TPTP Problem File: NUM925_2.p
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%------------------------------------------------------------------------------
% File : NUM925_2 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 192, 500 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : s2s_500_tff_l192 [Bla11]
% Status : Theorem
% Rating : 0.00 v5.4.0, 0.33 v5.3.0
% Syntax : Number of formulae : 767 ( 254 unt; 64 typ; 0 def)
% Number of atoms : 1438 ( 466 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 836 ( 101 ~; 19 |; 49 &)
% ( 195 <=>; 472 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 40 ( 27 >; 13 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 48 ( 48 usr; 22 con; 0-2 aty)
% Number of variables : 1269 (1257 !; 12 ?;1269 :)
% SPC : TF0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:25:32
%------------------------------------------------------------------------------
%----Should-be-implicit typings (15)
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__Int__Oint_J,type,
fun_int_int: $tType ).
tff(ty_ty_tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_J,type,
fun_int_fun_int_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
fun_nat_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__Int__Oint_J,type,
fun_nat_int: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
fun_nat_nat: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J,type,
fun_nat_real: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_J,type,
fun_nat_fun_nat_bool: $tType ).
tff(ty_ty_tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J,type,
fun_real_bool: $tType ).
tff(ty_ty_tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J,type,
fun_real_real: $tType ).
tff(ty_ty_tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J,type,
fun_re413263731l_bool: $tType ).
%----Explicit typings (49)
tff(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,
one_one_int: int ).
tff(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
one_one_nat: nat ).
tff(sy_c_Groups_Oone__class_Oone_000tc__RealDef__Oreal,type,
one_one_real: real ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
plus_plus_int: int > fun_int_int ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
plus_plus_nat: nat > fun_nat_nat ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__RealDef__Oreal,type,
plus_plus_real: real > fun_real_real ).
tff(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
times_times_int: ( int * int ) > int ).
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__Nat__Onat,type,
if_nat: ( bool * nat ) > fun_nat_nat ).
tff(sy_c_Int_OBit0,type,
bit0: int > int ).
tff(sy_c_Int_OBit1,type,
bit1: int > int ).
tff(sy_c_Int_OPls,type,
pls: int ).
tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Int__Oint,type,
number_number_of_int: int > int ).
tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Nat__Onat,type,
number_number_of_nat: int > nat ).
tff(sy_c_Int_Onumber__class_Onumber__of_000tc__RealDef__Oreal,type,
number267125858f_real: int > real ).
tff(sy_c_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_NthRoot_Osqrt,type,
sqrt: real > 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_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__Int__Oint,type,
hAPP_int_int: ( fun_int_int * int ) > int ).
tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
hAPP_i1948725293t_bool: ( fun_int_fun_int_bool * int ) > fun_int_bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool,type,
hAPP_nat_bool: ( fun_nat_bool * nat ) > bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Int__Oint,type,
hAPP_nat_int: ( fun_nat_int * nat ) > int ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Nat__Onat,type,
hAPP_nat_nat: ( fun_nat_nat * nat ) > nat ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__RealDef__Oreal,type,
hAPP_nat_real: ( fun_nat_real * nat ) > real ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
hAPP_n1699378549t_bool: ( fun_nat_fun_nat_bool * nat ) > fun_nat_bool ).
tff(sy_c_hAPP_000tc__RealDef__Oreal_000tc__HOL__Obool,type,
hAPP_real_bool: ( fun_real_bool * real ) > bool ).
tff(sy_c_hAPP_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
hAPP_real_real: ( fun_real_real * real ) > real ).
tff(sy_c_hAPP_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J,type,
hAPP_r1134773055l_bool: ( fun_re413263731l_bool * real ) > fun_real_bool ).
tff(sy_c_hBOOL,type,
hBOOL: bool > $o ).
tff(sy_v_m1____,type,
m1: int ).
tff(sy_v_n____,type,
n: nat ).
tff(sy_v_t____,type,
t: int ).
tff(sy_v_tn____,type,
tn: nat ).
%----Relevant facts (699)
tff(fact_0_n1pos,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_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] :
( ( hAPP_int_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] :
( ( hAPP_real_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_24: int] :
( ( hAPP_nat_int(power_power_int(A_24),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_int )
<=> ( A_24 = zero_zero_int ) ) ).
tff(fact_11_zero__eq__power2,axiom,
! [A_24: real] :
( ( hAPP_nat_real(power_power_real(A_24),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_real )
<=> ( A_24 = zero_zero_real ) ) ).
tff(fact_12_add__special_I2_J,axiom,
! [W_8: int] : hAPP_int_int(plus_plus_int(one_one_int),number_number_of_int(W_8)) = number_number_of_int(hAPP_int_int(plus_plus_int(bit1(pls)),W_8)) ).
tff(fact_13_add__special_I2_J,axiom,
! [W_8: int] : hAPP_real_real(plus_plus_real(one_one_real),number267125858f_real(W_8)) = number267125858f_real(hAPP_int_int(plus_plus_int(bit1(pls)),W_8)) ).
tff(fact_14_add__special_I3_J,axiom,
! [V_11: int] : hAPP_int_int(plus_plus_int(number_number_of_int(V_11)),one_one_int) = number_number_of_int(hAPP_int_int(plus_plus_int(V_11),bit1(pls))) ).
tff(fact_15_add__special_I3_J,axiom,
! [V_11: int] : hAPP_real_real(plus_plus_real(number267125858f_real(V_11)),one_one_real) = number267125858f_real(hAPP_int_int(plus_plus_int(V_11),bit1(pls))) ).
tff(fact_16_one__add__one__is__two,axiom,
hAPP_int_int(plus_plus_int(one_one_int),one_one_int) = number_number_of_int(bit0(bit1(pls))) ).
tff(fact_17_one__add__one__is__two,axiom,
hAPP_real_real(plus_plus_real(one_one_real),one_one_real) = number267125858f_real(bit0(bit1(pls))) ).
tff(fact_18_semiring__one__add__one__is__two,axiom,
hAPP_int_int(plus_plus_int(one_one_int),one_one_int) = number_number_of_int(bit0(bit1(pls))) ).
tff(fact_19_semiring__one__add__one__is__two,axiom,
hAPP_nat_nat(plus_plus_nat(one_one_nat),one_one_nat) = number_number_of_nat(bit0(bit1(pls))) ).
tff(fact_20_semiring__one__add__one__is__two,axiom,
hAPP_real_real(plus_plus_real(one_one_real),one_one_real) = number267125858f_real(bit0(bit1(pls))) ).
tff(fact_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_7: int] :
( ( ( number_number_of_nat(W_7) = zero_zero_nat )
=> ( hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(W_7)) = one_one_int ) )
& ( ( number_number_of_nat(W_7) != zero_zero_nat )
=> ( hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(W_7)) = zero_zero_int ) ) ) ).
tff(fact_23_power__0__left__number__of,axiom,
! [W_7: int] :
( ( ( number_number_of_nat(W_7) = zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(W_7)) = one_one_nat ) )
& ( ( number_number_of_nat(W_7) != zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(W_7)) = zero_zero_nat ) ) ) ).
tff(fact_24_power__0__left__number__of,axiom,
! [W_7: int] :
( ( ( number_number_of_nat(W_7) = zero_zero_nat )
=> ( hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(W_7)) = one_one_real ) )
& ( ( number_number_of_nat(W_7) != zero_zero_nat )
=> ( hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(W_7)) = 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_2: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(K_2)),number_number_of_int(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),L_1)) ) ).
tff(fact_32_plus__numeral__code_I9_J,axiom,
! [V_6: int,W: int] : hAPP_int_int(plus_plus_int(number_number_of_int(V_6)),number_number_of_int(W)) = number_number_of_int(hAPP_int_int(plus_plus_int(V_6),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_1: nat,N_1: nat] : hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,M_1)),N_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(M_1),N_1)) ).
tff(fact_37_int__power,axiom,
! [M_1: nat,N_1: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(M_1),N_1)) = hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,M_1)),N_1) ).
tff(fact_38_zadd__int__left,axiom,
! [M_1: nat,N_1: nat,Z_2: int] : hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,N_1)),Z_2)) = hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(plus_plus_nat(M_1),N_1))),Z_2) ).
tff(fact_39_zadd__int,axiom,
! [M_1: nat,N_1: nat] : hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(plus_plus_nat(M_1),N_1)) ).
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,
hAPP_nat_nat(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_2: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K_2)),bit1(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),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_2: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K_2)),bit0(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),L_1)) ) ).
tff(fact_51_zadd__strict__right__mono,axiom,
! [K: int,I_2: int,J_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),J_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(I_2),K)),hAPP_int_int(plus_plus_int(J_2),K))) ) ).
tff(fact_52_add__nat__number__of,axiom,
! [V_10: int,V_6: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_6),pls))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_6)),number_number_of_nat(V_10)) = number_number_of_nat(V_10) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_6),pls))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_10),pls))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_6)),number_number_of_nat(V_10)) = number_number_of_nat(V_6) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_10),pls))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_6)),number_number_of_nat(V_10)) = number_number_of_nat(hAPP_int_int(plus_plus_int(V_6),V_10)) ) ) ) ) ) ).
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_2: int,L_1: int] :
( ( bit1(K_2) = bit1(L_1) )
<=> ( K_2 = L_1 ) ) ).
tff(fact_62_rel__simps_I48_J,axiom,
! [K_2: int,L_1: int] :
( ( bit0(K_2) = bit0(L_1) )
<=> ( K_2 = L_1 ) ) ).
tff(fact_63_even__less__0__iff,axiom,
! [A_24: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_24),A_24)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_24),zero_zero_int)) ) ).
tff(fact_64_even__less__0__iff,axiom,
! [A_24: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_24),A_24)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_24),zero_zero_real)) ) ).
tff(fact_65_zadd__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(Z1),Z2)),Z3) = hAPP_int_int(plus_plus_int(Z1),hAPP_int_int(plus_plus_int(Z2),Z3)) ).
tff(fact_66_zadd__left__commute,axiom,
! [X: int,Y: int,Z_2: int] : hAPP_int_int(plus_plus_int(X),hAPP_int_int(plus_plus_int(Y),Z_2)) = hAPP_int_int(plus_plus_int(Y),hAPP_int_int(plus_plus_int(X),Z_2)) ).
tff(fact_67_zadd__commute,axiom,
! [Z_2: int,W: int] : hAPP_int_int(plus_plus_int(Z_2),W) = hAPP_int_int(plus_plus_int(W),Z_2) ).
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_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K_2)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),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_2: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K_2)),bit0(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),L_1)) ) ).
tff(fact_76_rel__simps_I10_J,axiom,
! [K_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K_2)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),pls)) ) ).
tff(fact_77_rel__simps_I4_J,axiom,
! [K_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),bit0(K_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),K_2)) ) ).
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_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),hAPP_int_int(plus_plus_int(Z_3),one_one_int)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_3))
| ( Wa = Z_3 ) ) ) ).
tff(fact_83_int__less__0__conv,axiom,
! [K: nat] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,K)),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_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),Z_3)),Z_3)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_3),zero_zero_int)) ) ).
tff(fact_89_double__eq__0__iff,axiom,
! [A_24: int] :
( ( hAPP_int_int(plus_plus_int(A_24),A_24) = zero_zero_int )
<=> ( A_24 = zero_zero_int ) ) ).
tff(fact_90_double__eq__0__iff,axiom,
! [A_24: real] :
( ( hAPP_real_real(plus_plus_real(A_24),A_24) = zero_zero_real )
<=> ( A_24 = zero_zero_real ) ) ).
tff(fact_91_rel__simps_I46_J,axiom,
! [K: int] : bit1(K) != pls ).
tff(fact_92_rel__simps_I39_J,axiom,
! [L: int] : pls != bit1(L) ).
tff(fact_93_rel__simps_I50_J,axiom,
! [K: int,L: int] : bit1(K) != bit0(L) ).
tff(fact_94_rel__simps_I49_J,axiom,
! [K: int,L: int] : bit0(K) != bit1(L) ).
tff(fact_95_rel__simps_I44_J,axiom,
! [K_2: int] :
( ( bit0(K_2) = pls )
<=> ( K_2 = 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: int] : hAPP_int_int(plus_plus_int(K),pls) = K ).
tff(fact_101_add__Pls,axiom,
! [K: int] : hAPP_int_int(plus_plus_int(pls),K) = K ).
tff(fact_102_add__Bit0__Bit0,axiom,
! [K: int,L: int] : hAPP_int_int(plus_plus_int(bit0(K)),bit0(L)) = bit0(hAPP_int_int(plus_plus_int(K),L)) ).
tff(fact_103_Bit0__def,axiom,
! [K: int] : bit0(K) = hAPP_int_int(plus_plus_int(K),K) ).
tff(fact_104_zadd__0__right,axiom,
! [Z_2: int] : hAPP_int_int(plus_plus_int(Z_2),zero_zero_int) = Z_2 ).
tff(fact_105_zadd__0,axiom,
! [Z_2: int] : hAPP_int_int(plus_plus_int(zero_zero_int),Z_2) = Z_2 ).
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_68: int] : hAPP_int_int(plus_plus_int(number_number_of_int(pls)),A_68) = A_68 ).
tff(fact_114_add__numeral__0,axiom,
! [A_68: real] : hAPP_real_real(plus_plus_real(number267125858f_real(pls)),A_68) = A_68 ).
tff(fact_115_add__numeral__0__right,axiom,
! [A_67: int] : hAPP_int_int(plus_plus_int(A_67),number_number_of_int(pls)) = A_67 ).
tff(fact_116_add__numeral__0__right,axiom,
! [A_67: real] : hAPP_real_real(plus_plus_real(A_67),number267125858f_real(pls)) = A_67 ).
tff(fact_117_power__eq__0__iff__number__of,axiom,
! [A_24: int,Wa: int] :
( ( hAPP_nat_int(power_power_int(A_24),number_number_of_nat(Wa)) = zero_zero_int )
<=> ( ( A_24 = zero_zero_int )
& ( number_number_of_nat(Wa) != zero_zero_nat ) ) ) ).
tff(fact_118_power__eq__0__iff__number__of,axiom,
! [A_24: nat,Wa: int] :
( ( hAPP_nat_nat(power_power_nat(A_24),number_number_of_nat(Wa)) = zero_zero_nat )
<=> ( ( A_24 = zero_zero_nat )
& ( number_number_of_nat(Wa) != zero_zero_nat ) ) ) ).
tff(fact_119_power__eq__0__iff__number__of,axiom,
! [A_24: real,Wa: int] :
( ( hAPP_nat_real(power_power_real(A_24),number_number_of_nat(Wa)) = zero_zero_real )
<=> ( ( A_24 = zero_zero_real )
& ( number_number_of_nat(Wa) != zero_zero_nat ) ) ) ).
tff(fact_120_add__number__of__left,axiom,
! [V_9: int,W_6: int,Z_5: int] : hAPP_int_int(plus_plus_int(number_number_of_int(V_9)),hAPP_int_int(plus_plus_int(number_number_of_int(W_6)),Z_5)) = hAPP_int_int(plus_plus_int(number_number_of_int(hAPP_int_int(plus_plus_int(V_9),W_6))),Z_5) ).
tff(fact_121_add__number__of__left,axiom,
! [V_9: int,W_6: int,Z_5: real] : hAPP_real_real(plus_plus_real(number267125858f_real(V_9)),hAPP_real_real(plus_plus_real(number267125858f_real(W_6)),Z_5)) = hAPP_real_real(plus_plus_real(number267125858f_real(hAPP_int_int(plus_plus_int(V_9),W_6))),Z_5) ).
tff(fact_122_add__number__of__eq,axiom,
! [V_8: int,W_5: int] : hAPP_int_int(plus_plus_int(number_number_of_int(V_8)),number_number_of_int(W_5)) = number_number_of_int(hAPP_int_int(plus_plus_int(V_8),W_5)) ).
tff(fact_123_add__number__of__eq,axiom,
! [V_8: int,W_5: int] : hAPP_real_real(plus_plus_real(number267125858f_real(V_8)),number267125858f_real(W_5)) = number267125858f_real(hAPP_int_int(plus_plus_int(V_8),W_5)) ).
tff(fact_124_number__of__add,axiom,
! [V_7: int,W_4: int] : number_number_of_int(hAPP_int_int(plus_plus_int(V_7),W_4)) = hAPP_int_int(plus_plus_int(number_number_of_int(V_7)),number_number_of_int(W_4)) ).
tff(fact_125_number__of__add,axiom,
! [V_7: int,W_4: int] : number267125858f_real(hAPP_int_int(plus_plus_int(V_7),W_4)) = hAPP_real_real(plus_plus_real(number267125858f_real(V_7)),number267125858f_real(W_4)) ).
tff(fact_126_add__Bit1__Bit0,axiom,
! [K: int,L: int] : hAPP_int_int(plus_plus_int(bit1(K)),bit0(L)) = bit1(hAPP_int_int(plus_plus_int(K),L)) ).
tff(fact_127_add__Bit0__Bit1,axiom,
! [K: int,L: int] : hAPP_int_int(plus_plus_int(bit0(K)),bit1(L)) = bit1(hAPP_int_int(plus_plus_int(K),L)) ).
tff(fact_128_Bit1__def,axiom,
! [K: int] : bit1(K) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),K)),K) ).
tff(fact_129_odd__nonzero,axiom,
! [Z_2: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),Z_2)),Z_2) != zero_zero_int ).
tff(fact_130_number__of__int,axiom,
! [N_29: nat] : number_number_of_nat(hAPP_nat_int(semiri1621563631at_int,N_29)) = hAPP_nat_nat(semiri984289939at_nat,N_29) ).
tff(fact_131_number__of__int,axiom,
! [N_29: nat] : number_number_of_int(hAPP_nat_int(semiri1621563631at_int,N_29)) = hAPP_nat_int(semiri1621563631at_int,N_29) ).
tff(fact_132_number__of__int,axiom,
! [N_29: nat] : number267125858f_real(hAPP_nat_int(semiri1621563631at_int,N_29)) = hAPP_nat_real(semiri132038758t_real,N_29) ).
tff(fact_133_zero__less__power2,axiom,
! [A_24: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(power_power_int(A_24),number_number_of_nat(bit0(bit1(pls))))))
<=> ( A_24 != zero_zero_int ) ) ).
tff(fact_134_zero__less__power2,axiom,
! [A_24: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(power_power_real(A_24),number_number_of_nat(bit0(bit1(pls))))))
<=> ( A_24 != zero_zero_real ) ) ).
tff(fact_135_power2__less__0,axiom,
! [A_66: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_66),number_number_of_nat(bit0(bit1(pls))))),zero_zero_int)) ).
tff(fact_136_power2__less__0,axiom,
! [A_66: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_66),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),hAPP_int_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),hAPP_real_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_16: int,Y_13: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X_16),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_13),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_int)) ).
tff(fact_140_not__sum__power2__lt__zero,axiom,
! [X_16: real,Y_13: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_16),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_13),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_real)) ).
tff(fact_141_number__of__Bit0,axiom,
! [W_3: int] : number_number_of_int(bit0(W_3)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(zero_zero_int),number_number_of_int(W_3))),number_number_of_int(W_3)) ).
tff(fact_142_number__of__Bit0,axiom,
! [W_3: int] : number267125858f_real(bit0(W_3)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(zero_zero_real),number267125858f_real(W_3))),number267125858f_real(W_3)) ).
tff(fact_143_number__of__Bit1,axiom,
! [W_2: int] : number_number_of_int(bit1(W_2)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),number_number_of_int(W_2))),number_number_of_int(W_2)) ).
tff(fact_144_number__of__Bit1,axiom,
! [W_2: int] : number267125858f_real(bit1(W_2)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(one_one_real),number267125858f_real(W_2))),number267125858f_real(W_2)) ).
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),hAPP_int_int(plus_plus_int(one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))) ).
tff(fact_149_of__nat__less__two__power,axiom,
! [N_28: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N_28)),hAPP_nat_int(power_power_int(number_number_of_int(bit0(bit1(pls)))),N_28))) ).
tff(fact_150_of__nat__less__two__power,axiom,
! [N_28: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,N_28)),hAPP_nat_real(power_power_real(number267125858f_real(bit0(bit1(pls)))),N_28))) ).
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] :
( ( hAPP_real_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_27: nat,A_65: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_65))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_27))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_nat_int(power_power_int(A_65),N_27))) ) ) ).
tff(fact_158_one__less__power,axiom,
! [N_27: nat,A_65: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_65))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_27))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(power_power_nat(A_65),N_27))) ) ) ).
tff(fact_159_one__less__power,axiom,
! [N_27: nat,A_65: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_65))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_27))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_nat_real(power_power_real(A_65),N_27))) ) ) ).
tff(fact_160_power__0__left,axiom,
! [N_26: nat] :
( ( ( N_26 = zero_zero_nat )
=> ( hAPP_nat_int(power_power_int(zero_zero_int),N_26) = one_one_int ) )
& ( ( N_26 != zero_zero_nat )
=> ( hAPP_nat_int(power_power_int(zero_zero_int),N_26) = zero_zero_int ) ) ) ).
tff(fact_161_power__0__left,axiom,
! [N_26: nat] :
( ( ( N_26 = zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(zero_zero_nat),N_26) = one_one_nat ) )
& ( ( N_26 != zero_zero_nat )
=> ( hAPP_nat_nat(power_power_nat(zero_zero_nat),N_26) = zero_zero_nat ) ) ) ).
tff(fact_162_power__0__left,axiom,
! [N_26: nat] :
( ( ( N_26 = zero_zero_nat )
=> ( hAPP_nat_real(power_power_real(zero_zero_real),N_26) = one_one_real ) )
& ( ( N_26 != zero_zero_nat )
=> ( hAPP_nat_real(power_power_real(zero_zero_real),N_26) = zero_zero_real ) ) ) ).
tff(fact_163_power__strict__decreasing,axiom,
! [A_64: int,N_25: nat,N_24: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_25),N_24))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_64))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_64),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_64),N_24)),hAPP_nat_int(power_power_int(A_64),N_25))) ) ) ) ).
tff(fact_164_power__strict__decreasing,axiom,
! [A_64: nat,N_25: nat,N_24: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_25),N_24))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_64))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_64),one_one_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_64),N_24)),hAPP_nat_nat(power_power_nat(A_64),N_25))) ) ) ) ).
tff(fact_165_power__strict__decreasing,axiom,
! [A_64: real,N_25: nat,N_24: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_25),N_24))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_64))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_64),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_64),N_24)),hAPP_nat_real(power_power_real(A_64),N_25))) ) ) ) ).
tff(fact_166_zero__less__two,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(one_one_int),one_one_int))) ).
tff(fact_167_zero__less__two,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(plus_plus_nat(one_one_nat),one_one_nat))) ).
tff(fact_168_zero__less__two,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(plus_plus_real(one_one_real),one_one_real))) ).
tff(fact_169_int__gr__induct,axiom,
! [P_1: fun_int_bool,K_2: int,I: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),I))
=> ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(K_2),one_one_int)))
=> ( ! [I_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),I_1))
=> ( hBOOL(hAPP_int_bool(P_1,I_1))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(I_1),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_1,I)) ) ) ) ).
tff(fact_170_tn0,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),tn)) ).
tff(fact_171_less__zeroE,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).
tff(fact_172_real__zero__not__eq__one,axiom,
zero_zero_real != one_one_real ).
tff(fact_173_less__not__refl,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1)) ).
tff(fact_174_not__add__less1,axiom,
! [I_2: nat,J_2: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_2),J_2)),I_2)) ).
tff(fact_175_not__add__less2,axiom,
! [J_2: nat,I_2: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(J_2),I_2)),I_2)) ).
tff(fact_176_number__of__is__id,axiom,
! [K: int] : number_number_of_int(K) = K ).
tff(fact_177_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_178_nat__add__commute,axiom,
! [M_1: nat,N_1: nat] : hAPP_nat_nat(plus_plus_nat(M_1),N_1) = hAPP_nat_nat(plus_plus_nat(N_1),M_1) ).
tff(fact_179_nat__add__left__commute,axiom,
! [X: nat,Y: nat,Z_2: nat] : hAPP_nat_nat(plus_plus_nat(X),hAPP_nat_nat(plus_plus_nat(Y),Z_2)) = hAPP_nat_nat(plus_plus_nat(Y),hAPP_nat_nat(plus_plus_nat(X),Z_2)) ).
tff(fact_180_nat__add__assoc,axiom,
! [M_1: nat,N_1: nat,K: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(M_1),N_1)),K) = hAPP_nat_nat(plus_plus_nat(M_1),hAPP_nat_nat(plus_plus_nat(N_1),K)) ).
tff(fact_181_nat__add__left__cancel,axiom,
! [K_2: nat,Ma: nat,Na: nat] :
( ( hAPP_nat_nat(plus_plus_nat(K_2),Ma) = hAPP_nat_nat(plus_plus_nat(K_2),Na) )
<=> ( Ma = Na ) ) ).
tff(fact_182_nat__add__right__cancel,axiom,
! [Ma: nat,K_2: nat,Na: nat] :
( ( hAPP_nat_nat(plus_plus_nat(Ma),K_2) = hAPP_nat_nat(plus_plus_nat(Na),K_2) )
<=> ( Ma = Na ) ) ).
tff(fact_183_nat__add__left__cancel__less,axiom,
! [K_2: nat,Ma: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(K_2),Ma)),hAPP_nat_nat(plus_plus_nat(K_2),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
tff(fact_184_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_185_less__irrefl__nat,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1)) ).
tff(fact_186_less__not__refl2,axiom,
! [N_1: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M_1))
=> ( M_1 != N_1 ) ) ).
tff(fact_187_less__not__refl3,axiom,
! [S: nat,T: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S),T))
=> ( S != T ) ) ).
tff(fact_188_trans__less__add1,axiom,
! [M_1: nat,I_2: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),hAPP_nat_nat(plus_plus_nat(J_2),M_1))) ) ).
tff(fact_189_trans__less__add2,axiom,
! [M_1: nat,I_2: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),hAPP_nat_nat(plus_plus_nat(M_1),J_2))) ) ).
tff(fact_190_add__less__mono1,axiom,
! [K: nat,I_2: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_2),K)),hAPP_nat_nat(plus_plus_nat(J_2),K))) ) ).
tff(fact_191_add__less__mono,axiom,
! [K: nat,L: nat,I_2: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_2),K)),hAPP_nat_nat(plus_plus_nat(J_2),L))) ) ) ).
tff(fact_192_less__add__eq__less,axiom,
! [M_1: nat,N_1: nat,K: nat,L: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),L))
=> ( ( hAPP_nat_nat(plus_plus_nat(M_1),L) = hAPP_nat_nat(plus_plus_nat(K),N_1) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ) ).
tff(fact_193_add__lessD1,axiom,
! [I_2: nat,J_2: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_2),J_2)),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),K)) ) ).
tff(fact_194_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_195_gr0I,axiom,
! [N_1: nat] :
( ( N_1 != zero_zero_nat )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).
tff(fact_196_gr__implies__not0,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
=> ( N_1 != zero_zero_nat ) ) ).
tff(fact_197_nat__power__less__imp__less,axiom,
! [M_1: nat,N_1: 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_1)),hAPP_nat_nat(power_power_nat(I_2),N_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ) ).
tff(fact_198_less__nat__zero__code,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).
tff(fact_199_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_200_add__gr__0,axiom,
! [Ma: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_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_201_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_202_not__less0,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).
tff(fact_203_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_204_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_205_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_206_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_207_less__nat__number__of,axiom,
! [Va: int,V: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,number_number_of_nat(Va)),number_number_of_nat(V)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Va),V))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V)) )
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Va),V)) ) ) ).
tff(fact_208_linorder__neqE__linordered__idom,axiom,
! [X_15: real,Y_12: real] :
( ( X_15 != Y_12 )
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_15),Y_12))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_12),X_15)) ) ) ).
tff(fact_209_linorder__neqE__linordered__idom,axiom,
! [X_15: int,Y_12: int] :
( ( X_15 != Y_12 )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_15),Y_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_12),X_15)) ) ) ).
tff(fact_210_add__eq__self__zero,axiom,
! [M_1: nat,N_1: nat] :
( ( hAPP_nat_nat(plus_plus_nat(M_1),N_1) = M_1 )
=> ( N_1 = zero_zero_nat ) ) ).
tff(fact_211_add__is__0,axiom,
! [Ma: nat,Na: nat] :
( ( hAPP_nat_nat(plus_plus_nat(Ma),Na) = zero_zero_nat )
<=> ( ( Ma = zero_zero_nat )
& ( Na = zero_zero_nat ) ) ) ).
tff(fact_212_Nat_Oadd__0__right,axiom,
! [M_1: nat] : hAPP_nat_nat(plus_plus_nat(M_1),zero_zero_nat) = M_1 ).
tff(fact_213_plus__nat_Oadd__0,axiom,
! [N_1: nat] : hAPP_nat_nat(plus_plus_nat(zero_zero_nat),N_1) = N_1 ).
tff(fact_214_power__one__right,axiom,
! [A_63: real] : hAPP_nat_real(power_power_real(A_63),one_one_nat) = A_63 ).
tff(fact_215_power__one__right,axiom,
! [A_63: nat] : hAPP_nat_nat(power_power_nat(A_63),one_one_nat) = A_63 ).
tff(fact_216_power__one__right,axiom,
! [A_63: int] : hAPP_nat_int(power_power_int(A_63),one_one_nat) = A_63 ).
tff(fact_217_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_218_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_219_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_220_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_221_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,hAPP_nat_nat(if_nat(P_1,Xa),Ya)) ) )
& ( ~ hBOOL(P_1)
=> ( hAPP_nat_int(semiri1621563631at_int,Ya) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(if_nat(P_1,Xa),Ya)) ) ) ) ).
tff(fact_222_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_223_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_224_one__neq__zero,axiom,
one_one_real != zero_zero_real ).
tff(fact_225_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
tff(fact_226_one__neq__zero,axiom,
one_one_int != zero_zero_int ).
tff(fact_227_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
tff(fact_228_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
tff(fact_229_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
tff(fact_230_field__power__not__zero,axiom,
! [N_23: nat,A_62: real] :
( ( A_62 != zero_zero_real )
=> ( hAPP_nat_real(power_power_real(A_62),N_23) != zero_zero_real ) ) ).
tff(fact_231_field__power__not__zero,axiom,
! [N_23: nat,A_62: int] :
( ( A_62 != zero_zero_int )
=> ( hAPP_nat_int(power_power_int(A_62),N_23) != zero_zero_int ) ) ).
tff(fact_232_power__one,axiom,
! [N_22: nat] : hAPP_nat_real(power_power_real(one_one_real),N_22) = one_one_real ).
tff(fact_233_power__one,axiom,
! [N_22: nat] : hAPP_nat_nat(power_power_nat(one_one_nat),N_22) = one_one_nat ).
tff(fact_234_power__one,axiom,
! [N_22: nat] : hAPP_nat_int(power_power_int(one_one_int),N_22) = one_one_int ).
tff(fact_235_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_236_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_237_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_238_less__imp__of__nat__less,axiom,
! [M_9: nat,N_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_21))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_9)),hAPP_nat_real(semiri132038758t_real,N_21))) ) ).
tff(fact_239_less__imp__of__nat__less,axiom,
! [M_9: nat,N_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_21))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_9)),hAPP_nat_nat(semiri984289939at_nat,N_21))) ) ).
tff(fact_240_less__imp__of__nat__less,axiom,
! [M_9: nat,N_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_21))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_9)),hAPP_nat_int(semiri1621563631at_int,N_21))) ) ).
tff(fact_241_of__nat__less__imp__less,axiom,
! [M_8: nat,N_20: nat] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_8)),hAPP_nat_real(semiri132038758t_real,N_20)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_8),N_20)) ) ).
tff(fact_242_of__nat__less__imp__less,axiom,
! [M_8: nat,N_20: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_8)),hAPP_nat_nat(semiri984289939at_nat,N_20)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_8),N_20)) ) ).
tff(fact_243_of__nat__less__imp__less,axiom,
! [M_8: nat,N_20: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_8)),hAPP_nat_int(semiri1621563631at_int,N_20)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_8),N_20)) ) ).
tff(fact_244_of__nat__add,axiom,
! [M_7: nat,N_19: nat] : hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(plus_plus_nat(M_7),N_19)) = hAPP_real_real(plus_plus_real(hAPP_nat_real(semiri132038758t_real,M_7)),hAPP_nat_real(semiri132038758t_real,N_19)) ).
tff(fact_245_of__nat__add,axiom,
! [M_7: nat,N_19: nat] : hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(plus_plus_nat(M_7),N_19)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(semiri984289939at_nat,M_7)),hAPP_nat_nat(semiri984289939at_nat,N_19)) ).
tff(fact_246_of__nat__add,axiom,
! [M_7: nat,N_19: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(plus_plus_nat(M_7),N_19)) = hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M_7)),hAPP_nat_int(semiri1621563631at_int,N_19)) ).
tff(fact_247_of__nat__1,axiom,
hAPP_nat_real(semiri132038758t_real,one_one_nat) = one_one_real ).
tff(fact_248_of__nat__1,axiom,
hAPP_nat_nat(semiri984289939at_nat,one_one_nat) = one_one_nat ).
tff(fact_249_of__nat__1,axiom,
hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int ).
tff(fact_250_of__nat__power,axiom,
! [M_6: nat,N_18: nat] : hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(power_power_nat(M_6),N_18)) = hAPP_nat_real(power_power_real(hAPP_nat_real(semiri132038758t_real,M_6)),N_18) ).
tff(fact_251_of__nat__power,axiom,
! [M_6: nat,N_18: nat] : hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(power_power_nat(M_6),N_18)) = hAPP_nat_nat(power_power_nat(hAPP_nat_nat(semiri984289939at_nat,M_6)),N_18) ).
tff(fact_252_of__nat__power,axiom,
! [M_6: nat,N_18: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(M_6),N_18)) = hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,M_6)),N_18) ).
tff(fact_253_transfer__int__nat__numerals_I2_J,axiom,
one_one_int = hAPP_nat_int(semiri1621563631at_int,one_one_nat) ).
tff(fact_254_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
! [X: nat,Y: nat] : hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(plus_plus_nat(X),Y)) ).
tff(fact_255_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
! [X: nat,N_1: nat] : hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,X)),N_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(X),N_1)) ).
tff(fact_256_pos__add__strict,axiom,
! [B_41: real,C_31: real,A_61: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_61))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_41),C_31))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_41),hAPP_real_real(plus_plus_real(A_61),C_31))) ) ) ).
tff(fact_257_pos__add__strict,axiom,
! [B_41: nat,C_31: nat,A_61: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_61))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_41),C_31))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_41),hAPP_nat_nat(plus_plus_nat(A_61),C_31))) ) ) ).
tff(fact_258_pos__add__strict,axiom,
! [B_41: int,C_31: int,A_61: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_61))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_41),C_31))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_41),hAPP_int_int(plus_plus_int(A_61),C_31))) ) ) ).
tff(fact_259_not__one__less__zero,axiom,
~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),zero_zero_real)) ).
tff(fact_260_not__one__less__zero,axiom,
~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).
tff(fact_261_not__one__less__zero,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),zero_zero_int)) ).
tff(fact_262_zero__less__one,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),one_one_real)) ).
tff(fact_263_zero__less__one,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).
tff(fact_264_zero__less__one,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
tff(fact_265_zero__less__power,axiom,
! [N_17: nat,A_60: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_60))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(power_power_real(A_60),N_17))) ) ).
tff(fact_266_zero__less__power,axiom,
! [N_17: nat,A_60: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_60))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(A_60),N_17))) ) ).
tff(fact_267_zero__less__power,axiom,
! [N_17: nat,A_60: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_60))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(power_power_int(A_60),N_17))) ) ).
tff(fact_268_less__add__one,axiom,
! [A_59: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_59),hAPP_real_real(plus_plus_real(A_59),one_one_real))) ).
tff(fact_269_less__add__one,axiom,
! [A_59: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_59),hAPP_nat_nat(plus_plus_nat(A_59),one_one_nat))) ).
tff(fact_270_less__add__one,axiom,
! [A_59: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_59),hAPP_int_int(plus_plus_int(A_59),one_one_int))) ).
tff(fact_271_power__inject__exp,axiom,
! [Ma: nat,Na: nat,A_24: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_24))
=> ( ( hAPP_nat_real(power_power_real(A_24),Ma) = hAPP_nat_real(power_power_real(A_24),Na) )
<=> ( Ma = Na ) ) ) ).
tff(fact_272_power__inject__exp,axiom,
! [Ma: nat,Na: nat,A_24: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_24))
=> ( ( hAPP_nat_nat(power_power_nat(A_24),Ma) = hAPP_nat_nat(power_power_nat(A_24),Na) )
<=> ( Ma = Na ) ) ) ).
tff(fact_273_power__inject__exp,axiom,
! [Ma: nat,Na: nat,A_24: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_24))
=> ( ( hAPP_nat_int(power_power_int(A_24),Ma) = hAPP_nat_int(power_power_int(A_24),Na) )
<=> ( Ma = Na ) ) ) ).
tff(fact_274_power__strict__increasing__iff,axiom,
! [Xa: nat,Ya: nat,B_21: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B_21))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(B_21),Xa)),hAPP_nat_real(power_power_real(B_21),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) ) ) ).
tff(fact_275_power__strict__increasing__iff,axiom,
! [Xa: nat,Ya: nat,B_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B_21))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(B_21),Xa)),hAPP_nat_nat(power_power_nat(B_21),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_21: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(B_21),Xa)),hAPP_nat_int(power_power_int(B_21),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) ) ) ).
tff(fact_277_power__less__imp__less__exp,axiom,
! [M_5: nat,N_16: nat,A_58: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_58))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_58),M_5)),hAPP_nat_real(power_power_real(A_58),N_16)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_5),N_16)) ) ) ).
tff(fact_278_power__less__imp__less__exp,axiom,
! [M_5: nat,N_16: nat,A_58: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_58))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_58),M_5)),hAPP_nat_nat(power_power_nat(A_58),N_16)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_5),N_16)) ) ) ).
tff(fact_279_power__less__imp__less__exp,axiom,
! [M_5: nat,N_16: nat,A_58: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_58))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_58),M_5)),hAPP_nat_int(power_power_int(A_58),N_16)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_5),N_16)) ) ) ).
tff(fact_280_power__strict__increasing,axiom,
! [A_57: real,N_15: nat,N_14: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_15),N_14))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_57))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_57),N_15)),hAPP_nat_real(power_power_real(A_57),N_14))) ) ) ).
tff(fact_281_power__strict__increasing,axiom,
! [A_57: nat,N_15: nat,N_14: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_15),N_14))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_57))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_57),N_15)),hAPP_nat_nat(power_power_nat(A_57),N_14))) ) ) ).
tff(fact_282_power__strict__increasing,axiom,
! [A_57: int,N_15: nat,N_14: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_15),N_14))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_57))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_57),N_15)),hAPP_nat_int(power_power_int(A_57),N_14))) ) ) ).
tff(fact_283_power__eq__0__iff,axiom,
! [A_24: real,Na: nat] :
( ( hAPP_nat_real(power_power_real(A_24),Na) = zero_zero_real )
<=> ( ( A_24 = zero_zero_real )
& ( Na != zero_zero_nat ) ) ) ).
tff(fact_284_power__eq__0__iff,axiom,
! [A_24: nat,Na: nat] :
( ( hAPP_nat_nat(power_power_nat(A_24),Na) = zero_zero_nat )
<=> ( ( A_24 = zero_zero_nat )
& ( Na != zero_zero_nat ) ) ) ).
tff(fact_285_power__eq__0__iff,axiom,
! [A_24: int,Na: nat] :
( ( hAPP_nat_int(power_power_int(A_24),Na) = zero_zero_int )
<=> ( ( A_24 = zero_zero_int )
& ( Na != zero_zero_nat ) ) ) ).
tff(fact_286_of__nat__less__0__iff,axiom,
! [M_4: nat] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_4)),zero_zero_real)) ).
tff(fact_287_of__nat__less__0__iff,axiom,
! [M_4: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_4)),zero_zero_nat)) ).
tff(fact_288_of__nat__less__0__iff,axiom,
! [M_4: nat] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_4)),zero_zero_int)) ).
tff(fact_289_power__0,axiom,
! [A_56: real] : hAPP_nat_real(power_power_real(A_56),zero_zero_nat) = one_one_real ).
tff(fact_290_power__0,axiom,
! [A_56: nat] : hAPP_nat_nat(power_power_nat(A_56),zero_zero_nat) = one_one_nat ).
tff(fact_291_power__0,axiom,
! [A_56: int] : hAPP_nat_int(power_power_int(A_56),zero_zero_nat) = one_one_int ).
tff(fact_292_of__nat__0,axiom,
hAPP_nat_real(semiri132038758t_real,zero_zero_nat) = zero_zero_real ).
tff(fact_293_of__nat__0,axiom,
hAPP_nat_nat(semiri984289939at_nat,zero_zero_nat) = zero_zero_nat ).
tff(fact_294_of__nat__0,axiom,
hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) = zero_zero_int ).
tff(fact_295_transfer__int__nat__numerals_I1_J,axiom,
zero_zero_int = hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) ).
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: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K))
=> ? [N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
& ( K = hAPP_nat_int(semiri1621563631at_int,N) ) ) ) ).
tff(fact_298_less__imp__add__positive,axiom,
! [I_2: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_2))
=> ? [K_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
& ( hAPP_nat_nat(plus_plus_nat(I_2),K_1) = J_2 ) ) ) ).
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_14: real] : hAPP_nat_real(power_power_real(X_14),zero_zero_nat) = one_one_real ).
tff(fact_301_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_14: nat] : hAPP_nat_nat(power_power_nat(X_14),zero_zero_nat) = one_one_nat ).
tff(fact_302_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_14: int] : hAPP_nat_int(power_power_int(X_14),zero_zero_nat) = one_one_int ).
tff(fact_303_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_24: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(plus_plus_real(A_24),A_24)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_24)) ) ).
tff(fact_304_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_24: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(A_24),A_24)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_24)) ) ).
tff(fact_305_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_24: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_24),A_24)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_24),zero_zero_real)) ) ).
tff(fact_306_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_24: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_24),A_24)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_24),zero_zero_int)) ) ).
tff(fact_307_zero__reorient,axiom,
! [Xa: real] :
( ( zero_zero_real = Xa )
<=> ( Xa = zero_zero_real ) ) ).
tff(fact_308_zero__reorient,axiom,
! [Xa: nat] :
( ( zero_zero_nat = Xa )
<=> ( Xa = zero_zero_nat ) ) ).
tff(fact_309_zero__reorient,axiom,
! [Xa: int] :
( ( zero_zero_int = Xa )
<=> ( Xa = zero_zero_int ) ) ).
tff(fact_310_add__right__imp__eq,axiom,
! [B_40: real,A_55: real,C_30: real] :
( ( hAPP_real_real(plus_plus_real(B_40),A_55) = hAPP_real_real(plus_plus_real(C_30),A_55) )
=> ( B_40 = C_30 ) ) ).
tff(fact_311_add__right__imp__eq,axiom,
! [B_40: nat,A_55: nat,C_30: nat] :
( ( hAPP_nat_nat(plus_plus_nat(B_40),A_55) = hAPP_nat_nat(plus_plus_nat(C_30),A_55) )
=> ( B_40 = C_30 ) ) ).
tff(fact_312_add__right__imp__eq,axiom,
! [B_40: int,A_55: int,C_30: int] :
( ( hAPP_int_int(plus_plus_int(B_40),A_55) = hAPP_int_int(plus_plus_int(C_30),A_55) )
=> ( B_40 = C_30 ) ) ).
tff(fact_313_add__imp__eq,axiom,
! [A_54: real,B_39: real,C_29: real] :
( ( hAPP_real_real(plus_plus_real(A_54),B_39) = hAPP_real_real(plus_plus_real(A_54),C_29) )
=> ( B_39 = C_29 ) ) ).
tff(fact_314_add__imp__eq,axiom,
! [A_54: nat,B_39: nat,C_29: nat] :
( ( hAPP_nat_nat(plus_plus_nat(A_54),B_39) = hAPP_nat_nat(plus_plus_nat(A_54),C_29) )
=> ( B_39 = C_29 ) ) ).
tff(fact_315_add__imp__eq,axiom,
! [A_54: int,B_39: int,C_29: int] :
( ( hAPP_int_int(plus_plus_int(A_54),B_39) = hAPP_int_int(plus_plus_int(A_54),C_29) )
=> ( B_39 = C_29 ) ) ).
tff(fact_316_add__left__imp__eq,axiom,
! [A_53: real,B_38: real,C_28: real] :
( ( hAPP_real_real(plus_plus_real(A_53),B_38) = hAPP_real_real(plus_plus_real(A_53),C_28) )
=> ( B_38 = C_28 ) ) ).
tff(fact_317_add__left__imp__eq,axiom,
! [A_53: nat,B_38: nat,C_28: nat] :
( ( hAPP_nat_nat(plus_plus_nat(A_53),B_38) = hAPP_nat_nat(plus_plus_nat(A_53),C_28) )
=> ( B_38 = C_28 ) ) ).
tff(fact_318_add__left__imp__eq,axiom,
! [A_53: int,B_38: int,C_28: int] :
( ( hAPP_int_int(plus_plus_int(A_53),B_38) = hAPP_int_int(plus_plus_int(A_53),C_28) )
=> ( B_38 = C_28 ) ) ).
tff(fact_319_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_52: real,B_37: real,C_27: real,D_6: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_52),B_37)),hAPP_real_real(plus_plus_real(C_27),D_6)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_52),C_27)),hAPP_real_real(plus_plus_real(B_37),D_6)) ).
tff(fact_320_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_52: nat,B_37: nat,C_27: nat,D_6: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_52),B_37)),hAPP_nat_nat(plus_plus_nat(C_27),D_6)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_52),C_27)),hAPP_nat_nat(plus_plus_nat(B_37),D_6)) ).
tff(fact_321_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_52: int,B_37: int,C_27: int,D_6: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_52),B_37)),hAPP_int_int(plus_plus_int(C_27),D_6)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_52),C_27)),hAPP_int_int(plus_plus_int(B_37),D_6)) ).
tff(fact_322_add__right__cancel,axiom,
! [B_21: real,A_24: real,C_15: real] :
( ( hAPP_real_real(plus_plus_real(B_21),A_24) = hAPP_real_real(plus_plus_real(C_15),A_24) )
<=> ( B_21 = C_15 ) ) ).
tff(fact_323_add__right__cancel,axiom,
! [B_21: nat,A_24: nat,C_15: nat] :
( ( hAPP_nat_nat(plus_plus_nat(B_21),A_24) = hAPP_nat_nat(plus_plus_nat(C_15),A_24) )
<=> ( B_21 = C_15 ) ) ).
tff(fact_324_add__right__cancel,axiom,
! [B_21: int,A_24: int,C_15: int] :
( ( hAPP_int_int(plus_plus_int(B_21),A_24) = hAPP_int_int(plus_plus_int(C_15),A_24) )
<=> ( B_21 = C_15 ) ) ).
tff(fact_325_add__left__cancel,axiom,
! [A_24: real,B_21: real,C_15: real] :
( ( hAPP_real_real(plus_plus_real(A_24),B_21) = hAPP_real_real(plus_plus_real(A_24),C_15) )
<=> ( B_21 = C_15 ) ) ).
tff(fact_326_add__left__cancel,axiom,
! [A_24: nat,B_21: nat,C_15: nat] :
( ( hAPP_nat_nat(plus_plus_nat(A_24),B_21) = hAPP_nat_nat(plus_plus_nat(A_24),C_15) )
<=> ( B_21 = C_15 ) ) ).
tff(fact_327_add__left__cancel,axiom,
! [A_24: int,B_21: int,C_15: int] :
( ( hAPP_int_int(plus_plus_int(A_24),B_21) = hAPP_int_int(plus_plus_int(A_24),C_15) )
<=> ( B_21 = C_15 ) ) ).
tff(fact_328_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_51: real,B_36: real,C_26: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_51),B_36)),C_26) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_51),C_26)),B_36) ).
tff(fact_329_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_51: nat,B_36: nat,C_26: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_51),B_36)),C_26) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_51),C_26)),B_36) ).
tff(fact_330_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_51: int,B_36: int,C_26: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_51),B_36)),C_26) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_51),C_26)),B_36) ).
tff(fact_331_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_50: real,B_35: real,C_25: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_50),B_35)),C_25) = hAPP_real_real(plus_plus_real(A_50),hAPP_real_real(plus_plus_real(B_35),C_25)) ).
tff(fact_332_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_50: nat,B_35: nat,C_25: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_50),B_35)),C_25) = hAPP_nat_nat(plus_plus_nat(A_50),hAPP_nat_nat(plus_plus_nat(B_35),C_25)) ).
tff(fact_333_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_50: int,B_35: int,C_25: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_50),B_35)),C_25) = hAPP_int_int(plus_plus_int(A_50),hAPP_int_int(plus_plus_int(B_35),C_25)) ).
tff(fact_334_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_49: real,B_34: real,C_24: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_49),B_34)),C_24) = hAPP_real_real(plus_plus_real(A_49),hAPP_real_real(plus_plus_real(B_34),C_24)) ).
tff(fact_335_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_49: nat,B_34: nat,C_24: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_49),B_34)),C_24) = hAPP_nat_nat(plus_plus_nat(A_49),hAPP_nat_nat(plus_plus_nat(B_34),C_24)) ).
tff(fact_336_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_49: int,B_34: int,C_24: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_49),B_34)),C_24) = hAPP_int_int(plus_plus_int(A_49),hAPP_int_int(plus_plus_int(B_34),C_24)) ).
tff(fact_337_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_48: real,C_23: real,D_5: real] : hAPP_real_real(plus_plus_real(A_48),hAPP_real_real(plus_plus_real(C_23),D_5)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_48),C_23)),D_5) ).
tff(fact_338_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_48: nat,C_23: nat,D_5: nat] : hAPP_nat_nat(plus_plus_nat(A_48),hAPP_nat_nat(plus_plus_nat(C_23),D_5)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_48),C_23)),D_5) ).
tff(fact_339_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_48: int,C_23: int,D_5: int] : hAPP_int_int(plus_plus_int(A_48),hAPP_int_int(plus_plus_int(C_23),D_5)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_48),C_23)),D_5) ).
tff(fact_340_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_47: real,C_22: real,D_4: real] : hAPP_real_real(plus_plus_real(A_47),hAPP_real_real(plus_plus_real(C_22),D_4)) = hAPP_real_real(plus_plus_real(C_22),hAPP_real_real(plus_plus_real(A_47),D_4)) ).
tff(fact_341_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_47: nat,C_22: nat,D_4: nat] : hAPP_nat_nat(plus_plus_nat(A_47),hAPP_nat_nat(plus_plus_nat(C_22),D_4)) = hAPP_nat_nat(plus_plus_nat(C_22),hAPP_nat_nat(plus_plus_nat(A_47),D_4)) ).
tff(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_47: int,C_22: int,D_4: int] : hAPP_int_int(plus_plus_int(A_47),hAPP_int_int(plus_plus_int(C_22),D_4)) = hAPP_int_int(plus_plus_int(C_22),hAPP_int_int(plus_plus_int(A_47),D_4)) ).
tff(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_46: real,C_21: real] : hAPP_real_real(plus_plus_real(A_46),C_21) = hAPP_real_real(plus_plus_real(C_21),A_46) ).
tff(fact_344_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_46: nat,C_21: nat] : hAPP_nat_nat(plus_plus_nat(A_46),C_21) = hAPP_nat_nat(plus_plus_nat(C_21),A_46) ).
tff(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_46: int,C_21: int] : hAPP_int_int(plus_plus_int(A_46),C_21) = hAPP_int_int(plus_plus_int(C_21),A_46) ).
tff(fact_346_one__reorient,axiom,
! [Xa: real] :
( ( one_one_real = Xa )
<=> ( Xa = one_one_real ) ) ).
tff(fact_347_one__reorient,axiom,
! [Xa: nat] :
( ( one_one_nat = Xa )
<=> ( Xa = one_one_nat ) ) ).
tff(fact_348_one__reorient,axiom,
! [Xa: int] :
( ( one_one_int = Xa )
<=> ( Xa = one_one_int ) ) ).
tff(fact_349_add__0__iff,axiom,
! [B_21: real,A_24: real] :
( ( B_21 = hAPP_real_real(plus_plus_real(B_21),A_24) )
<=> ( A_24 = zero_zero_real ) ) ).
tff(fact_350_add__0__iff,axiom,
! [B_21: nat,A_24: nat] :
( ( B_21 = hAPP_nat_nat(plus_plus_nat(B_21),A_24) )
<=> ( A_24 = zero_zero_nat ) ) ).
tff(fact_351_add__0__iff,axiom,
! [B_21: int,A_24: int] :
( ( B_21 = hAPP_int_int(plus_plus_int(B_21),A_24) )
<=> ( A_24 = zero_zero_int ) ) ).
tff(fact_352_add_Ocomm__neutral,axiom,
! [A_45: real] : hAPP_real_real(plus_plus_real(A_45),zero_zero_real) = A_45 ).
tff(fact_353_add_Ocomm__neutral,axiom,
! [A_45: nat] : hAPP_nat_nat(plus_plus_nat(A_45),zero_zero_nat) = A_45 ).
tff(fact_354_add_Ocomm__neutral,axiom,
! [A_45: int] : hAPP_int_int(plus_plus_int(A_45),zero_zero_int) = A_45 ).
tff(fact_355_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_44: real] : hAPP_real_real(plus_plus_real(A_44),zero_zero_real) = A_44 ).
tff(fact_356_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_44: nat] : hAPP_nat_nat(plus_plus_nat(A_44),zero_zero_nat) = A_44 ).
tff(fact_357_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_44: int] : hAPP_int_int(plus_plus_int(A_44),zero_zero_int) = A_44 ).
tff(fact_358_add__0__right,axiom,
! [A_43: real] : hAPP_real_real(plus_plus_real(A_43),zero_zero_real) = A_43 ).
tff(fact_359_add__0__right,axiom,
! [A_43: nat] : hAPP_nat_nat(plus_plus_nat(A_43),zero_zero_nat) = A_43 ).
tff(fact_360_add__0__right,axiom,
! [A_43: int] : hAPP_int_int(plus_plus_int(A_43),zero_zero_int) = A_43 ).
tff(fact_361_double__zero__sym,axiom,
! [A_24: real] :
( ( zero_zero_real = hAPP_real_real(plus_plus_real(A_24),A_24) )
<=> ( A_24 = zero_zero_real ) ) ).
tff(fact_362_double__zero__sym,axiom,
! [A_24: int] :
( ( zero_zero_int = hAPP_int_int(plus_plus_int(A_24),A_24) )
<=> ( A_24 = zero_zero_int ) ) ).
tff(fact_363_add__0,axiom,
! [A_42: real] : hAPP_real_real(plus_plus_real(zero_zero_real),A_42) = A_42 ).
tff(fact_364_add__0,axiom,
! [A_42: nat] : hAPP_nat_nat(plus_plus_nat(zero_zero_nat),A_42) = A_42 ).
tff(fact_365_add__0,axiom,
! [A_42: int] : hAPP_int_int(plus_plus_int(zero_zero_int),A_42) = A_42 ).
tff(fact_366_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_41: real] : hAPP_real_real(plus_plus_real(zero_zero_real),A_41) = A_41 ).
tff(fact_367_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_41: nat] : hAPP_nat_nat(plus_plus_nat(zero_zero_nat),A_41) = A_41 ).
tff(fact_368_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_41: int] : hAPP_int_int(plus_plus_int(zero_zero_int),A_41) = A_41 ).
tff(fact_369_add__0__left,axiom,
! [A_40: real] : hAPP_real_real(plus_plus_real(zero_zero_real),A_40) = A_40 ).
tff(fact_370_add__0__left,axiom,
! [A_40: nat] : hAPP_nat_nat(plus_plus_nat(zero_zero_nat),A_40) = A_40 ).
tff(fact_371_add__0__left,axiom,
! [A_40: int] : hAPP_int_int(plus_plus_int(zero_zero_int),A_40) = A_40 ).
tff(fact_372_add__less__imp__less__left,axiom,
! [C_20: real,A_39: real,B_33: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(C_20),A_39)),hAPP_real_real(plus_plus_real(C_20),B_33)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_39),B_33)) ) ).
tff(fact_373_add__less__imp__less__left,axiom,
! [C_20: nat,A_39: nat,B_33: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(C_20),A_39)),hAPP_nat_nat(plus_plus_nat(C_20),B_33)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_39),B_33)) ) ).
tff(fact_374_add__less__imp__less__left,axiom,
! [C_20: int,A_39: int,B_33: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(C_20),A_39)),hAPP_int_int(plus_plus_int(C_20),B_33)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_39),B_33)) ) ).
tff(fact_375_add__less__imp__less__right,axiom,
! [A_38: real,C_19: real,B_32: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_38),C_19)),hAPP_real_real(plus_plus_real(B_32),C_19)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_38),B_32)) ) ).
tff(fact_376_add__less__imp__less__right,axiom,
! [A_38: nat,C_19: nat,B_32: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_38),C_19)),hAPP_nat_nat(plus_plus_nat(B_32),C_19)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_38),B_32)) ) ).
tff(fact_377_add__less__imp__less__right,axiom,
! [A_38: int,C_19: int,B_32: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_38),C_19)),hAPP_int_int(plus_plus_int(B_32),C_19)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_38),B_32)) ) ).
tff(fact_378_add__strict__mono,axiom,
! [C_18: real,D_3: real,A_37: real,B_31: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_37),B_31))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_18),D_3))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_37),C_18)),hAPP_real_real(plus_plus_real(B_31),D_3))) ) ) ).
tff(fact_379_add__strict__mono,axiom,
! [C_18: nat,D_3: nat,A_37: nat,B_31: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_37),B_31))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_18),D_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_37),C_18)),hAPP_nat_nat(plus_plus_nat(B_31),D_3))) ) ) ).
tff(fact_380_add__strict__mono,axiom,
! [C_18: int,D_3: int,A_37: int,B_31: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_37),B_31))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_18),D_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_37),C_18)),hAPP_int_int(plus_plus_int(B_31),D_3))) ) ) ).
tff(fact_381_add__strict__left__mono,axiom,
! [C_17: real,A_36: real,B_30: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_36),B_30))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(C_17),A_36)),hAPP_real_real(plus_plus_real(C_17),B_30))) ) ).
tff(fact_382_add__strict__left__mono,axiom,
! [C_17: nat,A_36: nat,B_30: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_36),B_30))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(C_17),A_36)),hAPP_nat_nat(plus_plus_nat(C_17),B_30))) ) ).
tff(fact_383_add__strict__left__mono,axiom,
! [C_17: int,A_36: int,B_30: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_36),B_30))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(C_17),A_36)),hAPP_int_int(plus_plus_int(C_17),B_30))) ) ).
tff(fact_384_add__strict__right__mono,axiom,
! [C_16: real,A_35: real,B_29: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_35),B_29))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_35),C_16)),hAPP_real_real(plus_plus_real(B_29),C_16))) ) ).
tff(fact_385_add__strict__right__mono,axiom,
! [C_16: nat,A_35: nat,B_29: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_35),B_29))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_35),C_16)),hAPP_nat_nat(plus_plus_nat(B_29),C_16))) ) ).
tff(fact_386_add__strict__right__mono,axiom,
! [C_16: int,A_35: int,B_29: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_35),B_29))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_35),C_16)),hAPP_int_int(plus_plus_int(B_29),C_16))) ) ).
tff(fact_387_add__less__cancel__left,axiom,
! [C_15: real,A_24: real,B_21: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(C_15),A_24)),hAPP_real_real(plus_plus_real(C_15),B_21)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_24),B_21)) ) ).
tff(fact_388_add__less__cancel__left,axiom,
! [C_15: nat,A_24: nat,B_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(C_15),A_24)),hAPP_nat_nat(plus_plus_nat(C_15),B_21)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_24),B_21)) ) ).
tff(fact_389_add__less__cancel__left,axiom,
! [C_15: int,A_24: int,B_21: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(C_15),A_24)),hAPP_int_int(plus_plus_int(C_15),B_21)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_24),B_21)) ) ).
tff(fact_390_add__less__cancel__right,axiom,
! [A_24: real,C_15: real,B_21: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_24),C_15)),hAPP_real_real(plus_plus_real(B_21),C_15)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_24),B_21)) ) ).
tff(fact_391_add__less__cancel__right,axiom,
! [A_24: nat,C_15: nat,B_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_24),C_15)),hAPP_nat_nat(plus_plus_nat(B_21),C_15)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_24),B_21)) ) ).
tff(fact_392_add__less__cancel__right,axiom,
! [A_24: int,C_15: int,B_21: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_24),C_15)),hAPP_int_int(plus_plus_int(B_21),C_15)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_24),B_21)) ) ).
tff(fact_393_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_13: real] : hAPP_nat_real(power_power_real(X_13),one_one_nat) = X_13 ).
tff(fact_394_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_13: nat] : hAPP_nat_nat(power_power_nat(X_13),one_one_nat) = X_13 ).
tff(fact_395_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_13: int] : hAPP_nat_int(power_power_int(X_13),one_one_nat) = X_13 ).
tff(fact_396_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_397_add__neg__neg,axiom,
! [B_28: real,A_34: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_34),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_28),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_34),B_28)),zero_zero_real)) ) ) ).
tff(fact_398_add__neg__neg,axiom,
! [B_28: nat,A_34: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_34),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_28),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_34),B_28)),zero_zero_nat)) ) ) ).
tff(fact_399_add__neg__neg,axiom,
! [B_28: int,A_34: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_34),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_28),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_34),B_28)),zero_zero_int)) ) ) ).
tff(fact_400_add__pos__pos,axiom,
! [B_27: real,A_33: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_33))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_27))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(plus_plus_real(A_33),B_27))) ) ) ).
tff(fact_401_add__pos__pos,axiom,
! [B_27: nat,A_33: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_33))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_27))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(plus_plus_nat(A_33),B_27))) ) ) ).
tff(fact_402_add__pos__pos,axiom,
! [B_27: int,A_33: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_33))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_27))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(A_33),B_27))) ) ) ).
tff(fact_403_realpow__pos__nth__unique,axiom,
! [A: real,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A))
=> ? [X_12: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_12))
& ( hAPP_nat_real(power_power_real(X_12),N_1) = A )
& ! [Y_11: real] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_11))
& ( hAPP_nat_real(power_power_real(Y_11),N_1) = A ) )
=> ( Y_11 = X_12 ) ) ) ) ) ).
tff(fact_404_realpow__pos__nth,axiom,
! [A: real,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A))
=> ? [R: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R))
& ( hAPP_nat_real(power_power_real(R),N_1) = 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_6: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_6),pls))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_6)),one_one_nat) = one_one_nat ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_6),pls))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_6)),one_one_nat) = number_number_of_nat(succ(V_6)) ) ) ) ).
tff(fact_407_nat__1__add__number__of,axiom,
! [V_6: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_6),pls))
=> ( hAPP_nat_nat(plus_plus_nat(one_one_nat),number_number_of_nat(V_6)) = one_one_nat ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_6),pls))
=> ( hAPP_nat_nat(plus_plus_nat(one_one_nat),number_number_of_nat(V_6)) = number_number_of_nat(succ(V_6)) ) ) ) ).
tff(fact_408_add__le__cancel__right,axiom,
! [A_24: real,C_15: real,B_21: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_24),C_15)),hAPP_real_real(plus_plus_real(B_21),C_15)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_24),B_21)) ) ).
tff(fact_409_add__le__cancel__right,axiom,
! [A_24: nat,C_15: nat,B_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(A_24),C_15)),hAPP_nat_nat(plus_plus_nat(B_21),C_15)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_24),B_21)) ) ).
tff(fact_410_add__le__cancel__right,axiom,
! [A_24: int,C_15: int,B_21: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_24),C_15)),hAPP_int_int(plus_plus_int(B_21),C_15)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_24),B_21)) ) ).
tff(fact_411_add__le__cancel__left,axiom,
! [C_15: real,A_24: real,B_21: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(C_15),A_24)),hAPP_real_real(plus_plus_real(C_15),B_21)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_24),B_21)) ) ).
tff(fact_412_add__le__cancel__left,axiom,
! [C_15: nat,A_24: nat,B_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(C_15),A_24)),hAPP_nat_nat(plus_plus_nat(C_15),B_21)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_24),B_21)) ) ).
tff(fact_413_add__le__cancel__left,axiom,
! [C_15: int,A_24: int,B_21: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(C_15),A_24)),hAPP_int_int(plus_plus_int(C_15),B_21)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_24),B_21)) ) ).
tff(fact_414_add__right__mono,axiom,
! [C_14: real,A_32: real,B_26: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_32),B_26))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_32),C_14)),hAPP_real_real(plus_plus_real(B_26),C_14))) ) ).
tff(fact_415_add__right__mono,axiom,
! [C_14: nat,A_32: nat,B_26: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_32),B_26))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(A_32),C_14)),hAPP_nat_nat(plus_plus_nat(B_26),C_14))) ) ).
tff(fact_416_add__right__mono,axiom,
! [C_14: int,A_32: int,B_26: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_32),B_26))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_32),C_14)),hAPP_int_int(plus_plus_int(B_26),C_14))) ) ).
tff(fact_417_add__left__mono,axiom,
! [C_13: real,A_31: real,B_25: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_31),B_25))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(C_13),A_31)),hAPP_real_real(plus_plus_real(C_13),B_25))) ) ).
tff(fact_418_add__left__mono,axiom,
! [C_13: nat,A_31: nat,B_25: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_31),B_25))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(C_13),A_31)),hAPP_nat_nat(plus_plus_nat(C_13),B_25))) ) ).
tff(fact_419_add__left__mono,axiom,
! [C_13: int,A_31: int,B_25: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_31),B_25))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(C_13),A_31)),hAPP_int_int(plus_plus_int(C_13),B_25))) ) ).
tff(fact_420_add__mono,axiom,
! [C_12: real,D_2: real,A_30: real,B_24: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_30),B_24))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_12),D_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_30),C_12)),hAPP_real_real(plus_plus_real(B_24),D_2))) ) ) ).
tff(fact_421_add__mono,axiom,
! [C_12: nat,D_2: nat,A_30: nat,B_24: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_30),B_24))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_12),D_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(A_30),C_12)),hAPP_nat_nat(plus_plus_nat(B_24),D_2))) ) ) ).
tff(fact_422_add__mono,axiom,
! [C_12: int,D_2: int,A_30: int,B_24: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_30),B_24))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_12),D_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_30),C_12)),hAPP_int_int(plus_plus_int(B_24),D_2))) ) ) ).
tff(fact_423_add__le__imp__le__right,axiom,
! [A_29: real,C_11: real,B_23: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_29),C_11)),hAPP_real_real(plus_plus_real(B_23),C_11)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_29),B_23)) ) ).
tff(fact_424_add__le__imp__le__right,axiom,
! [A_29: nat,C_11: nat,B_23: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(A_29),C_11)),hAPP_nat_nat(plus_plus_nat(B_23),C_11)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_29),B_23)) ) ).
tff(fact_425_add__le__imp__le__right,axiom,
! [A_29: int,C_11: int,B_23: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_29),C_11)),hAPP_int_int(plus_plus_int(B_23),C_11)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_29),B_23)) ) ).
tff(fact_426_add__le__imp__le__left,axiom,
! [C_10: real,A_28: real,B_22: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(C_10),A_28)),hAPP_real_real(plus_plus_real(C_10),B_22)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_28),B_22)) ) ).
tff(fact_427_add__le__imp__le__left,axiom,
! [C_10: nat,A_28: nat,B_22: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(C_10),A_28)),hAPP_nat_nat(plus_plus_nat(C_10),B_22)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_28),B_22)) ) ).
tff(fact_428_add__le__imp__le__left,axiom,
! [C_10: int,A_28: int,B_22: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(C_10),A_28)),hAPP_int_int(plus_plus_int(C_10),B_22)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_28),B_22)) ) ).
tff(fact_429_power__increasing,axiom,
! [A_27: real,N_13: nat,N_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_13),N_12))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_27))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_27),N_13)),hAPP_nat_real(power_power_real(A_27),N_12))) ) ) ).
tff(fact_430_power__increasing,axiom,
! [A_27: nat,N_13: nat,N_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_13),N_12))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_27))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_27),N_13)),hAPP_nat_nat(power_power_nat(A_27),N_12))) ) ) ).
tff(fact_431_power__increasing,axiom,
! [A_27: int,N_13: nat,N_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_13),N_12))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_27))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_27),N_13)),hAPP_nat_int(power_power_int(A_27),N_12))) ) ) ).
tff(fact_432_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_433_rel__simps_I34_J,axiom,
! [K_2: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K_2)),bit1(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_2),L_1)) ) ).
tff(fact_434_rel__simps_I19_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),pls)) ).
tff(fact_435_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_436_rel__simps_I31_J,axiom,
! [K_2: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K_2)),bit0(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_2),L_1)) ) ).
tff(fact_437_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_438_zle__refl,axiom,
! [W: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),W)) ).
tff(fact_439_zle__linear,axiom,
! [Z_2: int,W: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),W))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_2)) ) ).
tff(fact_440_zle__trans,axiom,
! [K: int,I_2: int,J_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),J_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_2),K))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K)) ) ) ).
tff(fact_441_zle__antisym,axiom,
! [Z_2: int,W: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),W))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_2))
=> ( Z_2 = W ) ) ) ).
tff(fact_442_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_443_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_444_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_445_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_446_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_447_zless__le,axiom,
! [Z_3: int,Wa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_3),Wa))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_3),Wa))
& ( Z_3 != Wa ) ) ) ).
tff(fact_448_zadd__left__mono,axiom,
! [K: int,I_2: int,J_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),J_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(K),I_2)),hAPP_int_int(plus_plus_int(K),J_2))) ) ).
tff(fact_449_less__eq__number__of__int__code,axiom,
! [K_2: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(K_2)),number_number_of_int(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_2),L_1)) ) ).
tff(fact_450_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_451_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_452_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_453_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_454_power__decreasing,axiom,
! [A_26: real,N_11: nat,N_10: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_11),N_10))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_26))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_26),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_26),N_10)),hAPP_nat_real(power_power_real(A_26),N_11))) ) ) ) ).
tff(fact_455_power__decreasing,axiom,
! [A_26: nat,N_11: nat,N_10: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_11),N_10))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_26))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_26),one_one_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_26),N_10)),hAPP_nat_nat(power_power_nat(A_26),N_11))) ) ) ) ).
tff(fact_456_power__decreasing,axiom,
! [A_26: int,N_11: nat,N_10: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_11),N_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_26))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_26),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_26),N_10)),hAPP_nat_int(power_power_int(A_26),N_11))) ) ) ) ).
tff(fact_457_power__increasing__iff,axiom,
! [Xa: nat,Ya: nat,B_21: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B_21))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(B_21),Xa)),hAPP_nat_real(power_power_real(B_21),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) ) ) ).
tff(fact_458_power__increasing__iff,axiom,
! [Xa: nat,Ya: nat,B_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B_21))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(B_21),Xa)),hAPP_nat_nat(power_power_nat(B_21),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) ) ) ).
tff(fact_459_power__increasing__iff,axiom,
! [Xa: nat,Ya: nat,B_21: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(B_21),Xa)),hAPP_nat_int(power_power_int(B_21),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) ) ) ).
tff(fact_460_power__le__imp__le__exp,axiom,
! [M_3: nat,N_9: nat,A_25: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_25))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_25),M_3)),hAPP_nat_real(power_power_real(A_25),N_9)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),N_9)) ) ) ).
tff(fact_461_power__le__imp__le__exp,axiom,
! [M_3: nat,N_9: nat,A_25: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_25))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_25),M_3)),hAPP_nat_nat(power_power_nat(A_25),N_9)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),N_9)) ) ) ).
tff(fact_462_power__le__imp__le__exp,axiom,
! [M_3: nat,N_9: nat,A_25: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_25))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_25),M_3)),hAPP_nat_int(power_power_int(A_25),N_9)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),N_9)) ) ) ).
tff(fact_463_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_464_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_465_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_466_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_467_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A_24: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(plus_plus_real(A_24),A_24)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_24)) ) ).
tff(fact_468_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A_24: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(A_24),A_24)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_24)) ) ).
tff(fact_469_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A_24: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_24),A_24)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_24),zero_zero_real)) ) ).
tff(fact_470_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A_24: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_24),A_24)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_24),zero_zero_int)) ) ).
tff(fact_471_add__nonneg__nonneg,axiom,
! [B_20: real,A_23: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_23))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_20))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(plus_plus_real(A_23),B_20))) ) ) ).
tff(fact_472_add__nonneg__nonneg,axiom,
! [B_20: nat,A_23: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_23))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_20))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(plus_plus_nat(A_23),B_20))) ) ) ).
tff(fact_473_add__nonneg__nonneg,axiom,
! [B_20: int,A_23: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_23))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_20))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(A_23),B_20))) ) ) ).
tff(fact_474_add__nonneg__eq__0__iff,axiom,
! [Ya: real,Xa: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Ya))
=> ( ( hAPP_real_real(plus_plus_real(Xa),Ya) = zero_zero_real )
<=> ( ( Xa = zero_zero_real )
& ( Ya = zero_zero_real ) ) ) ) ) ).
tff(fact_475_add__nonneg__eq__0__iff,axiom,
! [Ya: nat,Xa: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Xa))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Ya))
=> ( ( hAPP_nat_nat(plus_plus_nat(Xa),Ya) = zero_zero_nat )
<=> ( ( Xa = zero_zero_nat )
& ( Ya = zero_zero_nat ) ) ) ) ) ).
tff(fact_476_add__nonneg__eq__0__iff,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))
=> ( ( hAPP_int_int(plus_plus_int(Xa),Ya) = zero_zero_int )
<=> ( ( Xa = zero_zero_int )
& ( Ya = zero_zero_int ) ) ) ) ) ).
tff(fact_477_add__increasing,axiom,
! [B_19: real,C_9: real,A_22: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_22))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_19),C_9))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_19),hAPP_real_real(plus_plus_real(A_22),C_9))) ) ) ).
tff(fact_478_add__increasing,axiom,
! [B_19: nat,C_9: nat,A_22: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_22))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_19),C_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_19),hAPP_nat_nat(plus_plus_nat(A_22),C_9))) ) ) ).
tff(fact_479_add__increasing,axiom,
! [B_19: int,C_9: int,A_22: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_22))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_19),C_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_19),hAPP_int_int(plus_plus_int(A_22),C_9))) ) ) ).
tff(fact_480_add__increasing2,axiom,
! [B_18: real,A_21: real,C_8: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_8))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_18),A_21))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_18),hAPP_real_real(plus_plus_real(A_21),C_8))) ) ) ).
tff(fact_481_add__increasing2,axiom,
! [B_18: nat,A_21: nat,C_8: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_8))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_18),A_21))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_18),hAPP_nat_nat(plus_plus_nat(A_21),C_8))) ) ) ).
tff(fact_482_add__increasing2,axiom,
! [B_18: int,A_21: int,C_8: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_8))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_18),A_21))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_18),hAPP_int_int(plus_plus_int(A_21),C_8))) ) ) ).
tff(fact_483_add__nonpos__nonpos,axiom,
! [B_17: real,A_20: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_20),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_17),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_20),B_17)),zero_zero_real)) ) ) ).
tff(fact_484_add__nonpos__nonpos,axiom,
! [B_17: nat,A_20: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_20),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_17),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(A_20),B_17)),zero_zero_nat)) ) ) ).
tff(fact_485_add__nonpos__nonpos,axiom,
! [B_17: int,A_20: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_20),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_17),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_20),B_17)),zero_zero_int)) ) ) ).
tff(fact_486_add__less__le__mono,axiom,
! [C_7: real,D_1: real,A_19: real,B_16: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_19),B_16))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_7),D_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_19),C_7)),hAPP_real_real(plus_plus_real(B_16),D_1))) ) ) ).
tff(fact_487_add__less__le__mono,axiom,
! [C_7: nat,D_1: nat,A_19: nat,B_16: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_19),B_16))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_7),D_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_19),C_7)),hAPP_nat_nat(plus_plus_nat(B_16),D_1))) ) ) ).
tff(fact_488_add__less__le__mono,axiom,
! [C_7: int,D_1: int,A_19: int,B_16: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_19),B_16))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_7),D_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_19),C_7)),hAPP_int_int(plus_plus_int(B_16),D_1))) ) ) ).
tff(fact_489_add__le__less__mono,axiom,
! [C_6: real,D: real,A_18: real,B_15: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_18),B_15))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_6),D))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_18),C_6)),hAPP_real_real(plus_plus_real(B_15),D))) ) ) ).
tff(fact_490_add__le__less__mono,axiom,
! [C_6: nat,D: nat,A_18: nat,B_15: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_18),B_15))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_6),D))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_18),C_6)),hAPP_nat_nat(plus_plus_nat(B_15),D))) ) ) ).
tff(fact_491_add__le__less__mono,axiom,
! [C_6: int,D: int,A_18: int,B_15: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_18),B_15))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_6),D))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_18),C_6)),hAPP_int_int(plus_plus_int(B_15),D))) ) ) ).
tff(fact_492_not__one__le__zero,axiom,
~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),zero_zero_real)) ).
tff(fact_493_not__one__le__zero,axiom,
~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),zero_zero_nat)) ).
tff(fact_494_not__one__le__zero,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),zero_zero_int)) ).
tff(fact_495_zero__le__one,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).
tff(fact_496_zero__le__one,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).
tff(fact_497_zero__le__one,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).
tff(fact_498_le__number__of__eq__not__less,axiom,
! [Va: int,Wa: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(Va)),number267125858f_real(Wa)))
<=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(Wa)),number267125858f_real(Va))) ) ).
tff(fact_499_le__number__of__eq__not__less,axiom,
! [Va: int,Wa: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,number_number_of_nat(Va)),number_number_of_nat(Wa)))
<=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,number_number_of_nat(Wa)),number_number_of_nat(Va))) ) ).
tff(fact_500_le__number__of__eq__not__less,axiom,
! [Va: int,Wa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(Va)),number_number_of_int(Wa)))
<=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(Wa)),number_number_of_int(Va))) ) ).
tff(fact_501_zero__le__power,axiom,
! [N_8: nat,A_17: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_17))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(power_power_real(A_17),N_8))) ) ).
tff(fact_502_zero__le__power,axiom,
! [N_8: nat,A_17: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_17))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(A_17),N_8))) ) ).
tff(fact_503_zero__le__power,axiom,
! [N_8: nat,A_17: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_17))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(A_17),N_8))) ) ).
tff(fact_504_power__mono,axiom,
! [N_7: nat,A_16: real,B_14: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_16),B_14))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_16))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_16),N_7)),hAPP_nat_real(power_power_real(B_14),N_7))) ) ) ).
tff(fact_505_power__mono,axiom,
! [N_7: nat,A_16: nat,B_14: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_16),B_14))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_16))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_16),N_7)),hAPP_nat_nat(power_power_nat(B_14),N_7))) ) ) ).
tff(fact_506_power__mono,axiom,
! [N_7: nat,A_16: int,B_14: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_16),B_14))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_16))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_16),N_7)),hAPP_nat_int(power_power_int(B_14),N_7))) ) ) ).
tff(fact_507_one__le__power,axiom,
! [N_6: nat,A_15: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_15))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(power_power_real(A_15),N_6))) ) ).
tff(fact_508_one__le__power,axiom,
! [N_6: nat,A_15: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_15))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(power_power_nat(A_15),N_6))) ) ).
tff(fact_509_one__le__power,axiom,
! [N_6: nat,A_15: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_15))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_nat_int(power_power_int(A_15),N_6))) ) ).
tff(fact_510_zero__le__imp__of__nat,axiom,
! [M_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,M_2))) ).
tff(fact_511_zero__le__imp__of__nat,axiom,
! [M_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,M_2))) ).
tff(fact_512_zero__le__imp__of__nat,axiom,
! [M_2: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,M_2))) ).
tff(fact_513_of__nat__0__le__iff,axiom,
! [N_5: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,N_5))) ).
tff(fact_514_of__nat__0__le__iff,axiom,
! [N_5: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,N_5))) ).
tff(fact_515_of__nat__0__le__iff,axiom,
! [N_5: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_5))) ).
tff(fact_516_rel__simps_I22_J,axiom,
! [K_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),bit1(K_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_2)) ) ).
tff(fact_517_rel__simps_I32_J,axiom,
! [K_2: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K_2)),bit1(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_2),L_1)) ) ).
tff(fact_518_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_519_rel__simps_I21_J,axiom,
! [K_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),bit0(K_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_2)) ) ).
tff(fact_520_rel__simps_I27_J,axiom,
! [K_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K_2)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_2),pls)) ) ).
tff(fact_521_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_522_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),hAPP_int_int(plus_plus_int(X),Y))) ) ) ).
tff(fact_523_zadd__zless__mono,axiom,
! [Z_4: int,Z_2: int,W_1: int,W: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),W))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_4),Z_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(W_1),Z_4)),hAPP_int_int(plus_plus_int(W),Z_2))) ) ) ).
tff(fact_524_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z_2: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Z_2))) ).
tff(fact_525_transfer__int__nat__quantifiers_I2_J,axiom,
! [P_1: fun_int_bool] :
( ? [X_12: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_12))
& hBOOL(hAPP_int_bool(P_1,X_12)) )
<=> ? [X_12: nat] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_12))) ) ).
tff(fact_526_transfer__int__nat__quantifiers_I1_J,axiom,
! [P_1: fun_int_bool] :
( ! [X_12: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_12))
=> hBOOL(hAPP_int_bool(P_1,X_12)) )
<=> ! [X_12: nat] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_12))) ) ).
tff(fact_527_zero__zle__int,axiom,
! [N_1: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_1))) ).
tff(fact_528_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [N_1: 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_1))) ) ).
tff(fact_529_zle__iff__zadd,axiom,
! [Wa: int,Z_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),Z_3))
<=> ? [N: nat] : Z_3 = hAPP_int_int(plus_plus_int(Wa),hAPP_nat_int(semiri1621563631at_int,N)) ) ).
tff(fact_530_semiring__1__add__number__of,axiom,
! [V_5: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_5))
=> ( hAPP_real_real(plus_plus_real(one_one_real),number267125858f_real(V_5)) = number267125858f_real(succ(V_5)) ) ) ).
tff(fact_531_semiring__1__add__number__of,axiom,
! [V_5: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_5))
=> ( hAPP_nat_nat(plus_plus_nat(one_one_nat),number_number_of_nat(V_5)) = number_number_of_nat(succ(V_5)) ) ) ).
tff(fact_532_semiring__1__add__number__of,axiom,
! [V_5: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_5))
=> ( hAPP_int_int(plus_plus_int(one_one_int),number_number_of_int(V_5)) = number_number_of_int(succ(V_5)) ) ) ).
tff(fact_533_semiring__number__of__add__1,axiom,
! [V_4: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_4))
=> ( hAPP_real_real(plus_plus_real(number267125858f_real(V_4)),one_one_real) = number267125858f_real(succ(V_4)) ) ) ).
tff(fact_534_semiring__number__of__add__1,axiom,
! [V_4: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_4))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_4)),one_one_nat) = number_number_of_nat(succ(V_4)) ) ) ).
tff(fact_535_semiring__number__of__add__1,axiom,
! [V_4: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_4))
=> ( hAPP_int_int(plus_plus_int(number_number_of_int(V_4)),one_one_int) = number_number_of_int(succ(V_4)) ) ) ).
tff(fact_536_add__pos__nonneg,axiom,
! [B_13: real,A_14: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_14))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_13))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(plus_plus_real(A_14),B_13))) ) ) ).
tff(fact_537_add__pos__nonneg,axiom,
! [B_13: nat,A_14: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_14))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_13))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(plus_plus_nat(A_14),B_13))) ) ) ).
tff(fact_538_add__pos__nonneg,axiom,
! [B_13: int,A_14: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_14))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(A_14),B_13))) ) ) ).
tff(fact_539_add__nonneg__pos,axiom,
! [B_12: real,A_13: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_13))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_12))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(plus_plus_real(A_13),B_12))) ) ) ).
tff(fact_540_add__nonneg__pos,axiom,
! [B_12: nat,A_13: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_13))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(plus_plus_nat(A_13),B_12))) ) ) ).
tff(fact_541_add__nonneg__pos,axiom,
! [B_12: int,A_13: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_13))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(A_13),B_12))) ) ) ).
tff(fact_542_add__strict__increasing,axiom,
! [B_11: real,C_5: real,A_12: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_12))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_11),C_5))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_11),hAPP_real_real(plus_plus_real(A_12),C_5))) ) ) ).
tff(fact_543_add__strict__increasing,axiom,
! [B_11: nat,C_5: nat,A_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_12))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_11),C_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_11),hAPP_nat_nat(plus_plus_nat(A_12),C_5))) ) ) ).
tff(fact_544_add__strict__increasing,axiom,
! [B_11: int,C_5: int,A_12: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_12))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_11),C_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_11),hAPP_int_int(plus_plus_int(A_12),C_5))) ) ) ).
tff(fact_545_add__strict__increasing2,axiom,
! [B_10: real,C_4: real,A_11: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_11))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_10),C_4))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_10),hAPP_real_real(plus_plus_real(A_11),C_4))) ) ) ).
tff(fact_546_add__strict__increasing2,axiom,
! [B_10: nat,C_4: nat,A_11: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_11))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_10),C_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_10),hAPP_nat_nat(plus_plus_nat(A_11),C_4))) ) ) ).
tff(fact_547_add__strict__increasing2,axiom,
! [B_10: int,C_4: int,A_11: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_11))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_10),C_4))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_10),hAPP_int_int(plus_plus_int(A_11),C_4))) ) ) ).
tff(fact_548_add__neg__nonpos,axiom,
! [B_9: real,A_10: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_10),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_9),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_10),B_9)),zero_zero_real)) ) ) ).
tff(fact_549_add__neg__nonpos,axiom,
! [B_9: nat,A_10: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_10),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_9),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_10),B_9)),zero_zero_nat)) ) ) ).
tff(fact_550_add__neg__nonpos,axiom,
! [B_9: int,A_10: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_10),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_9),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_10),B_9)),zero_zero_int)) ) ) ).
tff(fact_551_add__nonpos__neg,axiom,
! [B_8: real,A_9: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_9),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_8),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_9),B_8)),zero_zero_real)) ) ) ).
tff(fact_552_add__nonpos__neg,axiom,
! [B_8: nat,A_9: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_9),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_8),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_9),B_8)),zero_zero_nat)) ) ) ).
tff(fact_553_add__nonpos__neg,axiom,
! [B_8: int,A_9: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_9),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_8),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_9),B_8)),zero_zero_int)) ) ) ).
tff(fact_554_power__less__imp__less__base,axiom,
! [A_8: real,N_4: nat,B_7: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_8),N_4)),hAPP_nat_real(power_power_real(B_7),N_4)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_7))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_8),B_7)) ) ) ).
tff(fact_555_power__less__imp__less__base,axiom,
! [A_8: nat,N_4: nat,B_7: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_8),N_4)),hAPP_nat_nat(power_power_nat(B_7),N_4)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_8),B_7)) ) ) ).
tff(fact_556_power__less__imp__less__base,axiom,
! [A_8: int,N_4: nat,B_7: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_8),N_4)),hAPP_nat_int(power_power_int(B_7),N_4)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_8),B_7)) ) ) ).
tff(fact_557_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_558_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_559_rel__simps_I5_J,axiom,
! [K_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),bit1(K_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_2)) ) ).
tff(fact_560_rel__simps_I29_J,axiom,
! [K_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K_2)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),pls)) ) ).
tff(fact_561_rel__simps_I15_J,axiom,
! [K_2: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K_2)),bit1(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_2),L_1)) ) ).
tff(fact_562_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_563_rel__simps_I33_J,axiom,
! [K_2: int,L_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K_2)),bit0(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),L_1)) ) ).
tff(fact_564_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_565_int__one__le__iff__zero__less,axiom,
! [Z_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Z_3))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_3)) ) ).
tff(fact_566_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_567_succ__Pls,axiom,
succ(pls) = bit1(pls) ).
tff(fact_568_succ__Bit0,axiom,
! [K: int] : succ(bit0(K)) = bit1(K) ).
tff(fact_569_succ__Bit1,axiom,
! [K: int] : succ(bit1(K)) = bit0(succ(K)) ).
tff(fact_570_zle__add1__eq__le,axiom,
! [Wa: int,Z_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),hAPP_int_int(plus_plus_int(Z_3),one_one_int)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),Z_3)) ) ).
tff(fact_571_add1__zle__eq,axiom,
! [Wa: int,Z_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(Wa),one_one_int)),Z_3))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_3)) ) ).
tff(fact_572_zless__imp__add1__zle,axiom,
! [W: int,Z_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(W),one_one_int)),Z_2)) ) ).
tff(fact_573_succ__def,axiom,
! [K: int] : succ(K) = hAPP_int_int(plus_plus_int(K),one_one_int) ).
tff(fact_574_power__eq__imp__eq__base,axiom,
! [A_7: real,N_3: nat,B_6: real] :
( ( hAPP_nat_real(power_power_real(A_7),N_3) = hAPP_nat_real(power_power_real(B_6),N_3) )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_7))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_3))
=> ( A_7 = B_6 ) ) ) ) ) ).
tff(fact_575_power__eq__imp__eq__base,axiom,
! [A_7: nat,N_3: nat,B_6: nat] :
( ( hAPP_nat_nat(power_power_nat(A_7),N_3) = hAPP_nat_nat(power_power_nat(B_6),N_3) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_7))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_3))
=> ( A_7 = B_6 ) ) ) ) ) ).
tff(fact_576_power__eq__imp__eq__base,axiom,
! [A_7: int,N_3: nat,B_6: int] :
( ( hAPP_nat_int(power_power_int(A_7),N_3) = hAPP_nat_int(power_power_int(B_6),N_3) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_3))
=> ( A_7 = B_6 ) ) ) ) ) ).
tff(fact_577_semiring__add__number__of,axiom,
! [V_3: int,V_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_3))
=> ( hAPP_real_real(plus_plus_real(number267125858f_real(V_2)),number267125858f_real(V_3)) = number267125858f_real(hAPP_int_int(plus_plus_int(V_2),V_3)) ) ) ) ).
tff(fact_578_semiring__add__number__of,axiom,
! [V_3: int,V_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_3))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_2)),number_number_of_nat(V_3)) = number_number_of_nat(hAPP_int_int(plus_plus_int(V_2),V_3)) ) ) ) ).
tff(fact_579_semiring__add__number__of,axiom,
! [V_3: int,V_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_3))
=> ( hAPP_int_int(plus_plus_int(number_number_of_int(V_2)),number_number_of_int(V_3)) = number_number_of_int(hAPP_int_int(plus_plus_int(V_2),V_3)) ) ) ) ).
tff(fact_580_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_581_le__imp__0__less,axiom,
! [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_int,zero_zero_int),hAPP_int_int(plus_plus_int(one_one_int),Z_2))) ) ).
tff(fact_582_number__of__succ,axiom,
! [K_3: int] : number267125858f_real(succ(K_3)) = hAPP_real_real(plus_plus_real(one_one_real),number267125858f_real(K_3)) ).
tff(fact_583_number__of__succ,axiom,
! [K_3: int] : number_number_of_int(succ(K_3)) = hAPP_int_int(plus_plus_int(one_one_int),number_number_of_int(K_3)) ).
tff(fact_584_add__Bit1__Bit1,axiom,
! [K: int,L: int] : hAPP_int_int(plus_plus_int(bit1(K)),bit1(L)) = bit0(hAPP_int_int(plus_plus_int(K),succ(L))) ).
tff(fact_585_power__strict__mono,axiom,
! [N_2: nat,A_6: real,B_5: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_6),B_5))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_6),N_2)),hAPP_nat_real(power_power_real(B_5),N_2))) ) ) ) ).
tff(fact_586_power__strict__mono,axiom,
! [N_2: nat,A_6: nat,B_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_6),B_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_6),N_2)),hAPP_nat_nat(power_power_nat(B_5),N_2))) ) ) ) ).
tff(fact_587_power__strict__mono,axiom,
! [N_2: nat,A_6: int,B_5: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_6),B_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_6),N_2)),hAPP_nat_int(power_power_int(B_5),N_2))) ) ) ) ).
tff(fact_588_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_589_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_590_zero__le__power2,axiom,
! [A_5: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(power_power_real(A_5),number_number_of_nat(bit0(bit1(pls)))))) ).
tff(fact_591_zero__le__power2,axiom,
! [A_5: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(A_5),number_number_of_nat(bit0(bit1(pls)))))) ).
tff(fact_592_power2__le__imp__le,axiom,
! [X_11: real,Y_10: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(X_11),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_10),number_number_of_nat(bit0(bit1(pls))))))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_10))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_11),Y_10)) ) ) ).
tff(fact_593_power2__le__imp__le,axiom,
! [X_11: nat,Y_10: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(X_11),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_nat(power_power_nat(Y_10),number_number_of_nat(bit0(bit1(pls))))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_10))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_11),Y_10)) ) ) ).
tff(fact_594_power2__le__imp__le,axiom,
! [X_11: int,Y_10: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(X_11),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_10),number_number_of_nat(bit0(bit1(pls))))))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_10))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_11),Y_10)) ) ) ).
tff(fact_595_power2__eq__imp__eq,axiom,
! [X_10: real,Y_9: real] :
( ( hAPP_nat_real(power_power_real(X_10),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_real(power_power_real(Y_9),number_number_of_nat(bit0(bit1(pls)))) )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_10))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_9))
=> ( X_10 = Y_9 ) ) ) ) ).
tff(fact_596_power2__eq__imp__eq,axiom,
! [X_10: nat,Y_9: nat] :
( ( hAPP_nat_nat(power_power_nat(X_10),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_nat(power_power_nat(Y_9),number_number_of_nat(bit0(bit1(pls)))) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),X_10))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_9))
=> ( X_10 = Y_9 ) ) ) ) ).
tff(fact_597_power2__eq__imp__eq,axiom,
! [X_10: int,Y_9: int] :
( ( hAPP_nat_int(power_power_int(X_10),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_int(power_power_int(Y_9),number_number_of_nat(bit0(bit1(pls)))) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_9))
=> ( X_10 = Y_9 ) ) ) ) ).
tff(fact_598_of__nat__number__of__lemma,axiom,
! [V_1: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(V_1)))
=> ( hAPP_nat_real(semiri132038758t_real,number_number_of_nat(V_1)) = number267125858f_real(V_1) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(V_1)))
=> ( hAPP_nat_real(semiri132038758t_real,number_number_of_nat(V_1)) = zero_zero_real ) ) ) ).
tff(fact_599_of__nat__number__of__lemma,axiom,
! [V_1: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(V_1)))
=> ( hAPP_nat_int(semiri1621563631at_int,number_number_of_nat(V_1)) = number_number_of_int(V_1) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(V_1)))
=> ( hAPP_nat_int(semiri1621563631at_int,number_number_of_nat(V_1)) = zero_zero_int ) ) ) ).
tff(fact_600_power2__less__imp__less,axiom,
! [X_9: real,Y_8: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(X_9),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_8),number_number_of_nat(bit0(bit1(pls))))))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_8))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_9),Y_8)) ) ) ).
tff(fact_601_power2__less__imp__less,axiom,
! [X_9: nat,Y_8: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(X_9),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_nat(power_power_nat(Y_8),number_number_of_nat(bit0(bit1(pls))))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_9),Y_8)) ) ) ).
tff(fact_602_power2__less__imp__less,axiom,
! [X_9: int,Y_8: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(X_9),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_8),number_number_of_nat(bit0(bit1(pls))))))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_9),Y_8)) ) ) ).
tff(fact_603_sum__power2__ge__zero,axiom,
! [X_8: real,Y_7: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_8),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_7),number_number_of_nat(bit0(bit1(pls))))))) ).
tff(fact_604_sum__power2__ge__zero,axiom,
! [X_8: int,Y_7: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X_8),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_7),number_number_of_nat(bit0(bit1(pls))))))) ).
tff(fact_605_sum__power2__le__zero__iff,axiom,
! [Xa: real,Ya: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_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_606_sum__power2__le__zero__iff,axiom,
! [Xa: int,Ya: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_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_607_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_608_number__of1,axiom,
! [N_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(N_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit0(N_1))))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit1(N_1)))) ) ) ).
tff(fact_609_number__of2,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(pls))) ).
tff(fact_610_zero__le__imp__eq__int,axiom,
! [K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
=> ? [N: nat] : K = hAPP_nat_int(semiri1621563631at_int,N) ) ).
tff(fact_611_nonneg__int__cases,axiom,
! [K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
=> ~ ! [N: nat] : K != hAPP_nat_int(semiri1621563631at_int,N) ) ).
tff(fact_612_le0,axiom,
! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1)) ).
tff(fact_613_less__eq__nat_Osimps_I1_J,axiom,
! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1)) ).
tff(fact_614_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_615_less__or__eq__imp__le,axiom,
! [M_1: nat,N_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
| ( M_1 = N_1 ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ).
tff(fact_616_le__neq__implies__less,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
=> ( ( M_1 != N_1 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ) ).
tff(fact_617_less__imp__le__nat,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ).
tff(fact_618_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_619_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_620_le__refl,axiom,
! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),N_1)) ).
tff(fact_621_nat__le__linear,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1)) ) ).
tff(fact_622_eq__imp__le,axiom,
! [M_1: nat,N_1: nat] :
( ( M_1 = N_1 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ).
tff(fact_623_le__trans,axiom,
! [K: nat,I_2: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),K)) ) ) ).
tff(fact_624_le__antisym,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
=> ( M_1 = N_1 ) ) ) ).
tff(fact_625_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_626_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_627_le__add2,axiom,
! [N_1: nat,M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(plus_plus_nat(M_1),N_1))) ).
tff(fact_628_le__add1,axiom,
! [N_1: nat,M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(plus_plus_nat(N_1),M_1))) ).
tff(fact_629_le__iff__add,axiom,
! [Ma: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na))
<=> ? [K_1: nat] : Na = hAPP_nat_nat(plus_plus_nat(Ma),K_1) ) ).
tff(fact_630_nat__add__left__cancel__le,axiom,
! [K_2: nat,Ma: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(K_2),Ma)),hAPP_nat_nat(plus_plus_nat(K_2),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
tff(fact_631_trans__le__add1,axiom,
! [M_1: nat,I_2: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),hAPP_nat_nat(plus_plus_nat(J_2),M_1))) ) ).
tff(fact_632_trans__le__add2,axiom,
! [M_1: nat,I_2: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),hAPP_nat_nat(plus_plus_nat(M_1),J_2))) ) ).
tff(fact_633_add__le__mono1,axiom,
! [K: nat,I_2: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(I_2),K)),hAPP_nat_nat(plus_plus_nat(J_2),K))) ) ).
tff(fact_634_add__le__mono,axiom,
! [K: nat,L: nat,I_2: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(I_2),K)),hAPP_nat_nat(plus_plus_nat(J_2),L))) ) ) ).
tff(fact_635_add__leD2,axiom,
! [M_1: nat,K: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M_1),K)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_1)) ) ).
tff(fact_636_add__leD1,axiom,
! [M_1: nat,K: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M_1),K)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ).
tff(fact_637_add__leE,axiom,
! [M_1: nat,K: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M_1),K)),N_1))
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_1)) ) ) ).
tff(fact_638_real__add__left__mono,axiom,
! [Z_2: 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,hAPP_real_real(plus_plus_real(Z_2),X)),hAPP_real_real(plus_plus_real(Z_2),Y))) ) ).
tff(fact_639_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_640_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_641_le__nat__number__of,axiom,
! [Va: int,V: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,number_number_of_nat(Va)),number_number_of_nat(V)))
<=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),V))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls)) ) ) ).
tff(fact_642_two__realpow__ge__one,axiom,
! [N_1: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(power_power_real(number267125858f_real(bit0(bit1(pls)))),N_1))) ).
tff(fact_643_conj__le__cong,axiom,
! [P_2: 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_2) ) )
=> ( ( 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_2) ) ) ) ).
tff(fact_644_imp__le__cong,axiom,
! [P_2: 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_2) ) )
=> ( ( 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_2) ) ) ) ).
tff(fact_645_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_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),Na))
& ! [I_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),K_1))
=> ~ hBOOL(hAPP_nat_bool(P_1,I_1)) )
& hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(plus_plus_nat(K_1),one_one_nat))) ) ) ) ).
tff(fact_646_nonneg__eq__int,axiom,
! [Z_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2))
=> ~ ! [M: nat] : Z_2 != hAPP_nat_int(semiri1621563631at_int,M) ) ).
tff(fact_647_mono__nat__linear__lb,axiom,
! [Ma: nat,K_2: nat,F: fun_nat_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,hAPP_nat_nat(F,M)),hAPP_nat_nat(F,N))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(F,Ma)),K_2)),hAPP_nat_nat(F,hAPP_nat_nat(plus_plus_nat(Ma),K_2)))) ) ).
tff(fact_648_real__le__refl,axiom,
! [W: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),W)) ).
tff(fact_649_real__le__linear,axiom,
! [Z_2: real,W: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_2),W))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z_2)) ) ).
tff(fact_650_real__le__trans,axiom,
! [K: real,I_2: real,J_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I_2),J_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,J_2),K))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I_2),K)) ) ) ).
tff(fact_651_real__le__antisym,axiom,
! [Z_2: real,W: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_2),W))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z_2))
=> ( Z_2 = W ) ) ) ).
tff(fact_652_int__ge__induct,axiom,
! [P_1: fun_int_bool,K_2: int,I: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_2),I))
=> ( hBOOL(hAPP_int_bool(P_1,K_2))
=> ( ! [I_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_2),I_1))
=> ( hBOOL(hAPP_int_bool(P_1,I_1))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(I_1),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_1,I)) ) ) ) ).
tff(fact_653_ex__least__nat__le,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_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),Na))
& ! [I_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),K_1))
=> ~ hBOOL(hAPP_nat_bool(P_1,I_1)) )
& hBOOL(hAPP_nat_bool(P_1,K_1)) ) ) ) ).
tff(fact_654_less__mono__imp__le__mono,axiom,
! [I: nat,J: nat,F: fun_nat_nat] :
( ! [I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,I_1)),hAPP_nat_nat(F,J_1))) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,I)),hAPP_nat_nat(F,J))) ) ) ).
tff(fact_655_order__refl,axiom,
! [X_7: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_7),X_7)) ).
tff(fact_656_order__refl,axiom,
! [X_7: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_7),X_7)) ).
tff(fact_657_order__refl,axiom,
! [X_7: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_7),X_7)) ).
tff(fact_658_linorder__le__cases,axiom,
! [X_6: real,Y_6: real] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_6),Y_6))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_6),X_6)) ) ).
tff(fact_659_linorder__le__cases,axiom,
! [X_6: nat,Y_6: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_6),Y_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_6),X_6)) ) ).
tff(fact_660_linorder__le__cases,axiom,
! [X_6: int,Y_6: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_6),Y_6))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_6),X_6)) ) ).
tff(fact_661_xt1_I6_J,axiom,
! [Z_1: real,Y_5: real,X_5: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_5),X_5))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_1),Y_5))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_1),X_5)) ) ) ).
tff(fact_662_xt1_I6_J,axiom,
! [Z_1: nat,Y_5: nat,X_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_5),X_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_1),Y_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_1),X_5)) ) ) ).
tff(fact_663_xt1_I6_J,axiom,
! [Z_1: int,Y_5: int,X_5: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_5),X_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),Y_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),X_5)) ) ) ).
tff(fact_664_xt1_I5_J,axiom,
! [Y_4: real,X_4: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_4),X_4))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_4),Y_4))
=> ( X_4 = Y_4 ) ) ) ).
tff(fact_665_xt1_I5_J,axiom,
! [Y_4: nat,X_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_4),X_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_4),Y_4))
=> ( X_4 = Y_4 ) ) ) ).
tff(fact_666_xt1_I5_J,axiom,
! [Y_4: int,X_4: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_4),X_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_4),Y_4))
=> ( X_4 = Y_4 ) ) ) ).
tff(fact_667_order__trans,axiom,
! [Z: real,X_3: real,Y_3: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_3),Y_3))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_3),Z))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_3),Z)) ) ) ).
tff(fact_668_order__trans,axiom,
! [Z: nat,X_3: nat,Y_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_3),Y_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_3),Z))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_3),Z)) ) ) ).
tff(fact_669_order__trans,axiom,
! [Z: int,X_3: int,Y_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_3),Y_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_3),Z))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_3),Z)) ) ) ).
tff(fact_670_order__antisym,axiom,
! [X_2: real,Y_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),X_2))
=> ( X_2 = Y_2 ) ) ) ).
tff(fact_671_order__antisym,axiom,
! [X_2: nat,Y_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),X_2))
=> ( X_2 = Y_2 ) ) ) ).
tff(fact_672_order__antisym,axiom,
! [X_2: int,Y_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_2),X_2))
=> ( X_2 = Y_2 ) ) ) ).
tff(fact_673_xt1_I4_J,axiom,
! [C_3: real,B_4: real,A_4: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),A_4))
=> ( ( B_4 = C_3 )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_3),A_4)) ) ) ).
tff(fact_674_xt1_I4_J,axiom,
! [C_3: nat,B_4: nat,A_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_4),A_4))
=> ( ( B_4 = C_3 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_3),A_4)) ) ) ).
tff(fact_675_xt1_I4_J,axiom,
! [C_3: int,B_4: int,A_4: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_4),A_4))
=> ( ( B_4 = C_3 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_3),A_4)) ) ) ).
tff(fact_676_ord__le__eq__trans,axiom,
! [C_2: real,A_3: real,B_3: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_3),B_3))
=> ( ( B_3 = C_2 )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_3),C_2)) ) ) ).
tff(fact_677_ord__le__eq__trans,axiom,
! [C_2: nat,A_3: nat,B_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_3),B_3))
=> ( ( B_3 = C_2 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_3),C_2)) ) ) ).
tff(fact_678_ord__le__eq__trans,axiom,
! [C_2: int,A_3: int,B_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),B_3))
=> ( ( B_3 = C_2 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),C_2)) ) ) ).
tff(fact_679_xt1_I3_J,axiom,
! [C_1: real,A_2: real,B_2: real] :
( ( A_2 = B_2 )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),B_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),A_2)) ) ) ).
tff(fact_680_xt1_I3_J,axiom,
! [C_1: nat,A_2: nat,B_2: nat] :
( ( A_2 = B_2 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_1),B_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_1),A_2)) ) ) ).
tff(fact_681_xt1_I3_J,axiom,
! [C_1: int,A_2: int,B_2: int] :
( ( A_2 = B_2 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_1),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_1),A_2)) ) ) ).
tff(fact_682_ord__eq__le__trans,axiom,
! [C: real,A_1: real,B_1: real] :
( ( A_1 = B_1 )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_1),C))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),C)) ) ) ).
tff(fact_683_ord__eq__le__trans,axiom,
! [C: nat,A_1: nat,B_1: nat] :
( ( A_1 = B_1 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_1),C))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),C)) ) ) ).
tff(fact_684_ord__eq__le__trans,axiom,
! [C: int,A_1: int,B_1: int] :
( ( A_1 = B_1 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),C))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),C)) ) ) ).
tff(fact_685_order__antisym__conv,axiom,
! [Ya: real,Xa: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))
<=> ( Xa = Ya ) ) ) ).
tff(fact_686_order__antisym__conv,axiom,
! [Ya: nat,Xa: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ya),Xa))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya))
<=> ( Xa = Ya ) ) ) ).
tff(fact_687_order__antisym__conv,axiom,
! [Ya: int,Xa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ya),Xa))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya))
<=> ( Xa = Ya ) ) ) ).
tff(fact_688_order__eq__refl,axiom,
! [X_1: int,Y_1: int] :
( ( X_1 = Y_1 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1)) ) ).
tff(fact_689_real__lbound__gt__zero,axiom,
! [D2: real,D1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D2))
=> ? [E: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,E),D1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,E),D2)) ) ) ) ).
tff(fact_690_real__sqrt__sum__squares__ge__zero,axiom,
! [X: real,Y: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),sqrt(hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y),number_number_of_nat(bit0(bit1(pls)))))))) ).
tff(fact_691_zadd__power2,axiom,
! [A: int,B: int] : hAPP_nat_int(power_power_int(hAPP_int_int(plus_plus_int(A),B)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls))))),times_times_int(times_times_int(number_number_of_int(bit0(bit1(pls))),A),B))),hAPP_nat_int(power_power_int(B),number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_692_mult__Bit1,axiom,
! [K: int,L: int] : times_times_int(bit1(K),L) = hAPP_int_int(plus_plus_int(bit0(times_times_int(K,L))),L) ).
tff(fact_693_real__sqrt__gt__zero,axiom,
! [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),sqrt(X))) ) ).
tff(fact_694_real__sqrt__not__eq__zero,axiom,
! [X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> ( sqrt(X) != zero_zero_real ) ) ).
tff(fact_695_real__sqrt__lt__0__iff,axiom,
! [Xa: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,sqrt(Xa)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),zero_zero_real)) ) ).
tff(fact_696_real__sqrt__gt__0__iff,axiom,
! [Ya: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),sqrt(Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Ya)) ) ).
tff(fact_697_real__sqrt__lt__1__iff,axiom,
! [Xa: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,sqrt(Xa)),one_one_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),one_one_real)) ) ).
tff(fact_698_real__sqrt__gt__1__iff,axiom,
! [Ya: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),sqrt(Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Ya)) ) ).
%----Helper facts (3)
tff(help_If_1_1_If_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] : hAPP_nat_nat(if_nat(fTrue,X),Y) = X ).
tff(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] : hAPP_nat_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(hAPP_int_int(plus_plus_int(one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),number_number_of_nat(bit0(bit1(pls)))) != zero_zero_int ).
%------------------------------------------------------------------------------