TPTP Problem File: NUM925_30.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : NUM925_30 : TPTP v8.2.0. Released v8.2.0.
% Domain : Number Theory
% Problem : Sum of two squares line 192, 1000 axioms selected
% Version : NUM925_3 with the conjecture removed
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [TPTP]
% Names :
% Status : Satisfiable
% Rating : 1.00 v8.2.0
% Syntax : Number of formulae : 1285 ( 468 unt; 81 typ; 0 def)
% Number of atoms : 2604 ( 889 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 1618 ( 218 ~; 66 |; 148 &)
% ( 335 <=>; 851 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 4 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 60 ( 36 >; 24 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 67 ( 67 usr; 32 con; 0-3 aty)
% Number of variables : 2503 (2483 !; 20 ?;2503 :)
% SPC : TF0_SAT_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:48:35
%------------------------------------------------------------------------------
%----Should-be-implicit typings (13)
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__Int__Oint_Mtc__HOL__Obool_J,type,
fun_int_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__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J,type,
fun_re413263731l_bool: $tType ).
%----Explicit typings (68)
tff(sy_c_Groups_Oabs__class_Oabs_000tc__Int__Oint,type,
abs_abs_int: int > int ).
tff(sy_c_Groups_Oabs__class_Oabs_000tc__RealDef__Oreal,type,
abs_abs_real: real > real ).
tff(sy_c_Groups_Ominus__class_Ominus_000tc__Int__Oint,type,
minus_minus_int: ( int * int ) > int ).
tff(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
minus_minus_nat: ( nat * nat ) > nat ).
tff(sy_c_Groups_Ominus__class_Ominus_000tc__RealDef__Oreal,type,
minus_minus_real: ( real * real ) > real ).
tff(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,
one_one_int: int ).
tff(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
one_one_nat: nat ).
tff(sy_c_Groups_Oone__class_Oone_000tc__RealDef__Oreal,type,
one_one_real: real ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
plus_plus_int: ( int * int ) > int ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
plus_plus_nat: ( nat * nat ) > nat ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__RealDef__Oreal,type,
plus_plus_real: ( real * real ) > real ).
tff(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
times_times_int: ( int * int ) > int ).
tff(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
times_times_nat: ( nat * nat ) > nat ).
tff(sy_c_Groups_Otimes__class_Otimes_000tc__RealDef__Oreal,type,
times_times_real: ( real * real ) > real ).
tff(sy_c_Groups_Ozero__class_Ozero_000tc__Int__Oint,type,
zero_zero_int: int ).
tff(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_nat: nat ).
tff(sy_c_Groups_Ozero__class_Ozero_000tc__RealDef__Oreal,type,
zero_zero_real: real ).
tff(sy_c_If_000tc__Int__Oint,type,
if_int: ( bool * int * int ) > int ).
tff(sy_c_If_000tc__Nat__Onat,type,
if_nat: ( bool * nat * nat ) > nat ).
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_Onat,type,
nat_1: int > nat ).
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_Int_Osucc,type,
succ: int > int ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Int__Oint,type,
semiri1621563631at_int: fun_nat_int ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Nat__Onat,type,
semiri984289939at_nat: fun_nat_nat ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__RealDef__Oreal,type,
semiri132038758t_real: fun_nat_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_Residues_OLegendre,type,
legendre: ( int * int ) > int ).
tff(sy_c_Residues_OQuadRes,type,
quadRes: int > fun_int_bool ).
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_TwoSquares__Mirabelle__klrirdmhmq_Ois__sum2sq,type,
twoSqu820444569sum2sq: fun_int_bool ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fTrue,type,
fTrue: 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__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__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_v_m,type,
m: int ).
tff(sy_v_m1____,type,
m1: int ).
tff(sy_v_n____,type,
n: nat ).
tff(sy_v_s1____,type,
s1: int ).
tff(sy_v_s____,type,
s: int ).
tff(sy_v_t____,type,
t: int ).
tff(sy_v_tn____,type,
tn: nat ).
tff(sy_v_x____,type,
x: int ).
tff(sy_v_y____,type,
y: int ).
%----Relevant facts (1198)
tff(fact_0_n1pos,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n)))) ).
tff(fact_1_t1,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),t)) ).
tff(fact_2_sum__power2__eq__zero__iff,axiom,
! [Xa: int,Ya: int] :
( ( plus_plus_int(hAPP_nat_int(power_power_int(Xa),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(Ya),number_number_of_nat(bit0(bit1(pls))))) = zero_zero_int )
<=> ( ( Xa = zero_zero_int )
& ( Ya = zero_zero_int ) ) ) ).
tff(fact_3_sum__power2__eq__zero__iff,axiom,
! [Xa: real,Ya: real] :
( ( plus_plus_real(hAPP_nat_real(power_power_real(Xa),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_real(power_power_real(Ya),number_number_of_nat(bit0(bit1(pls))))) = zero_zero_real )
<=> ( ( Xa = zero_zero_real )
& ( Ya = zero_zero_real ) ) ) ).
tff(fact_4_one__power2,axiom,
hAPP_nat_int(power_power_int(one_one_int),number_number_of_nat(bit0(bit1(pls)))) = one_one_int ).
tff(fact_5_one__power2,axiom,
hAPP_nat_nat(power_power_nat(one_one_nat),number_number_of_nat(bit0(bit1(pls)))) = one_one_nat ).
tff(fact_6_one__power2,axiom,
hAPP_nat_real(power_power_real(one_one_real),number_number_of_nat(bit0(bit1(pls)))) = one_one_real ).
tff(fact_7_zero__power2,axiom,
hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_int ).
tff(fact_8_zero__power2,axiom,
hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_nat ).
tff(fact_9_zero__power2,axiom,
hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_real ).
tff(fact_10_zero__eq__power2,axiom,
! [A_1: int] :
( ( hAPP_nat_int(power_power_int(A_1),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_int )
<=> ( A_1 = zero_zero_int ) ) ).
tff(fact_11_zero__eq__power2,axiom,
! [A_1: real] :
( ( hAPP_nat_real(power_power_real(A_1),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_real )
<=> ( A_1 = zero_zero_real ) ) ).
tff(fact_12_add__special_I2_J,axiom,
! [W_13: int] : plus_plus_int(one_one_int,number_number_of_int(W_13)) = number_number_of_int(plus_plus_int(bit1(pls),W_13)) ).
tff(fact_13_add__special_I2_J,axiom,
! [W_13: int] : plus_plus_real(one_one_real,number267125858f_real(W_13)) = number267125858f_real(plus_plus_int(bit1(pls),W_13)) ).
tff(fact_14_add__special_I3_J,axiom,
! [V_17: int] : plus_plus_int(number_number_of_int(V_17),one_one_int) = number_number_of_int(plus_plus_int(V_17,bit1(pls))) ).
tff(fact_15_add__special_I3_J,axiom,
! [V_17: int] : plus_plus_real(number267125858f_real(V_17),one_one_real) = number267125858f_real(plus_plus_int(V_17,bit1(pls))) ).
tff(fact_16_one__add__one__is__two,axiom,
plus_plus_int(one_one_int,one_one_int) = number_number_of_int(bit0(bit1(pls))) ).
tff(fact_17_one__add__one__is__two,axiom,
plus_plus_real(one_one_real,one_one_real) = number267125858f_real(bit0(bit1(pls))) ).
tff(fact_18_semiring__one__add__one__is__two,axiom,
plus_plus_int(one_one_int,one_one_int) = number_number_of_int(bit0(bit1(pls))) ).
tff(fact_19_semiring__one__add__one__is__two,axiom,
plus_plus_nat(one_one_nat,one_one_nat) = number_number_of_nat(bit0(bit1(pls))) ).
tff(fact_20_semiring__one__add__one__is__two,axiom,
plus_plus_real(one_one_real,one_one_real) = number267125858f_real(bit0(bit1(pls))) ).
tff(fact_21_quartic__square__square,axiom,
! [X: int] : hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit1(pls))))),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit0(bit1(pls))))) ).
tff(fact_22_power__0__left__number__of,axiom,
! [W_12: int] :
( ( ( number_number_of_nat(W_12) = zero_zero_nat )
=> ( hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(W_12)) = one_one_int ) )
& ( ( number_number_of_nat(W_12) != zero_zero_nat )
=> ( hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(W_12)) = zero_zero_int ) ) ) ).
tff(fact_23_power__0__left__number__of,axiom,
! [W_12: int] :
( ( ( number_number_of_nat(W_12) = zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(W_12)) = one_one_nat ) )
& ( ( number_number_of_nat(W_12) != zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(W_12)) = zero_zero_nat ) ) ) ).
tff(fact_24_power__0__left__number__of,axiom,
! [W_12: int] :
( ( ( number_number_of_nat(W_12) = zero_zero_nat )
=> ( hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(W_12)) = one_one_real ) )
& ( ( number_number_of_nat(W_12) != zero_zero_nat )
=> ( hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(W_12)) = zero_zero_real ) ) ) ).
tff(fact_25_semiring__norm_I110_J,axiom,
one_one_int = number_number_of_int(bit1(pls)) ).
tff(fact_26_semiring__norm_I110_J,axiom,
one_one_real = number267125858f_real(bit1(pls)) ).
tff(fact_27_numeral__1__eq__1,axiom,
number_number_of_int(bit1(pls)) = one_one_int ).
tff(fact_28_numeral__1__eq__1,axiom,
number267125858f_real(bit1(pls)) = one_one_real ).
tff(fact_29_n0,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),n)) ).
tff(fact_30_zless__linear,axiom,
! [X: int,Y: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Y))
| ( X = Y )
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),X)) ) ).
tff(fact_31_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_32_plus__numeral__code_I9_J,axiom,
! [V_1: int,W: int] : plus_plus_int(number_number_of_int(V_1),number_number_of_int(W)) = number_number_of_int(plus_plus_int(V_1,W)) ).
tff(fact_33_less__number__of,axiom,
! [Xa: int,Ya: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(Xa)),number_number_of_int(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) ) ).
tff(fact_34_less__number__of,axiom,
! [Xa: int,Ya: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(Xa)),number267125858f_real(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) ) ).
tff(fact_35_zero__is__num__zero,axiom,
zero_zero_int = number_number_of_int(pls) ).
tff(fact_36_zpower__int,axiom,
! [M: nat,N: nat] : hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,M)),N) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(M),N)) ).
tff(fact_37_int__power,axiom,
! [M: nat,N: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(M),N)) = hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,M)),N) ).
tff(fact_38_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] : plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M),plus_plus_int(hAPP_nat_int(semiri1621563631at_int,N),Z)) = plus_plus_int(hAPP_nat_int(semiri1621563631at_int,plus_plus_nat(M,N)),Z) ).
tff(fact_39_zadd__int,axiom,
! [M: nat,N: nat] : plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,plus_plus_nat(M,N)) ).
tff(fact_40_int__1,axiom,
hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int ).
tff(fact_41_nat__number__of__Pls,axiom,
number_number_of_nat(pls) = zero_zero_nat ).
tff(fact_42_semiring__norm_I113_J,axiom,
zero_zero_nat = number_number_of_nat(pls) ).
tff(fact_43_int__eq__0__conv,axiom,
! [Na: nat] :
( ( hAPP_nat_int(semiri1621563631at_int,Na) = zero_zero_int )
<=> ( Na = zero_zero_nat ) ) ).
tff(fact_44_int__0,axiom,
hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) = zero_zero_int ).
tff(fact_45_nat__1__add__1,axiom,
plus_plus_nat(one_one_nat,one_one_nat) = number_number_of_nat(bit0(bit1(pls))) ).
tff(fact_46_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_47_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_48_rel__simps_I2_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),pls)) ).
tff(fact_49_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_50_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_51_zadd__strict__right__mono,axiom,
! [K_1: int,I_2: int,J_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),J_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(I_2,K_1)),plus_plus_int(J_1,K_1))) ) ).
tff(fact_52_add__nat__number__of,axiom,
! [V_2: int,V_1: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> ( plus_plus_nat(number_number_of_nat(V_1),number_number_of_nat(V_2)) = number_number_of_nat(V_2) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> ( plus_plus_nat(number_number_of_nat(V_1),number_number_of_nat(V_2)) = number_number_of_nat(V_1) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> ( plus_plus_nat(number_number_of_nat(V_1),number_number_of_nat(V_2)) = number_number_of_nat(plus_plus_int(V_1,V_2)) ) ) ) ) ) ).
tff(fact_53_one__is__num__one,axiom,
one_one_int = number_number_of_int(bit1(pls)) ).
tff(fact_54_nat__numeral__1__eq__1,axiom,
number_number_of_nat(bit1(pls)) = one_one_nat ).
tff(fact_55_Numeral1__eq1__nat,axiom,
one_one_nat = number_number_of_nat(bit1(pls)) ).
tff(fact_56_eq__number__of,axiom,
! [Xa: int,Ya: int] :
( ( number_number_of_int(Xa) = number_number_of_int(Ya) )
<=> ( Xa = Ya ) ) ).
tff(fact_57_eq__number__of,axiom,
! [Xa: int,Ya: int] :
( ( number267125858f_real(Xa) = number267125858f_real(Ya) )
<=> ( Xa = Ya ) ) ).
tff(fact_58_number__of__reorient,axiom,
! [Wa: int,Xa: nat] :
( ( number_number_of_nat(Wa) = Xa )
<=> ( Xa = number_number_of_nat(Wa) ) ) ).
tff(fact_59_number__of__reorient,axiom,
! [Wa: int,Xa: int] :
( ( number_number_of_int(Wa) = Xa )
<=> ( Xa = number_number_of_int(Wa) ) ) ).
tff(fact_60_number__of__reorient,axiom,
! [Wa: int,Xa: real] :
( ( number267125858f_real(Wa) = Xa )
<=> ( Xa = number267125858f_real(Wa) ) ) ).
tff(fact_61_rel__simps_I51_J,axiom,
! [K: int,L_1: int] :
( ( bit1(K) = bit1(L_1) )
<=> ( K = L_1 ) ) ).
tff(fact_62_rel__simps_I48_J,axiom,
! [K: int,L_1: int] :
( ( bit0(K) = bit0(L_1) )
<=> ( K = L_1 ) ) ).
tff(fact_63_even__less__0__iff,axiom,
! [A_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_1,A_1)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),zero_zero_int)) ) ).
tff(fact_64_even__less__0__iff,axiom,
! [A_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_1,A_1)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),zero_zero_real)) ) ).
tff(fact_65_zadd__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] : plus_plus_int(plus_plus_int(Z1,Z2),Z3) = plus_plus_int(Z1,plus_plus_int(Z2,Z3)) ).
tff(fact_66_zadd__left__commute,axiom,
! [X: int,Y: int,Z: int] : plus_plus_int(X,plus_plus_int(Y,Z)) = plus_plus_int(Y,plus_plus_int(X,Z)) ).
tff(fact_67_zadd__commute,axiom,
! [Z: int,W: int] : plus_plus_int(Z,W) = plus_plus_int(W,Z) ).
tff(fact_68_int__int__eq,axiom,
! [Ma: nat,Na: nat] :
( ( hAPP_nat_int(semiri1621563631at_int,Ma) = hAPP_nat_int(semiri1621563631at_int,Na) )
<=> ( Ma = Na ) ) ).
tff(fact_69_less__special_I3_J,axiom,
! [Xa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(Xa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),pls)) ) ).
tff(fact_70_less__special_I3_J,axiom,
! [Xa: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(Xa)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),pls)) ) ).
tff(fact_71_less__special_I1_J,axiom,
! [Ya: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),number_number_of_int(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Ya)) ) ).
tff(fact_72_less__special_I1_J,axiom,
! [Ya: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),number267125858f_real(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Ya)) ) ).
tff(fact_73_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_74_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_75_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_76_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_77_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_78_bin__less__0__simps_I4_J,axiom,
! [Wa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(Wa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),zero_zero_int)) ) ).
tff(fact_79_bin__less__0__simps_I1_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),zero_zero_int)) ).
tff(fact_80_bin__less__0__simps_I3_J,axiom,
! [Wa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(Wa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),zero_zero_int)) ) ).
tff(fact_81_int__0__less__1,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
tff(fact_82_zless__add1__eq,axiom,
! [Wa: int,Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),plus_plus_int(Z_1,one_one_int)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_1))
| ( Wa = Z_1 ) ) ) ).
tff(fact_83_int__less__0__conv,axiom,
! [K_1: nat] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,K_1)),zero_zero_int)) ).
tff(fact_84_less__special_I4_J,axiom,
! [Xa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(Xa)),one_one_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),bit1(pls))) ) ).
tff(fact_85_less__special_I4_J,axiom,
! [Xa: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(Xa)),one_one_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),bit1(pls))) ) ).
tff(fact_86_less__special_I2_J,axiom,
! [Ya: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),number_number_of_int(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(pls)),Ya)) ) ).
tff(fact_87_less__special_I2_J,axiom,
! [Ya: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),number267125858f_real(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(pls)),Ya)) ) ).
tff(fact_88_odd__less__0,axiom,
! [Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_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_89_double__eq__0__iff,axiom,
! [A_1: int] :
( ( plus_plus_int(A_1,A_1) = zero_zero_int )
<=> ( A_1 = zero_zero_int ) ) ).
tff(fact_90_double__eq__0__iff,axiom,
! [A_1: real] :
( ( plus_plus_real(A_1,A_1) = zero_zero_real )
<=> ( A_1 = zero_zero_real ) ) ).
tff(fact_91_rel__simps_I46_J,axiom,
! [K_1: int] : bit1(K_1) != pls ).
tff(fact_92_rel__simps_I39_J,axiom,
! [L: int] : pls != bit1(L) ).
tff(fact_93_rel__simps_I50_J,axiom,
! [K_1: int,L: int] : bit1(K_1) != bit0(L) ).
tff(fact_94_rel__simps_I49_J,axiom,
! [K_1: int,L: int] : bit0(K_1) != bit1(L) ).
tff(fact_95_rel__simps_I44_J,axiom,
! [K: int] :
( ( bit0(K) = pls )
<=> ( K = pls ) ) ).
tff(fact_96_rel__simps_I38_J,axiom,
! [L_1: int] :
( ( pls = bit0(L_1) )
<=> ( pls = L_1 ) ) ).
tff(fact_97_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_98_Pls__def,axiom,
pls = zero_zero_int ).
tff(fact_99_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
tff(fact_100_add__Pls__right,axiom,
! [K_1: int] : plus_plus_int(K_1,pls) = K_1 ).
tff(fact_101_add__Pls,axiom,
! [K_1: int] : plus_plus_int(pls,K_1) = K_1 ).
tff(fact_102_add__Bit0__Bit0,axiom,
! [K_1: int,L: int] : plus_plus_int(bit0(K_1),bit0(L)) = bit0(plus_plus_int(K_1,L)) ).
tff(fact_103_Bit0__def,axiom,
! [K_1: int] : bit0(K_1) = plus_plus_int(K_1,K_1) ).
tff(fact_104_zadd__0__right,axiom,
! [Z: int] : plus_plus_int(Z,zero_zero_int) = Z ).
tff(fact_105_zadd__0,axiom,
! [Z: int] : plus_plus_int(zero_zero_int,Z) = Z ).
tff(fact_106_semiring__numeral__0__eq__0,axiom,
number_number_of_int(pls) = zero_zero_int ).
tff(fact_107_semiring__numeral__0__eq__0,axiom,
number_number_of_nat(pls) = zero_zero_nat ).
tff(fact_108_semiring__numeral__0__eq__0,axiom,
number267125858f_real(pls) = zero_zero_real ).
tff(fact_109_number__of__Pls,axiom,
number_number_of_int(pls) = zero_zero_int ).
tff(fact_110_number__of__Pls,axiom,
number267125858f_real(pls) = zero_zero_real ).
tff(fact_111_semiring__norm_I112_J,axiom,
zero_zero_int = number_number_of_int(pls) ).
tff(fact_112_semiring__norm_I112_J,axiom,
zero_zero_real = number267125858f_real(pls) ).
tff(fact_113_add__numeral__0,axiom,
! [A_100: int] : plus_plus_int(number_number_of_int(pls),A_100) = A_100 ).
tff(fact_114_add__numeral__0,axiom,
! [A_100: real] : plus_plus_real(number267125858f_real(pls),A_100) = A_100 ).
tff(fact_115_add__numeral__0__right,axiom,
! [A_99: int] : plus_plus_int(A_99,number_number_of_int(pls)) = A_99 ).
tff(fact_116_add__numeral__0__right,axiom,
! [A_99: real] : plus_plus_real(A_99,number267125858f_real(pls)) = A_99 ).
tff(fact_117_power__eq__0__iff__number__of,axiom,
! [A_1: int,Wa: int] :
( ( hAPP_nat_int(power_power_int(A_1),number_number_of_nat(Wa)) = zero_zero_int )
<=> ( ( A_1 = zero_zero_int )
& ( number_number_of_nat(Wa) != zero_zero_nat ) ) ) ).
tff(fact_118_power__eq__0__iff__number__of,axiom,
! [A_1: nat,Wa: int] :
( ( hAPP_nat_nat(power_power_nat(A_1),number_number_of_nat(Wa)) = zero_zero_nat )
<=> ( ( A_1 = zero_zero_nat )
& ( number_number_of_nat(Wa) != zero_zero_nat ) ) ) ).
tff(fact_119_power__eq__0__iff__number__of,axiom,
! [A_1: real,Wa: int] :
( ( hAPP_nat_real(power_power_real(A_1),number_number_of_nat(Wa)) = zero_zero_real )
<=> ( ( A_1 = zero_zero_real )
& ( number_number_of_nat(Wa) != zero_zero_nat ) ) ) ).
tff(fact_120_add__number__of__left,axiom,
! [V_16: int,W_11: int,Z_6: int] : plus_plus_int(number_number_of_int(V_16),plus_plus_int(number_number_of_int(W_11),Z_6)) = plus_plus_int(number_number_of_int(plus_plus_int(V_16,W_11)),Z_6) ).
tff(fact_121_add__number__of__left,axiom,
! [V_16: int,W_11: int,Z_6: real] : plus_plus_real(number267125858f_real(V_16),plus_plus_real(number267125858f_real(W_11),Z_6)) = plus_plus_real(number267125858f_real(plus_plus_int(V_16,W_11)),Z_6) ).
tff(fact_122_add__number__of__eq,axiom,
! [V_15: int,W_10: int] : plus_plus_int(number_number_of_int(V_15),number_number_of_int(W_10)) = number_number_of_int(plus_plus_int(V_15,W_10)) ).
tff(fact_123_add__number__of__eq,axiom,
! [V_15: int,W_10: int] : plus_plus_real(number267125858f_real(V_15),number267125858f_real(W_10)) = number267125858f_real(plus_plus_int(V_15,W_10)) ).
tff(fact_124_number__of__add,axiom,
! [V_14: int,W_9: int] : number_number_of_int(plus_plus_int(V_14,W_9)) = plus_plus_int(number_number_of_int(V_14),number_number_of_int(W_9)) ).
tff(fact_125_number__of__add,axiom,
! [V_14: int,W_9: int] : number267125858f_real(plus_plus_int(V_14,W_9)) = plus_plus_real(number267125858f_real(V_14),number267125858f_real(W_9)) ).
tff(fact_126_add__Bit1__Bit0,axiom,
! [K_1: int,L: int] : plus_plus_int(bit1(K_1),bit0(L)) = bit1(plus_plus_int(K_1,L)) ).
tff(fact_127_add__Bit0__Bit1,axiom,
! [K_1: int,L: int] : plus_plus_int(bit0(K_1),bit1(L)) = bit1(plus_plus_int(K_1,L)) ).
tff(fact_128_Bit1__def,axiom,
! [K_1: int] : bit1(K_1) = plus_plus_int(plus_plus_int(one_one_int,K_1),K_1) ).
tff(fact_129_odd__nonzero,axiom,
! [Z: int] : plus_plus_int(plus_plus_int(one_one_int,Z),Z) != zero_zero_int ).
tff(fact_130_number__of__int,axiom,
! [N_26: nat] : number_number_of_nat(hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_nat(semiri984289939at_nat,N_26) ).
tff(fact_131_number__of__int,axiom,
! [N_26: nat] : number_number_of_int(hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_int(semiri1621563631at_int,N_26) ).
tff(fact_132_number__of__int,axiom,
! [N_26: nat] : number267125858f_real(hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_real(semiri132038758t_real,N_26) ).
tff(fact_133_zero__less__power2,axiom,
! [A_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(power_power_int(A_1),number_number_of_nat(bit0(bit1(pls))))))
<=> ( A_1 != zero_zero_int ) ) ).
tff(fact_134_zero__less__power2,axiom,
! [A_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(power_power_real(A_1),number_number_of_nat(bit0(bit1(pls))))))
<=> ( A_1 != zero_zero_real ) ) ).
tff(fact_135_power2__less__0,axiom,
! [A_98: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_98),number_number_of_nat(bit0(bit1(pls))))),zero_zero_int)) ).
tff(fact_136_power2__less__0,axiom,
! [A_98: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_98),number_number_of_nat(bit0(bit1(pls))))),zero_zero_real)) ).
tff(fact_137_sum__power2__gt__zero__iff,axiom,
! [Xa: int,Ya: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(hAPP_nat_int(power_power_int(Xa),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(Ya),number_number_of_nat(bit0(bit1(pls)))))))
<=> ( ( Xa != zero_zero_int )
| ( Ya != zero_zero_int ) ) ) ).
tff(fact_138_sum__power2__gt__zero__iff,axiom,
! [Xa: real,Ya: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),plus_plus_real(hAPP_nat_real(power_power_real(Xa),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_real(power_power_real(Ya),number_number_of_nat(bit0(bit1(pls)))))))
<=> ( ( Xa != zero_zero_real )
| ( Ya != zero_zero_real ) ) ) ).
tff(fact_139_not__sum__power2__lt__zero,axiom,
! [X_15: int,Y_11: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(hAPP_nat_int(power_power_int(X_15),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(Y_11),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_int)) ).
tff(fact_140_not__sum__power2__lt__zero,axiom,
! [X_15: real,Y_11: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(hAPP_nat_real(power_power_real(X_15),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_real(power_power_real(Y_11),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_real)) ).
tff(fact_141_number__of__Bit0,axiom,
! [W_8: int] : number_number_of_int(bit0(W_8)) = plus_plus_int(plus_plus_int(zero_zero_int,number_number_of_int(W_8)),number_number_of_int(W_8)) ).
tff(fact_142_number__of__Bit0,axiom,
! [W_8: int] : number267125858f_real(bit0(W_8)) = plus_plus_real(plus_plus_real(zero_zero_real,number267125858f_real(W_8)),number267125858f_real(W_8)) ).
tff(fact_143_number__of__Bit1,axiom,
! [W_7: int] : number_number_of_int(bit1(W_7)) = plus_plus_int(plus_plus_int(one_one_int,number_number_of_int(W_7)),number_number_of_int(W_7)) ).
tff(fact_144_number__of__Bit1,axiom,
! [W_7: int] : number267125858f_real(bit1(W_7)) = plus_plus_real(plus_plus_real(one_one_real,number267125858f_real(W_7)),number267125858f_real(W_7)) ).
tff(fact_145_semiring__numeral__1__eq__1,axiom,
number_number_of_int(bit1(pls)) = one_one_int ).
tff(fact_146_semiring__numeral__1__eq__1,axiom,
number_number_of_nat(bit1(pls)) = one_one_nat ).
tff(fact_147_semiring__numeral__1__eq__1,axiom,
number267125858f_real(bit1(pls)) = one_one_real ).
tff(fact_148_mn,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,m1),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n)))) ).
tff(fact_149_of__nat__less__two__power,axiom,
! [N_25: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N_25)),hAPP_nat_int(power_power_int(number_number_of_int(bit0(bit1(pls)))),N_25))) ).
tff(fact_150_of__nat__less__two__power,axiom,
! [N_25: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,N_25)),hAPP_nat_real(power_power_real(number267125858f_real(bit0(bit1(pls)))),N_25))) ).
tff(fact_151_transfer__int__nat__numerals_I3_J,axiom,
number_number_of_int(bit0(bit1(pls))) = hAPP_nat_int(semiri1621563631at_int,number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_152_transfer__int__nat__numerals_I4_J,axiom,
number_number_of_int(bit1(bit1(pls))) = hAPP_nat_int(semiri1621563631at_int,number_number_of_nat(bit1(bit1(pls)))) ).
tff(fact_153_realpow__two__sum__zero__iff,axiom,
! [Xa: real,Ya: real] :
( ( plus_plus_real(hAPP_nat_real(power_power_real(Xa),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_real(power_power_real(Ya),number_number_of_nat(bit0(bit1(pls))))) = zero_zero_real )
<=> ( ( Xa = zero_zero_real )
& ( Ya = zero_zero_real ) ) ) ).
tff(fact_154_of__nat__0__less__iff,axiom,
! [Na: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).
tff(fact_155_of__nat__0__less__iff,axiom,
! [Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).
tff(fact_156_of__nat__0__less__iff,axiom,
! [Na: nat] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).
tff(fact_157_one__less__power,axiom,
! [N_24: nat,A_97: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_97))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_24))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_nat_int(power_power_int(A_97),N_24))) ) ) ).
tff(fact_158_one__less__power,axiom,
! [N_24: nat,A_97: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_97))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_24))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(power_power_nat(A_97),N_24))) ) ) ).
tff(fact_159_one__less__power,axiom,
! [N_24: nat,A_97: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_97))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_24))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_nat_real(power_power_real(A_97),N_24))) ) ) ).
tff(fact_160_power__0__left,axiom,
! [N_23: nat] :
( ( ( N_23 = zero_zero_nat )
=> ( hAPP_nat_int(power_power_int(zero_zero_int),N_23) = one_one_int ) )
& ( ( N_23 != zero_zero_nat )
=> ( hAPP_nat_int(power_power_int(zero_zero_int),N_23) = zero_zero_int ) ) ) ).
tff(fact_161_power__0__left,axiom,
! [N_23: nat] :
( ( ( N_23 = zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(zero_zero_nat),N_23) = one_one_nat ) )
& ( ( N_23 != zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(zero_zero_nat),N_23) = zero_zero_nat ) ) ) ).
tff(fact_162_power__0__left,axiom,
! [N_23: nat] :
( ( ( N_23 = zero_zero_nat )
=> ( hAPP_nat_real(power_power_real(zero_zero_real),N_23) = one_one_real ) )
& ( ( N_23 != zero_zero_nat )
=> ( hAPP_nat_real(power_power_real(zero_zero_real),N_23) = zero_zero_real ) ) ) ).
tff(fact_163_power__strict__decreasing,axiom,
! [A_96: int,N_22: nat,N_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_22),N_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_96))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_96),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_96),N_21)),hAPP_nat_int(power_power_int(A_96),N_22))) ) ) ) ).
tff(fact_164_power__strict__decreasing,axiom,
! [A_96: nat,N_22: nat,N_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_22),N_21))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_96))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_96),one_one_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_96),N_21)),hAPP_nat_nat(power_power_nat(A_96),N_22))) ) ) ) ).
tff(fact_165_power__strict__decreasing,axiom,
! [A_96: real,N_22: nat,N_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_22),N_21))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_96))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_96),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_96),N_21)),hAPP_nat_real(power_power_real(A_96),N_22))) ) ) ) ).
tff(fact_166_zero__less__two,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(one_one_int,one_one_int))) ).
tff(fact_167_zero__less__two,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),plus_plus_nat(one_one_nat,one_one_nat))) ).
tff(fact_168_zero__less__two,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),plus_plus_real(one_one_real,one_one_real))) ).
tff(fact_169_int__gr__induct,axiom,
! [P_1: fun_int_bool,K: int,I_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),I_1))
=> ( hBOOL(hAPP_int_bool(P_1,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,plus_plus_int(I,one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).
tff(fact_170_transfer__int__nat__numerals_I1_J,axiom,
zero_zero_int = hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) ).
tff(fact_171_tn0,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),tn)) ).
tff(fact_172_less__zeroE,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
tff(fact_173_real__zero__not__eq__one,axiom,
zero_zero_real != one_one_real ).
tff(fact_174_less__not__refl,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).
tff(fact_175_not__add__less1,axiom,
! [I_2: nat,J_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I_2,J_1)),I_2)) ).
tff(fact_176_not__add__less2,axiom,
! [J_1: nat,I_2: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(J_1,I_2)),I_2)) ).
tff(fact_177_number__of__is__id,axiom,
! [K_1: int] : number_number_of_int(K_1) = K_1 ).
tff(fact_178_nat__neq__iff,axiom,
! [Ma: nat,Na: nat] :
( ( Ma != Na )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma)) ) ) ).
tff(fact_179_nat__add__commute,axiom,
! [M: nat,N: nat] : plus_plus_nat(M,N) = plus_plus_nat(N,M) ).
tff(fact_180_nat__add__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] : plus_plus_nat(X,plus_plus_nat(Y,Z)) = plus_plus_nat(Y,plus_plus_nat(X,Z)) ).
tff(fact_181_nat__add__assoc,axiom,
! [M: nat,N: nat,K_1: nat] : plus_plus_nat(plus_plus_nat(M,N),K_1) = plus_plus_nat(M,plus_plus_nat(N,K_1)) ).
tff(fact_182_nat__add__left__cancel,axiom,
! [K: nat,Ma: nat,Na: nat] :
( ( plus_plus_nat(K,Ma) = plus_plus_nat(K,Na) )
<=> ( Ma = Na ) ) ).
tff(fact_183_nat__add__right__cancel,axiom,
! [Ma: nat,K: nat,Na: nat] :
( ( plus_plus_nat(Ma,K) = plus_plus_nat(Na,K) )
<=> ( Ma = Na ) ) ).
tff(fact_184_nat__add__left__cancel__less,axiom,
! [K: nat,Ma: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(K,Ma)),plus_plus_nat(K,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
tff(fact_185_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y),X)) ) ) ).
tff(fact_186_less__irrefl__nat,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).
tff(fact_187_less__not__refl2,axiom,
! [N: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
=> ( M != N ) ) ).
tff(fact_188_less__not__refl3,axiom,
! [S_1: nat,T_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S_1),T_1))
=> ( S_1 != T_1 ) ) ).
tff(fact_189_trans__less__add1,axiom,
! [M: nat,I_2: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),plus_plus_nat(J_1,M))) ) ).
tff(fact_190_trans__less__add2,axiom,
! [M: nat,I_2: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),plus_plus_nat(M,J_1))) ) ).
tff(fact_191_add__less__mono1,axiom,
! [K_1: nat,I_2: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I_2,K_1)),plus_plus_nat(J_1,K_1))) ) ).
tff(fact_192_add__less__mono,axiom,
! [K_1: nat,L: nat,I_2: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),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,plus_plus_nat(I_2,K_1)),plus_plus_nat(J_1,L))) ) ) ).
tff(fact_193_less__add__eq__less,axiom,
! [M: nat,N: nat,K_1: nat,L: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),L))
=> ( ( plus_plus_nat(M,L) = plus_plus_nat(K_1,N) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
tff(fact_194_add__lessD1,axiom,
! [I_2: nat,J_1: nat,K_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I_2,J_1)),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),K_1)) ) ).
tff(fact_195_nat__less__cases,axiom,
! [P_1: fun_nat_fun_nat_bool,Ma: nat,Na: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma)) )
=> ( ( ( Ma = Na )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma)) ) ) ) ).
tff(fact_196_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
tff(fact_197_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ( N != zero_zero_nat ) ) ).
tff(fact_198_nat__power__less__imp__less,axiom,
! [M: nat,N: nat,I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),I_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(I_2),M)),hAPP_nat_nat(power_power_nat(I_2),N)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
tff(fact_199_less__nat__zero__code,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
tff(fact_200_nat__zero__less__power__iff,axiom,
! [Xa: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(Xa),Na)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Xa))
| ( Na = zero_zero_nat ) ) ) ).
tff(fact_201_add__gr__0,axiom,
! [Ma: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),plus_plus_nat(Ma,Na)))
<=> ( 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),Na)) ) ) ).
tff(fact_202_neq0__conv,axiom,
! [Na: nat] :
( ( Na != zero_zero_nat )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).
tff(fact_203_not__less0,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
tff(fact_204_zero__less__power__nat__eq,axiom,
! [Xa: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(Xa),Na)))
<=> ( ( Na = zero_zero_nat )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Xa)) ) ) ).
tff(fact_205_Nat__Transfer_Otransfer__int__nat__relations_I2_J,axiom,
! [Xa: nat,Ya: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) ) ).
tff(fact_206_zero__less__power__nat__eq__number__of,axiom,
! [Xa: nat,Wa: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(Xa),number_number_of_nat(Wa))))
<=> ( ( number_number_of_nat(Wa) = zero_zero_nat )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Xa)) ) ) ).
tff(fact_207_zless__int,axiom,
! [Ma: nat,Na: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
tff(fact_208_less__nat__number__of,axiom,
! [Va: int,V_3: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,number_number_of_nat(Va)),number_number_of_nat(V_3)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Va),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,Va),V_3)) ) ) ).
tff(fact_209_linorder__neqE__linordered__idom,axiom,
! [X_14: int,Y_10: int] :
( ( X_14 != Y_10 )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_14),Y_10))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_10),X_14)) ) ) ).
tff(fact_210_linorder__neqE__linordered__idom,axiom,
! [X_14: real,Y_10: real] :
( ( X_14 != Y_10 )
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_14),Y_10))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_10),X_14)) ) ) ).
tff(fact_211_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat(M,N) = M )
=> ( N = zero_zero_nat ) ) ).
tff(fact_212_add__is__0,axiom,
! [Ma: nat,Na: nat] :
( ( plus_plus_nat(Ma,Na) = zero_zero_nat )
<=> ( ( Ma = zero_zero_nat )
& ( Na = zero_zero_nat ) ) ) ).
tff(fact_213_Nat_Oadd__0__right,axiom,
! [M: nat] : plus_plus_nat(M,zero_zero_nat) = M ).
tff(fact_214_plus__nat_Oadd__0,axiom,
! [N: nat] : plus_plus_nat(zero_zero_nat,N) = N ).
tff(fact_215_power__one__right,axiom,
! [A_95: int] : hAPP_nat_int(power_power_int(A_95),one_one_nat) = A_95 ).
tff(fact_216_power__one__right,axiom,
! [A_95: nat] : hAPP_nat_nat(power_power_nat(A_95),one_one_nat) = A_95 ).
tff(fact_217_power__one__right,axiom,
! [A_95: real] : hAPP_nat_real(power_power_real(A_95),one_one_nat) = A_95 ).
tff(fact_218_of__nat__eq__iff,axiom,
! [Ma: nat,Na: nat] :
( ( hAPP_nat_real(semiri132038758t_real,Ma) = hAPP_nat_real(semiri132038758t_real,Na) )
<=> ( Ma = Na ) ) ).
tff(fact_219_of__nat__eq__iff,axiom,
! [Ma: nat,Na: nat] :
( ( hAPP_nat_nat(semiri984289939at_nat,Ma) = hAPP_nat_nat(semiri984289939at_nat,Na) )
<=> ( Ma = Na ) ) ).
tff(fact_220_of__nat__eq__iff,axiom,
! [Ma: nat,Na: nat] :
( ( hAPP_nat_int(semiri1621563631at_int,Ma) = hAPP_nat_int(semiri1621563631at_int,Na) )
<=> ( Ma = Na ) ) ).
tff(fact_221_Nat__Transfer_Otransfer__int__nat__relations_I1_J,axiom,
! [Xa: nat,Ya: nat] :
( ( hAPP_nat_int(semiri1621563631at_int,Xa) = hAPP_nat_int(semiri1621563631at_int,Ya) )
<=> ( Xa = Ya ) ) ).
tff(fact_222_int__if__cong,axiom,
! [Xa: nat,Ya: nat,P_1: bool] :
( ( hBOOL(P_1)
=> ( hAPP_nat_int(semiri1621563631at_int,Xa) = hAPP_nat_int(semiri1621563631at_int,if_nat(P_1,Xa,Ya)) ) )
& ( ~ hBOOL(P_1)
=> ( hAPP_nat_int(semiri1621563631at_int,Ya) = hAPP_nat_int(semiri1621563631at_int,if_nat(P_1,Xa,Ya)) ) ) ) ).
tff(fact_223_less__0__number__of,axiom,
! [Va: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),number_number_of_nat(Va)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Va)) ) ).
tff(fact_224_zero__less__int__conv,axiom,
! [Na: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).
tff(fact_225_one__neq__zero,axiom,
one_one_int != zero_zero_int ).
tff(fact_226_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
tff(fact_227_one__neq__zero,axiom,
one_one_real != zero_zero_real ).
tff(fact_228_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
tff(fact_229_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
tff(fact_230_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
tff(fact_231_field__power__not__zero,axiom,
! [N_20: nat,A_94: int] :
( ( A_94 != zero_zero_int )
=> ( hAPP_nat_int(power_power_int(A_94),N_20) != zero_zero_int ) ) ).
tff(fact_232_field__power__not__zero,axiom,
! [N_20: nat,A_94: real] :
( ( A_94 != zero_zero_real )
=> ( hAPP_nat_real(power_power_real(A_94),N_20) != zero_zero_real ) ) ).
tff(fact_233_power__one,axiom,
! [N_19: nat] : hAPP_nat_int(power_power_int(one_one_int),N_19) = one_one_int ).
tff(fact_234_power__one,axiom,
! [N_19: nat] : hAPP_nat_nat(power_power_nat(one_one_nat),N_19) = one_one_nat ).
tff(fact_235_power__one,axiom,
! [N_19: nat] : hAPP_nat_real(power_power_real(one_one_real),N_19) = one_one_real ).
tff(fact_236_of__nat__less__iff,axiom,
! [Ma: nat,Na: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
tff(fact_237_of__nat__less__iff,axiom,
! [Ma: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
tff(fact_238_of__nat__less__iff,axiom,
! [Ma: nat,Na: nat] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
tff(fact_239_less__imp__of__nat__less,axiom,
! [M_10: nat,N_18: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_18))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_10)),hAPP_nat_int(semiri1621563631at_int,N_18))) ) ).
tff(fact_240_less__imp__of__nat__less,axiom,
! [M_10: nat,N_18: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_18))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_10)),hAPP_nat_nat(semiri984289939at_nat,N_18))) ) ).
tff(fact_241_less__imp__of__nat__less,axiom,
! [M_10: nat,N_18: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_18))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_10)),hAPP_nat_real(semiri132038758t_real,N_18))) ) ).
tff(fact_242_of__nat__less__imp__less,axiom,
! [M_9: nat,N_17: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_9)),hAPP_nat_int(semiri1621563631at_int,N_17)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_17)) ) ).
tff(fact_243_of__nat__less__imp__less,axiom,
! [M_9: nat,N_17: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_9)),hAPP_nat_nat(semiri984289939at_nat,N_17)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_17)) ) ).
tff(fact_244_of__nat__less__imp__less,axiom,
! [M_9: nat,N_17: nat] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_9)),hAPP_nat_real(semiri132038758t_real,N_17)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_17)) ) ).
tff(fact_245_of__nat__add,axiom,
! [M_8: nat,N_16: nat] : hAPP_nat_int(semiri1621563631at_int,plus_plus_nat(M_8,N_16)) = plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M_8),hAPP_nat_int(semiri1621563631at_int,N_16)) ).
tff(fact_246_of__nat__add,axiom,
! [M_8: nat,N_16: nat] : hAPP_nat_nat(semiri984289939at_nat,plus_plus_nat(M_8,N_16)) = plus_plus_nat(hAPP_nat_nat(semiri984289939at_nat,M_8),hAPP_nat_nat(semiri984289939at_nat,N_16)) ).
tff(fact_247_of__nat__add,axiom,
! [M_8: nat,N_16: nat] : hAPP_nat_real(semiri132038758t_real,plus_plus_nat(M_8,N_16)) = plus_plus_real(hAPP_nat_real(semiri132038758t_real,M_8),hAPP_nat_real(semiri132038758t_real,N_16)) ).
tff(fact_248_of__nat__1,axiom,
hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int ).
tff(fact_249_of__nat__1,axiom,
hAPP_nat_nat(semiri984289939at_nat,one_one_nat) = one_one_nat ).
tff(fact_250_of__nat__1,axiom,
hAPP_nat_real(semiri132038758t_real,one_one_nat) = one_one_real ).
tff(fact_251_of__nat__power,axiom,
! [M_7: nat,N_15: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(M_7),N_15)) = hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,M_7)),N_15) ).
tff(fact_252_of__nat__power,axiom,
! [M_7: nat,N_15: nat] : hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(power_power_nat(M_7),N_15)) = hAPP_nat_nat(power_power_nat(hAPP_nat_nat(semiri984289939at_nat,M_7)),N_15) ).
tff(fact_253_of__nat__power,axiom,
! [M_7: nat,N_15: nat] : hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(power_power_nat(M_7),N_15)) = hAPP_nat_real(power_power_real(hAPP_nat_real(semiri132038758t_real,M_7)),N_15) ).
tff(fact_254_transfer__int__nat__numerals_I2_J,axiom,
one_one_int = hAPP_nat_int(semiri1621563631at_int,one_one_nat) ).
tff(fact_255_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
! [X: nat,Y: nat] : plus_plus_int(hAPP_nat_int(semiri1621563631at_int,X),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,plus_plus_nat(X,Y)) ).
tff(fact_256_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
! [X: nat,N: nat] : hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,X)),N) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(X),N)) ).
tff(fact_257_pos__add__strict,axiom,
! [B_61: int,C_45: int,A_93: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_93))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_61),C_45))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_61),plus_plus_int(A_93,C_45))) ) ) ).
tff(fact_258_pos__add__strict,axiom,
! [B_61: nat,C_45: nat,A_93: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_93))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_61),C_45))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_61),plus_plus_nat(A_93,C_45))) ) ) ).
tff(fact_259_pos__add__strict,axiom,
! [B_61: real,C_45: real,A_93: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_93))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_61),C_45))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_61),plus_plus_real(A_93,C_45))) ) ) ).
tff(fact_260_not__one__less__zero,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),zero_zero_int)) ).
tff(fact_261_not__one__less__zero,axiom,
~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).
tff(fact_262_not__one__less__zero,axiom,
~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),zero_zero_real)) ).
tff(fact_263_zero__less__one,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
tff(fact_264_zero__less__one,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).
tff(fact_265_zero__less__one,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),one_one_real)) ).
tff(fact_266_zero__less__power,axiom,
! [N_14: nat,A_92: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_92))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(power_power_int(A_92),N_14))) ) ).
tff(fact_267_zero__less__power,axiom,
! [N_14: nat,A_92: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_92))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(A_92),N_14))) ) ).
tff(fact_268_zero__less__power,axiom,
! [N_14: nat,A_92: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_92))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(power_power_real(A_92),N_14))) ) ).
tff(fact_269_less__add__one,axiom,
! [A_91: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_91),plus_plus_int(A_91,one_one_int))) ).
tff(fact_270_less__add__one,axiom,
! [A_91: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_91),plus_plus_nat(A_91,one_one_nat))) ).
tff(fact_271_less__add__one,axiom,
! [A_91: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_91),plus_plus_real(A_91,one_one_real))) ).
tff(fact_272_power__inject__exp,axiom,
! [Ma: nat,Na: nat,A_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_1))
=> ( ( hAPP_nat_int(power_power_int(A_1),Ma) = hAPP_nat_int(power_power_int(A_1),Na) )
<=> ( Ma = Na ) ) ) ).
tff(fact_273_power__inject__exp,axiom,
! [Ma: nat,Na: nat,A_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_1))
=> ( ( hAPP_nat_nat(power_power_nat(A_1),Ma) = hAPP_nat_nat(power_power_nat(A_1),Na) )
<=> ( Ma = Na ) ) ) ).
tff(fact_274_power__inject__exp,axiom,
! [Ma: nat,Na: nat,A_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_1))
=> ( ( hAPP_nat_real(power_power_real(A_1),Ma) = hAPP_nat_real(power_power_real(A_1),Na) )
<=> ( Ma = Na ) ) ) ).
tff(fact_275_power__strict__increasing__iff,axiom,
! [Xa: nat,Ya: nat,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(B),Xa)),hAPP_nat_int(power_power_int(B),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) ) ) ).
tff(fact_276_power__strict__increasing__iff,axiom,
! [Xa: nat,Ya: nat,B: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(B),Xa)),hAPP_nat_nat(power_power_nat(B),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) ) ) ).
tff(fact_277_power__strict__increasing__iff,axiom,
! [Xa: nat,Ya: nat,B: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(B),Xa)),hAPP_nat_real(power_power_real(B),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) ) ) ).
tff(fact_278_power__less__imp__less__exp,axiom,
! [M_6: nat,N_13: nat,A_90: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_90))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_90),M_6)),hAPP_nat_int(power_power_int(A_90),N_13)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_6),N_13)) ) ) ).
tff(fact_279_power__less__imp__less__exp,axiom,
! [M_6: nat,N_13: nat,A_90: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_90))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_90),M_6)),hAPP_nat_nat(power_power_nat(A_90),N_13)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_6),N_13)) ) ) ).
tff(fact_280_power__less__imp__less__exp,axiom,
! [M_6: nat,N_13: nat,A_90: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_90))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_90),M_6)),hAPP_nat_real(power_power_real(A_90),N_13)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_6),N_13)) ) ) ).
tff(fact_281_power__strict__increasing,axiom,
! [A_89: int,N_12: nat,N_11: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_12),N_11))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_89))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_89),N_12)),hAPP_nat_int(power_power_int(A_89),N_11))) ) ) ).
tff(fact_282_power__strict__increasing,axiom,
! [A_89: nat,N_12: nat,N_11: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_12),N_11))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_89))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_89),N_12)),hAPP_nat_nat(power_power_nat(A_89),N_11))) ) ) ).
tff(fact_283_power__strict__increasing,axiom,
! [A_89: real,N_12: nat,N_11: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_12),N_11))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_89))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_89),N_12)),hAPP_nat_real(power_power_real(A_89),N_11))) ) ) ).
tff(fact_284_power__eq__0__iff,axiom,
! [A_1: int,Na: nat] :
( ( hAPP_nat_int(power_power_int(A_1),Na) = zero_zero_int )
<=> ( ( A_1 = zero_zero_int )
& ( Na != zero_zero_nat ) ) ) ).
tff(fact_285_power__eq__0__iff,axiom,
! [A_1: nat,Na: nat] :
( ( hAPP_nat_nat(power_power_nat(A_1),Na) = zero_zero_nat )
<=> ( ( A_1 = zero_zero_nat )
& ( Na != zero_zero_nat ) ) ) ).
tff(fact_286_power__eq__0__iff,axiom,
! [A_1: real,Na: nat] :
( ( hAPP_nat_real(power_power_real(A_1),Na) = zero_zero_real )
<=> ( ( A_1 = zero_zero_real )
& ( Na != zero_zero_nat ) ) ) ).
tff(fact_287_of__nat__less__0__iff,axiom,
! [M_5: nat] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_5)),zero_zero_int)) ).
tff(fact_288_of__nat__less__0__iff,axiom,
! [M_5: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_5)),zero_zero_nat)) ).
tff(fact_289_of__nat__less__0__iff,axiom,
! [M_5: nat] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_5)),zero_zero_real)) ).
tff(fact_290_power__0,axiom,
! [A_88: int] : hAPP_nat_int(power_power_int(A_88),zero_zero_nat) = one_one_int ).
tff(fact_291_power__0,axiom,
! [A_88: nat] : hAPP_nat_nat(power_power_nat(A_88),zero_zero_nat) = one_one_nat ).
tff(fact_292_power__0,axiom,
! [A_88: real] : hAPP_nat_real(power_power_real(A_88),zero_zero_nat) = one_one_real ).
tff(fact_293_of__nat__0,axiom,
hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) = zero_zero_int ).
tff(fact_294_of__nat__0,axiom,
hAPP_nat_nat(semiri984289939at_nat,zero_zero_nat) = zero_zero_nat ).
tff(fact_295_of__nat__0,axiom,
hAPP_nat_real(semiri132038758t_real,zero_zero_nat) = zero_zero_real ).
tff(fact_296_pos2,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_297_zero__less__imp__eq__int,axiom,
! [K_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
=> ? [N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
& ( K_1 = hAPP_nat_int(semiri1621563631at_int,N_1) ) ) ) ).
tff(fact_298_less__imp__add__positive,axiom,
! [I_2: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1))
=> ? [K_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
& ( plus_plus_nat(I_2,K_2) = J_1 ) ) ) ).
tff(fact_299_exp__eq__1,axiom,
! [Xa: nat,Na: nat] :
( ( hAPP_nat_nat(power_power_nat(Xa),Na) = one_one_nat )
<=> ( ( Xa = one_one_nat )
| ( Na = zero_zero_nat ) ) ) ).
tff(fact_300_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_13: int] : hAPP_nat_int(power_power_int(X_13),zero_zero_nat) = one_one_int ).
tff(fact_301_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_13: nat] : hAPP_nat_nat(power_power_nat(X_13),zero_zero_nat) = one_one_nat ).
tff(fact_302_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_13: real] : hAPP_nat_real(power_power_real(X_13),zero_zero_nat) = one_one_real ).
tff(fact_303_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(A_1,A_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_1)) ) ).
tff(fact_304_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),plus_plus_real(A_1,A_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_1)) ) ).
tff(fact_305_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_1,A_1)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),zero_zero_int)) ) ).
tff(fact_306_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_1,A_1)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),zero_zero_real)) ) ).
tff(fact_307_add__pos__pos,axiom,
! [B_60: int,A_87: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_87))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_60))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(A_87,B_60))) ) ) ).
tff(fact_308_add__pos__pos,axiom,
! [B_60: nat,A_87: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_87))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_60))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),plus_plus_nat(A_87,B_60))) ) ) ).
tff(fact_309_add__pos__pos,axiom,
! [B_60: real,A_87: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_87))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_60))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),plus_plus_real(A_87,B_60))) ) ) ).
tff(fact_310_add__neg__neg,axiom,
! [B_59: int,A_86: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_86),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_59),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_86,B_59)),zero_zero_int)) ) ) ).
tff(fact_311_add__neg__neg,axiom,
! [B_59: nat,A_86: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_86),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_59),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_86,B_59)),zero_zero_nat)) ) ) ).
tff(fact_312_add__neg__neg,axiom,
! [B_59: real,A_86: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_86),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_59),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_86,B_59)),zero_zero_real)) ) ) ).
tff(fact_313_zero__reorient,axiom,
! [Xa: int] :
( ( zero_zero_int = Xa )
<=> ( Xa = zero_zero_int ) ) ).
tff(fact_314_zero__reorient,axiom,
! [Xa: nat] :
( ( zero_zero_nat = Xa )
<=> ( Xa = zero_zero_nat ) ) ).
tff(fact_315_zero__reorient,axiom,
! [Xa: real] :
( ( zero_zero_real = Xa )
<=> ( Xa = zero_zero_real ) ) ).
tff(fact_316_add__right__imp__eq,axiom,
! [B_58: int,A_85: int,C_44: int] :
( ( plus_plus_int(B_58,A_85) = plus_plus_int(C_44,A_85) )
=> ( B_58 = C_44 ) ) ).
tff(fact_317_add__right__imp__eq,axiom,
! [B_58: nat,A_85: nat,C_44: nat] :
( ( plus_plus_nat(B_58,A_85) = plus_plus_nat(C_44,A_85) )
=> ( B_58 = C_44 ) ) ).
tff(fact_318_add__right__imp__eq,axiom,
! [B_58: real,A_85: real,C_44: real] :
( ( plus_plus_real(B_58,A_85) = plus_plus_real(C_44,A_85) )
=> ( B_58 = C_44 ) ) ).
tff(fact_319_add__imp__eq,axiom,
! [A_84: int,B_57: int,C_43: int] :
( ( plus_plus_int(A_84,B_57) = plus_plus_int(A_84,C_43) )
=> ( B_57 = C_43 ) ) ).
tff(fact_320_add__imp__eq,axiom,
! [A_84: nat,B_57: nat,C_43: nat] :
( ( plus_plus_nat(A_84,B_57) = plus_plus_nat(A_84,C_43) )
=> ( B_57 = C_43 ) ) ).
tff(fact_321_add__imp__eq,axiom,
! [A_84: real,B_57: real,C_43: real] :
( ( plus_plus_real(A_84,B_57) = plus_plus_real(A_84,C_43) )
=> ( B_57 = C_43 ) ) ).
tff(fact_322_add__left__imp__eq,axiom,
! [A_83: int,B_56: int,C_42: int] :
( ( plus_plus_int(A_83,B_56) = plus_plus_int(A_83,C_42) )
=> ( B_56 = C_42 ) ) ).
tff(fact_323_add__left__imp__eq,axiom,
! [A_83: nat,B_56: nat,C_42: nat] :
( ( plus_plus_nat(A_83,B_56) = plus_plus_nat(A_83,C_42) )
=> ( B_56 = C_42 ) ) ).
tff(fact_324_add__left__imp__eq,axiom,
! [A_83: real,B_56: real,C_42: real] :
( ( plus_plus_real(A_83,B_56) = plus_plus_real(A_83,C_42) )
=> ( B_56 = C_42 ) ) ).
tff(fact_325_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_82: int,B_55: int,C_41: int,D_14: int] : plus_plus_int(plus_plus_int(A_82,B_55),plus_plus_int(C_41,D_14)) = plus_plus_int(plus_plus_int(A_82,C_41),plus_plus_int(B_55,D_14)) ).
tff(fact_326_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_82: nat,B_55: nat,C_41: nat,D_14: nat] : plus_plus_nat(plus_plus_nat(A_82,B_55),plus_plus_nat(C_41,D_14)) = plus_plus_nat(plus_plus_nat(A_82,C_41),plus_plus_nat(B_55,D_14)) ).
tff(fact_327_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_82: real,B_55: real,C_41: real,D_14: real] : plus_plus_real(plus_plus_real(A_82,B_55),plus_plus_real(C_41,D_14)) = plus_plus_real(plus_plus_real(A_82,C_41),plus_plus_real(B_55,D_14)) ).
tff(fact_328_add__right__cancel,axiom,
! [B: int,A_1: int,C_1: int] :
( ( plus_plus_int(B,A_1) = plus_plus_int(C_1,A_1) )
<=> ( B = C_1 ) ) ).
tff(fact_329_add__right__cancel,axiom,
! [B: nat,A_1: nat,C_1: nat] :
( ( plus_plus_nat(B,A_1) = plus_plus_nat(C_1,A_1) )
<=> ( B = C_1 ) ) ).
tff(fact_330_add__right__cancel,axiom,
! [B: real,A_1: real,C_1: real] :
( ( plus_plus_real(B,A_1) = plus_plus_real(C_1,A_1) )
<=> ( B = C_1 ) ) ).
tff(fact_331_add__left__cancel,axiom,
! [A_1: int,B: int,C_1: int] :
( ( plus_plus_int(A_1,B) = plus_plus_int(A_1,C_1) )
<=> ( B = C_1 ) ) ).
tff(fact_332_add__left__cancel,axiom,
! [A_1: nat,B: nat,C_1: nat] :
( ( plus_plus_nat(A_1,B) = plus_plus_nat(A_1,C_1) )
<=> ( B = C_1 ) ) ).
tff(fact_333_add__left__cancel,axiom,
! [A_1: real,B: real,C_1: real] :
( ( plus_plus_real(A_1,B) = plus_plus_real(A_1,C_1) )
<=> ( B = C_1 ) ) ).
tff(fact_334_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_81: int,B_54: int,C_40: int] : plus_plus_int(plus_plus_int(A_81,B_54),C_40) = plus_plus_int(plus_plus_int(A_81,C_40),B_54) ).
tff(fact_335_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_81: nat,B_54: nat,C_40: nat] : plus_plus_nat(plus_plus_nat(A_81,B_54),C_40) = plus_plus_nat(plus_plus_nat(A_81,C_40),B_54) ).
tff(fact_336_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_81: real,B_54: real,C_40: real] : plus_plus_real(plus_plus_real(A_81,B_54),C_40) = plus_plus_real(plus_plus_real(A_81,C_40),B_54) ).
tff(fact_337_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_80: int,B_53: int,C_39: int] : plus_plus_int(plus_plus_int(A_80,B_53),C_39) = plus_plus_int(A_80,plus_plus_int(B_53,C_39)) ).
tff(fact_338_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_80: nat,B_53: nat,C_39: nat] : plus_plus_nat(plus_plus_nat(A_80,B_53),C_39) = plus_plus_nat(A_80,plus_plus_nat(B_53,C_39)) ).
tff(fact_339_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_80: real,B_53: real,C_39: real] : plus_plus_real(plus_plus_real(A_80,B_53),C_39) = plus_plus_real(A_80,plus_plus_real(B_53,C_39)) ).
tff(fact_340_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_79: int,B_52: int,C_38: int] : plus_plus_int(plus_plus_int(A_79,B_52),C_38) = plus_plus_int(A_79,plus_plus_int(B_52,C_38)) ).
tff(fact_341_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_79: nat,B_52: nat,C_38: nat] : plus_plus_nat(plus_plus_nat(A_79,B_52),C_38) = plus_plus_nat(A_79,plus_plus_nat(B_52,C_38)) ).
tff(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_79: real,B_52: real,C_38: real] : plus_plus_real(plus_plus_real(A_79,B_52),C_38) = plus_plus_real(A_79,plus_plus_real(B_52,C_38)) ).
tff(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_78: int,C_37: int,D_13: int] : plus_plus_int(A_78,plus_plus_int(C_37,D_13)) = plus_plus_int(plus_plus_int(A_78,C_37),D_13) ).
tff(fact_344_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_78: nat,C_37: nat,D_13: nat] : plus_plus_nat(A_78,plus_plus_nat(C_37,D_13)) = plus_plus_nat(plus_plus_nat(A_78,C_37),D_13) ).
tff(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_78: real,C_37: real,D_13: real] : plus_plus_real(A_78,plus_plus_real(C_37,D_13)) = plus_plus_real(plus_plus_real(A_78,C_37),D_13) ).
tff(fact_346_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_77: int,C_36: int,D_12: int] : plus_plus_int(A_77,plus_plus_int(C_36,D_12)) = plus_plus_int(C_36,plus_plus_int(A_77,D_12)) ).
tff(fact_347_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_77: nat,C_36: nat,D_12: nat] : plus_plus_nat(A_77,plus_plus_nat(C_36,D_12)) = plus_plus_nat(C_36,plus_plus_nat(A_77,D_12)) ).
tff(fact_348_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_77: real,C_36: real,D_12: real] : plus_plus_real(A_77,plus_plus_real(C_36,D_12)) = plus_plus_real(C_36,plus_plus_real(A_77,D_12)) ).
tff(fact_349_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_76: int,C_35: int] : plus_plus_int(A_76,C_35) = plus_plus_int(C_35,A_76) ).
tff(fact_350_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_76: nat,C_35: nat] : plus_plus_nat(A_76,C_35) = plus_plus_nat(C_35,A_76) ).
tff(fact_351_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_76: real,C_35: real] : plus_plus_real(A_76,C_35) = plus_plus_real(C_35,A_76) ).
tff(fact_352_one__reorient,axiom,
! [Xa: int] :
( ( one_one_int = Xa )
<=> ( Xa = one_one_int ) ) ).
tff(fact_353_one__reorient,axiom,
! [Xa: nat] :
( ( one_one_nat = Xa )
<=> ( Xa = one_one_nat ) ) ).
tff(fact_354_one__reorient,axiom,
! [Xa: real] :
( ( one_one_real = Xa )
<=> ( Xa = one_one_real ) ) ).
tff(fact_355_add__0__iff,axiom,
! [B: int,A_1: int] :
( ( B = plus_plus_int(B,A_1) )
<=> ( A_1 = zero_zero_int ) ) ).
tff(fact_356_add__0__iff,axiom,
! [B: nat,A_1: nat] :
( ( B = plus_plus_nat(B,A_1) )
<=> ( A_1 = zero_zero_nat ) ) ).
tff(fact_357_add__0__iff,axiom,
! [B: real,A_1: real] :
( ( B = plus_plus_real(B,A_1) )
<=> ( A_1 = zero_zero_real ) ) ).
tff(fact_358_add_Ocomm__neutral,axiom,
! [A_75: int] : plus_plus_int(A_75,zero_zero_int) = A_75 ).
tff(fact_359_add_Ocomm__neutral,axiom,
! [A_75: nat] : plus_plus_nat(A_75,zero_zero_nat) = A_75 ).
tff(fact_360_add_Ocomm__neutral,axiom,
! [A_75: real] : plus_plus_real(A_75,zero_zero_real) = A_75 ).
tff(fact_361_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_74: int] : plus_plus_int(A_74,zero_zero_int) = A_74 ).
tff(fact_362_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_74: nat] : plus_plus_nat(A_74,zero_zero_nat) = A_74 ).
tff(fact_363_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_74: real] : plus_plus_real(A_74,zero_zero_real) = A_74 ).
tff(fact_364_add__0__right,axiom,
! [A_73: int] : plus_plus_int(A_73,zero_zero_int) = A_73 ).
tff(fact_365_add__0__right,axiom,
! [A_73: nat] : plus_plus_nat(A_73,zero_zero_nat) = A_73 ).
tff(fact_366_add__0__right,axiom,
! [A_73: real] : plus_plus_real(A_73,zero_zero_real) = A_73 ).
tff(fact_367_double__zero__sym,axiom,
! [A_1: int] :
( ( zero_zero_int = plus_plus_int(A_1,A_1) )
<=> ( A_1 = zero_zero_int ) ) ).
tff(fact_368_double__zero__sym,axiom,
! [A_1: real] :
( ( zero_zero_real = plus_plus_real(A_1,A_1) )
<=> ( A_1 = zero_zero_real ) ) ).
tff(fact_369_add__0,axiom,
! [A_72: int] : plus_plus_int(zero_zero_int,A_72) = A_72 ).
tff(fact_370_add__0,axiom,
! [A_72: nat] : plus_plus_nat(zero_zero_nat,A_72) = A_72 ).
tff(fact_371_add__0,axiom,
! [A_72: real] : plus_plus_real(zero_zero_real,A_72) = A_72 ).
tff(fact_372_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_71: int] : plus_plus_int(zero_zero_int,A_71) = A_71 ).
tff(fact_373_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_71: nat] : plus_plus_nat(zero_zero_nat,A_71) = A_71 ).
tff(fact_374_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_71: real] : plus_plus_real(zero_zero_real,A_71) = A_71 ).
tff(fact_375_add__0__left,axiom,
! [A_70: int] : plus_plus_int(zero_zero_int,A_70) = A_70 ).
tff(fact_376_add__0__left,axiom,
! [A_70: nat] : plus_plus_nat(zero_zero_nat,A_70) = A_70 ).
tff(fact_377_add__0__left,axiom,
! [A_70: real] : plus_plus_real(zero_zero_real,A_70) = A_70 ).
tff(fact_378_add__less__imp__less__left,axiom,
! [C_34: int,A_69: int,B_51: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(C_34,A_69)),plus_plus_int(C_34,B_51)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_69),B_51)) ) ).
tff(fact_379_add__less__imp__less__left,axiom,
! [C_34: nat,A_69: nat,B_51: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(C_34,A_69)),plus_plus_nat(C_34,B_51)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_69),B_51)) ) ).
tff(fact_380_add__less__imp__less__left,axiom,
! [C_34: real,A_69: real,B_51: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(C_34,A_69)),plus_plus_real(C_34,B_51)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_69),B_51)) ) ).
tff(fact_381_add__less__imp__less__right,axiom,
! [A_68: int,C_33: int,B_50: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_68,C_33)),plus_plus_int(B_50,C_33)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_68),B_50)) ) ).
tff(fact_382_add__less__imp__less__right,axiom,
! [A_68: nat,C_33: nat,B_50: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_68,C_33)),plus_plus_nat(B_50,C_33)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_68),B_50)) ) ).
tff(fact_383_add__less__imp__less__right,axiom,
! [A_68: real,C_33: real,B_50: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_68,C_33)),plus_plus_real(B_50,C_33)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_68),B_50)) ) ).
tff(fact_384_add__strict__mono,axiom,
! [C_32: int,D_11: int,A_67: int,B_49: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_67),B_49))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_32),D_11))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_67,C_32)),plus_plus_int(B_49,D_11))) ) ) ).
tff(fact_385_add__strict__mono,axiom,
! [C_32: nat,D_11: nat,A_67: nat,B_49: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_67),B_49))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_32),D_11))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_67,C_32)),plus_plus_nat(B_49,D_11))) ) ) ).
tff(fact_386_add__strict__mono,axiom,
! [C_32: real,D_11: real,A_67: real,B_49: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_67),B_49))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_32),D_11))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_67,C_32)),plus_plus_real(B_49,D_11))) ) ) ).
tff(fact_387_add__strict__left__mono,axiom,
! [C_31: int,A_66: int,B_48: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_66),B_48))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(C_31,A_66)),plus_plus_int(C_31,B_48))) ) ).
tff(fact_388_add__strict__left__mono,axiom,
! [C_31: nat,A_66: nat,B_48: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_66),B_48))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(C_31,A_66)),plus_plus_nat(C_31,B_48))) ) ).
tff(fact_389_add__strict__left__mono,axiom,
! [C_31: real,A_66: real,B_48: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_66),B_48))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(C_31,A_66)),plus_plus_real(C_31,B_48))) ) ).
tff(fact_390_add__strict__right__mono,axiom,
! [C_30: int,A_65: int,B_47: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_65),B_47))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_65,C_30)),plus_plus_int(B_47,C_30))) ) ).
tff(fact_391_add__strict__right__mono,axiom,
! [C_30: nat,A_65: nat,B_47: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_65),B_47))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_65,C_30)),plus_plus_nat(B_47,C_30))) ) ).
tff(fact_392_add__strict__right__mono,axiom,
! [C_30: real,A_65: real,B_47: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_65),B_47))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_65,C_30)),plus_plus_real(B_47,C_30))) ) ).
tff(fact_393_add__less__cancel__left,axiom,
! [C_1: int,A_1: int,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(C_1,A_1)),plus_plus_int(C_1,B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B)) ) ).
tff(fact_394_add__less__cancel__left,axiom,
! [C_1: nat,A_1: nat,B: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(C_1,A_1)),plus_plus_nat(C_1,B)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B)) ) ).
tff(fact_395_add__less__cancel__left,axiom,
! [C_1: real,A_1: real,B: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(C_1,A_1)),plus_plus_real(C_1,B)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B)) ) ).
tff(fact_396_add__less__cancel__right,axiom,
! [A_1: int,C_1: int,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_1,C_1)),plus_plus_int(B,C_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B)) ) ).
tff(fact_397_add__less__cancel__right,axiom,
! [A_1: nat,C_1: nat,B: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_1,C_1)),plus_plus_nat(B,C_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B)) ) ).
tff(fact_398_add__less__cancel__right,axiom,
! [A_1: real,C_1: real,B: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_1,C_1)),plus_plus_real(B,C_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B)) ) ).
tff(fact_399_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_12: int] : hAPP_nat_int(power_power_int(X_12),one_one_nat) = X_12 ).
tff(fact_400_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_12: nat] : hAPP_nat_nat(power_power_nat(X_12),one_one_nat) = X_12 ).
tff(fact_401_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_12: real] : hAPP_nat_real(power_power_real(X_12),one_one_nat) = X_12 ).
tff(fact_402_nat__power__eq__0__iff,axiom,
! [Ma: nat,Na: nat] :
( ( hAPP_nat_nat(power_power_nat(Ma),Na) = zero_zero_nat )
<=> ( ( Na != zero_zero_nat )
& ( Ma = zero_zero_nat ) ) ) ).
tff(fact_403_realpow__pos__nth__unique,axiom,
! [A: real,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A))
=> ? [X_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
& ( hAPP_nat_real(power_power_real(X_1),N) = A )
& ! [Y_1: real] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1))
& ( hAPP_nat_real(power_power_real(Y_1),N) = A ) )
=> ( Y_1 = X_1 ) ) ) ) ) ).
tff(fact_404_realpow__pos__nth,axiom,
! [A: real,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A))
=> ? [R: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R))
& ( hAPP_nat_real(power_power_real(R),N) = A ) ) ) ) ).
tff(fact_405_tpos,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),t)) ).
tff(fact_406_nat__number__of__add__1,axiom,
! [V_1: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> ( plus_plus_nat(number_number_of_nat(V_1),one_one_nat) = one_one_nat ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> ( plus_plus_nat(number_number_of_nat(V_1),one_one_nat) = number_number_of_nat(succ(V_1)) ) ) ) ).
tff(fact_407_nat__1__add__number__of,axiom,
! [V_1: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> ( plus_plus_nat(one_one_nat,number_number_of_nat(V_1)) = one_one_nat ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> ( plus_plus_nat(one_one_nat,number_number_of_nat(V_1)) = number_number_of_nat(succ(V_1)) ) ) ) ).
tff(fact_408_zadd__power3,axiom,
! [A: int,B_1: int] : hAPP_nat_int(power_power_int(plus_plus_int(A,B_1)),number_number_of_nat(bit1(bit1(pls)))) = plus_plus_int(plus_plus_int(plus_plus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit1(bit1(pls)))),times_times_int(times_times_int(number_number_of_int(bit1(bit1(pls))),hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls))))),B_1)),times_times_int(times_times_int(number_number_of_int(bit1(bit1(pls))),A),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit0(bit1(pls)))))),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit1(bit1(pls))))) ).
tff(fact_409_zadd__power2,axiom,
! [A: int,B_1: int] : hAPP_nat_int(power_power_int(plus_plus_int(A,B_1)),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_int(plus_plus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))),times_times_int(times_times_int(number_number_of_int(bit0(bit1(pls))),A),B_1)),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_410_tn,axiom,
tn = minus_minus_nat(nat_1(t),one_one_nat) ).
tff(fact_411__096_B_Bthesis_O_A_I_B_Btn_O_A_091_124_Atn_A_061_Anat_At_A_N_A1_059_A0_,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),minus_minus_nat(nat_1(t),one_one_nat))) ).
tff(fact_412_le__iff__diff__le__0,axiom,
! [A_1: int,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,minus_minus_int(A_1,B)),zero_zero_int)) ) ).
tff(fact_413_le__iff__diff__le__0,axiom,
! [A_1: real,B: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,minus_minus_real(A_1,B)),zero_zero_real)) ) ).
tff(fact_414_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_64: int,B_46: int] : times_times_int(A_64,B_46) = times_times_int(B_46,A_64) ).
tff(fact_415_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_64: nat,B_46: nat] : times_times_nat(A_64,B_46) = times_times_nat(B_46,A_64) ).
tff(fact_416_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_64: real,B_46: real] : times_times_real(A_64,B_46) = times_times_real(B_46,A_64) ).
tff(fact_417_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: int,Rx_6: int,Ry_4: int] : times_times_int(Lx_6,times_times_int(Rx_6,Ry_4)) = times_times_int(Rx_6,times_times_int(Lx_6,Ry_4)) ).
tff(fact_418_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: nat,Rx_6: nat,Ry_4: nat] : times_times_nat(Lx_6,times_times_nat(Rx_6,Ry_4)) = times_times_nat(Rx_6,times_times_nat(Lx_6,Ry_4)) ).
tff(fact_419_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: real,Rx_6: real,Ry_4: real] : times_times_real(Lx_6,times_times_real(Rx_6,Ry_4)) = times_times_real(Rx_6,times_times_real(Lx_6,Ry_4)) ).
tff(fact_420_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: int,Rx_5: int,Ry_3: int] : times_times_int(Lx_5,times_times_int(Rx_5,Ry_3)) = times_times_int(times_times_int(Lx_5,Rx_5),Ry_3) ).
tff(fact_421_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: nat,Rx_5: nat,Ry_3: nat] : times_times_nat(Lx_5,times_times_nat(Rx_5,Ry_3)) = times_times_nat(times_times_nat(Lx_5,Rx_5),Ry_3) ).
tff(fact_422_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: real,Rx_5: real,Ry_3: real] : times_times_real(Lx_5,times_times_real(Rx_5,Ry_3)) = times_times_real(times_times_real(Lx_5,Rx_5),Ry_3) ).
tff(fact_423_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_63: int,B_45: int,C_29: int] : times_times_int(times_times_int(A_63,B_45),C_29) = times_times_int(A_63,times_times_int(B_45,C_29)) ).
tff(fact_424_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_63: nat,B_45: nat,C_29: nat] : times_times_nat(times_times_nat(A_63,B_45),C_29) = times_times_nat(A_63,times_times_nat(B_45,C_29)) ).
tff(fact_425_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_63: real,B_45: real,C_29: real] : times_times_real(times_times_real(A_63,B_45),C_29) = times_times_real(A_63,times_times_real(B_45,C_29)) ).
tff(fact_426_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: int,Ly_4: int,Rx_4: int] : times_times_int(times_times_int(Lx_4,Ly_4),Rx_4) = times_times_int(Lx_4,times_times_int(Ly_4,Rx_4)) ).
tff(fact_427_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: nat,Ly_4: nat,Rx_4: nat] : times_times_nat(times_times_nat(Lx_4,Ly_4),Rx_4) = times_times_nat(Lx_4,times_times_nat(Ly_4,Rx_4)) ).
tff(fact_428_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: real,Ly_4: real,Rx_4: real] : times_times_real(times_times_real(Lx_4,Ly_4),Rx_4) = times_times_real(Lx_4,times_times_real(Ly_4,Rx_4)) ).
tff(fact_429_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: int,Ly_3: int,Rx_3: int] : times_times_int(times_times_int(Lx_3,Ly_3),Rx_3) = times_times_int(times_times_int(Lx_3,Rx_3),Ly_3) ).
tff(fact_430_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: nat,Ly_3: nat,Rx_3: nat] : times_times_nat(times_times_nat(Lx_3,Ly_3),Rx_3) = times_times_nat(times_times_nat(Lx_3,Rx_3),Ly_3) ).
tff(fact_431_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: real,Ly_3: real,Rx_3: real] : times_times_real(times_times_real(Lx_3,Ly_3),Rx_3) = times_times_real(times_times_real(Lx_3,Rx_3),Ly_3) ).
tff(fact_432_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: int,Ly_2: int,Rx_2: int,Ry_2: int] : times_times_int(times_times_int(Lx_2,Ly_2),times_times_int(Rx_2,Ry_2)) = times_times_int(Lx_2,times_times_int(Ly_2,times_times_int(Rx_2,Ry_2))) ).
tff(fact_433_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: nat,Ly_2: nat,Rx_2: nat,Ry_2: nat] : times_times_nat(times_times_nat(Lx_2,Ly_2),times_times_nat(Rx_2,Ry_2)) = times_times_nat(Lx_2,times_times_nat(Ly_2,times_times_nat(Rx_2,Ry_2))) ).
tff(fact_434_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: real,Ly_2: real,Rx_2: real,Ry_2: real] : times_times_real(times_times_real(Lx_2,Ly_2),times_times_real(Rx_2,Ry_2)) = times_times_real(Lx_2,times_times_real(Ly_2,times_times_real(Rx_2,Ry_2))) ).
tff(fact_435_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: real,Ly_1: real,Rx_1: real,Ry_1: real] : times_times_real(times_times_real(Lx_1,Ly_1),times_times_real(Rx_1,Ry_1)) = times_times_real(Rx_1,times_times_real(times_times_real(Lx_1,Ly_1),Ry_1)) ).
tff(fact_436_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: nat,Ly_1: nat,Rx_1: nat,Ry_1: nat] : times_times_nat(times_times_nat(Lx_1,Ly_1),times_times_nat(Rx_1,Ry_1)) = times_times_nat(Rx_1,times_times_nat(times_times_nat(Lx_1,Ly_1),Ry_1)) ).
tff(fact_437_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: int,Ly_1: int,Rx_1: int,Ry_1: int] : times_times_int(times_times_int(Lx_1,Ly_1),times_times_int(Rx_1,Ry_1)) = times_times_int(Rx_1,times_times_int(times_times_int(Lx_1,Ly_1),Ry_1)) ).
tff(fact_438_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: real,Ly: real,Rx: real,Ry: real] : times_times_real(times_times_real(Lx,Ly),times_times_real(Rx,Ry)) = times_times_real(times_times_real(Lx,Rx),times_times_real(Ly,Ry)) ).
tff(fact_439_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: nat,Ly: nat,Rx: nat,Ry: nat] : times_times_nat(times_times_nat(Lx,Ly),times_times_nat(Rx,Ry)) = times_times_nat(times_times_nat(Lx,Rx),times_times_nat(Ly,Ry)) ).
tff(fact_440_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: int,Ly: int,Rx: int,Ry: int] : times_times_int(times_times_int(Lx,Ly),times_times_int(Rx,Ry)) = times_times_int(times_times_int(Lx,Rx),times_times_int(Ly,Ry)) ).
tff(fact_441_diff__eq__diff__eq,axiom,
! [A_1: int,B: int,C_1: int,D: int] :
( ( minus_minus_int(A_1,B) = minus_minus_int(C_1,D) )
=> ( ( A_1 = B )
<=> ( C_1 = D ) ) ) ).
tff(fact_442_diff__eq__diff__eq,axiom,
! [A_1: real,B: real,C_1: real,D: real] :
( ( minus_minus_real(A_1,B) = minus_minus_real(C_1,D) )
=> ( ( A_1 = B )
<=> ( C_1 = D ) ) ) ).
tff(fact_443_diff__eq__diff__less__eq,axiom,
! [A_1: int,B: int,C_1: int,D: int] :
( ( minus_minus_int(A_1,B) = minus_minus_int(C_1,D) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_1),D)) ) ) ).
tff(fact_444_diff__eq__diff__less__eq,axiom,
! [A_1: real,B: real,C_1: real,D: real] :
( ( minus_minus_real(A_1,B) = minus_minus_real(C_1,D) )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),D)) ) ) ).
tff(fact_445_semiring__mult__number__of,axiom,
! [V_13: int,V_12: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_12))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_13))
=> ( times_times_real(number267125858f_real(V_12),number267125858f_real(V_13)) = number267125858f_real(times_times_int(V_12,V_13)) ) ) ) ).
tff(fact_446_semiring__mult__number__of,axiom,
! [V_13: int,V_12: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_12))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_13))
=> ( times_times_nat(number_number_of_nat(V_12),number_number_of_nat(V_13)) = number_number_of_nat(times_times_int(V_12,V_13)) ) ) ) ).
tff(fact_447_semiring__mult__number__of,axiom,
! [V_13: int,V_12: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_12))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_13))
=> ( times_times_int(number_number_of_int(V_12),number_number_of_int(V_13)) = number_number_of_int(times_times_int(V_12,V_13)) ) ) ) ).
tff(fact_448_zle__refl,axiom,
! [W: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),W)) ).
tff(fact_449_zmult__commute,axiom,
! [Z: int,W: int] : times_times_int(Z,W) = times_times_int(W,Z) ).
tff(fact_450_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_451_zmult__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] : times_times_int(times_times_int(Z1,Z2),Z3) = times_times_int(Z1,times_times_int(Z2,Z3)) ).
tff(fact_452_zle__trans,axiom,
! [K_1: int,I_2: int,J_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),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_2),K_1)) ) ) ).
tff(fact_453_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_454_le__number__of,axiom,
! [Xa: int,Ya: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(Xa)),number267125858f_real(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) ) ).
tff(fact_455_le__number__of,axiom,
! [Xa: int,Ya: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(Xa)),number_number_of_int(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) ) ).
tff(fact_456_number__of__mult,axiom,
! [V_11: int,W_6: int] : number267125858f_real(times_times_int(V_11,W_6)) = times_times_real(number267125858f_real(V_11),number267125858f_real(W_6)) ).
tff(fact_457_number__of__mult,axiom,
! [V_11: int,W_6: int] : number_number_of_int(times_times_int(V_11,W_6)) = times_times_int(number_number_of_int(V_11),number_number_of_int(W_6)) ).
tff(fact_458_arith__simps_I32_J,axiom,
! [V_10: int,W_5: int] : times_times_real(number267125858f_real(V_10),number267125858f_real(W_5)) = number267125858f_real(times_times_int(V_10,W_5)) ).
tff(fact_459_arith__simps_I32_J,axiom,
! [V_10: int,W_5: int] : times_times_int(number_number_of_int(V_10),number_number_of_int(W_5)) = number_number_of_int(times_times_int(V_10,W_5)) ).
tff(fact_460_mult__number__of__left,axiom,
! [V_9: int,W_4: int,Z_5: real] : times_times_real(number267125858f_real(V_9),times_times_real(number267125858f_real(W_4),Z_5)) = times_times_real(number267125858f_real(times_times_int(V_9,W_4)),Z_5) ).
tff(fact_461_mult__number__of__left,axiom,
! [V_9: int,W_4: int,Z_5: int] : times_times_int(number_number_of_int(V_9),times_times_int(number_number_of_int(W_4),Z_5)) = times_times_int(number_number_of_int(times_times_int(V_9,W_4)),Z_5) ).
tff(fact_462_eq__nat__nat__iff,axiom,
! [Z_4: int,Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_4))
=> ( ( nat_1(Z_1) = nat_1(Z_4) )
<=> ( Z_1 = Z_4 ) ) ) ) ).
tff(fact_463_right__diff__distrib__number__of,axiom,
! [V_8: int,B_44: int,C_28: int] : times_times_int(number_number_of_int(V_8),minus_minus_int(B_44,C_28)) = minus_minus_int(times_times_int(number_number_of_int(V_8),B_44),times_times_int(number_number_of_int(V_8),C_28)) ).
tff(fact_464_right__diff__distrib__number__of,axiom,
! [V_8: int,B_44: real,C_28: real] : times_times_real(number267125858f_real(V_8),minus_minus_real(B_44,C_28)) = minus_minus_real(times_times_real(number267125858f_real(V_8),B_44),times_times_real(number267125858f_real(V_8),C_28)) ).
tff(fact_465_left__diff__distrib__number__of,axiom,
! [A_62: int,B_43: int,V_7: int] : times_times_int(minus_minus_int(A_62,B_43),number_number_of_int(V_7)) = minus_minus_int(times_times_int(A_62,number_number_of_int(V_7)),times_times_int(B_43,number_number_of_int(V_7))) ).
tff(fact_466_left__diff__distrib__number__of,axiom,
! [A_62: real,B_43: real,V_7: int] : times_times_real(minus_minus_real(A_62,B_43),number267125858f_real(V_7)) = minus_minus_real(times_times_real(A_62,number267125858f_real(V_7)),times_times_real(B_43,number267125858f_real(V_7))) ).
tff(fact_467_number__of__diff,axiom,
! [V_6: int,W_3: int] : number_number_of_int(minus_minus_int(V_6,W_3)) = minus_minus_int(number_number_of_int(V_6),number_number_of_int(W_3)) ).
tff(fact_468_number__of__diff,axiom,
! [V_6: int,W_3: int] : number267125858f_real(minus_minus_int(V_6,W_3)) = minus_minus_real(number267125858f_real(V_6),number267125858f_real(W_3)) ).
tff(fact_469_of__nat__mult,axiom,
! [M_4: nat,N_10: nat] : hAPP_nat_real(semiri132038758t_real,times_times_nat(M_4,N_10)) = times_times_real(hAPP_nat_real(semiri132038758t_real,M_4),hAPP_nat_real(semiri132038758t_real,N_10)) ).
tff(fact_470_of__nat__mult,axiom,
! [M_4: nat,N_10: nat] : hAPP_nat_nat(semiri984289939at_nat,times_times_nat(M_4,N_10)) = times_times_nat(hAPP_nat_nat(semiri984289939at_nat,M_4),hAPP_nat_nat(semiri984289939at_nat,N_10)) ).
tff(fact_471_of__nat__mult,axiom,
! [M_4: nat,N_10: nat] : hAPP_nat_int(semiri1621563631at_int,times_times_nat(M_4,N_10)) = times_times_int(hAPP_nat_int(semiri1621563631at_int,M_4),hAPP_nat_int(semiri1621563631at_int,N_10)) ).
tff(fact_472_of__nat__le__iff,axiom,
! [Ma: nat,Na: nat] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
tff(fact_473_of__nat__le__iff,axiom,
! [Ma: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
tff(fact_474_of__nat__le__iff,axiom,
! [Ma: nat,Na: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
tff(fact_475_diff__commute,axiom,
! [I_2: nat,J_1: nat,K_1: nat] : minus_minus_nat(minus_minus_nat(I_2,J_1),K_1) = minus_minus_nat(minus_minus_nat(I_2,K_1),J_1) ).
tff(fact_476_nat__if__cong,axiom,
! [Xa: int,Ya: int,P_1: bool] :
( ( hBOOL(P_1)
=> ( nat_1(Xa) = nat_1(if_int(P_1,Xa,Ya)) ) )
& ( ~ hBOOL(P_1)
=> ( nat_1(Ya) = nat_1(if_int(P_1,Xa,Ya)) ) ) ) ).
tff(fact_477_split__mult__neg__le,axiom,
! [B_42: real,A_61: real] :
( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_61))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_42),zero_zero_real)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_61),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_42)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_61,B_42)),zero_zero_real)) ) ).
tff(fact_478_split__mult__neg__le,axiom,
! [B_42: nat,A_61: nat] :
( ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_61))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_42),zero_zero_nat)) )
| ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_61),zero_zero_nat))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_42)) ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_61,B_42)),zero_zero_nat)) ) ).
tff(fact_479_split__mult__neg__le,axiom,
! [B_42: int,A_61: int] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_61))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_42),zero_zero_int)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_61),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_42)) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_61,B_42)),zero_zero_int)) ) ).
tff(fact_480_split__mult__pos__le,axiom,
! [B_41: real,A_60: real] :
( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_60))
& 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,A_60),zero_zero_real))
& 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,zero_zero_real),times_times_real(A_60,B_41))) ) ).
tff(fact_481_split__mult__pos__le,axiom,
! [B_41: int,A_60: int] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_60))
& 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,A_60),zero_zero_int))
& 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,zero_zero_int),times_times_int(A_60,B_41))) ) ).
tff(fact_482_mult__mono,axiom,
! [C_27: real,D_10: real,A_59: real,B_40: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_59),B_40))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_27),D_10))
=> ( 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,zero_zero_real),C_27))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_59,C_27)),times_times_real(B_40,D_10))) ) ) ) ) ).
tff(fact_483_mult__mono,axiom,
! [C_27: nat,D_10: nat,A_59: nat,B_40: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_59),B_40))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_27),D_10))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_40))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_27))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_59,C_27)),times_times_nat(B_40,D_10))) ) ) ) ) ).
tff(fact_484_mult__mono,axiom,
! [C_27: int,D_10: int,A_59: int,B_40: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_59),B_40))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_27),D_10))
=> ( 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,zero_zero_int),C_27))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_59,C_27)),times_times_int(B_40,D_10))) ) ) ) ) ).
tff(fact_485_mult__mono_H,axiom,
! [C_26: real,D_9: real,A_58: real,B_39: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_58),B_39))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_26),D_9))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_58))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_26))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_58,C_26)),times_times_real(B_39,D_9))) ) ) ) ) ).
tff(fact_486_mult__mono_H,axiom,
! [C_26: nat,D_9: nat,A_58: nat,B_39: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_58),B_39))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_26),D_9))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_58))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_26))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_58,C_26)),times_times_nat(B_39,D_9))) ) ) ) ) ).
tff(fact_487_mult__mono_H,axiom,
! [C_26: int,D_9: int,A_58: int,B_39: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_58),B_39))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_26),D_9))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_58))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_26))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_58,C_26)),times_times_int(B_39,D_9))) ) ) ) ) ).
tff(fact_488_mult__left__mono__neg,axiom,
! [C_25: real,B_38: real,A_57: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_38),A_57))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_25),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(C_25,A_57)),times_times_real(C_25,B_38))) ) ) ).
tff(fact_489_mult__left__mono__neg,axiom,
! [C_25: int,B_38: int,A_57: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_38),A_57))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_25),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_25,A_57)),times_times_int(C_25,B_38))) ) ) ).
tff(fact_490_mult__right__mono__neg,axiom,
! [C_24: real,B_37: real,A_56: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_37),A_56))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_24),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_56,C_24)),times_times_real(B_37,C_24))) ) ) ).
tff(fact_491_mult__right__mono__neg,axiom,
! [C_24: int,B_37: int,A_56: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_37),A_56))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_24),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_56,C_24)),times_times_int(B_37,C_24))) ) ) ).
tff(fact_492_comm__mult__left__mono,axiom,
! [C_23: real,A_55: real,B_36: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_55),B_36))
=> ( 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,times_times_real(C_23,A_55)),times_times_real(C_23,B_36))) ) ) ).
tff(fact_493_comm__mult__left__mono,axiom,
! [C_23: nat,A_55: nat,B_36: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_55),B_36))
=> ( 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,times_times_nat(C_23,A_55)),times_times_nat(C_23,B_36))) ) ) ).
tff(fact_494_comm__mult__left__mono,axiom,
! [C_23: int,A_55: int,B_36: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_55),B_36))
=> ( 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,times_times_int(C_23,A_55)),times_times_int(C_23,B_36))) ) ) ).
tff(fact_495_mult__left__mono,axiom,
! [C_22: real,A_54: real,B_35: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_54),B_35))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_22))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(C_22,A_54)),times_times_real(C_22,B_35))) ) ) ).
tff(fact_496_mult__left__mono,axiom,
! [C_22: nat,A_54: nat,B_35: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_54),B_35))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_22))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(C_22,A_54)),times_times_nat(C_22,B_35))) ) ) ).
tff(fact_497_mult__left__mono,axiom,
! [C_22: int,A_54: int,B_35: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_54),B_35))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_22))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_22,A_54)),times_times_int(C_22,B_35))) ) ) ).
tff(fact_498_mult__right__mono,axiom,
! [C_21: real,A_53: real,B_34: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_53),B_34))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_21))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_53,C_21)),times_times_real(B_34,C_21))) ) ) ).
tff(fact_499_mult__right__mono,axiom,
! [C_21: nat,A_53: nat,B_34: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_53),B_34))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_21))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_53,C_21)),times_times_nat(B_34,C_21))) ) ) ).
tff(fact_500_mult__right__mono,axiom,
! [C_21: int,A_53: int,B_34: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_53),B_34))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_21))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_53,C_21)),times_times_int(B_34,C_21))) ) ) ).
tff(fact_501_mult__nonpos__nonpos,axiom,
! [B_33: real,A_52: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_52),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_33),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),times_times_real(A_52,B_33))) ) ) ).
tff(fact_502_mult__nonpos__nonpos,axiom,
! [B_33: int,A_52: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_52),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_33),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_52,B_33))) ) ) ).
tff(fact_503_mult__nonpos__nonneg,axiom,
! [B_32: real,A_51: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_51),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_32))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_51,B_32)),zero_zero_real)) ) ) ).
tff(fact_504_mult__nonpos__nonneg,axiom,
! [B_32: nat,A_51: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_51),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_32))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_51,B_32)),zero_zero_nat)) ) ) ).
tff(fact_505_mult__nonpos__nonneg,axiom,
! [B_32: int,A_51: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_51),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_32))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_51,B_32)),zero_zero_int)) ) ) ).
tff(fact_506_mult__nonneg__nonpos2,axiom,
! [B_31: 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_31),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(B_31,A_50)),zero_zero_real)) ) ) ).
tff(fact_507_mult__nonneg__nonpos2,axiom,
! [B_31: 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_31),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(B_31,A_50)),zero_zero_nat)) ) ) ).
tff(fact_508_mult__nonneg__nonpos2,axiom,
! [B_31: 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_31),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(B_31,A_50)),zero_zero_int)) ) ) ).
tff(fact_509_mult__nonneg__nonpos,axiom,
! [B_30: 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,B_30),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_49,B_30)),zero_zero_real)) ) ) ).
tff(fact_510_mult__nonneg__nonpos,axiom,
! [B_30: nat,A_49: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_49))
=> ( 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,times_times_nat(A_49,B_30)),zero_zero_nat)) ) ) ).
tff(fact_511_mult__nonneg__nonpos,axiom,
! [B_30: 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,B_30),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_49,B_30)),zero_zero_int)) ) ) ).
tff(fact_512_transfer__nat__int__relations_I1_J,axiom,
! [Ya: int,Xa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya))
=> ( ( nat_1(Xa) = nat_1(Ya) )
<=> ( Xa = Ya ) ) ) ) ).
tff(fact_513_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y: int,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(X,Y))) ) ) ).
tff(fact_514_mult__nonneg__nonneg,axiom,
! [B_29: real,A_48: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_48))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_29))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),times_times_real(A_48,B_29))) ) ) ).
tff(fact_515_mult__nonneg__nonneg,axiom,
! [B_29: nat,A_48: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_48))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_29))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),times_times_nat(A_48,B_29))) ) ) ).
tff(fact_516_mult__nonneg__nonneg,axiom,
! [B_29: int,A_48: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_48))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_29))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_48,B_29))) ) ) ).
tff(fact_517_le__add__iff1,axiom,
! [A_1: real,E_1: real,C_1: real,B: real,D: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(times_times_real(A_1,E_1),C_1)),plus_plus_real(times_times_real(B,E_1),D)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(times_times_real(minus_minus_real(A_1,B),E_1),C_1)),D)) ) ).
tff(fact_518_le__add__iff1,axiom,
! [A_1: int,E_1: int,C_1: int,B: int,D: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(A_1,E_1),C_1)),plus_plus_int(times_times_int(B,E_1),D)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(minus_minus_int(A_1,B),E_1),C_1)),D)) ) ).
tff(fact_519_eq__add__iff1,axiom,
! [A_1: real,E_1: real,C_1: real,B: real,D: real] :
( ( plus_plus_real(times_times_real(A_1,E_1),C_1) = plus_plus_real(times_times_real(B,E_1),D) )
<=> ( plus_plus_real(times_times_real(minus_minus_real(A_1,B),E_1),C_1) = D ) ) ).
tff(fact_520_eq__add__iff1,axiom,
! [A_1: int,E_1: int,C_1: int,B: int,D: int] :
( ( plus_plus_int(times_times_int(A_1,E_1),C_1) = plus_plus_int(times_times_int(B,E_1),D) )
<=> ( plus_plus_int(times_times_int(minus_minus_int(A_1,B),E_1),C_1) = D ) ) ).
tff(fact_521_le__add__iff2,axiom,
! [A_1: real,E_1: real,C_1: real,B: real,D: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(times_times_real(A_1,E_1),C_1)),plus_plus_real(times_times_real(B,E_1),D)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),plus_plus_real(times_times_real(minus_minus_real(B,A_1),E_1),D))) ) ).
tff(fact_522_le__add__iff2,axiom,
! [A_1: int,E_1: int,C_1: int,B: int,D: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(A_1,E_1),C_1)),plus_plus_int(times_times_int(B,E_1),D)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_1),plus_plus_int(times_times_int(minus_minus_int(B,A_1),E_1),D))) ) ).
tff(fact_523_eq__add__iff2,axiom,
! [A_1: real,E_1: real,C_1: real,B: real,D: real] :
( ( plus_plus_real(times_times_real(A_1,E_1),C_1) = plus_plus_real(times_times_real(B,E_1),D) )
<=> ( C_1 = plus_plus_real(times_times_real(minus_minus_real(B,A_1),E_1),D) ) ) ).
tff(fact_524_eq__add__iff2,axiom,
! [A_1: int,E_1: int,C_1: int,B: int,D: int] :
( ( plus_plus_int(times_times_int(A_1,E_1),C_1) = plus_plus_int(times_times_int(B,E_1),D) )
<=> ( C_1 = plus_plus_int(times_times_int(minus_minus_int(B,A_1),E_1),D) ) ) ).
tff(fact_525_mult__diff__mult,axiom,
! [X_11: real,Y_9: real,A_47: real,B_28: real] : minus_minus_real(times_times_real(X_11,Y_9),times_times_real(A_47,B_28)) = plus_plus_real(times_times_real(X_11,minus_minus_real(Y_9,B_28)),times_times_real(minus_minus_real(X_11,A_47),B_28)) ).
tff(fact_526_mult__diff__mult,axiom,
! [X_11: int,Y_9: int,A_47: int,B_28: int] : minus_minus_int(times_times_int(X_11,Y_9),times_times_int(A_47,B_28)) = plus_plus_int(times_times_int(X_11,minus_minus_int(Y_9,B_28)),times_times_int(minus_minus_int(X_11,A_47),B_28)) ).
tff(fact_527_mult__le__0__iff,axiom,
! [A_1: real,B: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_1,B)),zero_zero_real))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),zero_zero_real)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B)) ) ) ) ).
tff(fact_528_mult__le__0__iff,axiom,
! [A_1: int,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_1,B)),zero_zero_int))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B)) ) ) ) ).
tff(fact_529_zero__le__mult__iff,axiom,
! [A_1: real,B: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),times_times_real(A_1,B)))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),zero_zero_real)) ) ) ) ).
tff(fact_530_zero__le__mult__iff,axiom,
! [A_1: int,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_1,B)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) ) ) ) ).
tff(fact_531_all__nat,axiom,
! [P_1: fun_nat_bool] :
( ! [X1: nat] : hBOOL(hAPP_nat_bool(P_1,X1))
<=> ! [X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hBOOL(hAPP_nat_bool(P_1,nat_1(X_1))) ) ) ).
tff(fact_532_ex__nat,axiom,
! [P_1: fun_nat_bool] :
( ? [X1: nat] : hBOOL(hAPP_nat_bool(P_1,X1))
<=> ? [X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
& hBOOL(hAPP_nat_bool(P_1,nat_1(X_1))) ) ) ).
tff(fact_533_zero__le__square,axiom,
! [A_46: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),times_times_real(A_46,A_46))) ).
tff(fact_534_zero__le__square,axiom,
! [A_46: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_46,A_46))) ).
tff(fact_535_less__add__iff1,axiom,
! [A_1: real,E_1: real,C_1: real,B: real,D: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(times_times_real(A_1,E_1),C_1)),plus_plus_real(times_times_real(B,E_1),D)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(times_times_real(minus_minus_real(A_1,B),E_1),C_1)),D)) ) ).
tff(fact_536_less__add__iff1,axiom,
! [A_1: int,E_1: int,C_1: int,B: int,D: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(A_1,E_1),C_1)),plus_plus_int(times_times_int(B,E_1),D)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(minus_minus_int(A_1,B),E_1),C_1)),D)) ) ).
tff(fact_537_less__add__iff2,axiom,
! [A_1: real,E_1: real,C_1: real,B: real,D: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(times_times_real(A_1,E_1),C_1)),plus_plus_real(times_times_real(B,E_1),D)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_1),plus_plus_real(times_times_real(minus_minus_real(B,A_1),E_1),D))) ) ).
tff(fact_538_less__add__iff2,axiom,
! [A_1: int,E_1: int,C_1: int,B: int,D: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(A_1,E_1),C_1)),plus_plus_int(times_times_int(B,E_1),D)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),plus_plus_int(times_times_int(minus_minus_int(B,A_1),E_1),D))) ) ).
tff(fact_539_real__squared__diff__one__factored,axiom,
! [X_10: real] : minus_minus_real(times_times_real(X_10,X_10),one_one_real) = times_times_real(plus_plus_real(X_10,one_one_real),minus_minus_real(X_10,one_one_real)) ).
tff(fact_540_real__squared__diff__one__factored,axiom,
! [X_10: int] : minus_minus_int(times_times_int(X_10,X_10),one_one_int) = times_times_int(plus_plus_int(X_10,one_one_int),minus_minus_int(X_10,one_one_int)) ).
tff(fact_541_mult__left__le__imp__le,axiom,
! [C_20: real,A_45: real,B_27: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(C_20,A_45)),times_times_real(C_20,B_27)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_20))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_45),B_27)) ) ) ).
tff(fact_542_mult__left__le__imp__le,axiom,
! [C_20: nat,A_45: nat,B_27: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(C_20,A_45)),times_times_nat(C_20,B_27)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_20))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_45),B_27)) ) ) ).
tff(fact_543_mult__left__le__imp__le,axiom,
! [C_20: int,A_45: int,B_27: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_20,A_45)),times_times_int(C_20,B_27)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_20))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_45),B_27)) ) ) ).
tff(fact_544_mult__right__le__imp__le,axiom,
! [A_44: real,C_19: real,B_26: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_44,C_19)),times_times_real(B_26,C_19)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_19))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_44),B_26)) ) ) ).
tff(fact_545_mult__right__le__imp__le,axiom,
! [A_44: nat,C_19: nat,B_26: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_44,C_19)),times_times_nat(B_26,C_19)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_19))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_44),B_26)) ) ) ).
tff(fact_546_mult__right__le__imp__le,axiom,
! [A_44: int,C_19: int,B_26: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_44,C_19)),times_times_int(B_26,C_19)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_19))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_44),B_26)) ) ) ).
tff(fact_547_mult__less__imp__less__left,axiom,
! [C_18: real,A_43: real,B_25: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(C_18,A_43)),times_times_real(C_18,B_25)))
=> ( 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_real,A_43),B_25)) ) ) ).
tff(fact_548_mult__less__imp__less__left,axiom,
! [C_18: nat,A_43: nat,B_25: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(C_18,A_43)),times_times_nat(C_18,B_25)))
=> ( 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_nat,A_43),B_25)) ) ) ).
tff(fact_549_mult__less__imp__less__left,axiom,
! [C_18: int,A_43: int,B_25: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_18,A_43)),times_times_int(C_18,B_25)))
=> ( 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_int,A_43),B_25)) ) ) ).
tff(fact_550_mult__left__less__imp__less,axiom,
! [C_17: real,A_42: real,B_24: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(C_17,A_42)),times_times_real(C_17,B_24)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_17))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_42),B_24)) ) ) ).
tff(fact_551_mult__left__less__imp__less,axiom,
! [C_17: nat,A_42: nat,B_24: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(C_17,A_42)),times_times_nat(C_17,B_24)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_17))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_42),B_24)) ) ) ).
tff(fact_552_mult__left__less__imp__less,axiom,
! [C_17: int,A_42: int,B_24: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_17,A_42)),times_times_int(C_17,B_24)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_17))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_42),B_24)) ) ) ).
tff(fact_553_mult__less__imp__less__right,axiom,
! [A_41: real,C_16: real,B_23: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_41,C_16)),times_times_real(B_23,C_16)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_16))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_41),B_23)) ) ) ).
tff(fact_554_mult__less__imp__less__right,axiom,
! [A_41: nat,C_16: nat,B_23: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_41,C_16)),times_times_nat(B_23,C_16)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_16))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_41),B_23)) ) ) ).
tff(fact_555_mult__less__imp__less__right,axiom,
! [A_41: int,C_16: int,B_23: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_41,C_16)),times_times_int(B_23,C_16)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_16))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_41),B_23)) ) ) ).
tff(fact_556_mult__right__less__imp__less,axiom,
! [A_40: real,C_15: real,B_22: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_40,C_15)),times_times_real(B_22,C_15)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_15))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_40),B_22)) ) ) ).
tff(fact_557_mult__right__less__imp__less,axiom,
! [A_40: nat,C_15: nat,B_22: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_40,C_15)),times_times_nat(B_22,C_15)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_15))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_40),B_22)) ) ) ).
tff(fact_558_mult__right__less__imp__less,axiom,
! [A_40: int,C_15: int,B_22: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_40,C_15)),times_times_int(B_22,C_15)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_15))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_40),B_22)) ) ) ).
tff(fact_559_mult__le__less__imp__less,axiom,
! [C_14: real,D_8: real,A_39: real,B_21: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_39),B_21))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_14),D_8))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_39))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_14))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_39,C_14)),times_times_real(B_21,D_8))) ) ) ) ) ).
tff(fact_560_mult__le__less__imp__less,axiom,
! [C_14: nat,D_8: nat,A_39: nat,B_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_39),B_21))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_14),D_8))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_39))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_14))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_39,C_14)),times_times_nat(B_21,D_8))) ) ) ) ) ).
tff(fact_561_mult__le__less__imp__less,axiom,
! [C_14: int,D_8: int,A_39: int,B_21: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_39),B_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_14),D_8))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_39))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_14))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_39,C_14)),times_times_int(B_21,D_8))) ) ) ) ) ).
tff(fact_562_mult__less__le__imp__less,axiom,
! [C_13: real,D_7: real,A_38: real,B_20: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_38),B_20))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_13),D_7))
=> ( 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_real,zero_zero_real),C_13))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_38,C_13)),times_times_real(B_20,D_7))) ) ) ) ) ).
tff(fact_563_mult__less__le__imp__less,axiom,
! [C_13: nat,D_7: nat,A_38: nat,B_20: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_38),B_20))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_13),D_7))
=> ( 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_nat,zero_zero_nat),C_13))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_38,C_13)),times_times_nat(B_20,D_7))) ) ) ) ) ).
tff(fact_564_mult__less__le__imp__less,axiom,
! [C_13: int,D_7: int,A_38: int,B_20: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_38),B_20))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_13),D_7))
=> ( 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_int,zero_zero_int),C_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_38,C_13)),times_times_int(B_20,D_7))) ) ) ) ) ).
tff(fact_565_mult__strict__mono_H,axiom,
! [C_12: real,D_6: real,A_37: real,B_19: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_37),B_19))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_12),D_6))
=> ( 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),C_12))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_37,C_12)),times_times_real(B_19,D_6))) ) ) ) ) ).
tff(fact_566_mult__strict__mono_H,axiom,
! [C_12: nat,D_6: nat,A_37: nat,B_19: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_37),B_19))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_12),D_6))
=> ( 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),C_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_37,C_12)),times_times_nat(B_19,D_6))) ) ) ) ) ).
tff(fact_567_mult__strict__mono_H,axiom,
! [C_12: int,D_6: int,A_37: int,B_19: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_37),B_19))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_12),D_6))
=> ( 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),C_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_37,C_12)),times_times_int(B_19,D_6))) ) ) ) ) ).
tff(fact_568_mult__strict__mono,axiom,
! [C_11: real,D_5: real,A_36: real,B_18: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_36),B_18))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_11),D_5))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_18))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_11))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_36,C_11)),times_times_real(B_18,D_5))) ) ) ) ) ).
tff(fact_569_mult__strict__mono,axiom,
! [C_11: nat,D_5: nat,A_36: nat,B_18: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_36),B_18))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_11),D_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_18))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_11))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_36,C_11)),times_times_nat(B_18,D_5))) ) ) ) ) ).
tff(fact_570_mult__strict__mono,axiom,
! [C_11: int,D_5: int,A_36: int,B_18: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_36),B_18))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_11),D_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_18))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_11))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_36,C_11)),times_times_int(B_18,D_5))) ) ) ) ) ).
tff(fact_571_mult__le__cancel__left__neg,axiom,
! [A_1: real,B: 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,times_times_real(C_1,A_1)),times_times_real(C_1,B)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),A_1)) ) ) ).
tff(fact_572_mult__le__cancel__left__neg,axiom,
! [A_1: int,B: 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,times_times_int(C_1,A_1)),times_times_int(C_1,B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A_1)) ) ) ).
tff(fact_573_mult__le__cancel__left__pos,axiom,
! [A_1: real,B: 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,times_times_real(C_1,A_1)),times_times_real(C_1,B)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B)) ) ) ).
tff(fact_574_mult__le__cancel__left__pos,axiom,
! [A_1: int,B: 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,times_times_int(C_1,A_1)),times_times_int(C_1,B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B)) ) ) ).
tff(fact_575_sum__squares__ge__zero,axiom,
! [X_9: real,Y_8: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),plus_plus_real(times_times_real(X_9,X_9),times_times_real(Y_8,Y_8)))) ).
tff(fact_576_sum__squares__ge__zero,axiom,
! [X_9: int,Y_8: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(times_times_int(X_9,X_9),times_times_int(Y_8,Y_8)))) ).
tff(fact_577_sum__squares__le__zero__iff,axiom,
! [Xa: real,Ya: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(times_times_real(Xa,Xa),times_times_real(Ya,Ya))),zero_zero_real))
<=> ( ( Xa = zero_zero_real )
& ( Ya = zero_zero_real ) ) ) ).
tff(fact_578_sum__squares__le__zero__iff,axiom,
! [Xa: int,Ya: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(Xa,Xa),times_times_int(Ya,Ya))),zero_zero_int))
<=> ( ( Xa = zero_zero_int )
& ( Ya = zero_zero_int ) ) ) ).
tff(fact_579_mult__right__le__one__le,axiom,
! [Y_7: real,X_8: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_8))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_7))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_7),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(X_8,Y_7)),X_8)) ) ) ) ).
tff(fact_580_mult__right__le__one__le,axiom,
! [Y_7: int,X_8: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_8))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_7),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(X_8,Y_7)),X_8)) ) ) ) ).
tff(fact_581_mult__left__le__one__le,axiom,
! [Y_6: real,X_7: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_7))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_6))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_6),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(Y_6,X_7)),X_7)) ) ) ) ).
tff(fact_582_mult__left__le__one__le,axiom,
! [Y_6: int,X_7: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_6),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(Y_6,X_7)),X_7)) ) ) ) ).
tff(fact_583_nat__le__0,axiom,
! [Z: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),zero_zero_int))
=> ( nat_1(Z) = zero_zero_nat ) ) ).
tff(fact_584_nat__0__iff,axiom,
! [I_1: int] :
( ( nat_1(I_1) = zero_zero_nat )
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),zero_zero_int)) ) ).
tff(fact_585_power__increasing,axiom,
! [A_35: real,N_9: nat,N_8: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_9),N_8))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_35))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_35),N_9)),hAPP_nat_real(power_power_real(A_35),N_8))) ) ) ).
tff(fact_586_power__increasing,axiom,
! [A_35: nat,N_9: nat,N_8: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_9),N_8))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_35))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_35),N_9)),hAPP_nat_nat(power_power_nat(A_35),N_8))) ) ) ).
tff(fact_587_power__increasing,axiom,
! [A_35: int,N_9: nat,N_8: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_9),N_8))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_35))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_35),N_9)),hAPP_nat_int(power_power_int(A_35),N_8))) ) ) ).
tff(fact_588_int__nat__eq,axiom,
! [Z: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( hAPP_nat_int(semiri1621563631at_int,nat_1(Z)) = Z ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( hAPP_nat_int(semiri1621563631at_int,nat_1(Z)) = zero_zero_int ) ) ) ).
tff(fact_589_int__eq__iff,axiom,
! [Ma: nat,Z_1: int] :
( ( hAPP_nat_int(semiri1621563631at_int,Ma) = Z_1 )
<=> ( ( Ma = nat_1(Z_1) )
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) ) ) ).
tff(fact_590_nat__0__le,axiom,
! [Z: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( hAPP_nat_int(semiri1621563631at_int,nat_1(Z)) = Z ) ) ).
tff(fact_591_right__minus__eq,axiom,
! [A_1: real,B: real] :
( ( minus_minus_real(A_1,B) = zero_zero_real )
<=> ( A_1 = B ) ) ).
tff(fact_592_right__minus__eq,axiom,
! [A_1: int,B: int] :
( ( minus_minus_int(A_1,B) = zero_zero_int )
<=> ( A_1 = B ) ) ).
tff(fact_593_eq__iff__diff__eq__0,axiom,
! [A_1: real,B: real] :
( ( A_1 = B )
<=> ( minus_minus_real(A_1,B) = zero_zero_real ) ) ).
tff(fact_594_eq__iff__diff__eq__0,axiom,
! [A_1: int,B: int] :
( ( A_1 = B )
<=> ( minus_minus_int(A_1,B) = zero_zero_int ) ) ).
tff(fact_595_diff__self,axiom,
! [A_34: real] : minus_minus_real(A_34,A_34) = zero_zero_real ).
tff(fact_596_diff__self,axiom,
! [A_34: int] : minus_minus_int(A_34,A_34) = zero_zero_int ).
tff(fact_597_diff__0__right,axiom,
! [A_33: real] : minus_minus_real(A_33,zero_zero_real) = A_33 ).
tff(fact_598_diff__0__right,axiom,
! [A_33: int] : minus_minus_int(A_33,zero_zero_int) = A_33 ).
tff(fact_599_diff__eq__diff__less,axiom,
! [A_1: int,B: int,C_1: int,D: int] :
( ( minus_minus_int(A_1,B) = minus_minus_int(C_1,D) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),D)) ) ) ).
tff(fact_600_diff__eq__diff__less,axiom,
! [A_1: real,B: real,C_1: real,D: real] :
( ( minus_minus_real(A_1,B) = minus_minus_real(C_1,D) )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_1),D)) ) ) ).
tff(fact_601_add__diff__add,axiom,
! [A_32: real,C_10: real,B_17: real,D_4: real] : minus_minus_real(plus_plus_real(A_32,C_10),plus_plus_real(B_17,D_4)) = plus_plus_real(minus_minus_real(A_32,B_17),minus_minus_real(C_10,D_4)) ).
tff(fact_602_add__diff__add,axiom,
! [A_32: int,C_10: int,B_17: int,D_4: int] : minus_minus_int(plus_plus_int(A_32,C_10),plus_plus_int(B_17,D_4)) = plus_plus_int(minus_minus_int(A_32,B_17),minus_minus_int(C_10,D_4)) ).
tff(fact_603_add__diff__cancel,axiom,
! [A_31: real,B_16: real] : minus_minus_real(plus_plus_real(A_31,B_16),B_16) = A_31 ).
tff(fact_604_add__diff__cancel,axiom,
! [A_31: int,B_16: int] : minus_minus_int(plus_plus_int(A_31,B_16),B_16) = A_31 ).
tff(fact_605_diff__add__cancel,axiom,
! [A_30: real,B_15: real] : plus_plus_real(minus_minus_real(A_30,B_15),B_15) = A_30 ).
tff(fact_606_diff__add__cancel,axiom,
! [A_30: int,B_15: int] : plus_plus_int(minus_minus_int(A_30,B_15),B_15) = A_30 ).
tff(fact_607_mult__zero__left,axiom,
! [A_29: real] : times_times_real(zero_zero_real,A_29) = zero_zero_real ).
tff(fact_608_mult__zero__left,axiom,
! [A_29: nat] : times_times_nat(zero_zero_nat,A_29) = zero_zero_nat ).
tff(fact_609_mult__zero__left,axiom,
! [A_29: int] : times_times_int(zero_zero_int,A_29) = zero_zero_int ).
tff(fact_610_mult__zero__right,axiom,
! [A_28: real] : times_times_real(A_28,zero_zero_real) = zero_zero_real ).
tff(fact_611_mult__zero__right,axiom,
! [A_28: nat] : times_times_nat(A_28,zero_zero_nat) = zero_zero_nat ).
tff(fact_612_mult__zero__right,axiom,
! [A_28: int] : times_times_int(A_28,zero_zero_int) = zero_zero_int ).
tff(fact_613_mult__eq__0__iff,axiom,
! [A_1: real,B: real] :
( ( times_times_real(A_1,B) = zero_zero_real )
<=> ( ( A_1 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
tff(fact_614_mult__eq__0__iff,axiom,
! [A_1: int,B: int] :
( ( times_times_int(A_1,B) = zero_zero_int )
<=> ( ( A_1 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
tff(fact_615_no__zero__divisors,axiom,
! [B_14: real,A_27: real] :
( ( A_27 != zero_zero_real )
=> ( ( B_14 != zero_zero_real )
=> ( times_times_real(A_27,B_14) != zero_zero_real ) ) ) ).
tff(fact_616_no__zero__divisors,axiom,
! [B_14: nat,A_27: nat] :
( ( A_27 != zero_zero_nat )
=> ( ( B_14 != zero_zero_nat )
=> ( times_times_nat(A_27,B_14) != zero_zero_nat ) ) ) ).
tff(fact_617_no__zero__divisors,axiom,
! [B_14: int,A_27: int] :
( ( A_27 != zero_zero_int )
=> ( ( B_14 != zero_zero_int )
=> ( times_times_int(A_27,B_14) != zero_zero_int ) ) ) ).
tff(fact_618_divisors__zero,axiom,
! [A_26: real,B_13: real] :
( ( times_times_real(A_26,B_13) = zero_zero_real )
=> ( ( A_26 = zero_zero_real )
| ( B_13 = zero_zero_real ) ) ) ).
tff(fact_619_divisors__zero,axiom,
! [A_26: nat,B_13: nat] :
( ( times_times_nat(A_26,B_13) = zero_zero_nat )
=> ( ( A_26 = zero_zero_nat )
| ( B_13 = zero_zero_nat ) ) ) ).
tff(fact_620_divisors__zero,axiom,
! [A_26: int,B_13: int] :
( ( times_times_int(A_26,B_13) = zero_zero_int )
=> ( ( A_26 = zero_zero_int )
| ( B_13 = zero_zero_int ) ) ) ).
tff(fact_621_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_25: real] : times_times_real(A_25,zero_zero_real) = zero_zero_real ).
tff(fact_622_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_25: nat] : times_times_nat(A_25,zero_zero_nat) = zero_zero_nat ).
tff(fact_623_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_25: int] : times_times_int(A_25,zero_zero_int) = zero_zero_int ).
tff(fact_624_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_24: real] : times_times_real(zero_zero_real,A_24) = zero_zero_real ).
tff(fact_625_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_24: nat] : times_times_nat(zero_zero_nat,A_24) = zero_zero_nat ).
tff(fact_626_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_24: int] : times_times_int(zero_zero_int,A_24) = zero_zero_int ).
tff(fact_627_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat(M,N) = zero_zero_nat )
=> ( ( minus_minus_nat(N,M) = zero_zero_nat )
=> ( M = N ) ) ) ).
tff(fact_628_diff__self__eq__0,axiom,
! [M: nat] : minus_minus_nat(M,M) = zero_zero_nat ).
tff(fact_629_minus__nat_Odiff__0,axiom,
! [M: nat] : minus_minus_nat(M,zero_zero_nat) = M ).
tff(fact_630_diff__0__eq__0,axiom,
! [N: nat] : minus_minus_nat(zero_zero_nat,N) = zero_zero_nat ).
tff(fact_631_comm__semiring__class_Odistrib,axiom,
! [A_23: real,B_12: real,C_9: real] : times_times_real(plus_plus_real(A_23,B_12),C_9) = plus_plus_real(times_times_real(A_23,C_9),times_times_real(B_12,C_9)) ).
tff(fact_632_comm__semiring__class_Odistrib,axiom,
! [A_23: nat,B_12: nat,C_9: nat] : times_times_nat(plus_plus_nat(A_23,B_12),C_9) = plus_plus_nat(times_times_nat(A_23,C_9),times_times_nat(B_12,C_9)) ).
tff(fact_633_comm__semiring__class_Odistrib,axiom,
! [A_23: int,B_12: int,C_9: int] : times_times_int(plus_plus_int(A_23,B_12),C_9) = plus_plus_int(times_times_int(A_23,C_9),times_times_int(B_12,C_9)) ).
tff(fact_634_combine__common__factor,axiom,
! [A_22: real,E: real,B_11: real,C_8: real] : plus_plus_real(times_times_real(A_22,E),plus_plus_real(times_times_real(B_11,E),C_8)) = plus_plus_real(times_times_real(plus_plus_real(A_22,B_11),E),C_8) ).
tff(fact_635_combine__common__factor,axiom,
! [A_22: nat,E: nat,B_11: nat,C_8: nat] : plus_plus_nat(times_times_nat(A_22,E),plus_plus_nat(times_times_nat(B_11,E),C_8)) = plus_plus_nat(times_times_nat(plus_plus_nat(A_22,B_11),E),C_8) ).
tff(fact_636_combine__common__factor,axiom,
! [A_22: int,E: int,B_11: int,C_8: int] : plus_plus_int(times_times_int(A_22,E),plus_plus_int(times_times_int(B_11,E),C_8)) = plus_plus_int(times_times_int(plus_plus_int(A_22,B_11),E),C_8) ).
tff(fact_637_crossproduct__eq,axiom,
! [Wa: real,Ya: real,Xa: real,Z_1: real] :
( ( plus_plus_real(times_times_real(Wa,Ya),times_times_real(Xa,Z_1)) = plus_plus_real(times_times_real(Wa,Z_1),times_times_real(Xa,Ya)) )
<=> ( ( Wa = Xa )
| ( Ya = Z_1 ) ) ) ).
tff(fact_638_crossproduct__eq,axiom,
! [Wa: nat,Ya: nat,Xa: nat,Z_1: nat] :
( ( plus_plus_nat(times_times_nat(Wa,Ya),times_times_nat(Xa,Z_1)) = plus_plus_nat(times_times_nat(Wa,Z_1),times_times_nat(Xa,Ya)) )
<=> ( ( Wa = Xa )
| ( Ya = Z_1 ) ) ) ).
tff(fact_639_crossproduct__eq,axiom,
! [Wa: int,Ya: int,Xa: int,Z_1: int] :
( ( plus_plus_int(times_times_int(Wa,Ya),times_times_int(Xa,Z_1)) = plus_plus_int(times_times_int(Wa,Z_1),times_times_int(Xa,Ya)) )
<=> ( ( Wa = Xa )
| ( Ya = Z_1 ) ) ) ).
tff(fact_640_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_21: real,M_3: real,B_10: real] : plus_plus_real(times_times_real(A_21,M_3),times_times_real(B_10,M_3)) = times_times_real(plus_plus_real(A_21,B_10),M_3) ).
tff(fact_641_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_21: nat,M_3: nat,B_10: nat] : plus_plus_nat(times_times_nat(A_21,M_3),times_times_nat(B_10,M_3)) = times_times_nat(plus_plus_nat(A_21,B_10),M_3) ).
tff(fact_642_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_21: int,M_3: int,B_10: int] : plus_plus_int(times_times_int(A_21,M_3),times_times_int(B_10,M_3)) = times_times_int(plus_plus_int(A_21,B_10),M_3) ).
tff(fact_643_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_20: real,B_9: real,C_7: real] : times_times_real(plus_plus_real(A_20,B_9),C_7) = plus_plus_real(times_times_real(A_20,C_7),times_times_real(B_9,C_7)) ).
tff(fact_644_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_20: nat,B_9: nat,C_7: nat] : times_times_nat(plus_plus_nat(A_20,B_9),C_7) = plus_plus_nat(times_times_nat(A_20,C_7),times_times_nat(B_9,C_7)) ).
tff(fact_645_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_20: int,B_9: int,C_7: int] : times_times_int(plus_plus_int(A_20,B_9),C_7) = plus_plus_int(times_times_int(A_20,C_7),times_times_int(B_9,C_7)) ).
tff(fact_646_crossproduct__noteq,axiom,
! [C_1: real,D: real,A_1: real,B: real] :
( ( ( A_1 != B )
& ( C_1 != D ) )
<=> ( plus_plus_real(times_times_real(A_1,C_1),times_times_real(B,D)) != plus_plus_real(times_times_real(A_1,D),times_times_real(B,C_1)) ) ) ).
tff(fact_647_crossproduct__noteq,axiom,
! [C_1: nat,D: nat,A_1: nat,B: nat] :
( ( ( A_1 != B )
& ( C_1 != D ) )
<=> ( plus_plus_nat(times_times_nat(A_1,C_1),times_times_nat(B,D)) != plus_plus_nat(times_times_nat(A_1,D),times_times_nat(B,C_1)) ) ) ).
tff(fact_648_crossproduct__noteq,axiom,
! [C_1: int,D: int,A_1: int,B: int] :
( ( ( A_1 != B )
& ( C_1 != D ) )
<=> ( plus_plus_int(times_times_int(A_1,C_1),times_times_int(B,D)) != plus_plus_int(times_times_int(A_1,D),times_times_int(B,C_1)) ) ) ).
tff(fact_649_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_6: real,Y_5: real,Z_3: real] : times_times_real(X_6,plus_plus_real(Y_5,Z_3)) = plus_plus_real(times_times_real(X_6,Y_5),times_times_real(X_6,Z_3)) ).
tff(fact_650_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_6: nat,Y_5: nat,Z_3: nat] : times_times_nat(X_6,plus_plus_nat(Y_5,Z_3)) = plus_plus_nat(times_times_nat(X_6,Y_5),times_times_nat(X_6,Z_3)) ).
tff(fact_651_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_6: int,Y_5: int,Z_3: int] : times_times_int(X_6,plus_plus_int(Y_5,Z_3)) = plus_plus_int(times_times_int(X_6,Y_5),times_times_int(X_6,Z_3)) ).
tff(fact_652_nat__int,axiom,
! [N: nat] : nat_1(hAPP_nat_int(semiri1621563631at_int,N)) = N ).
tff(fact_653_mult_Ocomm__neutral,axiom,
! [A_19: real] : times_times_real(A_19,one_one_real) = A_19 ).
tff(fact_654_mult_Ocomm__neutral,axiom,
! [A_19: nat] : times_times_nat(A_19,one_one_nat) = A_19 ).
tff(fact_655_mult_Ocomm__neutral,axiom,
! [A_19: int] : times_times_int(A_19,one_one_int) = A_19 ).
tff(fact_656_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_18: real] : times_times_real(A_18,one_one_real) = A_18 ).
tff(fact_657_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_18: nat] : times_times_nat(A_18,one_one_nat) = A_18 ).
tff(fact_658_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_18: int] : times_times_int(A_18,one_one_int) = A_18 ).
tff(fact_659_mult__1__right,axiom,
! [A_17: real] : times_times_real(A_17,one_one_real) = A_17 ).
tff(fact_660_mult__1__right,axiom,
! [A_17: nat] : times_times_nat(A_17,one_one_nat) = A_17 ).
tff(fact_661_mult__1__right,axiom,
! [A_17: int] : times_times_int(A_17,one_one_int) = A_17 ).
tff(fact_662_mult__1,axiom,
! [A_16: real] : times_times_real(one_one_real,A_16) = A_16 ).
tff(fact_663_mult__1,axiom,
! [A_16: nat] : times_times_nat(one_one_nat,A_16) = A_16 ).
tff(fact_664_mult__1,axiom,
! [A_16: int] : times_times_int(one_one_int,A_16) = A_16 ).
tff(fact_665_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_15: real] : times_times_real(one_one_real,A_15) = A_15 ).
tff(fact_666_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_15: nat] : times_times_nat(one_one_nat,A_15) = A_15 ).
tff(fact_667_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_15: int] : times_times_int(one_one_int,A_15) = A_15 ).
tff(fact_668_mult__1__left,axiom,
! [A_14: real] : times_times_real(one_one_real,A_14) = A_14 ).
tff(fact_669_mult__1__left,axiom,
! [A_14: nat] : times_times_nat(one_one_nat,A_14) = A_14 ).
tff(fact_670_mult__1__left,axiom,
! [A_14: int] : times_times_int(one_one_int,A_14) = A_14 ).
tff(fact_671_add__le__imp__le__left,axiom,
! [C_6: real,A_13: real,B_8: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(C_6,A_13)),plus_plus_real(C_6,B_8)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_13),B_8)) ) ).
tff(fact_672_add__le__imp__le__left,axiom,
! [C_6: nat,A_13: nat,B_8: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(C_6,A_13)),plus_plus_nat(C_6,B_8)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_13),B_8)) ) ).
tff(fact_673_add__le__imp__le__left,axiom,
! [C_6: int,A_13: int,B_8: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(C_6,A_13)),plus_plus_int(C_6,B_8)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_13),B_8)) ) ).
tff(fact_674_add__le__imp__le__right,axiom,
! [A_12: real,C_5: real,B_7: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(A_12,C_5)),plus_plus_real(B_7,C_5)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_12),B_7)) ) ).
tff(fact_675_add__le__imp__le__right,axiom,
! [A_12: nat,C_5: nat,B_7: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_12,C_5)),plus_plus_nat(B_7,C_5)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_12),B_7)) ) ).
tff(fact_676_add__le__imp__le__right,axiom,
! [A_12: int,C_5: int,B_7: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_12,C_5)),plus_plus_int(B_7,C_5)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_12),B_7)) ) ).
tff(fact_677_add__mono,axiom,
! [C_4: real,D_3: real,A_11: real,B_6: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_11),B_6))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_4),D_3))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(A_11,C_4)),plus_plus_real(B_6,D_3))) ) ) ).
tff(fact_678_add__mono,axiom,
! [C_4: nat,D_3: nat,A_11: nat,B_6: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_11),B_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_4),D_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_11,C_4)),plus_plus_nat(B_6,D_3))) ) ) ).
tff(fact_679_add__mono,axiom,
! [C_4: int,D_3: int,A_11: int,B_6: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_11),B_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_4),D_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_11,C_4)),plus_plus_int(B_6,D_3))) ) ) ).
tff(fact_680_add__left__mono,axiom,
! [C_3: real,A_10: real,B_5: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_10),B_5))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(C_3,A_10)),plus_plus_real(C_3,B_5))) ) ).
tff(fact_681_add__left__mono,axiom,
! [C_3: nat,A_10: nat,B_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_10),B_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(C_3,A_10)),plus_plus_nat(C_3,B_5))) ) ).
tff(fact_682_add__left__mono,axiom,
! [C_3: int,A_10: int,B_5: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_10),B_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(C_3,A_10)),plus_plus_int(C_3,B_5))) ) ).
tff(fact_683_add__right__mono,axiom,
! [C_2: real,A_9: real,B_4: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_9),B_4))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(A_9,C_2)),plus_plus_real(B_4,C_2))) ) ).
tff(fact_684_add__right__mono,axiom,
! [C_2: nat,A_9: nat,B_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_9),B_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_9,C_2)),plus_plus_nat(B_4,C_2))) ) ).
tff(fact_685_add__right__mono,axiom,
! [C_2: int,A_9: int,B_4: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_9),B_4))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_9,C_2)),plus_plus_int(B_4,C_2))) ) ).
tff(fact_686_add__le__cancel__left,axiom,
! [C_1: real,A_1: real,B: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(C_1,A_1)),plus_plus_real(C_1,B)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B)) ) ).
tff(fact_687_add__le__cancel__left,axiom,
! [C_1: nat,A_1: nat,B: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(C_1,A_1)),plus_plus_nat(C_1,B)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),B)) ) ).
tff(fact_688_add__le__cancel__left,axiom,
! [C_1: int,A_1: int,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(C_1,A_1)),plus_plus_int(C_1,B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B)) ) ).
tff(fact_689_add__le__cancel__right,axiom,
! [A_1: real,C_1: real,B: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(A_1,C_1)),plus_plus_real(B,C_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B)) ) ).
tff(fact_690_add__le__cancel__right,axiom,
! [A_1: nat,C_1: nat,B: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_1,C_1)),plus_plus_nat(B,C_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),B)) ) ).
tff(fact_691_add__le__cancel__right,axiom,
! [A_1: int,C_1: int,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_1,C_1)),plus_plus_int(B,C_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B)) ) ).
tff(fact_692_less__imp__diff__less,axiom,
! [N: 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,minus_minus_nat(J_1,N)),K_1)) ) ).
tff(fact_693_diff__less__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,minus_minus_nat(L,N)),minus_minus_nat(L,M))) ) ) ).
tff(fact_694_power__commutes,axiom,
! [A_8: real,N_7: nat] : times_times_real(hAPP_nat_real(power_power_real(A_8),N_7),A_8) = times_times_real(A_8,hAPP_nat_real(power_power_real(A_8),N_7)) ).
tff(fact_695_power__commutes,axiom,
! [A_8: nat,N_7: nat] : times_times_nat(hAPP_nat_nat(power_power_nat(A_8),N_7),A_8) = times_times_nat(A_8,hAPP_nat_nat(power_power_nat(A_8),N_7)) ).
tff(fact_696_power__commutes,axiom,
! [A_8: int,N_7: nat] : times_times_int(hAPP_nat_int(power_power_int(A_8),N_7),A_8) = times_times_int(A_8,hAPP_nat_int(power_power_int(A_8),N_7)) ).
tff(fact_697_power__mult__distrib,axiom,
! [A_7: real,B_3: real,N_6: nat] : hAPP_nat_real(power_power_real(times_times_real(A_7,B_3)),N_6) = times_times_real(hAPP_nat_real(power_power_real(A_7),N_6),hAPP_nat_real(power_power_real(B_3),N_6)) ).
tff(fact_698_power__mult__distrib,axiom,
! [A_7: nat,B_3: nat,N_6: nat] : hAPP_nat_nat(power_power_nat(times_times_nat(A_7,B_3)),N_6) = times_times_nat(hAPP_nat_nat(power_power_nat(A_7),N_6),hAPP_nat_nat(power_power_nat(B_3),N_6)) ).
tff(fact_699_power__mult__distrib,axiom,
! [A_7: int,B_3: int,N_6: nat] : hAPP_nat_int(power_power_int(times_times_int(A_7,B_3)),N_6) = times_times_int(hAPP_nat_int(power_power_int(A_7),N_6),hAPP_nat_int(power_power_int(B_3),N_6)) ).
tff(fact_700_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_5: real,Y_4: real,Q_2: nat] : hAPP_nat_real(power_power_real(times_times_real(X_5,Y_4)),Q_2) = times_times_real(hAPP_nat_real(power_power_real(X_5),Q_2),hAPP_nat_real(power_power_real(Y_4),Q_2)) ).
tff(fact_701_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_5: nat,Y_4: nat,Q_2: nat] : hAPP_nat_nat(power_power_nat(times_times_nat(X_5,Y_4)),Q_2) = times_times_nat(hAPP_nat_nat(power_power_nat(X_5),Q_2),hAPP_nat_nat(power_power_nat(Y_4),Q_2)) ).
tff(fact_702_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_5: int,Y_4: int,Q_2: nat] : hAPP_nat_int(power_power_int(times_times_int(X_5,Y_4)),Q_2) = times_times_int(hAPP_nat_int(power_power_int(X_5),Q_2),hAPP_nat_int(power_power_int(Y_4),Q_2)) ).
tff(fact_703_diff__add__inverse2,axiom,
! [M: nat,N: nat] : minus_minus_nat(plus_plus_nat(M,N),N) = M ).
tff(fact_704_diff__add__inverse,axiom,
! [N: nat,M: nat] : minus_minus_nat(plus_plus_nat(N,M),N) = M ).
tff(fact_705_diff__diff__left,axiom,
! [I_2: nat,J_1: nat,K_1: nat] : minus_minus_nat(minus_minus_nat(I_2,J_1),K_1) = minus_minus_nat(I_2,plus_plus_nat(J_1,K_1)) ).
tff(fact_706_Nat_Odiff__cancel,axiom,
! [K_1: nat,M: nat,N: nat] : minus_minus_nat(plus_plus_nat(K_1,M),plus_plus_nat(K_1,N)) = minus_minus_nat(M,N) ).
tff(fact_707_diff__cancel2,axiom,
! [M: nat,K_1: nat,N: nat] : minus_minus_nat(plus_plus_nat(M,K_1),plus_plus_nat(N,K_1)) = minus_minus_nat(M,N) ).
tff(fact_708_mult__Pls,axiom,
! [W: int] : times_times_int(pls,W) = pls ).
tff(fact_709_mult__Bit0,axiom,
! [K_1: int,L: int] : times_times_int(bit0(K_1),L) = bit0(times_times_int(K_1,L)) ).
tff(fact_710_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_711_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_712_rel__simps_I19_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),pls)) ).
tff(fact_713_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_714_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_715_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_716_zmult__1,axiom,
! [Z: int] : times_times_int(one_one_int,Z) = Z ).
tff(fact_717_zmult__1__right,axiom,
! [Z: int] : times_times_int(Z,one_one_int) = Z ).
tff(fact_718_zless__le,axiom,
! [Z_1: int,Wa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),Wa))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),Wa))
& ( Z_1 != Wa ) ) ) ).
tff(fact_719_zadd__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] : times_times_int(W,plus_plus_int(Z1,Z2)) = plus_plus_int(times_times_int(W,Z1),times_times_int(W,Z2)) ).
tff(fact_720_zadd__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] : times_times_int(plus_plus_int(Z1,Z2),W) = plus_plus_int(times_times_int(Z1,W),times_times_int(Z2,W)) ).
tff(fact_721_zadd__left__mono,axiom,
! [K_1: int,I_2: int,J_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),J_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(K_1,I_2)),plus_plus_int(K_1,J_1))) ) ).
tff(fact_722_times__numeral__code_I5_J,axiom,
! [V_1: int,W: int] : times_times_int(number_number_of_int(V_1),number_number_of_int(W)) = number_number_of_int(times_times_int(V_1,W)) ).
tff(fact_723_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_724_convex__bound__le,axiom,
! [V_5: real,U_3: real,Y_3: real,X_4: real,A_6: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_4),A_6))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_3),A_6))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),U_3))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),V_5))
=> ( ( plus_plus_real(U_3,V_5) = one_one_real )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(times_times_real(U_3,X_4),times_times_real(V_5,Y_3))),A_6)) ) ) ) ) ) ).
tff(fact_725_convex__bound__le,axiom,
! [V_5: int,U_3: int,Y_3: int,X_4: int,A_6: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_4),A_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_3),A_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V_5))
=> ( ( plus_plus_int(U_3,V_5) = one_one_int )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(U_3,X_4),times_times_int(V_5,Y_3))),A_6)) ) ) ) ) ) ).
tff(fact_726_transfer__nat__int__relations_I2_J,axiom,
! [Ya: int,Xa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,nat_1(Xa)),nat_1(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) ) ) ) ).
tff(fact_727_nat__less__eq__zless,axiom,
! [Z_1: int,Wa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,nat_1(Wa)),nat_1(Z_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_1)) ) ) ).
tff(fact_728_le__special_I1_J,axiom,
! [Ya: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),number267125858f_real(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Ya)) ) ).
tff(fact_729_le__special_I1_J,axiom,
! [Ya: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Ya)) ) ).
tff(fact_730_le__special_I3_J,axiom,
! [Xa: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(Xa)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),pls)) ) ).
tff(fact_731_le__special_I3_J,axiom,
! [Xa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(Xa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),pls)) ) ).
tff(fact_732_power__decreasing,axiom,
! [A_5: real,N_5: nat,N_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_5),N_4))
=> ( 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,A_5),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_5),N_4)),hAPP_nat_real(power_power_real(A_5),N_5))) ) ) ) ).
tff(fact_733_power__decreasing,axiom,
! [A_5: nat,N_5: nat,N_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_5),N_4))
=> ( 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,A_5),one_one_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_5),N_4)),hAPP_nat_nat(power_power_nat(A_5),N_5))) ) ) ) ).
tff(fact_734_power__decreasing,axiom,
! [A_5: int,N_5: nat,N_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_5),N_4))
=> ( 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,A_5),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_5),N_4)),hAPP_nat_int(power_power_int(A_5),N_5))) ) ) ) ).
tff(fact_735_power__increasing__iff,axiom,
! [Xa: nat,Ya: nat,B: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(B),Xa)),hAPP_nat_real(power_power_real(B),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) ) ) ).
tff(fact_736_power__increasing__iff,axiom,
! [Xa: nat,Ya: nat,B: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(B),Xa)),hAPP_nat_nat(power_power_nat(B),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) ) ) ).
tff(fact_737_power__increasing__iff,axiom,
! [Xa: nat,Ya: nat,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(B),Xa)),hAPP_nat_int(power_power_int(B),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) ) ) ).
tff(fact_738_power__le__imp__le__exp,axiom,
! [M_2: nat,N_3: nat,A_4: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_4))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_4),M_2)),hAPP_nat_real(power_power_real(A_4),N_3)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_3)) ) ) ).
tff(fact_739_power__le__imp__le__exp,axiom,
! [M_2: nat,N_3: nat,A_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_4),M_2)),hAPP_nat_nat(power_power_nat(A_4),N_3)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_3)) ) ) ).
tff(fact_740_power__le__imp__le__exp,axiom,
! [M_2: nat,N_3: nat,A_4: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_4),M_2)),hAPP_nat_int(power_power_int(A_4),N_3)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_3)) ) ) ).
tff(fact_741_nat__eq__iff2,axiom,
! [Ma: nat,Wa: int] :
( ( Ma = nat_1(Wa) )
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa))
=> ( Wa = hAPP_nat_int(semiri1621563631at_int,Ma) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa))
=> ( Ma = zero_zero_nat ) ) ) ) ).
tff(fact_742_nat__eq__iff,axiom,
! [Wa: int,Ma: nat] :
( ( nat_1(Wa) = Ma )
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa))
=> ( Wa = hAPP_nat_int(semiri1621563631at_int,Ma) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa))
=> ( Ma = zero_zero_nat ) ) ) ) ).
tff(fact_743_Nat__Transfer_Otransfer__nat__int__functions_I1_J,axiom,
! [Y: int,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( plus_plus_nat(nat_1(X),nat_1(Y)) = nat_1(plus_plus_int(X,Y)) ) ) ) ).
tff(fact_744_nat__add__distrib,axiom,
! [Z_2: int,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_eq_int,zero_zero_int),Z_2))
=> ( nat_1(plus_plus_int(Z,Z_2)) = plus_plus_nat(nat_1(Z),nat_1(Z_2)) ) ) ) ).
tff(fact_745_int__eq__iff__number__of,axiom,
! [Ma: nat,Va: int] :
( ( hAPP_nat_int(semiri1621563631at_int,Ma) = number_number_of_int(Va) )
<=> ( ( Ma = nat_1(number_number_of_int(Va)) )
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(Va))) ) ) ).
tff(fact_746_Nat__Transfer_Otransfer__nat__int__functions_I4_J,axiom,
! [N: nat,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hAPP_nat_nat(power_power_nat(nat_1(X)),N) = nat_1(hAPP_nat_int(power_power_int(X),N)) ) ) ).
tff(fact_747_nat__power__eq,axiom,
! [N: nat,Z: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( nat_1(hAPP_nat_int(power_power_int(Z),N)) = hAPP_nat_nat(power_power_nat(nat_1(Z)),N) ) ) ).
tff(fact_748_convex__bound__lt,axiom,
! [V_4: real,U_2: real,Y_2: real,X_3: real,A_3: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_3),A_3))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),A_3))
=> ( 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_4))
=> ( ( plus_plus_real(U_2,V_4) = one_one_real )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(times_times_real(U_2,X_3),times_times_real(V_4,Y_2))),A_3)) ) ) ) ) ) ).
tff(fact_749_convex__bound__lt,axiom,
! [V_4: int,U_2: int,Y_2: int,X_3: int,A_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_3),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_2),A_3))
=> ( 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_4))
=> ( ( plus_plus_int(U_2,V_4) = one_one_int )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(U_2,X_3),times_times_int(V_4,Y_2))),A_3)) ) ) ) ) ) ).
tff(fact_750_nat__less__iff,axiom,
! [Ma: nat,Wa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,nat_1(Wa)),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),hAPP_nat_int(semiri1621563631at_int,Ma))) ) ) ).
tff(fact_751_le__special_I2_J,axiom,
! [Ya: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),number267125858f_real(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(pls)),Ya)) ) ).
tff(fact_752_le__special_I2_J,axiom,
! [Ya: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),number_number_of_int(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(pls)),Ya)) ) ).
tff(fact_753_le__special_I4_J,axiom,
! [Xa: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(Xa)),one_one_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),bit1(pls))) ) ).
tff(fact_754_le__special_I4_J,axiom,
! [Xa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(Xa)),one_one_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),bit1(pls))) ) ).
tff(fact_755_realpow__minus__mult,axiom,
! [X_2: real,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2))
=> ( times_times_real(hAPP_nat_real(power_power_real(X_2),minus_minus_nat(N_2,one_one_nat)),X_2) = hAPP_nat_real(power_power_real(X_2),N_2) ) ) ).
tff(fact_756_realpow__minus__mult,axiom,
! [X_2: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2))
=> ( times_times_nat(hAPP_nat_nat(power_power_nat(X_2),minus_minus_nat(N_2,one_one_nat)),X_2) = hAPP_nat_nat(power_power_nat(X_2),N_2) ) ) ).
tff(fact_757_realpow__minus__mult,axiom,
! [X_2: int,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2))
=> ( times_times_int(hAPP_nat_int(power_power_int(X_2),minus_minus_nat(N_2,one_one_nat)),X_2) = hAPP_nat_int(power_power_int(X_2),N_2) ) ) ).
tff(fact_758_less__iff__diff__less__0,axiom,
! [A_1: real,B: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,minus_minus_real(A_1,B)),zero_zero_real)) ) ).
tff(fact_759_less__iff__diff__less__0,axiom,
! [A_1: int,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,minus_minus_int(A_1,B)),zero_zero_int)) ) ).
tff(fact_760_transfer__nat__int__numerals_I1_J,axiom,
zero_zero_nat = nat_1(zero_zero_int) ).
tff(fact_761_nat__0,axiom,
nat_1(zero_zero_int) = zero_zero_nat ).
tff(fact_762_not__square__less__zero,axiom,
! [A_2: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_2,A_2)),zero_zero_real)) ).
tff(fact_763_not__square__less__zero,axiom,
! [A_2: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_2,A_2)),zero_zero_int)) ).
tff(fact_764_mult__less__cancel__right__disj,axiom,
! [A_1: real,C_1: real,B: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_1,C_1)),times_times_real(B,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_1),B)) )
| ( 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),A_1)) ) ) ) ).
tff(fact_765_mult__less__cancel__right__disj,axiom,
! [A_1: int,C_1: int,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_1,C_1)),times_times_int(B,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_1),B)) )
| ( 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),A_1)) ) ) ) ).
tff(fact_766_nat__number__of,axiom,
! [W: int] : nat_1(number_number_of_int(W)) = number_number_of_nat(W) ).
tff(fact_767_nat__number__of__def,axiom,
! [V_1: int] : number_number_of_nat(V_1) = nat_1(number_number_of_int(V_1)) ).
tff(fact_768_transfer__nat__int__numerals_I2_J,axiom,
one_one_nat = nat_1(one_one_int) ).
tff(fact_769_zero__less__diff,axiom,
! [Na: nat,Ma: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),minus_minus_nat(Na,Ma)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
tff(fact_770_diff__less,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,minus_minus_nat(M,N)),M)) ) ) ).
tff(fact_771_diff__add__0,axiom,
! [N: nat,M: nat] : minus_minus_nat(N,plus_plus_nat(N,M)) = zero_zero_nat ).
tff(fact_772_less__diff__conv,axiom,
! [I_1: nat,J: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),minus_minus_nat(J,K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I_1,K)),J)) ) ).
tff(fact_773_add__diff__inverse,axiom,
! [M: nat,N: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ( plus_plus_nat(N,minus_minus_nat(M,N)) = M ) ) ).
tff(fact_774_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_775_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_776_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_777_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_778_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_779_pos__zmult__pos,axiom,
! [B_1: 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),times_times_int(A,B_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1)) ) ) ).
tff(fact_780_zmult__zless__mono2,axiom,
! [K_1: int,I_2: int,J_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),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,times_times_int(K_1,I_2)),times_times_int(K_1,J_1))) ) ) ).
tff(fact_781_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_782_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y: int,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(X,Y))) ) ) ).
tff(fact_783_zadd__zless__mono,axiom,
! [Z_2: int,Z: int,W_2: int,W: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_2),W))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),Z))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(W_2,Z_2)),plus_plus_int(W,Z))) ) ) ).
tff(fact_784_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Z))) ).
tff(fact_785_transfer__int__nat__quantifiers_I2_J,axiom,
! [P_1: fun_int_bool] :
( ? [X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
& hBOOL(hAPP_int_bool(P_1,X_1)) )
<=> ? [X_1: nat] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_1))) ) ).
tff(fact_786_transfer__int__nat__quantifiers_I1_J,axiom,
! [P_1: fun_int_bool] :
( ! [X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hBOOL(hAPP_int_bool(P_1,X_1)) )
<=> ! [X_1: nat] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_1))) ) ).
tff(fact_787_zero__zle__int,axiom,
! [N: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N))) ).
tff(fact_788_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [N: nat,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(X),N))) ) ).
tff(fact_789_zle__iff__zadd,axiom,
! [Wa: int,Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),Z_1))
<=> ? [N_1: nat] : Z_1 = plus_plus_int(Wa,hAPP_nat_int(semiri1621563631at_int,N_1)) ) ).
tff(fact_790_zpower__zadd__distrib,axiom,
! [X: int,Y: nat,Z: nat] : hAPP_nat_int(power_power_int(X),plus_plus_nat(Y,Z)) = times_times_int(hAPP_nat_int(power_power_int(X),Y),hAPP_nat_int(power_power_int(X),Z)) ).
tff(fact_791_nat__mono__iff,axiom,
! [Wa: int,Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,nat_1(Wa)),nat_1(Z_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_1)) ) ) ).
tff(fact_792_zless__nat__conj,axiom,
! [Wa: int,Z_1: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,nat_1(Wa)),nat_1(Z_1)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_1)) ) ) ).
tff(fact_793_zless__nat__eq__int__zless,axiom,
! [Ma: nat,Z_1: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),nat_1(Z_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),Z_1)) ) ).
tff(fact_794_nat__diff__split__asm,axiom,
! [P_1: fun_nat_bool,A_1: nat,B: nat] :
( hBOOL(hAPP_nat_bool(P_1,minus_minus_nat(A_1,B)))
<=> ~ ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B))
& ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
| ? [D_2: nat] :
( ( A_1 = plus_plus_nat(B,D_2) )
& ~ hBOOL(hAPP_nat_bool(P_1,D_2)) ) ) ) ).
tff(fact_795_nat__diff__split,axiom,
! [P_1: fun_nat_bool,A_1: nat,B: nat] :
( hBOOL(hAPP_nat_bool(P_1,minus_minus_nat(A_1,B)))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B))
=> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
& ! [D_2: nat] :
( ( A_1 = plus_plus_nat(B,D_2) )
=> hBOOL(hAPP_nat_bool(P_1,D_2)) ) ) ) ).
tff(fact_796_eq__0__number__of,axiom,
! [Va: int] :
( ( zero_zero_nat = number_number_of_nat(Va) )
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls)) ) ).
tff(fact_797_eq__number__of__0,axiom,
! [Va: int] :
( ( number_number_of_nat(Va) = zero_zero_nat )
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls)) ) ).
tff(fact_798_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_799_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_800_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_801_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_802_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_803_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_804_mult__Bit1,axiom,
! [K_1: int,L: int] : times_times_int(bit1(K_1),L) = plus_plus_int(bit0(times_times_int(K_1,L)),L) ).
tff(fact_805_pos__zmult__eq__1__iff,axiom,
! [Na: int,Ma: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( ( times_times_int(Ma,Na) = one_one_int )
<=> ( ( Ma = one_one_int )
& ( Na = one_one_int ) ) ) ) ).
tff(fact_806_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_807_int__le__0__conv,axiom,
! [Na: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Na)),zero_zero_int))
<=> ( Na = zero_zero_nat ) ) ).
tff(fact_808_succ__Pls,axiom,
succ(pls) = bit1(pls) ).
tff(fact_809_succ__Bit0,axiom,
! [K_1: int] : succ(bit0(K_1)) = bit1(K_1) ).
tff(fact_810_succ__Bit1,axiom,
! [K_1: int] : succ(bit1(K_1)) = bit0(succ(K_1)) ).
tff(fact_811_zle__add1__eq__le,axiom,
! [Wa: int,Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),plus_plus_int(Z_1,one_one_int)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),Z_1)) ) ).
tff(fact_812_add1__zle__eq,axiom,
! [Wa: int,Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(Wa,one_one_int)),Z_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_1)) ) ).
tff(fact_813_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,plus_plus_int(W,one_one_int)),Z)) ) ).
tff(fact_814_succ__def,axiom,
! [K_1: int] : succ(K_1) = plus_plus_int(K_1,one_one_int) ).
tff(fact_815_zero__less__nat__eq,axiom,
! [Z_1: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),nat_1(Z_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1)) ) ).
tff(fact_816_transfer__nat__int__numerals_I4_J,axiom,
number_number_of_nat(bit1(bit1(pls))) = nat_1(number_number_of_int(bit1(bit1(pls)))) ).
tff(fact_817_split__nat,axiom,
! [P_1: fun_nat_bool,I_1: int] :
( hBOOL(hAPP_nat_bool(P_1,nat_1(I_1)))
<=> ( ! [N_1: nat] :
( ( I_1 = hAPP_nat_int(semiri1621563631at_int,N_1) )
=> hBOOL(hAPP_nat_bool(P_1,N_1)) )
& ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),zero_zero_int))
=> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) ) ) ) ).
tff(fact_818_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_819_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),plus_plus_int(one_one_int,Z))) ) ).
tff(fact_820_zmult__zless__mono2__lemma,axiom,
! [K_1: nat,I_2: int,J_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),J_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(hAPP_nat_int(semiri1621563631at_int,K_1),I_2)),times_times_int(hAPP_nat_int(semiri1621563631at_int,K_1),J_1))) ) ) ).
tff(fact_821_add__Bit1__Bit1,axiom,
! [K_1: int,L: int] : plus_plus_int(bit1(K_1),bit1(L)) = bit0(plus_plus_int(K_1,succ(L))) ).
tff(fact_822_transfer__nat__int__numerals_I3_J,axiom,
number_number_of_nat(bit0(bit1(pls))) = nat_1(number_number_of_int(bit0(bit1(pls)))) ).
tff(fact_823_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_824_cube__square,axiom,
! [A: 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_825_power2__ge__self,axiom,
! [X: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit1(pls)))))) ).
tff(fact_826_int__pos__lt__two__imp__zero__or__one,axiom,
! [X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),number_number_of_int(bit0(bit1(pls)))))
=> ( ( X = zero_zero_int )
| ( X = one_one_int ) ) ) ) ).
tff(fact_827_Euler_Oaux__1,axiom,
! [A: int,P_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_2))
=> ( hAPP_nat_int(power_power_int(A),nat_1(P_2)) = times_times_int(A,hAPP_nat_int(power_power_int(A),minus_minus_nat(nat_1(P_2),one_one_nat))) ) ) ).
tff(fact_828_self__quotient__aux2,axiom,
! [R_1: int,Q: int,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( ( A = plus_plus_int(R_1,times_times_int(A,Q)) )
=> ( 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),one_one_int)) ) ) ) ).
tff(fact_829_self__quotient__aux1,axiom,
! [R_1: int,Q: int,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( ( A = plus_plus_int(R_1,times_times_int(A,Q)) )
=> ( 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)) ) ) ) ).
tff(fact_830_smaller_I2_J,axiom,
~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)))
=> ~ hBOOL(hAPP_int_bool(twoSqu820444569sum2sq,times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))))) ) ).
tff(fact_831_q__pos__lemma,axiom,
! [B_2: int,Q_1: int,R_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(times_times_int(B_2,Q_1),R_2)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Q_1)) ) ) ) ).
tff(fact_832_q__neg__lemma,axiom,
! [B_2: int,Q_1: int,R_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(B_2,Q_1),R_2)),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),zero_zero_int)) ) ) ) ).
tff(fact_833_nQ1,axiom,
~ hBOOL(hAPP_int_bool(twoSqu820444569sum2sq,times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,zero_zero_nat))))) ).
tff(fact_834_le0,axiom,
! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).
tff(fact_835_p0,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) ).
tff(fact_836_t__l__p,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,t),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) ).
tff(fact_837_qf1pt,axiom,
hBOOL(hAPP_int_bool(twoSqu820444569sum2sq,times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),t))) ).
tff(fact_838_IH,axiom,
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)))
& hBOOL(hAPP_int_bool(twoSqu820444569sum2sq,times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))))) ) ).
tff(fact_839__096_B_Bthesis_O_A_I_B_Bx_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_061_A_I4_A,axiom,
~ ! [X_1: int,Y_1: 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_1),number_number_of_nat(bit0(bit1(pls))))) != times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))) ).
tff(fact_840_t,axiom,
plus_plus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls)))),one_one_int) = times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),t) ).
tff(fact_841_p,axiom,
hBOOL(hAPP_int_bool(zprime,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) ).
tff(fact_842__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_,axiom,
( ( t = one_one_int )
=> ? [X_1: int,Y_1: 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_1),number_number_of_nat(bit0(bit1(pls))))) = plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int) ) ).
tff(fact_843_le__refl,axiom,
! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),N)) ).
tff(fact_844_le__square,axiom,
! [M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),times_times_nat(M,M))) ).
tff(fact_845_le__cube,axiom,
! [M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),times_times_nat(M,times_times_nat(M,M)))) ).
tff(fact_846_nat__mult__commute,axiom,
! [M: nat,N: nat] : times_times_nat(M,N) = times_times_nat(N,M) ).
tff(fact_847_nat__le__linear,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) ) ).
tff(fact_848_nat__mult__assoc,axiom,
! [M: nat,N: nat,K_1: nat] : times_times_nat(times_times_nat(M,N),K_1) = times_times_nat(M,times_times_nat(N,K_1)) ).
tff(fact_849_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_850_mult__le__mono1,axiom,
! [K_1: nat,I_2: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(I_2,K_1)),times_times_nat(J_1,K_1))) ) ).
tff(fact_851_mult__le__mono2,axiom,
! [K_1: nat,I_2: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(K_1,I_2)),times_times_nat(K_1,J_1))) ) ).
tff(fact_852_le__trans,axiom,
! [K_1: nat,I_2: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),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_2),K_1)) ) ) ).
tff(fact_853_le__antisym,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> ( M = N ) ) ) ).
tff(fact_854_mult__le__mono,axiom,
! [K_1: nat,L: nat,I_2: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),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,times_times_nat(I_2,K_1)),times_times_nat(J_1,L))) ) ) ).
tff(fact_855_real__le__eq__diff,axiom,
! [Xa: real,Ya: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,minus_minus_real(Xa,Ya)),zero_zero_real)) ) ).
tff(fact_856_mult__le__cancel2,axiom,
! [Ma: nat,K: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(Ma,K)),times_times_nat(Na,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),Na)) ) ) ).
tff(fact_857_mult__le__cancel1,axiom,
! [K: nat,Ma: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(K,Ma)),times_times_nat(K,Na)))
<=> ( 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),Na)) ) ) ).
tff(fact_858_real__mult__le__cancel__iff1,axiom,
! [Xa: real,Ya: 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,times_times_real(Xa,Z_1)),times_times_real(Ya,Z_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) ) ) ).
tff(fact_859_real__mult__le__cancel__iff2,axiom,
! [Xa: real,Ya: 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,times_times_real(Z_1,Xa)),times_times_real(Z_1,Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) ) ) ).
tff(fact_860_zdiff__int,axiom,
! [N: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> ( minus_minus_int(hAPP_nat_int(semiri1621563631at_int,M),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,minus_minus_nat(M,N)) ) ) ).
tff(fact_861_diff__bin__simps_I1_J,axiom,
! [K_1: int] : minus_minus_int(K_1,pls) = K_1 ).
tff(fact_862_diff__bin__simps_I7_J,axiom,
! [K_1: int,L: int] : minus_minus_int(bit0(K_1),bit0(L)) = bit0(minus_minus_int(K_1,L)) ).
tff(fact_863_zdiff__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] : times_times_int(minus_minus_int(Z1,Z2),W) = minus_minus_int(times_times_int(Z1,W),times_times_int(Z2,W)) ).
tff(fact_864_zdiff__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] : times_times_int(W,minus_minus_int(Z1,Z2)) = minus_minus_int(times_times_int(W,Z1),times_times_int(W,Z2)) ).
tff(fact_865_mult__0,axiom,
! [N: nat] : times_times_nat(zero_zero_nat,N) = zero_zero_nat ).
tff(fact_866_mult__0__right,axiom,
! [M: nat] : times_times_nat(M,zero_zero_nat) = zero_zero_nat ).
tff(fact_867_mult__is__0,axiom,
! [Ma: nat,Na: nat] :
( ( times_times_nat(Ma,Na) = zero_zero_nat )
<=> ( ( Ma = zero_zero_nat )
| ( Na = zero_zero_nat ) ) ) ).
tff(fact_868_mult__cancel1,axiom,
! [K: nat,Ma: nat,Na: nat] :
( ( times_times_nat(K,Ma) = times_times_nat(K,Na) )
<=> ( ( Ma = Na )
| ( K = zero_zero_nat ) ) ) ).
tff(fact_869_mult__cancel2,axiom,
! [Ma: nat,K: nat,Na: nat] :
( ( times_times_nat(Ma,K) = times_times_nat(Na,K) )
<=> ( ( Ma = Na )
| ( K = zero_zero_nat ) ) ) ).
tff(fact_870_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).
tff(fact_871_le__0__eq,axiom,
! [Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),zero_zero_nat))
<=> ( Na = zero_zero_nat ) ) ).
tff(fact_872_real__mult__left__cancel,axiom,
! [A_1: real,B: real,C_1: real] :
( ( C_1 != zero_zero_real )
=> ( ( times_times_real(C_1,A_1) = times_times_real(C_1,B) )
<=> ( A_1 = B ) ) ) ).
tff(fact_873_real__mult__right__cancel,axiom,
! [A_1: real,B: real,C_1: real] :
( ( C_1 != zero_zero_real )
=> ( ( times_times_real(A_1,C_1) = times_times_real(B,C_1) )
<=> ( A_1 = B ) ) ) ).
tff(fact_874_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
| ( M = N ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_875_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ( ( M != N )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
tff(fact_876_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_877_le__eq__less__or__eq,axiom,
! [Ma: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na))
| ( Ma = Na ) ) ) ).
tff(fact_878_nat__less__le,axiom,
! [Ma: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na))
& ( Ma != Na ) ) ) ).
tff(fact_879_real__less__def,axiom,
! [Xa: real,Ya: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))
& ( Xa != Ya ) ) ) ).
tff(fact_880_less__eq__real__def,axiom,
! [Xa: real,Ya: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))
| ( Xa = Ya ) ) ) ).
tff(fact_881_add__mult__distrib,axiom,
! [M: nat,N: nat,K_1: nat] : times_times_nat(plus_plus_nat(M,N),K_1) = plus_plus_nat(times_times_nat(M,K_1),times_times_nat(N,K_1)) ).
tff(fact_882_add__mult__distrib2,axiom,
! [K_1: nat,M: nat,N: nat] : times_times_nat(K_1,plus_plus_nat(M,N)) = plus_plus_nat(times_times_nat(K_1,M),times_times_nat(K_1,N)) ).
tff(fact_883_add__leE,axiom,
! [M: nat,K_1: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(M,K_1)),N))
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N)) ) ) ).
tff(fact_884_add__leD1,axiom,
! [M: nat,K_1: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(M,K_1)),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_885_add__leD2,axiom,
! [M: nat,K_1: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(M,K_1)),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N)) ) ).
tff(fact_886_add__le__mono,axiom,
! [K_1: nat,L: nat,I_2: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),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,plus_plus_nat(I_2,K_1)),plus_plus_nat(J_1,L))) ) ) ).
tff(fact_887_add__le__mono1,axiom,
! [K_1: nat,I_2: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(I_2,K_1)),plus_plus_nat(J_1,K_1))) ) ).
tff(fact_888_trans__le__add2,axiom,
! [M: nat,I_2: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),plus_plus_nat(M,J_1))) ) ).
tff(fact_889_trans__le__add1,axiom,
! [M: nat,I_2: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),plus_plus_nat(J_1,M))) ) ).
tff(fact_890_nat__add__left__cancel__le,axiom,
! [K: nat,Ma: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(K,Ma)),plus_plus_nat(K,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
tff(fact_891_le__iff__add,axiom,
! [Ma: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na))
<=> ? [K_2: nat] : Na = plus_plus_nat(Ma,K_2) ) ).
tff(fact_892_le__add1,axiom,
! [N: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),plus_plus_nat(N,M))) ).
tff(fact_893_le__add2,axiom,
! [N: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),plus_plus_nat(M,N))) ).
tff(fact_894_nat__mult__eq__one,axiom,
! [Na: nat,Ma: nat] :
( ( times_times_nat(Na,Ma) = one_one_nat )
<=> ( ( Na = one_one_nat )
& ( Ma = one_one_nat ) ) ) ).
tff(fact_895_nat__mult__eq__1__iff,axiom,
! [Ma: nat,Na: nat] :
( ( times_times_nat(Ma,Na) = one_one_nat )
<=> ( ( Ma = one_one_nat )
& ( Na = one_one_nat ) ) ) ).
tff(fact_896_nat__mult__1__right,axiom,
! [N: nat] : times_times_nat(N,one_one_nat) = N ).
tff(fact_897_nat__1__eq__mult__iff,axiom,
! [Ma: nat,Na: nat] :
( ( one_one_nat = times_times_nat(Ma,Na) )
<=> ( ( Ma = one_one_nat )
& ( Na = one_one_nat ) ) ) ).
tff(fact_898_nat__mult__1,axiom,
! [N: nat] : times_times_nat(one_one_nat,N) = N ).
tff(fact_899_diff__mult__distrib,axiom,
! [M: nat,N: nat,K_1: nat] : times_times_nat(minus_minus_nat(M,N),K_1) = minus_minus_nat(times_times_nat(M,K_1),times_times_nat(N,K_1)) ).
tff(fact_900_diff__mult__distrib2,axiom,
! [K_1: nat,M: nat,N: nat] : times_times_nat(K_1,minus_minus_nat(M,N)) = minus_minus_nat(times_times_nat(K_1,M),times_times_nat(K_1,N)) ).
tff(fact_901_real__mult__1,axiom,
! [Z: real] : times_times_real(one_one_real,Z) = Z ).
tff(fact_902_le__diff__iff,axiom,
! [Na: 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),Na))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(Ma,K)),minus_minus_nat(Na,K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ) ) ).
tff(fact_903_Nat_Odiff__diff__eq,axiom,
! [N: 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))
=> ( minus_minus_nat(minus_minus_nat(M,K_1),minus_minus_nat(N,K_1)) = minus_minus_nat(M,N) ) ) ) ).
tff(fact_904_eq__diff__iff,axiom,
! [Na: 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),Na))
=> ( ( minus_minus_nat(Ma,K) = minus_minus_nat(Na,K) )
<=> ( Ma = Na ) ) ) ) ).
tff(fact_905_diff__diff__cancel,axiom,
! [I_2: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N))
=> ( minus_minus_nat(N,minus_minus_nat(N,I_2)) = I_2 ) ) ).
tff(fact_906_diff__le__mono,axiom,
! [L: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(M,L)),minus_minus_nat(N,L))) ) ).
tff(fact_907_diff__le__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(L,N)),minus_minus_nat(L,M))) ) ).
tff(fact_908_Nat_Odiff__le__self,axiom,
! [M: nat,N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(M,N)),M)) ).
tff(fact_909_real__add__mult__distrib,axiom,
! [Z1: real,Z2: real,W: real] : times_times_real(plus_plus_real(Z1,Z2),W) = plus_plus_real(times_times_real(Z1,W),times_times_real(Z2,W)) ).
tff(fact_910_zpower__zpower,axiom,
! [X: int,Y: nat,Z: nat] : hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(X),Y)),Z) = hAPP_nat_int(power_power_int(X),times_times_nat(Y,Z)) ).
tff(fact_911_real__add__left__mono,axiom,
! [Z: real,X: real,Y: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(Z,X)),plus_plus_real(Z,Y))) ) ).
tff(fact_912_diff__bin__simps_I10_J,axiom,
! [K_1: int,L: int] : minus_minus_int(bit1(K_1),bit1(L)) = bit0(minus_minus_int(K_1,L)) ).
tff(fact_913_diff__bin__simps_I9_J,axiom,
! [K_1: int,L: int] : minus_minus_int(bit1(K_1),bit0(L)) = bit1(minus_minus_int(K_1,L)) ).
tff(fact_914_diff__bin__simps_I3_J,axiom,
! [L: int] : minus_minus_int(pls,bit0(L)) = bit0(minus_minus_int(pls,L)) ).
tff(fact_915_Euler_Oaux1,axiom,
! [A: int,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),A))
=> ( ( X != minus_minus_int(A,one_one_int) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),minus_minus_int(A,one_one_int))) ) ) ) ).
tff(fact_916_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,minus_minus_int(K,L_1)),zero_zero_int)) ) ).
tff(fact_917_mult__less__mono2,axiom,
! [K_1: nat,I_2: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),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,times_times_nat(K_1,I_2)),times_times_nat(K_1,J_1))) ) ) ).
tff(fact_918_mult__less__mono1,axiom,
! [K_1: nat,I_2: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),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,times_times_nat(I_2,K_1)),times_times_nat(J_1,K_1))) ) ) ).
tff(fact_919_mult__less__cancel2,axiom,
! [Ma: nat,K: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(Ma,K)),times_times_nat(Na,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),Na)) ) ) ).
tff(fact_920_mult__less__cancel1,axiom,
! [K: nat,Ma: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(K,Ma)),times_times_nat(K,Na)))
<=> ( 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),Na)) ) ) ).
tff(fact_921_nat__0__less__mult__iff,axiom,
! [Ma: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),times_times_nat(Ma,Na)))
<=> ( 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),Na)) ) ) ).
tff(fact_922_not__real__square__gt__zero,axiom,
! [Xa: real] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),times_times_real(Xa,Xa)))
<=> ( Xa = zero_zero_real ) ) ).
tff(fact_923_real__mult__less__mono2,axiom,
! [X: 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),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(Z,X)),times_times_real(Z,Y))) ) ) ).
tff(fact_924_real__mult__order,axiom,
! [Y: real,X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> ( 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),times_times_real(X,Y))) ) ) ).
tff(fact_925_real__mult__less__iff1,axiom,
! [Xa: real,Ya: 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,times_times_real(Xa,Z_1)),times_times_real(Ya,Z_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) ) ) ).
tff(fact_926_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M = times_times_nat(M,N) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
tff(fact_927_diff__is__0__eq,axiom,
! [Ma: nat,Na: nat] :
( ( minus_minus_nat(Ma,Na) = zero_zero_nat )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
tff(fact_928_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ( minus_minus_nat(M,N) = zero_zero_nat ) ) ).
tff(fact_929_real__two__squares__add__zero__iff,axiom,
! [Xa: real,Ya: real] :
( ( plus_plus_real(times_times_real(Xa,Xa),times_times_real(Ya,Ya)) = zero_zero_real )
<=> ( ( Xa = zero_zero_real )
& ( Ya = zero_zero_real ) ) ) ).
tff(fact_930_diff__less__mono,axiom,
! [C: nat,A: nat,B_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C),A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,minus_minus_nat(A,C)),minus_minus_nat(B_1,C))) ) ) ).
tff(fact_931_less__diff__iff,axiom,
! [Na: 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),Na))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,minus_minus_nat(Ma,K)),minus_minus_nat(Na,K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ) ) ).
tff(fact_932_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
! [X: nat,Y: nat] : times_times_int(hAPP_nat_int(semiri1621563631at_int,X),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,times_times_nat(X,Y)) ).
tff(fact_933_zmult__int,axiom,
! [M: nat,N: nat] : times_times_int(hAPP_nat_int(semiri1621563631at_int,M),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,times_times_nat(M,N)) ).
tff(fact_934_int__mult,axiom,
! [M: nat,N: nat] : hAPP_nat_int(semiri1621563631at_int,times_times_nat(M,N)) = times_times_int(hAPP_nat_int(semiri1621563631at_int,M),hAPP_nat_int(semiri1621563631at_int,N)) ).
tff(fact_935_diff__diff__right,axiom,
! [I_2: nat,K_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1))
=> ( minus_minus_nat(I_2,minus_minus_nat(J_1,K_1)) = minus_minus_nat(plus_plus_nat(I_2,K_1),J_1) ) ) ).
tff(fact_936_le__diff__conv,axiom,
! [J: nat,K: nat,I_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(J,K)),I_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),plus_plus_nat(I_1,K))) ) ).
tff(fact_937_le__add__diff,axiom,
! [M: nat,K_1: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),minus_minus_nat(plus_plus_nat(N,M),K_1))) ) ).
tff(fact_938_le__add__diff__inverse,axiom,
! [N: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> ( plus_plus_nat(N,minus_minus_nat(M,N)) = M ) ) ).
tff(fact_939_add__diff__assoc,axiom,
! [I_2: nat,K_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1))
=> ( plus_plus_nat(I_2,minus_minus_nat(J_1,K_1)) = minus_minus_nat(plus_plus_nat(I_2,J_1),K_1) ) ) ).
tff(fact_940_le__diff__conv2,axiom,
! [I_1: nat,K: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),minus_minus_nat(J,K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(I_1,K)),J)) ) ) ).
tff(fact_941_le__add__diff__inverse2,axiom,
! [N: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> ( plus_plus_nat(minus_minus_nat(M,N),N) = M ) ) ).
tff(fact_942_le__imp__diff__is__add,axiom,
! [K: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> ( ( minus_minus_nat(J,I_1) = K )
<=> ( J = plus_plus_nat(K,I_1) ) ) ) ).
tff(fact_943_diff__add__assoc,axiom,
! [I_2: nat,K_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1))
=> ( minus_minus_nat(plus_plus_nat(I_2,J_1),K_1) = plus_plus_nat(I_2,minus_minus_nat(J_1,K_1)) ) ) ).
tff(fact_944_add__diff__assoc2,axiom,
! [I_2: nat,K_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1))
=> ( plus_plus_nat(minus_minus_nat(J_1,K_1),I_2) = minus_minus_nat(plus_plus_nat(J_1,I_2),K_1) ) ) ).
tff(fact_945_diff__add__assoc2,axiom,
! [I_2: nat,K_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1))
=> ( minus_minus_nat(plus_plus_nat(J_1,I_2),K_1) = plus_plus_nat(minus_minus_nat(J_1,K_1),I_2) ) ) ).
tff(fact_946_Nat__Transfer_Otransfer__int__nat__relations_I3_J,axiom,
! [Xa: nat,Ya: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) ) ).
tff(fact_947_zle__int,axiom,
! [Ma: nat,Na: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
tff(fact_948_is__mult__sum2sq,axiom,
! [Y: int,X: int] :
( hBOOL(hAPP_int_bool(twoSqu820444569sum2sq,X))
=> ( hBOOL(hAPP_int_bool(twoSqu820444569sum2sq,Y))
=> hBOOL(hAPP_int_bool(twoSqu820444569sum2sq,times_times_int(X,Y))) ) ) ).
tff(fact_949_zle__diff1__eq,axiom,
! [Wa: int,Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),minus_minus_int(Z_1,one_one_int)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_1)) ) ).
tff(fact_950_le__nat__number__of,axiom,
! [Va: int,V_3: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,number_number_of_nat(Va)),number_number_of_nat(V_3)))
<=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),V_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls)) ) ) ).
tff(fact_951_transfer__nat__int__relations_I3_J,axiom,
! [Ya: int,Xa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,nat_1(Xa)),nat_1(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) ) ) ) ).
tff(fact_952_nat__diff__distrib,axiom,
! [Z: int,Z_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),Z))
=> ( nat_1(minus_minus_int(Z,Z_2)) = minus_minus_nat(nat_1(Z),nat_1(Z_2)) ) ) ) ).
tff(fact_953_nat__mult__distrib,axiom,
! [Z_2: int,Z: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( nat_1(times_times_int(Z,Z_2)) = times_times_nat(nat_1(Z),nat_1(Z_2)) ) ) ).
tff(fact_954_Nat__Transfer_Otransfer__nat__int__functions_I2_J,axiom,
! [Y: int,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( times_times_nat(nat_1(X),nat_1(Y)) = nat_1(times_times_int(X,Y)) ) ) ) ).
tff(fact_955_mult__eq__if,axiom,
! [N: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( times_times_nat(M,N) = zero_zero_nat ) )
& ( ( M != zero_zero_nat )
=> ( times_times_nat(M,N) = plus_plus_nat(N,times_times_nat(minus_minus_nat(M,one_one_nat),N)) ) ) ) ).
tff(fact_956_nat__le__eq__zle,axiom,
! [Z_1: int,Wa: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Wa))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,nat_1(Wa)),nat_1(Z_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),Z_1)) ) ) ).
tff(fact_957_power__eq__if,axiom,
! [P_2: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(P_2),M) = one_one_nat ) )
& ( ( M != zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(P_2),M) = times_times_nat(P_2,hAPP_nat_nat(power_power_nat(P_2),minus_minus_nat(M,one_one_nat))) ) ) ) ).
tff(fact_958_nat__mult__2,axiom,
! [Z: nat] : times_times_nat(number_number_of_nat(bit0(bit1(pls))),Z) = plus_plus_nat(Z,Z) ).
tff(fact_959_nat__mult__2__right,axiom,
! [Z: nat] : times_times_nat(Z,number_number_of_nat(bit0(bit1(pls)))) = plus_plus_nat(Z,Z) ).
tff(fact_960_nat__number__of__mult__left,axiom,
! [V_2: int,K_1: nat,V_1: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> ( times_times_nat(number_number_of_nat(V_1),times_times_nat(number_number_of_nat(V_2),K_1)) = zero_zero_nat ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> ( times_times_nat(number_number_of_nat(V_1),times_times_nat(number_number_of_nat(V_2),K_1)) = times_times_nat(number_number_of_nat(times_times_int(V_1,V_2)),K_1) ) ) ) ).
tff(fact_961_mult__nat__number__of,axiom,
! [V_2: int,V_1: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> ( times_times_nat(number_number_of_nat(V_1),number_number_of_nat(V_2)) = zero_zero_nat ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> ( times_times_nat(number_number_of_nat(V_1),number_number_of_nat(V_2)) = number_number_of_nat(times_times_int(V_1,V_2)) ) ) ) ).
tff(fact_962_four__x__squared,axiom,
! [X: real] : times_times_real(number267125858f_real(bit0(bit0(bit1(pls)))),hAPP_nat_real(power_power_real(X),number_number_of_nat(bit0(bit1(pls))))) = hAPP_nat_real(power_power_real(times_times_real(number267125858f_real(bit0(bit1(pls))),X)),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_963_two__realpow__ge__one,axiom,
! [N: 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))) ).
tff(fact_964_Euler_Oaux2,axiom,
! [B_1: 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_1),C))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B_1))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),A)) ) ) ) ).
tff(fact_965_zspecial__product,axiom,
! [A: int,B_1: int] : times_times_int(plus_plus_int(A,B_1),minus_minus_int(A,B_1)) = minus_minus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_966_diff__square,axiom,
! [X: nat,Y: nat] : minus_minus_nat(hAPP_nat_nat(power_power_nat(X),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_nat(power_power_nat(Y),number_number_of_nat(bit0(bit1(pls))))) = times_times_nat(plus_plus_nat(X,Y),minus_minus_nat(X,Y)) ).
tff(fact_967_zdiff__power3,axiom,
! [A: int,B_1: int] : hAPP_nat_int(power_power_int(minus_minus_int(A,B_1)),number_number_of_nat(bit1(bit1(pls)))) = minus_minus_int(plus_plus_int(minus_minus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit1(bit1(pls)))),times_times_int(times_times_int(number_number_of_int(bit1(bit1(pls))),hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls))))),B_1)),times_times_int(times_times_int(number_number_of_int(bit1(bit1(pls))),A),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit0(bit1(pls)))))),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit1(bit1(pls))))) ).
tff(fact_968_zdiff__power2,axiom,
! [A: int,B_1: int] : hAPP_nat_int(power_power_int(minus_minus_int(A,B_1)),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_int(minus_minus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))),times_times_int(times_times_int(number_number_of_int(bit0(bit1(pls))),A),B_1)),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_969_zdiv__mono2__neg__lemma,axiom,
! [B_1: int,Q: int,R_1: int,B_2: int,Q_1: int,R_2: int] :
( ( plus_plus_int(times_times_int(B_1,Q),R_1) = plus_plus_int(times_times_int(B_2,Q_1),R_2) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(B_2,Q_1),R_2)),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),B_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q)) ) ) ) ) ) ) ).
tff(fact_970_unique__quotient__lemma__neg,axiom,
! [B_1: int,Q_1: int,R_2: int,Q: int,R_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(B_1,Q_1),R_2)),plus_plus_int(times_times_int(B_1,Q),R_1)))
=> ( 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_1),R_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),R_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q),Q_1)) ) ) ) ) ).
tff(fact_971_zdiv__mono2__lemma,axiom,
! [B_1: int,Q: int,R_1: int,B_2: int,Q_1: int,R_2: int] :
( ( plus_plus_int(times_times_int(B_1,Q),R_1) = plus_plus_int(times_times_int(B_2,Q_1),R_2) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(times_times_int(B_2,Q_1),R_2)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_2))
=> ( 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_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),B_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q),Q_1)) ) ) ) ) ) ) ).
tff(fact_972_unique__quotient__lemma,axiom,
! [B_1: int,Q_1: int,R_2: int,Q: int,R_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(B_1,Q_1),R_2)),plus_plus_int(times_times_int(B_1,Q),R_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q)) ) ) ) ) ).
tff(fact_973_xy,axiom,
plus_plus_int(hAPP_nat_int(power_power_int(x),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(y),number_number_of_nat(bit0(bit1(pls))))) = times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))) ).
tff(fact_974_Int2_Oaux__1,axiom,
! [P_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_2))
=> ( minus_minus_nat(nat_1(P_2),number_number_of_nat(bit0(bit1(pls)))) = nat_1(minus_minus_int(P_2,number_number_of_int(bit0(bit1(pls))))) ) ) ).
tff(fact_975_Int2_Oaux__2,axiom,
! [P_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),nat_1(minus_minus_int(P_2,number_number_of_int(bit0(bit1(pls))))))) ) ).
tff(fact_976__096_B_Bthesis_O_A_I_B_Bt_O_As_____A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_,axiom,
~ ! [T: int] : plus_plus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls)))),one_one_int) != times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),T) ).
tff(fact_977_real__le__refl,axiom,
! [W: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),W)) ).
tff(fact_978_real__mult__commute,axiom,
! [Z: real,W: real] : times_times_real(Z,W) = times_times_real(W,Z) ).
tff(fact_979_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_980_real__mult__assoc,axiom,
! [Z1: real,Z2: real,Z3: real] : times_times_real(times_times_real(Z1,Z2),Z3) = times_times_real(Z1,times_times_real(Z2,Z3)) ).
tff(fact_981_real__le__trans,axiom,
! [K_1: real,I_2: real,J_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I_2),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_2),K_1)) ) ) ).
tff(fact_982_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_983_zprime__2,axiom,
hBOOL(hAPP_int_bool(zprime,number_number_of_int(bit0(bit1(pls))))) ).
tff(fact_984_Int2_Oaux1,axiom,
! [A: int,B_1: int,C: int] :
( ( minus_minus_int(A,B_1) = C )
=> ( A = plus_plus_int(C,B_1) ) ) ).
tff(fact_985__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,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)),plus_plus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls)))),one_one_int))) ).
tff(fact_986_real__sum__squared__expand,axiom,
! [X: real,Y: real] : hAPP_nat_real(power_power_real(plus_plus_real(X,Y)),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_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),number_number_of_nat(bit0(bit1(pls))))),times_times_real(times_times_real(number267125858f_real(bit0(bit1(pls))),X),Y)) ).
tff(fact_987__096s_____A_094_A2_A_N_A_N1_A_061_As_____A_094_A2_A_L_A1_096,axiom,
minus_minus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls)))),number_number_of_int(min)) = plus_plus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls)))),one_one_int) ).
tff(fact_988__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,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_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_989__096Legendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_061_A1_096,axiom,
legendre(number_number_of_int(min),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) = one_one_int ).
tff(fact_990_zdvd__zdiffD,axiom,
! [K_1: int,M: int,N: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),minus_minus_int(M,N)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),M)) ) ) ).
tff(fact_991_zprime__zdvd__zmult__better,axiom,
! [M: int,N: int,P_2: int] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),times_times_int(M,N)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),M))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),N)) ) ) ) ).
tff(fact_992_rel__simps_I24_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),min)) ).
tff(fact_993_zdvd__bounds,axiom,
! [N: int,M: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),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),M)) ) ) ).
tff(fact_994_zprime__zdvd__power,axiom,
! [A: int,N: nat,P_2: int] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),hAPP_nat_int(power_power_int(A),N)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),A)) ) ) ).
tff(fact_995_zdvd__not__zless,axiom,
! [N: 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))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M)) ) ) ).
tff(fact_996_zdvd__mult__cancel,axiom,
! [K_1: int,M: int,N: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,times_times_int(K_1,M)),times_times_int(K_1,N)))
=> ( ( K_1 != zero_zero_int )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N)) ) ) ).
tff(fact_997_zdvd__antisym__nonneg,axiom,
! [N: 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))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M))
=> ( M = N ) ) ) ) ) ).
tff(fact_998_zdvd__reduce,axiom,
! [K: int,Na: int,Ma: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),plus_plus_int(Na,times_times_int(K,Ma))))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),Na)) ) ).
tff(fact_999_zdvd__period,axiom,
! [C_1: int,Xa: int,Ta: int,A_1: int,D: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),D))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),plus_plus_int(Xa,Ta)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),plus_plus_int(plus_plus_int(Xa,times_times_int(C_1,D)),Ta))) ) ) ).
tff(fact_1000_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_1001_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_1002_rel__simps_I6_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),pls)) ).
tff(fact_1003_rel__simps_I3_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),min)) ).
tff(fact_1004_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_1005_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_1006_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_1007_bin__less__0__simps_I2_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),zero_zero_int)) ).
tff(fact_1008_rel__simps_I23_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),pls)) ).
tff(fact_1009_rel__simps_I20_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),min)) ).
tff(fact_1010_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_1011_eq__number__of__Pls__Min,axiom,
number_number_of_int(pls) != number_number_of_int(min) ).
tff(fact_1012_rel__simps_I7_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),min)) ).
tff(fact_1013_rel__simps_I42_J,axiom,
! [L: int] : min != bit0(L) ).
tff(fact_1014_rel__simps_I45_J,axiom,
! [K_1: int] : bit0(K_1) != min ).
tff(fact_1015_rel__simps_I40_J,axiom,
min != pls ).
tff(fact_1016_rel__simps_I37_J,axiom,
pls != min ).
tff(fact_1017_Bit1__Min,axiom,
bit1(min) = min ).
tff(fact_1018_rel__simps_I43_J,axiom,
! [L_1: int] :
( ( min = bit1(L_1) )
<=> ( min = L_1 ) ) ).
tff(fact_1019_rel__simps_I47_J,axiom,
! [K: int] :
( ( bit1(K) = min )
<=> ( K = min ) ) ).
tff(fact_1020_succ__Min,axiom,
succ(min) = pls ).
tff(fact_1021_diff__bin__simps_I2_J,axiom,
! [K_1: int] : minus_minus_int(K_1,min) = succ(K_1) ).
tff(fact_1022_zdvd__imp__le,axiom,
! [Z: int,N: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z),N))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),N)) ) ) ).
tff(fact_1023_zpower__zdvd__prop1,axiom,
! [P_2: int,Y: int,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),Y))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),hAPP_nat_int(power_power_int(Y),N))) ) ) ).
tff(fact_1024_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_1025_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_1026_diff__bin__simps_I4_J,axiom,
! [L: int] : minus_minus_int(pls,bit1(L)) = bit1(minus_minus_int(min,L)) ).
tff(fact_1027_diff__bin__simps_I6_J,axiom,
! [L: int] : minus_minus_int(min,bit1(L)) = bit0(minus_minus_int(min,L)) ).
tff(fact_1028_diff__bin__simps_I5_J,axiom,
! [L: int] : minus_minus_int(min,bit0(L)) = bit1(minus_minus_int(min,L)) ).
tff(fact_1029_zmult__eq__1__iff,axiom,
! [Ma: int,Na: int] :
( ( times_times_int(Ma,Na) = one_one_int )
<=> ( ( ( Ma = one_one_int )
& ( Na = one_one_int ) )
| ( ( Ma = number_number_of_int(min) )
& ( Na = number_number_of_int(min) ) ) ) ) ).
tff(fact_1030_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( times_times_int(M,N) = one_one_int )
=> ( ( M = one_one_int )
| ( M = number_number_of_int(min) ) ) ) ).
tff(fact_1031_zprime__power__zdvd__cancel__right,axiom,
! [N: nat,A: int,B_1: int,P_2: int] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_2),N)),times_times_int(A,B_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_2),N)),A)) ) ) ) ).
tff(fact_1032_zprime__power__zdvd__cancel__left,axiom,
! [N: nat,B_1: int,A: int,P_2: int] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_2),N)),times_times_int(A,B_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_2),N)),B_1)) ) ) ) ).
tff(fact_1033_zpower__zdvd__prop2,axiom,
! [Y: int,N: nat,P_2: int] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),hAPP_nat_int(power_power_int(Y),N)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),Y)) ) ) ) ).
tff(fact_1034__096QuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_096,axiom,
hBOOL(hAPP_int_bool(quadRes(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)),number_number_of_int(min))) ).
tff(fact_1035_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)),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) ).
tff(fact_1036__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_,axiom,
( ~ hBOOL(hAPP_int_bool(quadRes(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)),number_number_of_int(min)))
=> ( legendre(number_number_of_int(min),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) != one_one_int ) ) ).
tff(fact_1037_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)),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) ).
tff(fact_1038_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),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)))
& hBOOL(hAPP_int_bool(zcong(s1,s),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) ) ).
tff(fact_1039__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)),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) ).
tff(fact_1040_divides__mul__r,axiom,
! [C: nat,A: nat,B_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(A,C)),times_times_nat(B_1,C))) ) ).
tff(fact_1041_divides__mul__l,axiom,
! [C: nat,A: nat,B_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(C,A)),times_times_nat(C,B_1))) ) ).
tff(fact_1042_dvd__diff__nat,axiom,
! [N: 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))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),minus_minus_nat(M,N))) ) ) ).
tff(fact_1043_zcong__eq__zdvd__prop,axiom,
! [Xa: int,P_3: int] :
( hBOOL(hAPP_int_bool(zcong(Xa,zero_zero_int),P_3))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Xa)) ) ).
tff(fact_1044_zcong__zero__equiv__div,axiom,
! [A_1: int,Ma: int] :
( hBOOL(hAPP_int_bool(zcong(A_1,zero_zero_int),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),A_1)) ) ).
tff(fact_1045_nat__dvd__not__less,axiom,
! [N: 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))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M)) ) ) ).
tff(fact_1046_divides__ge,axiom,
! [A: nat,B_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
=> ( ( B_1 = zero_zero_nat )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B_1)) ) ) ).
tff(fact_1047_nat__mult__dvd__cancel__disj_H,axiom,
! [Ma: nat,K: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(Ma,K)),times_times_nat(Na,K)))
<=> ( ( K = zero_zero_nat )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),Na)) ) ) ).
tff(fact_1048_dvd__diffD,axiom,
! [K_1: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),minus_minus_nat(M,N)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),M)) ) ) ) ).
tff(fact_1049_dvd__diffD1,axiom,
! [K_1: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),minus_minus_nat(M,N)))
=> ( 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),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N)) ) ) ) ).
tff(fact_1050_divides__rev,axiom,
! [A: nat,N: nat,B_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(A),N)),hAPP_nat_nat(power_power_nat(B_1),N)))
=> ( ( N != zero_zero_nat )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1)) ) ) ).
tff(fact_1051_divides__exp2,axiom,
! [X: nat,Y: nat,N: nat] :
( ( N != zero_zero_nat )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(X),N)),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).
tff(fact_1052_Nat__Transfer_Otransfer__int__nat__relations_I4_J,axiom,
! [Xa: nat,Ya: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) ) ).
tff(fact_1053_zdvd__int,axiom,
! [Xa: nat,Ya: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya))) ) ).
tff(fact_1054_divides__exp,axiom,
! [N: nat,X: nat,Y: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(X),N)),hAPP_nat_nat(power_power_nat(Y),N))) ) ).
tff(fact_1055_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_1056_dvd__reduce,axiom,
! [K: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),plus_plus_nat(Na,K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),Na)) ) ).
tff(fact_1057_divides__add__revr,axiom,
! [B_1: nat,D_1: nat,A: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_1),A))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_1),plus_plus_nat(A,B_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_1),B_1)) ) ) ).
tff(fact_1058_zcong__id,axiom,
! [M: int] : hBOOL(hAPP_int_bool(zcong(M,zero_zero_int),M)) ).
tff(fact_1059_zcong__shift,axiom,
! [C: int,A: int,B_1: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(A,B_1),M))
=> hBOOL(hAPP_int_bool(zcong(plus_plus_int(A,C),plus_plus_int(B_1,C)),M)) ) ).
tff(fact_1060_zcong__eq__trans,axiom,
! [D_1: int,C: int,A: int,B_1: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(A,B_1),M))
=> ( ( B_1 = C )
=> ( hBOOL(hAPP_int_bool(zcong(C,D_1),M))
=> hBOOL(hAPP_int_bool(zcong(A,D_1),M)) ) ) ) ).
tff(fact_1061_zcong__zpower,axiom,
! [Z: nat,X: int,Y: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(X,Y),M))
=> hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X),Z),hAPP_nat_int(power_power_int(Y),Z)),M)) ) ).
tff(fact_1062_zcong__less__eq,axiom,
! [M: int,Y: int,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
=> ( 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,Y),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),M))
=> ( X = Y ) ) ) ) ) ) ) ).
tff(fact_1063_zcong__not__zero,axiom,
! [M: int,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M))
=> ~ hBOOL(hAPP_int_bool(zcong(X,zero_zero_int),M)) ) ) ).
tff(fact_1064_zcong__zmult__prop1,axiom,
! [C_1: int,D: int,A_1: int,B: int,Ma: int] :
( hBOOL(hAPP_int_bool(zcong(A_1,B),Ma))
=> ( hBOOL(hAPP_int_bool(zcong(C_1,times_times_int(A_1,D)),Ma))
<=> hBOOL(hAPP_int_bool(zcong(C_1,times_times_int(B,D)),Ma)) ) ) ).
tff(fact_1065_zcong__zmult__prop2,axiom,
! [C_1: int,D: int,A_1: int,B: int,Ma: int] :
( hBOOL(hAPP_int_bool(zcong(A_1,B),Ma))
=> ( hBOOL(hAPP_int_bool(zcong(C_1,times_times_int(D,A_1)),Ma))
<=> hBOOL(hAPP_int_bool(zcong(C_1,times_times_int(D,B)),Ma)) ) ) ).
tff(fact_1066_Int2_Ozcong__zero,axiom,
! [M: int,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M))
=> ( hBOOL(hAPP_int_bool(zcong(X,zero_zero_int),M))
=> ( X = zero_zero_int ) ) ) ) ).
tff(fact_1067_zcong__zmult__prop3,axiom,
! [Y: int,X: int,P_2: int] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( ~ hBOOL(hAPP_int_bool(zcong(X,zero_zero_int),P_2))
=> ( ~ hBOOL(hAPP_int_bool(zcong(Y,zero_zero_int),P_2))
=> ~ hBOOL(hAPP_int_bool(zcong(times_times_int(X,Y),zero_zero_int),P_2)) ) ) ) ).
tff(fact_1068_dvd__imp__le,axiom,
! [K_1: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N)) ) ) ).
tff(fact_1069_dvd__mult__cancel,axiom,
! [K_1: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(K_1,M)),times_times_nat(K_1,N)))
=> ( 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)) ) ) ).
tff(fact_1070_zcong__zprime__prod__zero,axiom,
! [B_1: int,A: int,P_2: int] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(zcong(times_times_int(A,B_1),zero_zero_int),P_2))
=> ( hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_2))
| hBOOL(hAPP_int_bool(zcong(B_1,zero_zero_int),P_2)) ) ) ) ) ).
tff(fact_1071_zcong__zprime__prod__zero__contra,axiom,
! [B_1: int,A: int,P_2: int] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( ( ~ hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_2))
& ~ hBOOL(hAPP_int_bool(zcong(B_1,zero_zero_int),P_2)) )
=> ~ hBOOL(hAPP_int_bool(zcong(times_times_int(A,B_1),zero_zero_int),P_2)) ) ) ) ).
tff(fact_1072_divides__div__not,axiom,
! [X: nat,Q: nat,N: nat,R_1: nat] :
( ( X = plus_plus_nat(times_times_nat(Q,N),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))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),X)) ) ) ) ).
tff(fact_1073_dvd__mult__cancel1,axiom,
! [Na: 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,times_times_nat(Ma,Na)),Ma))
<=> ( Na = one_one_nat ) ) ) ).
tff(fact_1074_dvd__mult__cancel2,axiom,
! [Na: 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,times_times_nat(Na,Ma)),Ma))
<=> ( Na = one_one_nat ) ) ) ).
tff(fact_1075_transfer__nat__int__relations_I4_J,axiom,
! [Ya: int,Xa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,nat_1(Xa)),nat_1(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Xa),Ya)) ) ) ) ).
tff(fact_1076_power__dvd__imp__le,axiom,
! [I_2: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(I_2),M)),hAPP_nat_nat(power_power_nat(I_2),N)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),I_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ) ).
tff(fact_1077_nat__dvd__iff,axiom,
! [Z_1: int,Ma: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,nat_1(Z_1)),Ma))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_1),hAPP_nat_int(semiri1621563631at_int,Ma))) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
=> ( Ma = zero_zero_nat ) ) ) ) ).
tff(fact_1078_zcong__neg__1__impl__ne__1,axiom,
! [X: int,P_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_2))
=> ( hBOOL(hAPP_int_bool(zcong(X,number_number_of_int(min)),P_2))
=> ~ hBOOL(hAPP_int_bool(zcong(X,one_one_int),P_2)) ) ) ).
tff(fact_1079_Euler_Oaux____1,axiom,
! [Y: int,X: int,P_2: int] :
( ~ hBOOL(hAPP_int_bool(zcong(X,zero_zero_int),P_2))
=> ( hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(Y),number_number_of_nat(bit0(bit1(pls)))),X),P_2))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),Y)) ) ) ).
tff(fact_1080_divides__cases,axiom,
! [N: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M))
=> ( ( M = zero_zero_nat )
| ( M = N )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N)),M)) ) ) ).
tff(fact_1081_Legendre__1mod4,axiom,
! [M: int] :
( hBOOL(hAPP_int_bool(zprime,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),M),one_one_int)))
=> ( legendre(number_number_of_int(min),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),M),one_one_int)) = one_one_int ) ) ).
tff(fact_1082__096_091s_____A_094_A2_A_061_As1_A_094_A2_093_A_Imod_A4_A_K_Am_A_L_A1_,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))))),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) ).
tff(fact_1083__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s,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),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)))
& hBOOL(hAPP_int_bool(zcong(s1,X_1),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)))
& ! [Y_1: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)))
& hBOOL(hAPP_int_bool(zcong(s1,Y_1),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) )
=> ( Y_1 = X_1 ) ) ) ).
tff(fact_1084_dvd_Oorder__refl,axiom,
! [X: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X)) ).
tff(fact_1085__096_B_Bthesis_O_A_I_B_Bs_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L,axiom,
~ ! [S: int] :
~ ( 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),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)))
& hBOOL(hAPP_int_bool(zcong(s1,S),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) ) ).
tff(fact_1086_dvd_Oless__asym,axiom,
! [X: nat,Y: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).
tff(fact_1087_dvd_Oless__trans,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> ( ( 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),Z))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),X)) ) ) ) ).
tff(fact_1088_dvd_Oless__asym_H,axiom,
! [A: nat,B_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1)) ) ) ).
tff(fact_1089_dvd_Oless__le__trans,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),X)) ) ) ) ).
tff(fact_1090_dvd_Oord__less__eq__trans,axiom,
! [C: nat,A: nat,B_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A)) )
=> ( ( B_1 = 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_1091_dvd_Oless__imp__triv,axiom,
! [P_1: bool,Xa: nat,Ya: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) )
=> hBOOL(P_1) ) ) ).
tff(fact_1092_dvd_Oless__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> ( Y != X ) ) ).
tff(fact_1093_dvd_Oless__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> ( X != Y ) ) ).
tff(fact_1094_dvd_Oless__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).
tff(fact_1095_dvd_Oless__imp__le,axiom,
! [X: nat,Y: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ).
tff(fact_1096_dvd_Oless__not__sym,axiom,
! [X: nat,Y: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).
tff(fact_1097_dvd_Oless__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> ( X != Y ) ) ).
tff(fact_1098_dvd_Ole__less__trans,axiom,
! [Z: nat,X: nat,Y: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),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),Z))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),X)) ) ) ) ).
tff(fact_1099_dvd_Oord__eq__less__trans,axiom,
! [C: nat,A: nat,B_1: nat] :
( ( A = B_1 )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),C))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C),B_1)) )
=> ( 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_1100_dvd_Oorder__trans,axiom,
! [Z: nat,X: nat,Y: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z)) ) ) ).
tff(fact_1101_dvd_Oantisym,axiom,
! [X: nat,Y: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
=> ( X = Y ) ) ) ).
tff(fact_1102_dvd__antisym,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M))
=> ( M = N ) ) ) ).
tff(fact_1103_dvd_Oord__le__eq__trans,axiom,
! [C: nat,A: nat,B_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
=> ( ( B_1 = C )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C)) ) ) ).
tff(fact_1104_dvd_Oord__eq__le__trans,axiom,
! [C: nat,A: nat,B_1: nat] :
( ( A = B_1 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),C))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C)) ) ) ).
tff(fact_1105_dvd_Ole__neq__trans,axiom,
! [A: nat,B_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
=> ( ( A != B_1 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A)) ) ) ) ).
tff(fact_1106_dvd_Ole__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
| ( X = Y ) ) ) ).
tff(fact_1107_dvd_Oantisym__conv,axiom,
! [Ya: nat,Xa: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
<=> ( Xa = Ya ) ) ) ).
tff(fact_1108_dvd_Oeq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ).
tff(fact_1109_dvd_Oneq__le__trans,axiom,
! [A: nat,B_1: nat] :
( ( A != B_1 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A)) ) ) ) ).
tff(fact_1110_dvd_Oless__le__not__le,axiom,
! [Xa: nat,Ya: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) ) ) ).
tff(fact_1111_dvd_Oless__le,axiom,
! [Xa: nat,Ya: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& ( Xa != Ya ) ) ) ).
tff(fact_1112_dvd_Ole__less,axiom,
! [Xa: nat,Ya: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) )
| ( Xa = Ya ) ) ) ).
tff(fact_1113_dvd_Oeq__iff,axiom,
! [Xa: nat,Ya: nat] :
( ( Xa = Ya )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) ) ) ).
tff(fact_1114_dvd_Oless__irrefl,axiom,
! [X: nat] :
~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X)) ) ).
tff(fact_1115_divides__antisym,axiom,
! [Xa: nat,Ya: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) )
<=> ( Xa = Ya ) ) ).
tff(fact_1116_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_1117_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_1118_neg__one__power,axiom,
! [N: nat] :
( ( hAPP_nat_int(power_power_int(number_number_of_int(min)),N) = one_one_int )
| ( hAPP_nat_int(power_power_int(number_number_of_int(min)),N) = number_number_of_int(min) ) ) ).
tff(fact_1119_Legendre__def,axiom,
! [A: int,P_2: int] :
( ( hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_2))
=> ( legendre(A,P_2) = zero_zero_int ) )
& ( ~ hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_2))
=> ( ( hBOOL(hAPP_int_bool(quadRes(P_2),A))
=> ( legendre(A,P_2) = one_one_int ) )
& ( ~ hBOOL(hAPP_int_bool(quadRes(P_2),A))
=> ( legendre(A,P_2) = number_number_of_int(min) ) ) ) ) ) ).
tff(fact_1120_QuadRes__def,axiom,
! [Ma: int,Xa: int] :
( hBOOL(hAPP_int_bool(quadRes(Ma),Xa))
<=> ? [Y_1: int] : hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(Y_1),number_number_of_nat(bit0(bit1(pls)))),Xa),Ma)) ) ).
tff(fact_1121_Little__Fermat,axiom,
! [X: int,P_2: int] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),X))
=> hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X),nat_1(minus_minus_int(P_2,one_one_int))),one_one_int),P_2)) ) ) ).
tff(fact_1122_zcong__square__zless,axiom,
! [A: int,P_2: int] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( 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_2))
=> ( hBOOL(hAPP_int_bool(zcong(times_times_int(A,A),one_one_int),P_2))
=> ( ( A = one_one_int )
| ( A = minus_minus_int(P_2,one_one_int) ) ) ) ) ) ) ).
tff(fact_1123_zcong__trans,axiom,
! [C: int,A: int,B_1: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(A,B_1),M))
=> ( hBOOL(hAPP_int_bool(zcong(B_1,C),M))
=> hBOOL(hAPP_int_bool(zcong(A,C),M)) ) ) ).
tff(fact_1124_zcong__refl,axiom,
! [K_1: int,M: int] : hBOOL(hAPP_int_bool(zcong(K_1,K_1),M)) ).
tff(fact_1125_zcong__sym,axiom,
! [A_1: int,B: int,Ma: int] :
( hBOOL(hAPP_int_bool(zcong(A_1,B),Ma))
<=> hBOOL(hAPP_int_bool(zcong(B,A_1),Ma)) ) ).
tff(fact_1126_IntPrimes_Ozcong__zero,axiom,
! [A_1: int,B: int] :
( hBOOL(hAPP_int_bool(zcong(A_1,B),zero_zero_int))
<=> ( A_1 = B ) ) ).
tff(fact_1127_zcong__1,axiom,
! [A: int,B_1: int] : hBOOL(hAPP_int_bool(zcong(A,B_1),one_one_int)) ).
tff(fact_1128_zcong__zmult__self,axiom,
! [A: int,M: int,B_1: int] : hBOOL(hAPP_int_bool(zcong(times_times_int(A,M),times_times_int(B_1,M)),M)) ).
tff(fact_1129_zcong__scalar,axiom,
! [K_1: int,A: int,B_1: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(A,B_1),M))
=> hBOOL(hAPP_int_bool(zcong(times_times_int(A,K_1),times_times_int(B_1,K_1)),M)) ) ).
tff(fact_1130_zcong__scalar2,axiom,
! [K_1: int,A: int,B_1: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(A,B_1),M))
=> hBOOL(hAPP_int_bool(zcong(times_times_int(K_1,A),times_times_int(K_1,B_1)),M)) ) ).
tff(fact_1131_zcong__zmult,axiom,
! [C: int,D_1: int,A: int,B_1: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(A,B_1),M))
=> ( hBOOL(hAPP_int_bool(zcong(C,D_1),M))
=> hBOOL(hAPP_int_bool(zcong(times_times_int(A,C),times_times_int(B_1,D_1)),M)) ) ) ).
tff(fact_1132_zcong__zadd,axiom,
! [C: int,D_1: int,A: int,B_1: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(A,B_1),M))
=> ( hBOOL(hAPP_int_bool(zcong(C,D_1),M))
=> hBOOL(hAPP_int_bool(zcong(plus_plus_int(A,C),plus_plus_int(B_1,D_1)),M)) ) ) ).
tff(fact_1133_zcong__zdiff,axiom,
! [C: int,D_1: int,A: int,B_1: int,M: int] :
( hBOOL(hAPP_int_bool(zcong(A,B_1),M))
=> ( hBOOL(hAPP_int_bool(zcong(C,D_1),M))
=> hBOOL(hAPP_int_bool(zcong(minus_minus_int(A,C),minus_minus_int(B_1,D_1)),M)) ) ) ).
tff(fact_1134_zcong__not,axiom,
! [B_1: 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_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),A))
=> ~ hBOOL(hAPP_int_bool(zcong(A,B_1),M)) ) ) ) ) ).
tff(fact_1135_xzgcda__linear__aux1,axiom,
! [A: int,R_1: int,B_1: int,M: int,C: int,D_1: int,N: int] : plus_plus_int(times_times_int(minus_minus_int(A,times_times_int(R_1,B_1)),M),times_times_int(minus_minus_int(C,times_times_int(R_1,D_1)),N)) = minus_minus_int(plus_plus_int(times_times_int(A,M),times_times_int(C,N)),times_times_int(R_1,plus_plus_int(times_times_int(B_1,M),times_times_int(D_1,N)))) ).
tff(fact_1136_zcong__iff__lin,axiom,
! [A_1: int,B: int,Ma: int] :
( hBOOL(hAPP_int_bool(zcong(A_1,B),Ma))
<=> ? [K_2: int] : B = plus_plus_int(A_1,times_times_int(Ma,K_2)) ) ).
tff(fact_1137_zcong__def,axiom,
! [A_1: int,B: int,Ma: int] :
( hBOOL(hAPP_int_bool(zcong(A_1,B),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),minus_minus_int(A_1,B))) ) ).
tff(fact_1138_norR__mem__unique__aux,axiom,
! [A: int,B_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),minus_minus_int(B_1,one_one_int)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B_1)) ) ).
tff(fact_1139_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_1140_zcong__zless__imp__eq,axiom,
! [B_1: 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_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),M))
=> ( hBOOL(hAPP_int_bool(zcong(A,B_1),M))
=> ( A = B_1 ) ) ) ) ) ) ).
tff(fact_1141_zprime__zdvd__zmult,axiom,
! [N: int,P_2: int,M: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M))
=> ( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),times_times_int(M,N)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),M))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),N)) ) ) ) ) ).
tff(fact_1142_zprime__def,axiom,
! [P_3: int] :
( hBOOL(hAPP_int_bool(zprime,P_3))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),P_3))
& ! [M_1: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M_1),P_3)) )
=> ( ( M_1 = one_one_int )
| ( M_1 = P_3 ) ) ) ) ) ).
tff(fact_1143_zcong__square,axiom,
! [A: int,P_2: int] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(zcong(times_times_int(A,A),one_one_int),P_2))
=> ( hBOOL(hAPP_int_bool(zcong(A,one_one_int),P_2))
| hBOOL(hAPP_int_bool(zcong(A,minus_minus_int(P_2,one_one_int)),P_2)) ) ) ) ) ).
tff(fact_1144_zdiff__int__split,axiom,
! [P_1: fun_int_bool,Xa: nat,Ya: nat] :
( hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,minus_minus_nat(Xa,Ya))))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ya),Xa))
=> hBOOL(hAPP_int_bool(P_1,minus_minus_int(hAPP_nat_int(semiri1621563631at_int,Xa),hAPP_nat_int(semiri1621563631at_int,Ya)))) )
& ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya))
=> hBOOL(hAPP_int_bool(P_1,zero_zero_int)) ) ) ) ).
tff(fact_1145_prime__g__5,axiom,
! [P_2: int] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( ( P_2 != number_number_of_int(bit0(bit1(pls))) )
=> ( ( P_2 != 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_2)) ) ) ) ).
tff(fact_1146_conj__le__cong,axiom,
! [P_4: bool,P_1: bool,Xa: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(P_1)
<=> hBOOL(P_4) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
& hBOOL(P_1) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
& hBOOL(P_4) ) ) ) ).
tff(fact_1147_imp__le__cong,axiom,
! [P_4: bool,P_1: bool,Xa: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(P_1)
<=> hBOOL(P_4) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> hBOOL(P_1) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> hBOOL(P_4) ) ) ) ).
tff(fact_1148_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,times_times_int(K,Ma)),times_times_int(K,Ta))) ) ) ).
tff(fact_1149_number__of2,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(pls))) ).
tff(fact_1150_inv__not__p__minus__1__aux,axiom,
! [A_1: int,P_3: int] :
( hBOOL(hAPP_int_bool(zcong(times_times_int(A_1,minus_minus_int(P_3,one_one_int)),one_one_int),P_3))
<=> hBOOL(hAPP_int_bool(zcong(A_1,minus_minus_int(P_3,one_one_int)),P_3)) ) ).
tff(fact_1151_zcong__zpower__zmult,axiom,
! [Z: nat,X: int,Y: nat,P_2: int] :
( hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X),Y),one_one_int),P_2))
=> hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X),times_times_nat(Y,Z)),one_one_int),P_2)) ) ).
tff(fact_1152_number__of1,axiom,
! [N: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(N)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit0(N))))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit1(N)))) ) ) ).
tff(fact_1153__096_B_Bthesis_O_A_I_B_Bs_Aw_O_Aw_A_061_Ay_A_N_As_A_K_A_I1_A_L_Aint_An,axiom,
~ ! [S: int,W_1: int] :
~ ( ( W_1 = minus_minus_int(y,times_times_int(S,plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n)))) )
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(number_number_of_int(bit0(bit1(pls))),abs_abs_int(W_1))),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n)))) ) ).
tff(fact_1154__096_B_Bthesis_O_A_I_B_Br_Av_O_Av_A_061_Ax_A_N_Ar_A_K_A_I1_A_L_Aint_An,axiom,
~ ! [R: int,V: int] :
~ ( ( V = minus_minus_int(x,times_times_int(R,plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n)))) )
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(number_number_of_int(bit0(bit1(pls))),abs_abs_int(V))),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n)))) ) ).
tff(fact_1155_dvd__imp__le__int,axiom,
! [D_1: int,I_2: int] :
( ( I_2 != zero_zero_int )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),I_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(D_1)),abs_abs_int(I_2))) ) ) ).
tff(fact_1156_nat__abs__mult__distrib,axiom,
! [W: int,Z: int] : nat_1(abs_abs_int(times_times_int(W,Z))) = times_times_nat(nat_1(abs_abs_int(W)),nat_1(abs_abs_int(Z))) ).
tff(fact_1157_zdvd__antisym__abs,axiom,
! [A: int,B_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_1),A))
=> ( abs_abs_int(A) = abs_abs_int(B_1) ) ) ) ).
tff(fact_1158_zdvd1__eq,axiom,
! [Xa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Xa),one_one_int))
<=> ( abs_abs_int(Xa) = one_one_int ) ) ).
tff(fact_1159_abs__int__eq,axiom,
! [M: nat] : abs_abs_int(hAPP_nat_int(semiri1621563631at_int,M)) = hAPP_nat_int(semiri1621563631at_int,M) ).
tff(fact_1160_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( abs_abs_int(times_times_int(M,N)) = one_one_int )
=> ( abs_abs_int(M) = one_one_int ) ) ).
tff(fact_1161_zero__le__zpower__abs,axiom,
! [X: int,N: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(abs_abs_int(X)),N))) ).
tff(fact_1162_zabs__less__one__iff,axiom,
! [Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,abs_abs_int(Z_1)),one_one_int))
<=> ( Z_1 = zero_zero_int ) ) ).
tff(fact_1163_abs__eq__1__iff,axiom,
! [Z_1: int] :
( ( abs_abs_int(Z_1) = one_one_int )
<=> ( ( Z_1 = one_one_int )
| ( Z_1 = number_number_of_int(min) ) ) ) ).
tff(fact_1164_abs__power3__distrib,axiom,
! [X: int] : abs_abs_int(hAPP_nat_int(power_power_int(X),number_number_of_nat(bit1(bit1(pls))))) = hAPP_nat_int(power_power_int(abs_abs_int(X)),number_number_of_nat(bit1(bit1(pls)))) ).
tff(fact_1165_zero__less__zpower__abs__iff,axiom,
! [Xa: int,Na: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(power_power_int(abs_abs_int(Xa)),Na)))
<=> ( ( Xa != zero_zero_int )
| ( Na = zero_zero_nat ) ) ) ).
tff(fact_1166_zdvd__mult__cancel1,axiom,
! [Na: int,Ma: int] :
( ( Ma != zero_zero_int )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,times_times_int(Ma,Na)),Ma))
<=> ( abs_abs_int(Na) = one_one_int ) ) ) ).
tff(fact_1167_int__dvd__iff,axiom,
! [Ma: nat,Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,Ma)),Z_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),nat_1(abs_abs_int(Z_1)))) ) ).
tff(fact_1168_dvd__int__iff,axiom,
! [Z_1: int,Ma: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_1),hAPP_nat_int(semiri1621563631at_int,Ma)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,nat_1(abs_abs_int(Z_1))),Ma)) ) ).
tff(fact_1169_abs__power2__distrib,axiom,
! [A: int] : abs_abs_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls))))) = hAPP_nat_int(power_power_int(abs_abs_int(A)),number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_1170_power2__eq__iff__abs__eq,axiom,
! [A_1: int,B: int] :
( ( hAPP_nat_int(power_power_int(A_1),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_int(power_power_int(B),number_number_of_nat(bit0(bit1(pls)))) )
<=> ( abs_abs_int(A_1) = abs_abs_int(B) ) ) ).
tff(fact_1171_power2__eq1__iff,axiom,
! [A: int] :
( ( hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))) = one_one_int )
=> ( abs_abs_int(A) = one_one_int ) ) ).
tff(fact_1172_incr__lemma,axiom,
! [Z: int,X: int,D_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),plus_plus_int(X,times_times_int(plus_plus_int(abs_abs_int(minus_minus_int(X,Z)),one_one_int),D_1)))) ) ).
tff(fact_1173_decr__lemma,axiom,
! [X: int,Z: int,D_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,minus_minus_int(X,times_times_int(plus_plus_int(abs_abs_int(minus_minus_int(X,Z)),one_one_int),D_1))),Z)) ) ).
tff(fact_1174_best__division__abs,axiom,
! [Y: int,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
=> ? [N_1: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(number_number_of_int(bit0(bit1(pls))),abs_abs_int(minus_minus_int(Y,times_times_int(N_1,X))))),X)) ) ).
tff(fact_1175_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_1: int] :
( hBOOL(hAPP_int_bool(P_1,X_1))
=> hBOOL(hAPP_int_bool(P_1,minus_minus_int(X_1,D))) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
=> ! [X_1: int] :
( hBOOL(hAPP_int_bool(P_1,X_1))
=> hBOOL(hAPP_int_bool(P_1,minus_minus_int(X_1,times_times_int(K,D)))) ) ) ) ) ).
tff(fact_1176_abs__add__one__not__less__self,axiom,
! [X: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(abs_abs_real(X),one_one_real)),X)) ).
tff(fact_1177_abs__add__one__gt__zero,axiom,
! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),plus_plus_real(one_one_real,abs_abs_real(X)))) ).
tff(fact_1178_less__one__imp__sqr__less__one,axiom,
! [X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,abs_abs_real(X)),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(X),number_number_of_nat(bit0(bit1(pls))))),one_one_real)) ) ).
tff(fact_1179_ex__least__nat__less,axiom,
! [Na: nat,P_1: fun_nat_bool] :
( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(P_1,Na))
=> ? [K_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),Na))
& ! [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,plus_plus_nat(K_2,one_one_nat))) ) ) ) ).
tff(fact_1180_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_1: int] :
( hBOOL(hAPP_int_bool(P_1,X_1))
=> hBOOL(hAPP_int_bool(P_1,plus_plus_int(X_1,D))) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
=> ! [X_1: int] :
( hBOOL(hAPP_int_bool(P_1,X_1))
=> hBOOL(hAPP_int_bool(P_1,plus_plus_int(X_1,times_times_int(K,D)))) ) ) ) ) ).
tff(fact_1181_nat__less__add__iff1,axiom,
! [U: nat,Ma: nat,Na: nat,J: nat,I_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(times_times_nat(I_1,U),Ma)),plus_plus_nat(times_times_nat(J,U),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(times_times_nat(minus_minus_nat(I_1,J),U),Ma)),Na)) ) ) ).
tff(fact_1182_nat__less__add__iff2,axiom,
! [U: nat,Ma: nat,Na: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(times_times_nat(I_1,U),Ma)),plus_plus_nat(times_times_nat(J,U),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),plus_plus_nat(times_times_nat(minus_minus_nat(J,I_1),U),Na))) ) ) ).
tff(fact_1183_nat__mult__eq__cancel__disj,axiom,
! [K: nat,Ma: nat,Na: nat] :
( ( times_times_nat(K,Ma) = times_times_nat(K,Na) )
<=> ( ( K = zero_zero_nat )
| ( Ma = Na ) ) ) ).
tff(fact_1184_left__add__mult__distrib,axiom,
! [I_2: nat,U_1: nat,J_1: nat,K_1: nat] : plus_plus_nat(times_times_nat(I_2,U_1),plus_plus_nat(times_times_nat(J_1,U_1),K_1)) = plus_plus_nat(times_times_nat(plus_plus_nat(I_2,J_1),U_1),K_1) ).
tff(fact_1185_nat__mult__less__cancel1,axiom,
! [Ma: nat,Na: 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,times_times_nat(K,Ma)),times_times_nat(K,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ) ).
tff(fact_1186_nat__mult__eq__cancel1,axiom,
! [Ma: nat,Na: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( ( times_times_nat(K,Ma) = times_times_nat(K,Na) )
<=> ( Ma = Na ) ) ) ).
tff(fact_1187_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,Ma: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(K,Ma)),times_times_nat(K,Na)))
<=> ( ( K = zero_zero_nat )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),Na)) ) ) ).
tff(fact_1188_nat__mult__dvd__cancel1,axiom,
! [Ma: nat,Na: 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,times_times_nat(K,Ma)),times_times_nat(K,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),Na)) ) ) ).
tff(fact_1189_nat__mult__le__cancel1,axiom,
! [Ma: nat,Na: 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,times_times_nat(K,Ma)),times_times_nat(K,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ) ).
tff(fact_1190_nat__le__add__iff1,axiom,
! [U: nat,Ma: nat,Na: nat,J: nat,I_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(times_times_nat(I_1,U),Ma)),plus_plus_nat(times_times_nat(J,U),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(times_times_nat(minus_minus_nat(I_1,J),U),Ma)),Na)) ) ) ).
tff(fact_1191_nat__diff__add__eq1,axiom,
! [U_1: nat,M: nat,N: nat,J_1: nat,I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),I_2))
=> ( minus_minus_nat(plus_plus_nat(times_times_nat(I_2,U_1),M),plus_plus_nat(times_times_nat(J_1,U_1),N)) = minus_minus_nat(plus_plus_nat(times_times_nat(minus_minus_nat(I_2,J_1),U_1),M),N) ) ) ).
tff(fact_1192_nat__eq__add__iff1,axiom,
! [U: nat,Ma: nat,Na: nat,J: nat,I_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1))
=> ( ( plus_plus_nat(times_times_nat(I_1,U),Ma) = plus_plus_nat(times_times_nat(J,U),Na) )
<=> ( plus_plus_nat(times_times_nat(minus_minus_nat(I_1,J),U),Ma) = Na ) ) ) ).
tff(fact_1193_nat__le__add__iff2,axiom,
! [U: nat,Ma: nat,Na: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(times_times_nat(I_1,U),Ma)),plus_plus_nat(times_times_nat(J,U),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),plus_plus_nat(times_times_nat(minus_minus_nat(J,I_1),U),Na))) ) ) ).
tff(fact_1194_nat__diff__add__eq2,axiom,
! [U_1: nat,M: nat,N: nat,I_2: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
=> ( minus_minus_nat(plus_plus_nat(times_times_nat(I_2,U_1),M),plus_plus_nat(times_times_nat(J_1,U_1),N)) = minus_minus_nat(M,plus_plus_nat(times_times_nat(minus_minus_nat(J_1,I_2),U_1),N)) ) ) ).
tff(fact_1195_nat__eq__add__iff2,axiom,
! [U: nat,Ma: nat,Na: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> ( ( plus_plus_nat(times_times_nat(I_1,U),Ma) = plus_plus_nat(times_times_nat(J,U),Na) )
<=> ( Ma = plus_plus_nat(times_times_nat(minus_minus_nat(J,I_1),U),Na) ) ) ) ).
tff(fact_1196_nat0__intermed__int__val,axiom,
! [K: int,F: fun_nat_int,Na: nat] :
( ! [I: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),Na))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(minus_minus_int(hAPP_nat_int(F,plus_plus_nat(I,one_one_nat)),hAPP_nat_int(F,I)))),one_one_int)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),hAPP_nat_int(F,Na)))
=> ? [I: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),Na))
& ( hAPP_nat_int(F,I) = K ) ) ) ) ) ).
tff(fact_1197_int__val__lemma,axiom,
! [K: int,F: fun_nat_int,Na: nat] :
( ! [I: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),Na))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(minus_minus_int(hAPP_nat_int(F,plus_plus_nat(I,one_one_nat)),hAPP_nat_int(F,I)))),one_one_int)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),hAPP_nat_int(F,Na)))
=> ? [I: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),Na))
& ( hAPP_nat_int(F,I) = K ) ) ) ) ) ).
%----Helper facts (6)
tff(help_If_1_1_If_000tc__Int__Oint_T,axiom,
! [X: int,Y: int] : if_int(fTrue,X,Y) = X ).
tff(help_If_2_1_If_000tc__Int__Oint_T,axiom,
! [X: int,Y: int] : if_int(fFalse,X,Y) = Y ).
tff(help_If_3_1_If_000tc__Int__Oint_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_If_1_1_If_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] : if_nat(fTrue,X,Y) = X ).
tff(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] : if_nat(fFalse,X,Y) = Y ).
tff(help_If_3_1_If_000tc__Nat__Onat_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
%----Conjectures (1)
% tff(conj_0,conjecture,
% hAPP_nat_int(power_power_int(plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))),number_number_of_nat(bit0(bit1(pls)))) != zero_zero_int ).
%------------------------------------------------------------------------------