Ma)) ) ).
tff(fact_1192_StandardRes__prop3,axiom,
! [X: int,P_2: int] :
( ~ hBOOL(hAPP_int_bool(zcong(X,zero_zero_int),P_2))
<=> ( standardRes(P_2,X) != zero_zero_int ) ) ).
tff(fact_1193_StandardRes__prop1,axiom,
! [X_1: int,M: int] : hBOOL(hAPP_int_bool(zcong(X_1,standardRes(M,X_1)),M)) ).
tff(fact_1194_StandardRes__ubound,axiom,
! [X_1: int,P_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,standardRes(P_3,X_1)),P_3)) ) ).
tff(fact_1195_StandardRes__SR__prop,axiom,
! [X: int,P_2: int] :
( hBOOL(member_int(X,sr(P_2)))
=> ( standardRes(P_2,X) = X ) ) ).
tff(fact_1196_split__mod,axiom,
! [P_1: fun_nat_bool,N: nat,K: nat] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(div_mod_nat(N),K)))
<=> ( ( ( K = zero_zero_nat )
=> hBOOL(hAPP_nat_bool(P_1,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_1,J)) ) ) ) ) ) ).
tff(fact_1197_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))) ) ) ).
%----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,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls))))),one_one_int)),zero_zero_int)) ).
%------------------------------------------------------------------------------