TPTP Problem File: NUM923_30.p
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%------------------------------------------------------------------------------
% File : NUM923_30 : TPTP v8.2.0. Released v8.2.0.
% Domain : Number Theory
% Problem : Sum of two squares line 23, 1000 axioms selected
% Version : NUM923_3 with the conjecture removed
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [TPTP]
% Names :
% Status : Satisfiable
% Rating : 1.00 v8.2.0
% Syntax : Number of formulae : 1298 ( 391 unt; 96 typ; 0 def)
% Number of atoms : 2657 ( 856 equ)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 1701 ( 246 ~; 69 |; 116 &)
% ( 307 <=>; 963 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 81 ( 49 >; 32 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 75 ( 75 usr; 27 con; 0-8 aty)
% Number of variables : 2747 (2702 !; 45 ?;2747 :)
% SPC : TF0_SAT_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:33:46
%------------------------------------------------------------------------------
%----Should-be-implicit typings (20)
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__RealDef__Oreal_J,type,
fun_int_real: $tType ).
tff(ty_ty_tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_J,type,
fun_int_fun_int_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
fun_nat_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__Int__Oint_J,type,
fun_nat_int: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
fun_nat_nat: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J,type,
fun_nat_real: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_J,type,
fun_nat_fun_nat_bool: $tType ).
tff(ty_ty_tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J,type,
fun_real_bool: $tType ).
tff(ty_ty_tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J,type,
fun_real_real: $tType ).
tff(ty_ty_tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J,type,
fun_re413263731l_bool: $tType ).
tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__HOL__Obool_J,type,
fun_Pr974702441t_bool: $tType ).
tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
fun_Pr1503267378t_bool: $tType ).
tff(ty_ty_tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
product_prod_int_int: $tType ).
tff(ty_ty_tc__prod_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_J,type,
produc393999548nt_int: $tType ).
%----Explicit typings (76)
tff(sy_c_Archimedean__Field_Oceiling_000tc__RealDef__Oreal,type,
archim856651990g_real: real > int ).
tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_000tc__RealDef__Oreal,type,
archim1246769320r_real: real > int ).
tff(sy_c_Divides_Odivmod__int__rel,type,
divmod_int_rel: ( int * int ) > fun_Pr974702441t_bool ).
tff(sy_c_Divides_Opdivmod,type,
pdivmod: ( int * int ) > product_prod_int_int ).
tff(sy_c_Divides_OposDivAlg,type,
posDivAlg: ( int * int ) > product_prod_int_int ).
tff(sy_c_Groups_Oabs__class_Oabs_000tc__Int__Oint,type,
abs_abs_int: int > int ).
tff(sy_c_Groups_Oabs__class_Oabs_000tc__RealDef__Oreal,type,
abs_abs_real: real > real ).
tff(sy_c_Groups_Ominus__class_Ominus_000tc__Int__Oint,type,
minus_minus_int: int > fun_int_int ).
tff(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
minus_minus_nat: nat > fun_nat_nat ).
tff(sy_c_Groups_Ominus__class_Ominus_000tc__RealDef__Oreal,type,
minus_minus_real: real > fun_real_real ).
tff(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,
one_one_int: int ).
tff(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
one_one_nat: nat ).
tff(sy_c_Groups_Oone__class_Oone_000tc__RealDef__Oreal,type,
one_one_real: real ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
plus_plus_int: int > fun_int_int ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
plus_plus_nat: nat > fun_nat_nat ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__RealDef__Oreal,type,
plus_plus_real: real > fun_real_real ).
tff(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
times_times_int: int > fun_int_int ).
tff(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
times_times_nat: nat > fun_nat_nat ).
tff(sy_c_Groups_Otimes__class_Otimes_000tc__RealDef__Oreal,type,
times_times_real: real > fun_real_real ).
tff(sy_c_Groups_Ozero__class_Ozero_000tc__Int__Oint,type,
zero_zero_int: int ).
tff(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_nat: nat ).
tff(sy_c_Groups_Ozero__class_Ozero_000tc__RealDef__Oreal,type,
zero_zero_real: real ).
tff(sy_c_If_000tc__Nat__Onat,type,
if_nat: ( bool * nat ) > fun_nat_nat ).
tff(sy_c_IntFact_Ozfact,type,
zfact: int > int ).
tff(sy_c_IntPrimes_Oxzgcd,type,
xzgcd: ( int * int ) > produc393999548nt_int ).
tff(sy_c_IntPrimes_Oxzgcda,type,
xzgcda: ( int * int * int * int * int * int * int * int ) > produc393999548nt_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_Opred,type,
pred: int > int ).
tff(sy_c_Int_Oring__1__class_OInts_000tc__RealDef__Oreal,type,
ring_1_Ints_real: fun_real_bool ).
tff(sy_c_Int_Osucc,type,
succ: int > int ).
tff(sy_c_Nat_OSuc,type,
suc: fun_nat_nat ).
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__RealDef__Oreal,type,
semiri132038758t_real: fun_nat_real ).
tff(sy_c_Nat__Numeral_Oneg,type,
nat_neg: fun_int_bool ).
tff(sy_c_Nat__Transfer_Otsub,type,
nat_tsub: ( int * int ) > int ).
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_Product__Type_OPair_000tc__Int__Oint_000tc__Int__Oint,type,
product_Pair_int_int: ( int * int ) > product_prod_int_int ).
tff(sy_c_Product__Type_OPair_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__O,type,
produc282740534nt_int: ( int * product_prod_int_int ) > produc393999548nt_int ).
tff(sy_c_Product__Type_Ocurry_000tc__Int__Oint_000tc__Int__Oint_000tc__HOL__Obool,type,
produc262399358t_bool: ( fun_Pr974702441t_bool * int ) > fun_int_bool ).
tff(sy_c_Product__Type_Ocurry_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__,type,
produc97574839t_bool: ( fun_Pr1503267378t_bool * int ) > fun_Pr974702441t_bool ).
tff(sy_c_RComplete_Onatceiling,type,
natceiling: real > nat ).
tff(sy_c_RComplete_Onatfloor,type,
natfloor: real > nat ).
tff(sy_c_RealDef_Oreal_000tc__Int__Oint,type,
real_int: fun_int_real ).
tff(sy_c_RealDef_Oreal_000tc__Nat__Onat,type,
real_nat: fun_nat_real ).
tff(sy_c_TwoSquares__Mirabelle__klrirdmhmq_Ois__sum2sq,type,
twoSqu820444569sum2sq: fun_int_bool ).
tff(sy_c_TwoSquares__Mirabelle__klrirdmhmq_Osum2sq,type,
twoSqu949963151sum2sq: product_prod_int_int > int ).
tff(sy_c_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__RealDef__Oreal,type,
hAPP_int_real: ( fun_int_real * int ) > real ).
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_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__HOL__Obool,type,
hAPP_P603027463t_bool: ( fun_Pr974702441t_bool * product_prod_int_int ) > bool ).
tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_J,type,
hAPP_P178748804t_bool: ( fun_Pr1503267378t_bool * produc393999548nt_int ) > bool ).
tff(sy_c_hBOOL,type,
hBOOL: bool > $o ).
tff(sy_c_member_000tc__RealDef__Oreal,type,
member_real: ( real * fun_real_bool ) > bool ).
tff(sy_v_a,type,
a: int ).
tff(sy_v_b,type,
b: int ).
tff(sy_v_p,type,
p: int ).
tff(sy_v_q,type,
q: int ).
%----Relevant facts (1199)
tff(fact_0_xzgcda__linear__aux1,axiom,
! [A: int,R: int,B: int,M: int,C: int,D_1: int,N_1: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(A),hAPP_int_int(times_times_int(R),B))),M)),hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(C),hAPP_int_int(times_times_int(R),D_1))),N_1)) = hAPP_int_int(minus_minus_int(hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A),M)),hAPP_int_int(times_times_int(C),N_1))),hAPP_int_int(times_times_int(R),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),M)),hAPP_int_int(times_times_int(D_1),N_1)))) ).
tff(fact_1_mult__diff__mult,axiom,
! [X_46: int,Y_46: int,A_117: int,B_92: int] : hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(X_46),Y_46)),hAPP_int_int(times_times_int(A_117),B_92)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(X_46),hAPP_int_int(minus_minus_int(Y_46),B_92))),hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(X_46),A_117)),B_92)) ).
tff(fact_2_mult__diff__mult,axiom,
! [X_46: real,Y_46: real,A_117: real,B_92: real] : hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(X_46),Y_46)),hAPP_real_real(times_times_real(A_117),B_92)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_46),hAPP_real_real(minus_minus_real(Y_46),B_92))),hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(X_46),A_117)),B_92)) ).
tff(fact_3_mult_Oprod__diff__prod,axiom,
! [X_1: real,Y_1: real,A_116: real,B_91: real] : hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(X_1),Y_1)),hAPP_real_real(times_times_real(A_116),B_91)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(X_1),A_116)),hAPP_real_real(minus_minus_real(Y_1),B_91))),hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(X_1),A_116)),B_91))),hAPP_real_real(times_times_real(A_116),hAPP_real_real(minus_minus_real(Y_1),B_91))) ).
tff(fact_4_eq__add__iff2,axiom,
! [Aa: int,E_1: int,C_1: int,Ba: int,D_2: int] :
( ( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Aa),E_1)),C_1) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Ba),E_1)),D_2) )
<=> ( C_1 = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(Ba),Aa)),E_1)),D_2) ) ) ).
tff(fact_5_eq__add__iff2,axiom,
! [Aa: real,E_1: real,C_1: real,Ba: real,D_2: real] :
( ( hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Aa),E_1)),C_1) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Ba),E_1)),D_2) )
<=> ( C_1 = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(Ba),Aa)),E_1)),D_2) ) ) ).
tff(fact_6_eq__add__iff1,axiom,
! [Aa: int,E_1: int,C_1: int,Ba: int,D_2: int] :
( ( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Aa),E_1)),C_1) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Ba),E_1)),D_2) )
<=> ( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(Aa),Ba)),E_1)),C_1) = D_2 ) ) ).
tff(fact_7_eq__add__iff1,axiom,
! [Aa: real,E_1: real,C_1: real,Ba: real,D_2: real] :
( ( hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Aa),E_1)),C_1) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Ba),E_1)),D_2) )
<=> ( hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(Aa),Ba)),E_1)),C_1) = D_2 ) ) ).
tff(fact_8_is__sum2sq__def,axiom,
! [X_1: int] :
( hBOOL(hAPP_int_bool(twoSqu820444569sum2sq,X_1))
<=> ? [A_49: int,B_65: int] : twoSqu949963151sum2sq(product_Pair_int_int(A_49,B_65)) = X_1 ) ).
tff(fact_9_Int2_Oaux1,axiom,
! [A: int,B: int,C: int] :
( ( hAPP_int_int(minus_minus_int(A),B) = C )
=> ( A = hAPP_int_int(plus_plus_int(C),B) ) ) ).
tff(fact_10_zdiff__zmult__distrib2,axiom,
! [W_1: int,Z1: int,Z2: int] : hAPP_int_int(times_times_int(W_1),hAPP_int_int(minus_minus_int(Z1),Z2)) = hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(W_1),Z1)),hAPP_int_int(times_times_int(W_1),Z2)) ).
tff(fact_11_zdiff__zmult__distrib,axiom,
! [Z1: int,Z2: int,W_1: int] : hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(Z1),Z2)),W_1) = hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(Z1),W_1)),hAPP_int_int(times_times_int(Z2),W_1)) ).
tff(fact_12_zadd__zmult__distrib2,axiom,
! [W_1: int,Z1: int,Z2: int] : hAPP_int_int(times_times_int(W_1),hAPP_int_int(plus_plus_int(Z1),Z2)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(W_1),Z1)),hAPP_int_int(times_times_int(W_1),Z2)) ).
tff(fact_13_zadd__zmult__distrib,axiom,
! [Z1: int,Z2: int,W_1: int] : hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(Z1),Z2)),W_1) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Z1),W_1)),hAPP_int_int(times_times_int(Z2),W_1)) ).
tff(fact_14_diff__add__cancel,axiom,
! [A_115: int,B_90: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(minus_minus_int(A_115),B_90)),B_90) = A_115 ).
tff(fact_15_diff__add__cancel,axiom,
! [A_115: real,B_90: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(minus_minus_real(A_115),B_90)),B_90) = A_115 ).
tff(fact_16_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_114: int,B_89: int,C_54: int] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(A_114),B_89)),C_54) = hAPP_int_int(times_times_int(A_114),hAPP_int_int(times_times_int(B_89),C_54)) ).
tff(fact_17_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_114: real,B_89: real,C_54: real] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(A_114),B_89)),C_54) = hAPP_real_real(times_times_real(A_114),hAPP_real_real(times_times_real(B_89),C_54)) ).
tff(fact_18_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_114: nat,B_89: nat,C_54: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(A_114),B_89)),C_54) = hAPP_nat_nat(times_times_nat(A_114),hAPP_nat_nat(times_times_nat(B_89),C_54)) ).
tff(fact_19_add__right__imp__eq,axiom,
! [B_88: int,A_113: int,C_53: int] :
( ( hAPP_int_int(plus_plus_int(B_88),A_113) = hAPP_int_int(plus_plus_int(C_53),A_113) )
=> ( B_88 = C_53 ) ) ).
tff(fact_20_add__right__imp__eq,axiom,
! [B_88: real,A_113: real,C_53: real] :
( ( hAPP_real_real(plus_plus_real(B_88),A_113) = hAPP_real_real(plus_plus_real(C_53),A_113) )
=> ( B_88 = C_53 ) ) ).
tff(fact_21_add__right__imp__eq,axiom,
! [B_88: nat,A_113: nat,C_53: nat] :
( ( hAPP_nat_nat(plus_plus_nat(B_88),A_113) = hAPP_nat_nat(plus_plus_nat(C_53),A_113) )
=> ( B_88 = C_53 ) ) ).
tff(fact_22_add__imp__eq,axiom,
! [A_112: int,B_87: int,C_52: int] :
( ( hAPP_int_int(plus_plus_int(A_112),B_87) = hAPP_int_int(plus_plus_int(A_112),C_52) )
=> ( B_87 = C_52 ) ) ).
tff(fact_23_add__imp__eq,axiom,
! [A_112: real,B_87: real,C_52: real] :
( ( hAPP_real_real(plus_plus_real(A_112),B_87) = hAPP_real_real(plus_plus_real(A_112),C_52) )
=> ( B_87 = C_52 ) ) ).
tff(fact_24_add__imp__eq,axiom,
! [A_112: nat,B_87: nat,C_52: nat] :
( ( hAPP_nat_nat(plus_plus_nat(A_112),B_87) = hAPP_nat_nat(plus_plus_nat(A_112),C_52) )
=> ( B_87 = C_52 ) ) ).
tff(fact_25_add__left__imp__eq,axiom,
! [A_111: int,B_86: int,C_51: int] :
( ( hAPP_int_int(plus_plus_int(A_111),B_86) = hAPP_int_int(plus_plus_int(A_111),C_51) )
=> ( B_86 = C_51 ) ) ).
tff(fact_26_add__left__imp__eq,axiom,
! [A_111: real,B_86: real,C_51: real] :
( ( hAPP_real_real(plus_plus_real(A_111),B_86) = hAPP_real_real(plus_plus_real(A_111),C_51) )
=> ( B_86 = C_51 ) ) ).
tff(fact_27_add__left__imp__eq,axiom,
! [A_111: nat,B_86: nat,C_51: nat] :
( ( hAPP_nat_nat(plus_plus_nat(A_111),B_86) = hAPP_nat_nat(plus_plus_nat(A_111),C_51) )
=> ( B_86 = C_51 ) ) ).
tff(fact_28_add__right__cancel,axiom,
! [Ba: int,Aa: int,C_1: int] :
( ( hAPP_int_int(plus_plus_int(Ba),Aa) = hAPP_int_int(plus_plus_int(C_1),Aa) )
<=> ( Ba = C_1 ) ) ).
tff(fact_29_add__right__cancel,axiom,
! [Ba: real,Aa: real,C_1: real] :
( ( hAPP_real_real(plus_plus_real(Ba),Aa) = hAPP_real_real(plus_plus_real(C_1),Aa) )
<=> ( Ba = C_1 ) ) ).
tff(fact_30_add__right__cancel,axiom,
! [Ba: nat,Aa: nat,C_1: nat] :
( ( hAPP_nat_nat(plus_plus_nat(Ba),Aa) = hAPP_nat_nat(plus_plus_nat(C_1),Aa) )
<=> ( Ba = C_1 ) ) ).
tff(fact_31_add__left__cancel,axiom,
! [Aa: int,Ba: int,C_1: int] :
( ( hAPP_int_int(plus_plus_int(Aa),Ba) = hAPP_int_int(plus_plus_int(Aa),C_1) )
<=> ( Ba = C_1 ) ) ).
tff(fact_32_add__left__cancel,axiom,
! [Aa: real,Ba: real,C_1: real] :
( ( hAPP_real_real(plus_plus_real(Aa),Ba) = hAPP_real_real(plus_plus_real(Aa),C_1) )
<=> ( Ba = C_1 ) ) ).
tff(fact_33_add__left__cancel,axiom,
! [Aa: nat,Ba: nat,C_1: nat] :
( ( hAPP_nat_nat(plus_plus_nat(Aa),Ba) = hAPP_nat_nat(plus_plus_nat(Aa),C_1) )
<=> ( Ba = C_1 ) ) ).
tff(fact_34_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_110: int,B_85: int,C_50: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_110),B_85)),C_50) = hAPP_int_int(plus_plus_int(A_110),hAPP_int_int(plus_plus_int(B_85),C_50)) ).
tff(fact_35_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_110: real,B_85: real,C_50: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_110),B_85)),C_50) = hAPP_real_real(plus_plus_real(A_110),hAPP_real_real(plus_plus_real(B_85),C_50)) ).
tff(fact_36_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_110: nat,B_85: nat,C_50: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_110),B_85)),C_50) = hAPP_nat_nat(plus_plus_nat(A_110),hAPP_nat_nat(plus_plus_nat(B_85),C_50)) ).
tff(fact_37_diff__eq__diff__eq,axiom,
! [Aa: int,Ba: int,C_1: int,D_2: int] :
( ( hAPP_int_int(minus_minus_int(Aa),Ba) = hAPP_int_int(minus_minus_int(C_1),D_2) )
=> ( ( Aa = Ba )
<=> ( C_1 = D_2 ) ) ) ).
tff(fact_38_diff__eq__diff__eq,axiom,
! [Aa: real,Ba: real,C_1: real,D_2: real] :
( ( hAPP_real_real(minus_minus_real(Aa),Ba) = hAPP_real_real(minus_minus_real(C_1),D_2) )
=> ( ( Aa = Ba )
<=> ( C_1 = D_2 ) ) ) ).
tff(fact_39_zmult__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Z1),Z2)),Z3) = hAPP_int_int(times_times_int(Z1),hAPP_int_int(times_times_int(Z2),Z3)) ).
tff(fact_40_zmult__commute,axiom,
! [Z_1: int,W_1: int] : hAPP_int_int(times_times_int(Z_1),W_1) = hAPP_int_int(times_times_int(W_1),Z_1) ).
tff(fact_41_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_42_zadd__left__commute,axiom,
! [X: int,Y: int,Z_1: int] : hAPP_int_int(plus_plus_int(X),hAPP_int_int(plus_plus_int(Y),Z_1)) = hAPP_int_int(plus_plus_int(Y),hAPP_int_int(plus_plus_int(X),Z_1)) ).
tff(fact_43_zadd__commute,axiom,
! [Z_1: int,W_1: int] : hAPP_int_int(plus_plus_int(Z_1),W_1) = hAPP_int_int(plus_plus_int(W_1),Z_1) ).
tff(fact_44_combine__common__factor,axiom,
! [A_109: int,E_2: int,B_84: int,C_49: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_109),E_2)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_84),E_2)),C_49)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_109),B_84)),E_2)),C_49) ).
tff(fact_45_combine__common__factor,axiom,
! [A_109: real,E_2: real,B_84: real,C_49: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_109),E_2)),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(B_84),E_2)),C_49)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_109),B_84)),E_2)),C_49) ).
tff(fact_46_combine__common__factor,axiom,
! [A_109: nat,E_2: nat,B_84: nat,C_49: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_109),E_2)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(B_84),E_2)),C_49)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_109),B_84)),E_2)),C_49) ).
tff(fact_47_mult__left_Oadd,axiom,
! [X_45: real,Y_45: real,Ya_1: real] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(X_45),Y_45)),Ya_1) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_45),Ya_1)),hAPP_real_real(times_times_real(Y_45),Ya_1)) ).
tff(fact_48_mult_Oadd__left,axiom,
! [A_108: real,A_107: real,B_83: real] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_108),A_107)),B_83) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_108),B_83)),hAPP_real_real(times_times_real(A_107),B_83)) ).
tff(fact_49_comm__semiring__class_Odistrib,axiom,
! [A_106: int,B_82: int,C_48: int] : hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_106),B_82)),C_48) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_106),C_48)),hAPP_int_int(times_times_int(B_82),C_48)) ).
tff(fact_50_comm__semiring__class_Odistrib,axiom,
! [A_106: real,B_82: real,C_48: real] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_106),B_82)),C_48) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_106),C_48)),hAPP_real_real(times_times_real(B_82),C_48)) ).
tff(fact_51_comm__semiring__class_Odistrib,axiom,
! [A_106: nat,B_82: nat,C_48: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_106),B_82)),C_48) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_106),C_48)),hAPP_nat_nat(times_times_nat(B_82),C_48)) ).
tff(fact_52_mult__right_Oadd,axiom,
! [Xa_1: real,X_44: real,Y_44: real] : hAPP_real_real(times_times_real(Xa_1),hAPP_real_real(plus_plus_real(X_44),Y_44)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Xa_1),X_44)),hAPP_real_real(times_times_real(Xa_1),Y_44)) ).
tff(fact_53_mult_Oadd__right,axiom,
! [A_105: real,B_81: real,B_80: real] : hAPP_real_real(times_times_real(A_105),hAPP_real_real(plus_plus_real(B_81),B_80)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_105),B_81)),hAPP_real_real(times_times_real(A_105),B_80)) ).
tff(fact_54_mult__left_Odiff,axiom,
! [X_43: real,Y_43: real,Ya: real] : hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(X_43),Y_43)),Ya) = hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(X_43),Ya)),hAPP_real_real(times_times_real(Y_43),Ya)) ).
tff(fact_55_mult_Odiff__left,axiom,
! [A_104: real,A_103: real,B_79: real] : hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(A_104),A_103)),B_79) = hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(A_104),B_79)),hAPP_real_real(times_times_real(A_103),B_79)) ).
tff(fact_56_mult__right_Odiff,axiom,
! [Xa: real,X_42: real,Y_42: real] : hAPP_real_real(times_times_real(Xa),hAPP_real_real(minus_minus_real(X_42),Y_42)) = hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(Xa),X_42)),hAPP_real_real(times_times_real(Xa),Y_42)) ).
tff(fact_57_mult_Odiff__right,axiom,
! [A_102: real,B_78: real,B_77: real] : hAPP_real_real(times_times_real(A_102),hAPP_real_real(minus_minus_real(B_78),B_77)) = hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(A_102),B_78)),hAPP_real_real(times_times_real(A_102),B_77)) ).
tff(fact_58_add__diff__add,axiom,
! [A_101: int,C_47: int,B_76: int,D_12: int] : hAPP_int_int(minus_minus_int(hAPP_int_int(plus_plus_int(A_101),C_47)),hAPP_int_int(plus_plus_int(B_76),D_12)) = hAPP_int_int(plus_plus_int(hAPP_int_int(minus_minus_int(A_101),B_76)),hAPP_int_int(minus_minus_int(C_47),D_12)) ).
tff(fact_59_add__diff__add,axiom,
! [A_101: real,C_47: real,B_76: real,D_12: real] : hAPP_real_real(minus_minus_real(hAPP_real_real(plus_plus_real(A_101),C_47)),hAPP_real_real(plus_plus_real(B_76),D_12)) = hAPP_real_real(plus_plus_real(hAPP_real_real(minus_minus_real(A_101),B_76)),hAPP_real_real(minus_minus_real(C_47),D_12)) ).
tff(fact_60_add__diff__cancel,axiom,
! [A_100: int,B_75: int] : hAPP_int_int(minus_minus_int(hAPP_int_int(plus_plus_int(A_100),B_75)),B_75) = A_100 ).
tff(fact_61_add__diff__cancel,axiom,
! [A_100: real,B_75: real] : hAPP_real_real(minus_minus_real(hAPP_real_real(plus_plus_real(A_100),B_75)),B_75) = A_100 ).
tff(fact_62_crossproduct__eq,axiom,
! [W: int,Y_1: int,X_1: int,Z: int] :
( ( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(W),Y_1)),hAPP_int_int(times_times_int(X_1),Z)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(W),Z)),hAPP_int_int(times_times_int(X_1),Y_1)) )
<=> ( ( W = X_1 )
| ( Y_1 = Z ) ) ) ).
tff(fact_63_crossproduct__eq,axiom,
! [W: real,Y_1: real,X_1: real,Z: real] :
( ( hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(W),Y_1)),hAPP_real_real(times_times_real(X_1),Z)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(W),Z)),hAPP_real_real(times_times_real(X_1),Y_1)) )
<=> ( ( W = X_1 )
| ( Y_1 = Z ) ) ) ).
tff(fact_64_crossproduct__eq,axiom,
! [W: nat,Y_1: nat,X_1: nat,Z: nat] :
( ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(W),Y_1)),hAPP_nat_nat(times_times_nat(X_1),Z)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(W),Z)),hAPP_nat_nat(times_times_nat(X_1),Y_1)) )
<=> ( ( W = X_1 )
| ( Y_1 = Z ) ) ) ).
tff(fact_65_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_99: int,M_7: int,B_74: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_99),M_7)),hAPP_int_int(times_times_int(B_74),M_7)) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_99),B_74)),M_7) ).
tff(fact_66_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_99: real,M_7: real,B_74: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_99),M_7)),hAPP_real_real(times_times_real(B_74),M_7)) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_99),B_74)),M_7) ).
tff(fact_67_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_99: nat,M_7: nat,B_74: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_99),M_7)),hAPP_nat_nat(times_times_nat(B_74),M_7)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_99),B_74)),M_7) ).
tff(fact_68_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_98: int,B_73: int,C_46: int] : hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_98),B_73)),C_46) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_98),C_46)),hAPP_int_int(times_times_int(B_73),C_46)) ).
tff(fact_69_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_98: real,B_73: real,C_46: real] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_98),B_73)),C_46) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_98),C_46)),hAPP_real_real(times_times_real(B_73),C_46)) ).
tff(fact_70_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_98: nat,B_73: nat,C_46: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_98),B_73)),C_46) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_98),C_46)),hAPP_nat_nat(times_times_nat(B_73),C_46)) ).
tff(fact_71_crossproduct__noteq,axiom,
! [C_1: int,D_2: int,Aa: int,Ba: int] :
( ( ( Aa != Ba )
& ( C_1 != D_2 ) )
<=> ( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Aa),C_1)),hAPP_int_int(times_times_int(Ba),D_2)) != hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Aa),D_2)),hAPP_int_int(times_times_int(Ba),C_1)) ) ) ).
tff(fact_72_crossproduct__noteq,axiom,
! [C_1: real,D_2: real,Aa: real,Ba: real] :
( ( ( Aa != Ba )
& ( C_1 != D_2 ) )
<=> ( hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Aa),C_1)),hAPP_real_real(times_times_real(Ba),D_2)) != hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Aa),D_2)),hAPP_real_real(times_times_real(Ba),C_1)) ) ) ).
tff(fact_73_crossproduct__noteq,axiom,
! [C_1: nat,D_2: nat,Aa: nat,Ba: nat] :
( ( ( Aa != Ba )
& ( C_1 != D_2 ) )
<=> ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(Aa),C_1)),hAPP_nat_nat(times_times_nat(Ba),D_2)) != hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(Aa),D_2)),hAPP_nat_nat(times_times_nat(Ba),C_1)) ) ) ).
tff(fact_74_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_41: int,Y_41: int,Z_17: int] : hAPP_int_int(times_times_int(X_41),hAPP_int_int(plus_plus_int(Y_41),Z_17)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(X_41),Y_41)),hAPP_int_int(times_times_int(X_41),Z_17)) ).
tff(fact_75_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_41: real,Y_41: real,Z_17: real] : hAPP_real_real(times_times_real(X_41),hAPP_real_real(plus_plus_real(Y_41),Z_17)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_41),Y_41)),hAPP_real_real(times_times_real(X_41),Z_17)) ).
tff(fact_76_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_41: nat,Y_41: nat,Z_17: nat] : hAPP_nat_nat(times_times_nat(X_41),hAPP_nat_nat(plus_plus_nat(Y_41),Z_17)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(X_41),Y_41)),hAPP_nat_nat(times_times_nat(X_41),Z_17)) ).
tff(fact_77_Pair__inject,axiom,
! [A_97: int,B_72: int,A_96: int,B_71: int] :
( ( product_Pair_int_int(A_97,B_72) = product_Pair_int_int(A_96,B_71) )
=> ~ ( ( A_97 = A_96 )
=> ( B_72 != B_71 ) ) ) ).
tff(fact_78_Pair__inject,axiom,
! [A_97: int,B_72: product_prod_int_int,A_96: int,B_71: product_prod_int_int] :
( ( produc282740534nt_int(A_97,B_72) = produc282740534nt_int(A_96,B_71) )
=> ~ ( ( A_97 = A_96 )
=> ( B_72 != B_71 ) ) ) ).
tff(fact_79_Pair__eq,axiom,
! [Aa: int,Ba: int,A_95: int,B_70: int] :
( ( product_Pair_int_int(Aa,Ba) = product_Pair_int_int(A_95,B_70) )
<=> ( ( Aa = A_95 )
& ( Ba = B_70 ) ) ) ).
tff(fact_80_Pair__eq,axiom,
! [Aa: int,Ba: product_prod_int_int,A_95: int,B_70: product_prod_int_int] :
( ( produc282740534nt_int(Aa,Ba) = produc282740534nt_int(A_95,B_70) )
<=> ( ( Aa = A_95 )
& ( Ba = B_70 ) ) ) ).
tff(fact_81_split__paired__All,axiom,
! [P_1: fun_Pr974702441t_bool] :
( ! [X1: product_prod_int_int] : hBOOL(hAPP_P603027463t_bool(P_1,X1))
<=> ! [A_49: int,B_65: int] : hBOOL(hAPP_P603027463t_bool(P_1,product_Pair_int_int(A_49,B_65))) ) ).
tff(fact_82_split__paired__All,axiom,
! [P_1: fun_Pr1503267378t_bool] :
( ! [X1: produc393999548nt_int] : hBOOL(hAPP_P178748804t_bool(P_1,X1))
<=> ! [A_49: int,B_65: product_prod_int_int] : hBOOL(hAPP_P178748804t_bool(P_1,produc282740534nt_int(A_49,B_65))) ) ).
tff(fact_83_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_94: int,B_69: int,C_45: int,D_11: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_94),B_69)),hAPP_int_int(plus_plus_int(C_45),D_11)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_94),C_45)),hAPP_int_int(plus_plus_int(B_69),D_11)) ).
tff(fact_84_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_94: real,B_69: real,C_45: real,D_11: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_94),B_69)),hAPP_real_real(plus_plus_real(C_45),D_11)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_94),C_45)),hAPP_real_real(plus_plus_real(B_69),D_11)) ).
tff(fact_85_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_94: nat,B_69: nat,C_45: nat,D_11: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_94),B_69)),hAPP_nat_nat(plus_plus_nat(C_45),D_11)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_94),C_45)),hAPP_nat_nat(plus_plus_nat(B_69),D_11)) ).
tff(fact_86_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_93: int,B_68: int,C_44: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_93),B_68)),C_44) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_93),C_44)),B_68) ).
tff(fact_87_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_93: real,B_68: real,C_44: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_93),B_68)),C_44) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_93),C_44)),B_68) ).
tff(fact_88_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_93: nat,B_68: nat,C_44: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_93),B_68)),C_44) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_93),C_44)),B_68) ).
tff(fact_89_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_92: int,B_67: int] : hAPP_int_int(times_times_int(A_92),B_67) = hAPP_int_int(times_times_int(B_67),A_92) ).
tff(fact_90_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_92: real,B_67: real] : hAPP_real_real(times_times_real(A_92),B_67) = hAPP_real_real(times_times_real(B_67),A_92) ).
tff(fact_91_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_92: nat,B_67: nat] : hAPP_nat_nat(times_times_nat(A_92),B_67) = hAPP_nat_nat(times_times_nat(B_67),A_92) ).
tff(fact_92_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: int,Rx_6: int,Ry_4: int] : hAPP_int_int(times_times_int(Lx_6),hAPP_int_int(times_times_int(Rx_6),Ry_4)) = hAPP_int_int(times_times_int(Rx_6),hAPP_int_int(times_times_int(Lx_6),Ry_4)) ).
tff(fact_93_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: real,Rx_6: real,Ry_4: real] : hAPP_real_real(times_times_real(Lx_6),hAPP_real_real(times_times_real(Rx_6),Ry_4)) = hAPP_real_real(times_times_real(Rx_6),hAPP_real_real(times_times_real(Lx_6),Ry_4)) ).
tff(fact_94_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: nat,Rx_6: nat,Ry_4: nat] : hAPP_nat_nat(times_times_nat(Lx_6),hAPP_nat_nat(times_times_nat(Rx_6),Ry_4)) = hAPP_nat_nat(times_times_nat(Rx_6),hAPP_nat_nat(times_times_nat(Lx_6),Ry_4)) ).
tff(fact_95_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: int,Rx_5: int,Ry_3: int] : hAPP_int_int(times_times_int(Lx_5),hAPP_int_int(times_times_int(Rx_5),Ry_3)) = hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_5),Rx_5)),Ry_3) ).
tff(fact_96_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: real,Rx_5: real,Ry_3: real] : hAPP_real_real(times_times_real(Lx_5),hAPP_real_real(times_times_real(Rx_5),Ry_3)) = hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_5),Rx_5)),Ry_3) ).
tff(fact_97_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: nat,Rx_5: nat,Ry_3: nat] : hAPP_nat_nat(times_times_nat(Lx_5),hAPP_nat_nat(times_times_nat(Rx_5),Ry_3)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_5),Rx_5)),Ry_3) ).
tff(fact_98_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: int,Ly_4: int,Rx_4: int] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_4),Ly_4)),Rx_4) = hAPP_int_int(times_times_int(Lx_4),hAPP_int_int(times_times_int(Ly_4),Rx_4)) ).
tff(fact_99_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: real,Ly_4: real,Rx_4: real] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_4),Ly_4)),Rx_4) = hAPP_real_real(times_times_real(Lx_4),hAPP_real_real(times_times_real(Ly_4),Rx_4)) ).
tff(fact_100_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: nat,Ly_4: nat,Rx_4: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_4),Ly_4)),Rx_4) = hAPP_nat_nat(times_times_nat(Lx_4),hAPP_nat_nat(times_times_nat(Ly_4),Rx_4)) ).
tff(fact_101_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: int,Ly_3: int,Rx_3: int] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_3),Ly_3)),Rx_3) = hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_3),Rx_3)),Ly_3) ).
tff(fact_102_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: real,Ly_3: real,Rx_3: real] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_3),Ly_3)),Rx_3) = hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_3),Rx_3)),Ly_3) ).
tff(fact_103_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: nat,Ly_3: nat,Rx_3: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_3),Ly_3)),Rx_3) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_3),Rx_3)),Ly_3) ).
tff(fact_104_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: int,Ly_2: int,Rx_2: int,Ry_2: int] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_2),Ly_2)),hAPP_int_int(times_times_int(Rx_2),Ry_2)) = hAPP_int_int(times_times_int(Lx_2),hAPP_int_int(times_times_int(Ly_2),hAPP_int_int(times_times_int(Rx_2),Ry_2))) ).
tff(fact_105_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: real,Ly_2: real,Rx_2: real,Ry_2: real] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_2),Ly_2)),hAPP_real_real(times_times_real(Rx_2),Ry_2)) = hAPP_real_real(times_times_real(Lx_2),hAPP_real_real(times_times_real(Ly_2),hAPP_real_real(times_times_real(Rx_2),Ry_2))) ).
tff(fact_106_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: nat,Ly_2: nat,Rx_2: nat,Ry_2: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_2),Ly_2)),hAPP_nat_nat(times_times_nat(Rx_2),Ry_2)) = hAPP_nat_nat(times_times_nat(Lx_2),hAPP_nat_nat(times_times_nat(Ly_2),hAPP_nat_nat(times_times_nat(Rx_2),Ry_2))) ).
tff(fact_107_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: int,Ly_1: int,Rx_1: int,Ry_1: int] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_1),Ly_1)),hAPP_int_int(times_times_int(Rx_1),Ry_1)) = hAPP_int_int(times_times_int(Rx_1),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_1),Ly_1)),Ry_1)) ).
tff(fact_108_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: real,Ly_1: real,Rx_1: real,Ry_1: real] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_1),Ly_1)),hAPP_real_real(times_times_real(Rx_1),Ry_1)) = hAPP_real_real(times_times_real(Rx_1),hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_1),Ly_1)),Ry_1)) ).
tff(fact_109_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: nat,Ly_1: nat,Rx_1: nat,Ry_1: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_1),Ly_1)),hAPP_nat_nat(times_times_nat(Rx_1),Ry_1)) = hAPP_nat_nat(times_times_nat(Rx_1),hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_1),Ly_1)),Ry_1)) ).
tff(fact_110_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: int,Ly: int,Rx: int,Ry: int] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx),Ly)),hAPP_int_int(times_times_int(Rx),Ry)) = hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx),Rx)),hAPP_int_int(times_times_int(Ly),Ry)) ).
tff(fact_111_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: real,Ly: real,Rx: real,Ry: real] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx),Ly)),hAPP_real_real(times_times_real(Rx),Ry)) = hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx),Rx)),hAPP_real_real(times_times_real(Ly),Ry)) ).
tff(fact_112_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: nat,Ly: nat,Rx: nat,Ry: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx),Ly)),hAPP_nat_nat(times_times_nat(Rx),Ry)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx),Rx)),hAPP_nat_nat(times_times_nat(Ly),Ry)) ).
tff(fact_113_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_91: int,C_43: int] : hAPP_int_int(plus_plus_int(A_91),C_43) = hAPP_int_int(plus_plus_int(C_43),A_91) ).
tff(fact_114_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_91: real,C_43: real] : hAPP_real_real(plus_plus_real(A_91),C_43) = hAPP_real_real(plus_plus_real(C_43),A_91) ).
tff(fact_115_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_91: nat,C_43: nat] : hAPP_nat_nat(plus_plus_nat(A_91),C_43) = hAPP_nat_nat(plus_plus_nat(C_43),A_91) ).
tff(fact_116_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_90: int,C_42: int,D_10: int] : hAPP_int_int(plus_plus_int(A_90),hAPP_int_int(plus_plus_int(C_42),D_10)) = hAPP_int_int(plus_plus_int(C_42),hAPP_int_int(plus_plus_int(A_90),D_10)) ).
tff(fact_117_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_90: real,C_42: real,D_10: real] : hAPP_real_real(plus_plus_real(A_90),hAPP_real_real(plus_plus_real(C_42),D_10)) = hAPP_real_real(plus_plus_real(C_42),hAPP_real_real(plus_plus_real(A_90),D_10)) ).
tff(fact_118_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_90: nat,C_42: nat,D_10: nat] : hAPP_nat_nat(plus_plus_nat(A_90),hAPP_nat_nat(plus_plus_nat(C_42),D_10)) = hAPP_nat_nat(plus_plus_nat(C_42),hAPP_nat_nat(plus_plus_nat(A_90),D_10)) ).
tff(fact_119_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_89: int,C_41: int,D_9: int] : hAPP_int_int(plus_plus_int(A_89),hAPP_int_int(plus_plus_int(C_41),D_9)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_89),C_41)),D_9) ).
tff(fact_120_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_89: real,C_41: real,D_9: real] : hAPP_real_real(plus_plus_real(A_89),hAPP_real_real(plus_plus_real(C_41),D_9)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_89),C_41)),D_9) ).
tff(fact_121_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_89: nat,C_41: nat,D_9: nat] : hAPP_nat_nat(plus_plus_nat(A_89),hAPP_nat_nat(plus_plus_nat(C_41),D_9)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_89),C_41)),D_9) ).
tff(fact_122_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_88: int,B_66: int,C_40: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_88),B_66)),C_40) = hAPP_int_int(plus_plus_int(A_88),hAPP_int_int(plus_plus_int(B_66),C_40)) ).
tff(fact_123_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_88: real,B_66: real,C_40: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_88),B_66)),C_40) = hAPP_real_real(plus_plus_real(A_88),hAPP_real_real(plus_plus_real(B_66),C_40)) ).
tff(fact_124_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_88: nat,B_66: nat,C_40: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_88),B_66)),C_40) = hAPP_nat_nat(plus_plus_nat(A_88),hAPP_nat_nat(plus_plus_nat(B_66),C_40)) ).
tff(fact_125_split__paired__Ex,axiom,
! [P_1: fun_Pr974702441t_bool] :
( ? [X1: product_prod_int_int] : hBOOL(hAPP_P603027463t_bool(P_1,X1))
<=> ? [A_49: int,B_65: int] : hBOOL(hAPP_P603027463t_bool(P_1,product_Pair_int_int(A_49,B_65))) ) ).
tff(fact_126_split__paired__Ex,axiom,
! [P_1: fun_Pr1503267378t_bool] :
( ? [X1: produc393999548nt_int] : hBOOL(hAPP_P178748804t_bool(P_1,X1))
<=> ? [A_49: int,B_65: product_prod_int_int] : hBOOL(hAPP_P178748804t_bool(P_1,produc282740534nt_int(A_49,B_65))) ) ).
tff(fact_127_prod__induct3,axiom,
! [X_1: produc393999548nt_int,P_1: fun_Pr1503267378t_bool] :
( ! [A_49: int,B_65: int,C_39: int] : hBOOL(hAPP_P178748804t_bool(P_1,produc282740534nt_int(A_49,product_Pair_int_int(B_65,C_39))))
=> hBOOL(hAPP_P178748804t_bool(P_1,X_1)) ) ).
tff(fact_128_prod__cases3,axiom,
! [Y_40: produc393999548nt_int] :
~ ! [A_49: int,B_65: int,C_39: int] : Y_40 != produc282740534nt_int(A_49,product_Pair_int_int(B_65,C_39)) ).
tff(fact_129_PairE,axiom,
! [P_3: product_prod_int_int] :
~ ! [X_2: int,Y_2: int] : P_3 != product_Pair_int_int(X_2,Y_2) ).
tff(fact_130_PairE,axiom,
! [P_3: produc393999548nt_int] :
~ ! [X_2: int,Y_2: product_prod_int_int] : P_3 != produc282740534nt_int(X_2,Y_2) ).
tff(fact_131_mem__def,axiom,
! [X_1: real,A_87: fun_real_bool] :
( hBOOL(member_real(X_1,A_87))
<=> hBOOL(hAPP_real_bool(A_87,X_1)) ) ).
tff(fact_132_prod_Oexhaust,axiom,
! [Y_39: product_prod_int_int] :
~ ! [A_49: int,B_65: int] : Y_39 != product_Pair_int_int(A_49,B_65) ).
tff(fact_133_prod_Oexhaust,axiom,
! [Y_39: produc393999548nt_int] :
~ ! [A_49: int,B_65: product_prod_int_int] : Y_39 != produc282740534nt_int(A_49,B_65) ).
tff(fact_134_curryI,axiom,
! [F: fun_Pr974702441t_bool,Aa: int,Ba: int] :
( hBOOL(hAPP_P603027463t_bool(F,product_Pair_int_int(Aa,Ba)))
=> hBOOL(hAPP_int_bool(produc262399358t_bool(F,Aa),Ba)) ) ).
tff(fact_135_curryI,axiom,
! [F: fun_Pr1503267378t_bool,Aa: int,Ba: product_prod_int_int] :
( hBOOL(hAPP_P178748804t_bool(F,produc282740534nt_int(Aa,Ba)))
=> hBOOL(hAPP_P603027463t_bool(produc97574839t_bool(F,Aa),Ba)) ) ).
tff(fact_136_le__add__iff1,axiom,
! [Aa: int,E_1: int,C_1: int,Ba: int,D_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Aa),E_1)),C_1)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Ba),E_1)),D_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(Aa),Ba)),E_1)),C_1)),D_2)) ) ).
tff(fact_137_le__add__iff1,axiom,
! [Aa: real,E_1: real,C_1: real,Ba: real,D_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Aa),E_1)),C_1)),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Ba),E_1)),D_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(Aa),Ba)),E_1)),C_1)),D_2)) ) ).
tff(fact_138_le__add__iff2,axiom,
! [Aa: int,E_1: int,C_1: int,Ba: int,D_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Aa),E_1)),C_1)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Ba),E_1)),D_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_1),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(Ba),Aa)),E_1)),D_2))) ) ).
tff(fact_139_le__add__iff2,axiom,
! [Aa: real,E_1: real,C_1: real,Ba: real,D_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Aa),E_1)),C_1)),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Ba),E_1)),D_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(Ba),Aa)),E_1)),D_2))) ) ).
tff(fact_140_real__squared__diff__one__factored,axiom,
! [X_40: int] : hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(X_40),X_40)),one_one_int) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(X_40),one_one_int)),hAPP_int_int(minus_minus_int(X_40),one_one_int)) ).
tff(fact_141_real__squared__diff__one__factored,axiom,
! [X_40: real] : hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(X_40),X_40)),one_one_real) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(X_40),one_one_real)),hAPP_real_real(minus_minus_real(X_40),one_one_real)) ).
tff(fact_142_less__add__iff1,axiom,
! [Aa: int,E_1: int,C_1: int,Ba: int,D_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Aa),E_1)),C_1)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Ba),E_1)),D_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(Aa),Ba)),E_1)),C_1)),D_2)) ) ).
tff(fact_143_less__add__iff1,axiom,
! [Aa: real,E_1: real,C_1: real,Ba: real,D_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Aa),E_1)),C_1)),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Ba),E_1)),D_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(Aa),Ba)),E_1)),C_1)),D_2)) ) ).
tff(fact_144_less__add__iff2,axiom,
! [Aa: int,E_1: int,C_1: int,Ba: int,D_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Aa),E_1)),C_1)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Ba),E_1)),D_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(Ba),Aa)),E_1)),D_2))) ) ).
tff(fact_145_less__add__iff2,axiom,
! [Aa: real,E_1: real,C_1: real,Ba: real,D_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Aa),E_1)),C_1)),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Ba),E_1)),D_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_1),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(Ba),Aa)),E_1)),D_2))) ) ).
tff(fact_146_add__number__of__diff1,axiom,
! [V_17: int,W_11: int,C_38: int] : hAPP_int_int(plus_plus_int(number_number_of_int(V_17)),hAPP_int_int(minus_minus_int(number_number_of_int(W_11)),C_38)) = hAPP_int_int(minus_minus_int(number_number_of_int(hAPP_int_int(plus_plus_int(V_17),W_11))),C_38) ).
tff(fact_147_add__number__of__diff1,axiom,
! [V_17: int,W_11: int,C_38: real] : hAPP_real_real(plus_plus_real(number267125858f_real(V_17)),hAPP_real_real(minus_minus_real(number267125858f_real(W_11)),C_38)) = hAPP_real_real(minus_minus_real(number267125858f_real(hAPP_int_int(plus_plus_int(V_17),W_11))),C_38) ).
tff(fact_148_curry__conv,axiom,
! [F: fun_Pr974702441t_bool,Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(produc262399358t_bool(F,Aa),Ba))
<=> hBOOL(hAPP_P603027463t_bool(F,product_Pair_int_int(Aa,Ba))) ) ).
tff(fact_149_curry__conv,axiom,
! [F: fun_Pr1503267378t_bool,Aa: int,Ba: product_prod_int_int] :
( hBOOL(hAPP_P603027463t_bool(produc97574839t_bool(F,Aa),Ba))
<=> hBOOL(hAPP_P178748804t_bool(F,produc282740534nt_int(Aa,Ba))) ) ).
tff(fact_150_less__1__mult,axiom,
! [N_3: int,M_6: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),N_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(times_times_int(M_6),N_3))) ) ) ).
tff(fact_151_less__1__mult,axiom,
! [N_3: real,M_6: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),M_6))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),N_3))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(times_times_real(M_6),N_3))) ) ) ).
tff(fact_152_less__1__mult,axiom,
! [N_3: nat,M_6: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),M_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(times_times_nat(M_6),N_3))) ) ) ).
tff(fact_153_less__add__one,axiom,
! [A_86: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_86),hAPP_int_int(plus_plus_int(A_86),one_one_int))) ).
tff(fact_154_less__add__one,axiom,
! [A_86: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_86),hAPP_real_real(plus_plus_real(A_86),one_one_real))) ).
tff(fact_155_less__add__one,axiom,
! [A_86: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_86),hAPP_nat_nat(plus_plus_nat(A_86),one_one_nat))) ).
tff(fact_156_add__less__le__mono,axiom,
! [C_37: int,D_8: int,A_85: int,B_64: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_85),B_64))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_37),D_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_85),C_37)),hAPP_int_int(plus_plus_int(B_64),D_8))) ) ) ).
tff(fact_157_add__less__le__mono,axiom,
! [C_37: real,D_8: real,A_85: real,B_64: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_85),B_64))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_37),D_8))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_85),C_37)),hAPP_real_real(plus_plus_real(B_64),D_8))) ) ) ).
tff(fact_158_add__less__le__mono,axiom,
! [C_37: nat,D_8: nat,A_85: nat,B_64: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_85),B_64))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_37),D_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_85),C_37)),hAPP_nat_nat(plus_plus_nat(B_64),D_8))) ) ) ).
tff(fact_159_add__le__less__mono,axiom,
! [C_36: int,D_7: int,A_84: int,B_63: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_84),B_63))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_36),D_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_84),C_36)),hAPP_int_int(plus_plus_int(B_63),D_7))) ) ) ).
tff(fact_160_add__le__less__mono,axiom,
! [C_36: real,D_7: real,A_84: real,B_63: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_84),B_63))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_36),D_7))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_84),C_36)),hAPP_real_real(plus_plus_real(B_63),D_7))) ) ) ).
tff(fact_161_add__le__less__mono,axiom,
! [C_36: nat,D_7: nat,A_84: nat,B_63: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_84),B_63))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_36),D_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_84),C_36)),hAPP_nat_nat(plus_plus_nat(B_63),D_7))) ) ) ).
tff(fact_162_one__reorient,axiom,
! [X_1: int] :
( ( one_one_int = X_1 )
<=> ( X_1 = one_one_int ) ) ).
tff(fact_163_one__reorient,axiom,
! [X_1: real] :
( ( one_one_real = X_1 )
<=> ( X_1 = one_one_real ) ) ).
tff(fact_164_one__reorient,axiom,
! [X_1: nat] :
( ( one_one_nat = X_1 )
<=> ( X_1 = one_one_nat ) ) ).
tff(fact_165_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),hAPP_int_int(plus_plus_int(Z),one_one_int)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z)) ) ).
tff(fact_166_number__of__reorient,axiom,
! [W: int,X_1: int] :
( ( number_number_of_int(W) = X_1 )
<=> ( X_1 = number_number_of_int(W) ) ) ).
tff(fact_167_number__of__reorient,axiom,
! [W: int,X_1: real] :
( ( number267125858f_real(W) = X_1 )
<=> ( X_1 = number267125858f_real(W) ) ) ).
tff(fact_168_number__of__reorient,axiom,
! [W: int,X_1: nat] :
( ( number_number_of_nat(W) = X_1 )
<=> ( X_1 = number_number_of_nat(W) ) ) ).
tff(fact_169_le__number__of__eq__not__less,axiom,
! [V_1: int,W: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(V_1)),number_number_of_int(W)))
<=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(W)),number_number_of_int(V_1))) ) ).
tff(fact_170_le__number__of__eq__not__less,axiom,
! [V_1: int,W: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(V_1)),number267125858f_real(W)))
<=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(W)),number267125858f_real(V_1))) ) ).
tff(fact_171_le__number__of__eq__not__less,axiom,
! [V_1: int,W: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,number_number_of_nat(V_1)),number_number_of_nat(W)))
<=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,number_number_of_nat(W)),number_number_of_nat(V_1))) ) ).
tff(fact_172_eq__number__of,axiom,
! [X_1: int,Y_1: int] :
( ( number_number_of_int(X_1) = number_number_of_int(Y_1) )
<=> ( X_1 = Y_1 ) ) ).
tff(fact_173_eq__number__of,axiom,
! [X_1: int,Y_1: int] :
( ( number267125858f_real(X_1) = number267125858f_real(Y_1) )
<=> ( X_1 = Y_1 ) ) ).
tff(fact_174_less__number__of,axiom,
! [X_1: int,Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(X_1)),number_number_of_int(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1)) ) ).
tff(fact_175_less__number__of,axiom,
! [X_1: int,Y_1: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(X_1)),number267125858f_real(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1)) ) ).
tff(fact_176_le__number__of,axiom,
! [X_1: int,Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(X_1)),number_number_of_int(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1)) ) ).
tff(fact_177_le__number__of,axiom,
! [X_1: int,Y_1: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(X_1)),number267125858f_real(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1)) ) ).
tff(fact_178_add1__zle__eq,axiom,
! [W: int,Z: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(W),one_one_int)),Z))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z)) ) ).
tff(fact_179_linorder__neqE__linordered__idom,axiom,
! [X_39: int,Y_38: int] :
( ( X_39 != Y_38 )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_39),Y_38))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_38),X_39)) ) ) ).
tff(fact_180_linorder__neqE__linordered__idom,axiom,
! [X_39: real,Y_38: real] :
( ( X_39 != Y_38 )
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_39),Y_38))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_38),X_39)) ) ) ).
tff(fact_181_zless__imp__add1__zle,axiom,
! [W_1: int,Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(W_1),one_one_int)),Z_1)) ) ).
tff(fact_182_order__le__neq__implies__less,axiom,
! [X_38: int,Y_37: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_38),Y_37))
=> ( ( X_38 != Y_37 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_38),Y_37)) ) ) ).
tff(fact_183_order__le__neq__implies__less,axiom,
! [X_38: real,Y_37: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_38),Y_37))
=> ( ( X_38 != Y_37 )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_38),Y_37)) ) ) ).
tff(fact_184_order__le__neq__implies__less,axiom,
! [X_38: nat,Y_37: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_38),Y_37))
=> ( ( X_38 != Y_37 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_38),Y_37)) ) ) ).
tff(fact_185_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),hAPP_int_int(minus_minus_int(Z),one_one_int)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z)) ) ).
tff(fact_186_zadd__zless__mono,axiom,
! [Z_16: int,Z_1: int,W_10: int,W_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_10),W_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_16),Z_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(W_10),Z_16)),hAPP_int_int(plus_plus_int(W_1),Z_1))) ) ) ).
tff(fact_187_zless__add1__eq,axiom,
! [W: int,Z: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),hAPP_int_int(plus_plus_int(Z),one_one_int)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z))
| ( W = Z ) ) ) ).
tff(fact_188_add__less__cancel__right,axiom,
! [Aa: int,C_1: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(Aa),C_1)),hAPP_int_int(plus_plus_int(Ba),C_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) ) ).
tff(fact_189_add__less__cancel__right,axiom,
! [Aa: real,C_1: real,Ba: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(Aa),C_1)),hAPP_real_real(plus_plus_real(Ba),C_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba)) ) ).
tff(fact_190_add__less__cancel__right,axiom,
! [Aa: nat,C_1: nat,Ba: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(Aa),C_1)),hAPP_nat_nat(plus_plus_nat(Ba),C_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba)) ) ).
tff(fact_191_add__less__cancel__left,axiom,
! [C_1: int,Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(C_1),Aa)),hAPP_int_int(plus_plus_int(C_1),Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) ) ).
tff(fact_192_add__less__cancel__left,axiom,
! [C_1: real,Aa: real,Ba: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(C_1),Aa)),hAPP_real_real(plus_plus_real(C_1),Ba)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba)) ) ).
tff(fact_193_add__less__cancel__left,axiom,
! [C_1: nat,Aa: nat,Ba: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(C_1),Aa)),hAPP_nat_nat(plus_plus_nat(C_1),Ba)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba)) ) ).
tff(fact_194_add__strict__right__mono,axiom,
! [C_35: int,A_83: int,B_62: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_83),B_62))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_83),C_35)),hAPP_int_int(plus_plus_int(B_62),C_35))) ) ).
tff(fact_195_add__strict__right__mono,axiom,
! [C_35: real,A_83: real,B_62: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_83),B_62))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_83),C_35)),hAPP_real_real(plus_plus_real(B_62),C_35))) ) ).
tff(fact_196_add__strict__right__mono,axiom,
! [C_35: nat,A_83: nat,B_62: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_83),B_62))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_83),C_35)),hAPP_nat_nat(plus_plus_nat(B_62),C_35))) ) ).
tff(fact_197_add__strict__left__mono,axiom,
! [C_34: int,A_82: int,B_61: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_82),B_61))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(C_34),A_82)),hAPP_int_int(plus_plus_int(C_34),B_61))) ) ).
tff(fact_198_add__strict__left__mono,axiom,
! [C_34: real,A_82: real,B_61: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_82),B_61))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(C_34),A_82)),hAPP_real_real(plus_plus_real(C_34),B_61))) ) ).
tff(fact_199_add__strict__left__mono,axiom,
! [C_34: nat,A_82: nat,B_61: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_82),B_61))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(C_34),A_82)),hAPP_nat_nat(plus_plus_nat(C_34),B_61))) ) ).
tff(fact_200_add__strict__mono,axiom,
! [C_33: int,D_6: int,A_81: int,B_60: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_81),B_60))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_33),D_6))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_81),C_33)),hAPP_int_int(plus_plus_int(B_60),D_6))) ) ) ).
tff(fact_201_add__strict__mono,axiom,
! [C_33: real,D_6: real,A_81: real,B_60: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_81),B_60))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_33),D_6))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_81),C_33)),hAPP_real_real(plus_plus_real(B_60),D_6))) ) ) ).
tff(fact_202_add__strict__mono,axiom,
! [C_33: nat,D_6: nat,A_81: nat,B_60: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_81),B_60))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_33),D_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_81),C_33)),hAPP_nat_nat(plus_plus_nat(B_60),D_6))) ) ) ).
tff(fact_203_add__less__imp__less__right,axiom,
! [A_80: int,C_32: int,B_59: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_80),C_32)),hAPP_int_int(plus_plus_int(B_59),C_32)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_80),B_59)) ) ).
tff(fact_204_add__less__imp__less__right,axiom,
! [A_80: real,C_32: real,B_59: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_80),C_32)),hAPP_real_real(plus_plus_real(B_59),C_32)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_80),B_59)) ) ).
tff(fact_205_add__less__imp__less__right,axiom,
! [A_80: nat,C_32: nat,B_59: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_80),C_32)),hAPP_nat_nat(plus_plus_nat(B_59),C_32)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_80),B_59)) ) ).
tff(fact_206_add__less__imp__less__left,axiom,
! [C_31: int,A_79: int,B_58: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(C_31),A_79)),hAPP_int_int(plus_plus_int(C_31),B_58)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_79),B_58)) ) ).
tff(fact_207_add__less__imp__less__left,axiom,
! [C_31: real,A_79: real,B_58: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(C_31),A_79)),hAPP_real_real(plus_plus_real(C_31),B_58)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_79),B_58)) ) ).
tff(fact_208_add__less__imp__less__left,axiom,
! [C_31: nat,A_79: nat,B_58: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(C_31),A_79)),hAPP_nat_nat(plus_plus_nat(C_31),B_58)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_79),B_58)) ) ).
tff(fact_209_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_78: int] : hAPP_int_int(times_times_int(one_one_int),A_78) = A_78 ).
tff(fact_210_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_78: real] : hAPP_real_real(times_times_real(one_one_real),A_78) = A_78 ).
tff(fact_211_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_78: nat] : hAPP_nat_nat(times_times_nat(one_one_nat),A_78) = A_78 ).
tff(fact_212_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_77: int] : hAPP_int_int(times_times_int(A_77),one_one_int) = A_77 ).
tff(fact_213_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_77: real] : hAPP_real_real(times_times_real(A_77),one_one_real) = A_77 ).
tff(fact_214_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_77: nat] : hAPP_nat_nat(times_times_nat(A_77),one_one_nat) = A_77 ).
tff(fact_215_mult__1__left,axiom,
! [A_76: int] : hAPP_int_int(times_times_int(one_one_int),A_76) = A_76 ).
tff(fact_216_mult__1__left,axiom,
! [A_76: real] : hAPP_real_real(times_times_real(one_one_real),A_76) = A_76 ).
tff(fact_217_mult__1__left,axiom,
! [A_76: nat] : hAPP_nat_nat(times_times_nat(one_one_nat),A_76) = A_76 ).
tff(fact_218_mult__1,axiom,
! [A_75: int] : hAPP_int_int(times_times_int(one_one_int),A_75) = A_75 ).
tff(fact_219_mult__1,axiom,
! [A_75: real] : hAPP_real_real(times_times_real(one_one_real),A_75) = A_75 ).
tff(fact_220_mult__1,axiom,
! [A_75: nat] : hAPP_nat_nat(times_times_nat(one_one_nat),A_75) = A_75 ).
tff(fact_221_mult__1__right,axiom,
! [A_74: int] : hAPP_int_int(times_times_int(A_74),one_one_int) = A_74 ).
tff(fact_222_mult__1__right,axiom,
! [A_74: real] : hAPP_real_real(times_times_real(A_74),one_one_real) = A_74 ).
tff(fact_223_mult__1__right,axiom,
! [A_74: nat] : hAPP_nat_nat(times_times_nat(A_74),one_one_nat) = A_74 ).
tff(fact_224_mult_Ocomm__neutral,axiom,
! [A_73: int] : hAPP_int_int(times_times_int(A_73),one_one_int) = A_73 ).
tff(fact_225_mult_Ocomm__neutral,axiom,
! [A_73: real] : hAPP_real_real(times_times_real(A_73),one_one_real) = A_73 ).
tff(fact_226_mult_Ocomm__neutral,axiom,
! [A_73: nat] : hAPP_nat_nat(times_times_nat(A_73),one_one_nat) = A_73 ).
tff(fact_227_diff__eq__diff__less,axiom,
! [Aa: int,Ba: int,C_1: int,D_2: int] :
( ( hAPP_int_int(minus_minus_int(Aa),Ba) = hAPP_int_int(minus_minus_int(C_1),D_2) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),D_2)) ) ) ).
tff(fact_228_diff__eq__diff__less,axiom,
! [Aa: real,Ba: real,C_1: real,D_2: real] :
( ( hAPP_real_real(minus_minus_real(Aa),Ba) = hAPP_real_real(minus_minus_real(C_1),D_2) )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_1),D_2)) ) ) ).
tff(fact_229_add__le__cancel__right,axiom,
! [Aa: int,C_1: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(Aa),C_1)),hAPP_int_int(plus_plus_int(Ba),C_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba)) ) ).
tff(fact_230_add__le__cancel__right,axiom,
! [Aa: real,C_1: real,Ba: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(Aa),C_1)),hAPP_real_real(plus_plus_real(Ba),C_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba)) ) ).
tff(fact_231_add__le__cancel__right,axiom,
! [Aa: nat,C_1: nat,Ba: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(Aa),C_1)),hAPP_nat_nat(plus_plus_nat(Ba),C_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Aa),Ba)) ) ).
tff(fact_232_add__le__cancel__left,axiom,
! [C_1: int,Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(C_1),Aa)),hAPP_int_int(plus_plus_int(C_1),Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba)) ) ).
tff(fact_233_add__le__cancel__left,axiom,
! [C_1: real,Aa: real,Ba: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(C_1),Aa)),hAPP_real_real(plus_plus_real(C_1),Ba)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba)) ) ).
tff(fact_234_add__le__cancel__left,axiom,
! [C_1: nat,Aa: nat,Ba: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(C_1),Aa)),hAPP_nat_nat(plus_plus_nat(C_1),Ba)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Aa),Ba)) ) ).
tff(fact_235_add__right__mono,axiom,
! [C_30: int,A_72: int,B_57: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_72),B_57))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_72),C_30)),hAPP_int_int(plus_plus_int(B_57),C_30))) ) ).
tff(fact_236_add__right__mono,axiom,
! [C_30: real,A_72: real,B_57: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_72),B_57))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_72),C_30)),hAPP_real_real(plus_plus_real(B_57),C_30))) ) ).
tff(fact_237_add__right__mono,axiom,
! [C_30: nat,A_72: nat,B_57: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_72),B_57))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(A_72),C_30)),hAPP_nat_nat(plus_plus_nat(B_57),C_30))) ) ).
tff(fact_238_add__left__mono,axiom,
! [C_29: int,A_71: int,B_56: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_71),B_56))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(C_29),A_71)),hAPP_int_int(plus_plus_int(C_29),B_56))) ) ).
tff(fact_239_add__left__mono,axiom,
! [C_29: real,A_71: real,B_56: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_71),B_56))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(C_29),A_71)),hAPP_real_real(plus_plus_real(C_29),B_56))) ) ).
tff(fact_240_add__left__mono,axiom,
! [C_29: nat,A_71: nat,B_56: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_71),B_56))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(C_29),A_71)),hAPP_nat_nat(plus_plus_nat(C_29),B_56))) ) ).
tff(fact_241_add__mono,axiom,
! [C_28: int,D_5: int,A_70: int,B_55: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_70),B_55))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_28),D_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_70),C_28)),hAPP_int_int(plus_plus_int(B_55),D_5))) ) ) ).
tff(fact_242_add__mono,axiom,
! [C_28: real,D_5: real,A_70: real,B_55: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_70),B_55))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_28),D_5))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_70),C_28)),hAPP_real_real(plus_plus_real(B_55),D_5))) ) ) ).
tff(fact_243_add__mono,axiom,
! [C_28: nat,D_5: nat,A_70: nat,B_55: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_70),B_55))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_28),D_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(A_70),C_28)),hAPP_nat_nat(plus_plus_nat(B_55),D_5))) ) ) ).
tff(fact_244_add__le__imp__le__right,axiom,
! [A_69: int,C_27: int,B_54: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_69),C_27)),hAPP_int_int(plus_plus_int(B_54),C_27)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_69),B_54)) ) ).
tff(fact_245_add__le__imp__le__right,axiom,
! [A_69: real,C_27: real,B_54: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_69),C_27)),hAPP_real_real(plus_plus_real(B_54),C_27)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_69),B_54)) ) ).
tff(fact_246_add__le__imp__le__right,axiom,
! [A_69: nat,C_27: nat,B_54: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(A_69),C_27)),hAPP_nat_nat(plus_plus_nat(B_54),C_27)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_69),B_54)) ) ).
tff(fact_247_add__le__imp__le__left,axiom,
! [C_26: int,A_68: int,B_53: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(C_26),A_68)),hAPP_int_int(plus_plus_int(C_26),B_53)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_68),B_53)) ) ).
tff(fact_248_add__le__imp__le__left,axiom,
! [C_26: real,A_68: real,B_53: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(C_26),A_68)),hAPP_real_real(plus_plus_real(C_26),B_53)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_68),B_53)) ) ).
tff(fact_249_add__le__imp__le__left,axiom,
! [C_26: nat,A_68: nat,B_53: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(C_26),A_68)),hAPP_nat_nat(plus_plus_nat(C_26),B_53)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_68),B_53)) ) ).
tff(fact_250_diff__eq__diff__less__eq,axiom,
! [Aa: int,Ba: int,C_1: int,D_2: int] :
( ( hAPP_int_int(minus_minus_int(Aa),Ba) = hAPP_int_int(minus_minus_int(C_1),D_2) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_1),D_2)) ) ) ).
tff(fact_251_diff__eq__diff__less__eq,axiom,
! [Aa: real,Ba: real,C_1: real,D_2: real] :
( ( hAPP_real_real(minus_minus_real(Aa),Ba) = hAPP_real_real(minus_minus_real(C_1),D_2) )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),D_2)) ) ) ).
tff(fact_252_times__numeral__code_I5_J,axiom,
! [V: int,W_1: int] : hAPP_int_int(times_times_int(number_number_of_int(V)),number_number_of_int(W_1)) = number_number_of_int(hAPP_int_int(times_times_int(V),W_1)) ).
tff(fact_253_plus__numeral__code_I9_J,axiom,
! [V: int,W_1: int] : hAPP_int_int(plus_plus_int(number_number_of_int(V)),number_number_of_int(W_1)) = number_number_of_int(hAPP_int_int(plus_plus_int(V),W_1)) ).
tff(fact_254_zadd__strict__right__mono,axiom,
! [K: int,I: int,J: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(I),K)),hAPP_int_int(plus_plus_int(J),K))) ) ).
tff(fact_255_zadd__left__mono,axiom,
! [K: int,I: int,J: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(K),I)),hAPP_int_int(plus_plus_int(K),J))) ) ).
tff(fact_256_zmult__1,axiom,
! [Z_1: int] : hAPP_int_int(times_times_int(one_one_int),Z_1) = Z_1 ).
tff(fact_257_zmult__1__right,axiom,
! [Z_1: int] : hAPP_int_int(times_times_int(Z_1),one_one_int) = Z_1 ).
tff(fact_258_right__distrib__number__of,axiom,
! [V_16: int,B_52: int,C_25: int] : hAPP_int_int(times_times_int(number_number_of_int(V_16)),hAPP_int_int(plus_plus_int(B_52),C_25)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(V_16)),B_52)),hAPP_int_int(times_times_int(number_number_of_int(V_16)),C_25)) ).
tff(fact_259_right__distrib__number__of,axiom,
! [V_16: int,B_52: real,C_25: real] : hAPP_real_real(times_times_real(number267125858f_real(V_16)),hAPP_real_real(plus_plus_real(B_52),C_25)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(number267125858f_real(V_16)),B_52)),hAPP_real_real(times_times_real(number267125858f_real(V_16)),C_25)) ).
tff(fact_260_right__distrib__number__of,axiom,
! [V_16: int,B_52: nat,C_25: nat] : hAPP_nat_nat(times_times_nat(number_number_of_nat(V_16)),hAPP_nat_nat(plus_plus_nat(B_52),C_25)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(number_number_of_nat(V_16)),B_52)),hAPP_nat_nat(times_times_nat(number_number_of_nat(V_16)),C_25)) ).
tff(fact_261_left__distrib__number__of,axiom,
! [A_67: int,B_51: int,V_15: int] : hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_67),B_51)),number_number_of_int(V_15)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_67),number_number_of_int(V_15))),hAPP_int_int(times_times_int(B_51),number_number_of_int(V_15))) ).
tff(fact_262_left__distrib__number__of,axiom,
! [A_67: real,B_51: real,V_15: int] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_67),B_51)),number267125858f_real(V_15)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_67),number267125858f_real(V_15))),hAPP_real_real(times_times_real(B_51),number267125858f_real(V_15))) ).
tff(fact_263_left__distrib__number__of,axiom,
! [A_67: nat,B_51: nat,V_15: int] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_67),B_51)),number_number_of_nat(V_15)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_67),number_number_of_nat(V_15))),hAPP_nat_nat(times_times_nat(B_51),number_number_of_nat(V_15))) ).
tff(fact_264_right__diff__distrib__number__of,axiom,
! [V_14: int,B_50: int,C_24: int] : hAPP_int_int(times_times_int(number_number_of_int(V_14)),hAPP_int_int(minus_minus_int(B_50),C_24)) = hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(number_number_of_int(V_14)),B_50)),hAPP_int_int(times_times_int(number_number_of_int(V_14)),C_24)) ).
tff(fact_265_right__diff__distrib__number__of,axiom,
! [V_14: int,B_50: real,C_24: real] : hAPP_real_real(times_times_real(number267125858f_real(V_14)),hAPP_real_real(minus_minus_real(B_50),C_24)) = hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(number267125858f_real(V_14)),B_50)),hAPP_real_real(times_times_real(number267125858f_real(V_14)),C_24)) ).
tff(fact_266_left__diff__distrib__number__of,axiom,
! [A_66: int,B_49: int,V_13: int] : hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(A_66),B_49)),number_number_of_int(V_13)) = hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(A_66),number_number_of_int(V_13))),hAPP_int_int(times_times_int(B_49),number_number_of_int(V_13))) ).
tff(fact_267_left__diff__distrib__number__of,axiom,
! [A_66: real,B_49: real,V_13: int] : hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(A_66),B_49)),number267125858f_real(V_13)) = hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(A_66),number267125858f_real(V_13))),hAPP_real_real(times_times_real(B_49),number267125858f_real(V_13))) ).
tff(fact_268_number__of__mult,axiom,
! [V_12: int,W_9: int] : number_number_of_int(hAPP_int_int(times_times_int(V_12),W_9)) = hAPP_int_int(times_times_int(number_number_of_int(V_12)),number_number_of_int(W_9)) ).
tff(fact_269_number__of__mult,axiom,
! [V_12: int,W_9: int] : number267125858f_real(hAPP_int_int(times_times_int(V_12),W_9)) = hAPP_real_real(times_times_real(number267125858f_real(V_12)),number267125858f_real(W_9)) ).
tff(fact_270_arith__simps_I32_J,axiom,
! [V_11: int,W_8: int] : hAPP_int_int(times_times_int(number_number_of_int(V_11)),number_number_of_int(W_8)) = number_number_of_int(hAPP_int_int(times_times_int(V_11),W_8)) ).
tff(fact_271_arith__simps_I32_J,axiom,
! [V_11: int,W_8: int] : hAPP_real_real(times_times_real(number267125858f_real(V_11)),number267125858f_real(W_8)) = number267125858f_real(hAPP_int_int(times_times_int(V_11),W_8)) ).
tff(fact_272_mult__number__of__left,axiom,
! [V_10: int,W_7: int,Z_15: int] : hAPP_int_int(times_times_int(number_number_of_int(V_10)),hAPP_int_int(times_times_int(number_number_of_int(W_7)),Z_15)) = hAPP_int_int(times_times_int(number_number_of_int(hAPP_int_int(times_times_int(V_10),W_7))),Z_15) ).
tff(fact_273_mult__number__of__left,axiom,
! [V_10: int,W_7: int,Z_15: real] : hAPP_real_real(times_times_real(number267125858f_real(V_10)),hAPP_real_real(times_times_real(number267125858f_real(W_7)),Z_15)) = hAPP_real_real(times_times_real(number267125858f_real(hAPP_int_int(times_times_int(V_10),W_7))),Z_15) ).
tff(fact_274_number__of__add,axiom,
! [V_9: int,W_6: int] : number_number_of_int(hAPP_int_int(plus_plus_int(V_9),W_6)) = hAPP_int_int(plus_plus_int(number_number_of_int(V_9)),number_number_of_int(W_6)) ).
tff(fact_275_number__of__add,axiom,
! [V_9: int,W_6: int] : number267125858f_real(hAPP_int_int(plus_plus_int(V_9),W_6)) = hAPP_real_real(plus_plus_real(number267125858f_real(V_9)),number267125858f_real(W_6)) ).
tff(fact_276_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_277_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_278_add__number__of__left,axiom,
! [V_7: int,W_4: int,Z_14: int] : hAPP_int_int(plus_plus_int(number_number_of_int(V_7)),hAPP_int_int(plus_plus_int(number_number_of_int(W_4)),Z_14)) = hAPP_int_int(plus_plus_int(number_number_of_int(hAPP_int_int(plus_plus_int(V_7),W_4))),Z_14) ).
tff(fact_279_add__number__of__left,axiom,
! [V_7: int,W_4: int,Z_14: real] : hAPP_real_real(plus_plus_real(number267125858f_real(V_7)),hAPP_real_real(plus_plus_real(number267125858f_real(W_4)),Z_14)) = hAPP_real_real(plus_plus_real(number267125858f_real(hAPP_int_int(plus_plus_int(V_7),W_4))),Z_14) ).
tff(fact_280_number__of__diff,axiom,
! [V_6: int,W_3: int] : number_number_of_int(hAPP_int_int(minus_minus_int(V_6),W_3)) = hAPP_int_int(minus_minus_int(number_number_of_int(V_6)),number_number_of_int(W_3)) ).
tff(fact_281_number__of__diff,axiom,
! [V_6: int,W_3: int] : number267125858f_real(hAPP_int_int(minus_minus_int(V_6),W_3)) = hAPP_real_real(minus_minus_real(number267125858f_real(V_6)),number267125858f_real(W_3)) ).
tff(fact_282_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_5: int] : hAPP_int_int(plus_plus_int(M_5),M_5) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(one_one_int),one_one_int)),M_5) ).
tff(fact_283_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_5: real] : hAPP_real_real(plus_plus_real(M_5),M_5) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(one_one_real),one_one_real)),M_5) ).
tff(fact_284_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_5: nat] : hAPP_nat_nat(plus_plus_nat(M_5),M_5) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(one_one_nat),one_one_nat)),M_5) ).
tff(fact_285_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_4: int,A_65: int] : hAPP_int_int(plus_plus_int(M_4),hAPP_int_int(times_times_int(A_65),M_4)) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_65),one_one_int)),M_4) ).
tff(fact_286_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_4: real,A_65: real] : hAPP_real_real(plus_plus_real(M_4),hAPP_real_real(times_times_real(A_65),M_4)) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_65),one_one_real)),M_4) ).
tff(fact_287_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_4: nat,A_65: nat] : hAPP_nat_nat(plus_plus_nat(M_4),hAPP_nat_nat(times_times_nat(A_65),M_4)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_65),one_one_nat)),M_4) ).
tff(fact_288_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_64: int,M_3: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_64),M_3)),M_3) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_64),one_one_int)),M_3) ).
tff(fact_289_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_64: real,M_3: real] : hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_64),M_3)),M_3) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_64),one_one_real)),M_3) ).
tff(fact_290_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_64: nat,M_3: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_64),M_3)),M_3) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_64),one_one_nat)),M_3) ).
tff(fact_291_curryD,axiom,
! [F: fun_Pr974702441t_bool,Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(produc262399358t_bool(F,Aa),Ba))
=> hBOOL(hAPP_P603027463t_bool(F,product_Pair_int_int(Aa,Ba))) ) ).
tff(fact_292_curryD,axiom,
! [F: fun_Pr1503267378t_bool,Aa: int,Ba: product_prod_int_int] :
( hBOOL(hAPP_P603027463t_bool(produc97574839t_bool(F,Aa),Ba))
=> hBOOL(hAPP_P178748804t_bool(F,produc282740534nt_int(Aa,Ba))) ) ).
tff(fact_293_curryE,axiom,
! [F: fun_Pr974702441t_bool,Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(produc262399358t_bool(F,Aa),Ba))
=> hBOOL(hAPP_P603027463t_bool(F,product_Pair_int_int(Aa,Ba))) ) ).
tff(fact_294_curryE,axiom,
! [F: fun_Pr1503267378t_bool,Aa: int,Ba: product_prod_int_int] :
( hBOOL(hAPP_P603027463t_bool(produc97574839t_bool(F,Aa),Ba))
=> hBOOL(hAPP_P178748804t_bool(F,produc282740534nt_int(Aa,Ba))) ) ).
tff(fact_295_order__refl,axiom,
! [X_37: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_37),X_37)) ).
tff(fact_296_order__refl,axiom,
! [X_37: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_37),X_37)) ).
tff(fact_297_order__refl,axiom,
! [X_37: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_37),X_37)) ).
tff(fact_298_xt1_I8_J,axiom,
! [Z_13: int,Y_36: int,X_36: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_36),X_36))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_13),Y_36))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_13),X_36)) ) ) ).
tff(fact_299_xt1_I8_J,axiom,
! [Z_13: real,Y_36: real,X_36: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_36),X_36))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_13),Y_36))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_13),X_36)) ) ) ).
tff(fact_300_xt1_I8_J,axiom,
! [Z_13: nat,Y_36: nat,X_36: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_36),X_36))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_13),Y_36))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_13),X_36)) ) ) ).
tff(fact_301_order__le__less__trans,axiom,
! [Z_12: int,X_35: int,Y_35: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_35),Y_35))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_35),Z_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_35),Z_12)) ) ) ).
tff(fact_302_order__le__less__trans,axiom,
! [Z_12: real,X_35: real,Y_35: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_35),Y_35))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_35),Z_12))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_35),Z_12)) ) ) ).
tff(fact_303_order__le__less__trans,axiom,
! [Z_12: nat,X_35: nat,Y_35: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_35),Y_35))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_35),Z_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_35),Z_12)) ) ) ).
tff(fact_304_xt1_I7_J,axiom,
! [Z_11: int,Y_34: int,X_34: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_34),X_34))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_11),Y_34))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_11),X_34)) ) ) ).
tff(fact_305_xt1_I7_J,axiom,
! [Z_11: real,Y_34: real,X_34: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_34),X_34))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_11),Y_34))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_11),X_34)) ) ) ).
tff(fact_306_xt1_I7_J,axiom,
! [Z_11: nat,Y_34: nat,X_34: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_34),X_34))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_11),Y_34))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_11),X_34)) ) ) ).
tff(fact_307_order__less__le__trans,axiom,
! [Z_10: int,X_33: int,Y_33: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_33),Y_33))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_33),Z_10))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_33),Z_10)) ) ) ).
tff(fact_308_order__less__le__trans,axiom,
! [Z_10: real,X_33: real,Y_33: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_33),Y_33))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_33),Z_10))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_33),Z_10)) ) ) ).
tff(fact_309_order__less__le__trans,axiom,
! [Z_10: nat,X_33: nat,Y_33: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_33),Y_33))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_33),Z_10))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_33),Z_10)) ) ) ).
tff(fact_310_xt1_I11_J,axiom,
! [B_48: int,A_63: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_48),A_63))
=> ( ( A_63 != B_48 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_48),A_63)) ) ) ).
tff(fact_311_xt1_I11_J,axiom,
! [B_48: real,A_63: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_48),A_63))
=> ( ( A_63 != B_48 )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_48),A_63)) ) ) ).
tff(fact_312_xt1_I11_J,axiom,
! [B_48: nat,A_63: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_48),A_63))
=> ( ( A_63 != B_48 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_48),A_63)) ) ) ).
tff(fact_313_order__le__neq__trans,axiom,
! [A_62: int,B_47: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_62),B_47))
=> ( ( A_62 != B_47 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_62),B_47)) ) ) ).
tff(fact_314_order__le__neq__trans,axiom,
! [A_62: real,B_47: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_62),B_47))
=> ( ( A_62 != B_47 )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_62),B_47)) ) ) ).
tff(fact_315_order__le__neq__trans,axiom,
! [A_62: nat,B_47: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_62),B_47))
=> ( ( A_62 != B_47 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_62),B_47)) ) ) ).
tff(fact_316_order__le__imp__less__or__eq,axiom,
! [X_32: int,Y_32: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_32),Y_32))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_32),Y_32))
| ( X_32 = Y_32 ) ) ) ).
tff(fact_317_order__le__imp__less__or__eq,axiom,
! [X_32: real,Y_32: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_32),Y_32))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_32),Y_32))
| ( X_32 = Y_32 ) ) ) ).
tff(fact_318_order__le__imp__less__or__eq,axiom,
! [X_32: nat,Y_32: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_32),Y_32))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_32),Y_32))
| ( X_32 = Y_32 ) ) ) ).
tff(fact_319_linorder__antisym__conv2,axiom,
! [X_1: int,Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
<=> ( X_1 = Y_1 ) ) ) ).
tff(fact_320_linorder__antisym__conv2,axiom,
! [X_1: real,Y_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
<=> ( X_1 = Y_1 ) ) ) ).
tff(fact_321_linorder__antisym__conv2,axiom,
! [X_1: nat,Y_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
<=> ( X_1 = Y_1 ) ) ) ).
tff(fact_322_order__less__imp__le,axiom,
! [X_31: int,Y_31: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_31),Y_31))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_31),Y_31)) ) ).
tff(fact_323_order__less__imp__le,axiom,
! [X_31: real,Y_31: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_31),Y_31))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_31),Y_31)) ) ).
tff(fact_324_order__less__imp__le,axiom,
! [X_31: nat,Y_31: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_31),Y_31))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_31),Y_31)) ) ).
tff(fact_325_zle__refl,axiom,
! [W_1: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),W_1)) ).
tff(fact_326_number__of__is__id,axiom,
! [K: int] : number_number_of_int(K) = K ).
tff(fact_327_zle__linear,axiom,
! [Z_1: int,W_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W_1))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_1)) ) ).
tff(fact_328_zless__le,axiom,
! [Z: int,W: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),W))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W))
& ( Z != W ) ) ) ).
tff(fact_329_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_330_less__number__of__int__code,axiom,
! [K_1: int,L: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(K_1)),number_number_of_int(L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) ) ).
tff(fact_331_less__eq__number__of__int__code,axiom,
! [K_1: int,L: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(K_1)),number_number_of_int(L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) ) ).
tff(fact_332_zle__trans,axiom,
! [K: int,I: int,J: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J),K))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),K)) ) ) ).
tff(fact_333_zle__antisym,axiom,
! [Z_1: int,W_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_1))
=> ( Z_1 = W_1 ) ) ) ).
tff(fact_334_linorder__linear,axiom,
! [X_30: int,Y_30: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_30),Y_30))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_30),X_30)) ) ).
tff(fact_335_linorder__linear,axiom,
! [X_30: real,Y_30: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_30),Y_30))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_30),X_30)) ) ).
tff(fact_336_linorder__linear,axiom,
! [X_30: nat,Y_30: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_30),Y_30))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_30),X_30)) ) ).
tff(fact_337_order__eq__iff,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))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1)) ) ) ).
tff(fact_338_order__eq__iff,axiom,
! [X_1: real,Y_1: real] :
( ( X_1 = Y_1 )
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),X_1)) ) ) ).
tff(fact_339_order__eq__iff,axiom,
! [X_1: nat,Y_1: nat] :
( ( X_1 = Y_1 )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1)) ) ) ).
tff(fact_340_order__eq__refl,axiom,
! [X_29: int,Y_29: int] :
( ( X_29 = Y_29 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_29),Y_29)) ) ).
tff(fact_341_order__eq__refl,axiom,
! [X_29: real,Y_29: real] :
( ( X_29 = Y_29 )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_29),Y_29)) ) ).
tff(fact_342_order__eq__refl,axiom,
! [X_29: nat,Y_29: nat] :
( ( X_29 = Y_29 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_29),Y_29)) ) ).
tff(fact_343_order__antisym__conv,axiom,
! [Y_1: int,X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
<=> ( X_1 = Y_1 ) ) ) ).
tff(fact_344_order__antisym__conv,axiom,
! [Y_1: real,X_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
<=> ( X_1 = Y_1 ) ) ) ).
tff(fact_345_order__antisym__conv,axiom,
! [Y_1: nat,X_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
<=> ( X_1 = Y_1 ) ) ) ).
tff(fact_346_ord__eq__le__trans,axiom,
! [C_23: int,A_61: int,B_46: int] :
( ( A_61 = B_46 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_46),C_23))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_61),C_23)) ) ) ).
tff(fact_347_ord__eq__le__trans,axiom,
! [C_23: real,A_61: real,B_46: real] :
( ( A_61 = B_46 )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_46),C_23))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_61),C_23)) ) ) ).
tff(fact_348_ord__eq__le__trans,axiom,
! [C_23: nat,A_61: nat,B_46: nat] :
( ( A_61 = B_46 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_46),C_23))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_61),C_23)) ) ) ).
tff(fact_349_xt1_I3_J,axiom,
! [C_22: int,A_60: int,B_45: int] :
( ( A_60 = B_45 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_22),B_45))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_22),A_60)) ) ) ).
tff(fact_350_xt1_I3_J,axiom,
! [C_22: real,A_60: real,B_45: real] :
( ( A_60 = B_45 )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_22),B_45))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_22),A_60)) ) ) ).
tff(fact_351_xt1_I3_J,axiom,
! [C_22: nat,A_60: nat,B_45: nat] :
( ( A_60 = B_45 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_22),B_45))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_22),A_60)) ) ) ).
tff(fact_352_ord__le__eq__trans,axiom,
! [C_21: int,A_59: int,B_44: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_59),B_44))
=> ( ( B_44 = C_21 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_59),C_21)) ) ) ).
tff(fact_353_ord__le__eq__trans,axiom,
! [C_21: real,A_59: real,B_44: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_59),B_44))
=> ( ( B_44 = C_21 )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_59),C_21)) ) ) ).
tff(fact_354_ord__le__eq__trans,axiom,
! [C_21: nat,A_59: nat,B_44: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_59),B_44))
=> ( ( B_44 = C_21 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_59),C_21)) ) ) ).
tff(fact_355_xt1_I4_J,axiom,
! [C_20: int,B_43: int,A_58: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_43),A_58))
=> ( ( B_43 = C_20 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_20),A_58)) ) ) ).
tff(fact_356_xt1_I4_J,axiom,
! [C_20: real,B_43: real,A_58: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_43),A_58))
=> ( ( B_43 = C_20 )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_20),A_58)) ) ) ).
tff(fact_357_xt1_I4_J,axiom,
! [C_20: nat,B_43: nat,A_58: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_43),A_58))
=> ( ( B_43 = C_20 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_20),A_58)) ) ) ).
tff(fact_358_order__antisym,axiom,
! [X_28: int,Y_28: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_28),Y_28))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_28),X_28))
=> ( X_28 = Y_28 ) ) ) ).
tff(fact_359_order__antisym,axiom,
! [X_28: real,Y_28: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_28),Y_28))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_28),X_28))
=> ( X_28 = Y_28 ) ) ) ).
tff(fact_360_order__antisym,axiom,
! [X_28: nat,Y_28: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_28),Y_28))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_28),X_28))
=> ( X_28 = Y_28 ) ) ) ).
tff(fact_361_order__trans,axiom,
! [Z_9: int,X_27: int,Y_27: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_27),Y_27))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_27),Z_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_27),Z_9)) ) ) ).
tff(fact_362_order__trans,axiom,
! [Z_9: real,X_27: real,Y_27: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_27),Y_27))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_27),Z_9))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_27),Z_9)) ) ) ).
tff(fact_363_order__trans,axiom,
! [Z_9: nat,X_27: nat,Y_27: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_27),Y_27))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_27),Z_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_27),Z_9)) ) ) ).
tff(fact_364_xt1_I5_J,axiom,
! [Y_26: int,X_26: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_26),X_26))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_26),Y_26))
=> ( X_26 = Y_26 ) ) ) ).
tff(fact_365_xt1_I5_J,axiom,
! [Y_26: real,X_26: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_26),X_26))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_26),Y_26))
=> ( X_26 = Y_26 ) ) ) ).
tff(fact_366_xt1_I5_J,axiom,
! [Y_26: nat,X_26: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_26),X_26))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_26),Y_26))
=> ( X_26 = Y_26 ) ) ) ).
tff(fact_367_xt1_I6_J,axiom,
! [Z_8: int,Y_25: int,X_25: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_25),X_25))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_8),Y_25))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_8),X_25)) ) ) ).
tff(fact_368_xt1_I6_J,axiom,
! [Z_8: real,Y_25: real,X_25: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_25),X_25))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_8),Y_25))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_8),X_25)) ) ) ).
tff(fact_369_xt1_I6_J,axiom,
! [Z_8: nat,Y_25: nat,X_25: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_25),X_25))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_8),Y_25))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_8),X_25)) ) ) ).
tff(fact_370_linorder__le__cases,axiom,
! [X_24: int,Y_24: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_24),Y_24))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_24),X_24)) ) ).
tff(fact_371_linorder__le__cases,axiom,
! [X_24: real,Y_24: real] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_24),Y_24))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_24),X_24)) ) ).
tff(fact_372_linorder__le__cases,axiom,
! [X_24: nat,Y_24: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_24),Y_24))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_24),X_24)) ) ).
tff(fact_373_order__less__irrefl,axiom,
! [X_23: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_23),X_23)) ).
tff(fact_374_order__less__irrefl,axiom,
! [X_23: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_23),X_23)) ).
tff(fact_375_order__less__irrefl,axiom,
! [X_23: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_23),X_23)) ).
tff(fact_376_linorder__neq__iff,axiom,
! [X_1: int,Y_1: int] :
( ( X_1 != Y_1 )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),X_1)) ) ) ).
tff(fact_377_linorder__neq__iff,axiom,
! [X_1: real,Y_1: real] :
( ( X_1 != Y_1 )
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),X_1)) ) ) ).
tff(fact_378_linorder__neq__iff,axiom,
! [X_1: nat,Y_1: nat] :
( ( X_1 != Y_1 )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_1),X_1)) ) ) ).
tff(fact_379_not__less__iff__gr__or__eq,axiom,
! [X_1: int,Y_1: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),X_1))
| ( X_1 = Y_1 ) ) ) ).
tff(fact_380_not__less__iff__gr__or__eq,axiom,
! [X_1: real,Y_1: real] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),X_1))
| ( X_1 = Y_1 ) ) ) ).
tff(fact_381_not__less__iff__gr__or__eq,axiom,
! [X_1: nat,Y_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_1),X_1))
| ( X_1 = Y_1 ) ) ) ).
tff(fact_382_linorder__less__linear,axiom,
! [X_22: int,Y_23: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_22),Y_23))
| ( X_22 = Y_23 )
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_23),X_22)) ) ).
tff(fact_383_linorder__less__linear,axiom,
! [X_22: real,Y_23: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_22),Y_23))
| ( X_22 = Y_23 )
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_23),X_22)) ) ).
tff(fact_384_linorder__less__linear,axiom,
! [X_22: nat,Y_23: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_22),Y_23))
| ( X_22 = Y_23 )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_23),X_22)) ) ).
tff(fact_385_linorder__antisym__conv3,axiom,
! [Y_1: int,X_1: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),X_1))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
<=> ( X_1 = Y_1 ) ) ) ).
tff(fact_386_linorder__antisym__conv3,axiom,
! [Y_1: real,X_1: real] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),X_1))
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
<=> ( X_1 = Y_1 ) ) ) ).
tff(fact_387_linorder__antisym__conv3,axiom,
! [Y_1: nat,X_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_1),X_1))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
<=> ( X_1 = Y_1 ) ) ) ).
tff(fact_388_linorder__neqE,axiom,
! [X_21: int,Y_22: int] :
( ( X_21 != Y_22 )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_21),Y_22))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_22),X_21)) ) ) ).
tff(fact_389_linorder__neqE,axiom,
! [X_21: real,Y_22: real] :
( ( X_21 != Y_22 )
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_21),Y_22))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_22),X_21)) ) ) ).
tff(fact_390_linorder__neqE,axiom,
! [X_21: nat,Y_22: nat] :
( ( X_21 != Y_22 )
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_21),Y_22))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_22),X_21)) ) ) ).
tff(fact_391_less__imp__neq,axiom,
! [X_20: int,Y_21: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_20),Y_21))
=> ( X_20 != Y_21 ) ) ).
tff(fact_392_less__imp__neq,axiom,
! [X_20: real,Y_21: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_20),Y_21))
=> ( X_20 != Y_21 ) ) ).
tff(fact_393_less__imp__neq,axiom,
! [X_20: nat,Y_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_20),Y_21))
=> ( X_20 != Y_21 ) ) ).
tff(fact_394_order__less__not__sym,axiom,
! [X_19: int,Y_20: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_19),Y_20))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_20),X_19)) ) ).
tff(fact_395_order__less__not__sym,axiom,
! [X_19: real,Y_20: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_19),Y_20))
=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_20),X_19)) ) ).
tff(fact_396_order__less__not__sym,axiom,
! [X_19: nat,Y_20: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_19),Y_20))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_20),X_19)) ) ).
tff(fact_397_order__less__imp__not__less,axiom,
! [X_18: int,Y_19: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_18),Y_19))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_19),X_18)) ) ).
tff(fact_398_order__less__imp__not__less,axiom,
! [X_18: real,Y_19: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_18),Y_19))
=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_19),X_18)) ) ).
tff(fact_399_order__less__imp__not__less,axiom,
! [X_18: nat,Y_19: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_18),Y_19))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_19),X_18)) ) ).
tff(fact_400_order__less__imp__not__eq,axiom,
! [X_17: int,Y_18: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_17),Y_18))
=> ( X_17 != Y_18 ) ) ).
tff(fact_401_order__less__imp__not__eq,axiom,
! [X_17: real,Y_18: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_17),Y_18))
=> ( X_17 != Y_18 ) ) ).
tff(fact_402_order__less__imp__not__eq,axiom,
! [X_17: nat,Y_18: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_17),Y_18))
=> ( X_17 != Y_18 ) ) ).
tff(fact_403_order__less__imp__not__eq2,axiom,
! [X_16: int,Y_17: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_16),Y_17))
=> ( Y_17 != X_16 ) ) ).
tff(fact_404_order__less__imp__not__eq2,axiom,
! [X_16: real,Y_17: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_16),Y_17))
=> ( Y_17 != X_16 ) ) ).
tff(fact_405_order__less__imp__not__eq2,axiom,
! [X_16: nat,Y_17: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_16),Y_17))
=> ( Y_17 != X_16 ) ) ).
tff(fact_406_order__less__imp__triv,axiom,
! [P_1: bool,X_1: int,Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),X_1))
=> hBOOL(P_1) ) ) ).
tff(fact_407_order__less__imp__triv,axiom,
! [P_1: bool,X_1: real,Y_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),X_1))
=> hBOOL(P_1) ) ) ).
tff(fact_408_order__less__imp__triv,axiom,
! [P_1: bool,X_1: nat,Y_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_1),X_1))
=> hBOOL(P_1) ) ) ).
tff(fact_409_order__less__asym_H,axiom,
! [A_57: int,B_42: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_57),B_42))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_42),A_57)) ) ).
tff(fact_410_order__less__asym_H,axiom,
! [A_57: real,B_42: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_57),B_42))
=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_42),A_57)) ) ).
tff(fact_411_order__less__asym_H,axiom,
! [A_57: nat,B_42: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_57),B_42))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_42),A_57)) ) ).
tff(fact_412_xt1_I9_J,axiom,
! [B_41: int,A_56: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_41),A_56))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_56),B_41)) ) ).
tff(fact_413_xt1_I9_J,axiom,
! [B_41: real,A_56: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_41),A_56))
=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_56),B_41)) ) ).
tff(fact_414_xt1_I9_J,axiom,
! [B_41: nat,A_56: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_41),A_56))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_56),B_41)) ) ).
tff(fact_415_ord__eq__less__trans,axiom,
! [C_19: int,A_55: int,B_40: int] :
( ( A_55 = B_40 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_40),C_19))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_55),C_19)) ) ) ).
tff(fact_416_ord__eq__less__trans,axiom,
! [C_19: real,A_55: real,B_40: real] :
( ( A_55 = B_40 )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_40),C_19))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_55),C_19)) ) ) ).
tff(fact_417_ord__eq__less__trans,axiom,
! [C_19: nat,A_55: nat,B_40: nat] :
( ( A_55 = B_40 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_40),C_19))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_55),C_19)) ) ) ).
tff(fact_418_xt1_I1_J,axiom,
! [C_18: int,A_54: int,B_39: int] :
( ( A_54 = B_39 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_18),B_39))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_18),A_54)) ) ) ).
tff(fact_419_xt1_I1_J,axiom,
! [C_18: real,A_54: real,B_39: real] :
( ( A_54 = B_39 )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_18),B_39))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_18),A_54)) ) ) ).
tff(fact_420_xt1_I1_J,axiom,
! [C_18: nat,A_54: nat,B_39: nat] :
( ( A_54 = B_39 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_18),B_39))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_18),A_54)) ) ) ).
tff(fact_421_ord__less__eq__trans,axiom,
! [C_17: int,A_53: int,B_38: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_53),B_38))
=> ( ( B_38 = C_17 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_53),C_17)) ) ) ).
tff(fact_422_ord__less__eq__trans,axiom,
! [C_17: real,A_53: real,B_38: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_53),B_38))
=> ( ( B_38 = C_17 )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_53),C_17)) ) ) ).
tff(fact_423_ord__less__eq__trans,axiom,
! [C_17: nat,A_53: nat,B_38: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_53),B_38))
=> ( ( B_38 = C_17 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_53),C_17)) ) ) ).
tff(fact_424_xt1_I2_J,axiom,
! [C_16: int,B_37: int,A_52: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_37),A_52))
=> ( ( B_37 = C_16 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_16),A_52)) ) ) ).
tff(fact_425_xt1_I2_J,axiom,
! [C_16: real,B_37: real,A_52: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_37),A_52))
=> ( ( B_37 = C_16 )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_16),A_52)) ) ) ).
tff(fact_426_xt1_I2_J,axiom,
! [C_16: nat,B_37: nat,A_52: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_37),A_52))
=> ( ( B_37 = C_16 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_16),A_52)) ) ) ).
tff(fact_427_order__less__trans,axiom,
! [Z_7: int,X_15: int,Y_16: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_15),Y_16))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_16),Z_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_15),Z_7)) ) ) ).
tff(fact_428_order__less__trans,axiom,
! [Z_7: real,X_15: real,Y_16: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_15),Y_16))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_16),Z_7))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_15),Z_7)) ) ) ).
tff(fact_429_order__less__trans,axiom,
! [Z_7: nat,X_15: nat,Y_16: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_15),Y_16))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_16),Z_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_15),Z_7)) ) ) ).
tff(fact_430_xt1_I10_J,axiom,
! [Z_6: int,Y_15: int,X_14: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_15),X_14))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_6),Y_15))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_6),X_14)) ) ) ).
tff(fact_431_xt1_I10_J,axiom,
! [Z_6: real,Y_15: real,X_14: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_15),X_14))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_6),Y_15))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_6),X_14)) ) ) ).
tff(fact_432_xt1_I10_J,axiom,
! [Z_6: nat,Y_15: nat,X_14: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_15),X_14))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_6),Y_15))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_6),X_14)) ) ) ).
tff(fact_433_order__less__asym,axiom,
! [X_13: int,Y_14: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_13),Y_14))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_14),X_13)) ) ).
tff(fact_434_order__less__asym,axiom,
! [X_13: real,Y_14: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_13),Y_14))
=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_14),X_13)) ) ).
tff(fact_435_order__less__asym,axiom,
! [X_13: nat,Y_14: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_13),Y_14))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_14),X_13)) ) ).
tff(fact_436_linorder__cases,axiom,
! [X_12: int,Y_13: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_12),Y_13))
=> ( ( X_12 != Y_13 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_13),X_12)) ) ) ).
tff(fact_437_linorder__cases,axiom,
! [X_12: real,Y_13: real] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_12),Y_13))
=> ( ( X_12 != Y_13 )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_13),X_12)) ) ) ).
tff(fact_438_linorder__cases,axiom,
! [X_12: nat,Y_13: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_12),Y_13))
=> ( ( X_12 != Y_13 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_13),X_12)) ) ) ).
tff(fact_439_linorder__not__less,axiom,
! [X_1: int,Y_1: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1)) ) ).
tff(fact_440_linorder__not__less,axiom,
! [X_1: real,Y_1: real] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),X_1)) ) ).
tff(fact_441_linorder__not__less,axiom,
! [X_1: nat,Y_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1)) ) ).
tff(fact_442_linorder__not__le,axiom,
! [X_1: int,Y_1: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),X_1)) ) ).
tff(fact_443_linorder__not__le,axiom,
! [X_1: real,Y_1: real] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),X_1)) ) ).
tff(fact_444_linorder__not__le,axiom,
! [X_1: nat,Y_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_1),X_1)) ) ).
tff(fact_445_linorder__le__less__linear,axiom,
! [X_11: int,Y_12: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_11),Y_12))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_12),X_11)) ) ).
tff(fact_446_linorder__le__less__linear,axiom,
! [X_11: real,Y_12: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_11),Y_12))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_12),X_11)) ) ).
tff(fact_447_linorder__le__less__linear,axiom,
! [X_11: nat,Y_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_11),Y_12))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_12),X_11)) ) ).
tff(fact_448_order__less__le,axiom,
! [X_1: int,Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
& ( X_1 != Y_1 ) ) ) ).
tff(fact_449_order__less__le,axiom,
! [X_1: real,Y_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
& ( X_1 != Y_1 ) ) ) ).
tff(fact_450_order__less__le,axiom,
! [X_1: nat,Y_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
& ( X_1 != Y_1 ) ) ) ).
tff(fact_451_less__le__not__le,axiom,
! [X_1: int,Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
& ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1)) ) ) ).
tff(fact_452_less__le__not__le,axiom,
! [X_1: real,Y_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
& ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),X_1)) ) ) ).
tff(fact_453_less__le__not__le,axiom,
! [X_1: nat,Y_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1)) ) ) ).
tff(fact_454_order__le__less,axiom,
! [X_1: int,Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
| ( X_1 = Y_1 ) ) ) ).
tff(fact_455_order__le__less,axiom,
! [X_1: real,Y_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
| ( X_1 = Y_1 ) ) ) ).
tff(fact_456_order__le__less,axiom,
! [X_1: nat,Y_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
| ( X_1 = Y_1 ) ) ) ).
tff(fact_457_leI,axiom,
! [X_10: int,Y_11: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_10),Y_11))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_11),X_10)) ) ).
tff(fact_458_leI,axiom,
! [X_10: real,Y_11: real] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_10),Y_11))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_11),X_10)) ) ).
tff(fact_459_leI,axiom,
! [X_10: nat,Y_11: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_10),Y_11))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_11),X_10)) ) ).
tff(fact_460_not__leE,axiom,
! [Y_10: int,X_9: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_10),X_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_9),Y_10)) ) ).
tff(fact_461_not__leE,axiom,
! [Y_10: real,X_9: real] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_10),X_9))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_9),Y_10)) ) ).
tff(fact_462_not__leE,axiom,
! [Y_10: nat,X_9: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_10),X_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_9),Y_10)) ) ).
tff(fact_463_linorder__antisym__conv1,axiom,
! [X_1: int,Y_1: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
<=> ( X_1 = Y_1 ) ) ) ).
tff(fact_464_linorder__antisym__conv1,axiom,
! [X_1: real,Y_1: real] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
<=> ( X_1 = Y_1 ) ) ) ).
tff(fact_465_linorder__antisym__conv1,axiom,
! [X_1: nat,Y_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
<=> ( X_1 = Y_1 ) ) ) ).
tff(fact_466_order__neq__le__trans,axiom,
! [A_51: int,B_36: int] :
( ( A_51 != B_36 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_51),B_36))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_51),B_36)) ) ) ).
tff(fact_467_order__neq__le__trans,axiom,
! [A_51: real,B_36: real] :
( ( A_51 != B_36 )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_51),B_36))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_51),B_36)) ) ) ).
tff(fact_468_order__neq__le__trans,axiom,
! [A_51: nat,B_36: nat] :
( ( A_51 != B_36 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_51),B_36))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_51),B_36)) ) ) ).
tff(fact_469_xt1_I12_J,axiom,
! [A_50: int,B_35: int] :
( ( A_50 != B_35 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_35),A_50))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_35),A_50)) ) ) ).
tff(fact_470_xt1_I12_J,axiom,
! [A_50: real,B_35: real] :
( ( A_50 != B_35 )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_35),A_50))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_35),A_50)) ) ) ).
tff(fact_471_xt1_I12_J,axiom,
! [A_50: nat,B_35: nat] :
( ( A_50 != B_35 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_35),A_50))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_35),A_50)) ) ) ).
tff(fact_472_leD,axiom,
! [Y_9: int,X_8: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_9),X_8))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_8),Y_9)) ) ).
tff(fact_473_leD,axiom,
! [Y_9: real,X_8: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_9),X_8))
=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_8),Y_9)) ) ).
tff(fact_474_leD,axiom,
! [Y_9: nat,X_8: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_9),X_8))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_8),Y_9)) ) ).
tff(fact_475_norR__mem__unique__aux,axiom,
! [A: int,B: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),hAPP_int_int(minus_minus_int(B),one_one_int)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) ) ).
tff(fact_476_int__le__induct,axiom,
! [P_1: fun_int_bool,I_1: int,K_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),K_1))
=> ( hBOOL(hAPP_int_bool(P_1,K_1))
=> ( ! [I_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1))
=> ( hBOOL(hAPP_int_bool(P_1,I_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(I_2),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).
tff(fact_477_int__less__induct,axiom,
! [P_1: fun_int_bool,I_1: int,K_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),K_1))
=> ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(K_1),one_one_int)))
=> ( ! [I_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),K_1))
=> ( hBOOL(hAPP_int_bool(P_1,I_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(I_2),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).
tff(fact_478_int__ge__induct,axiom,
! [P_1: fun_int_bool,K_1: int,I_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_1))
=> ( hBOOL(hAPP_int_bool(P_1,K_1))
=> ( ! [I_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2))
=> ( hBOOL(hAPP_int_bool(P_1,I_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(I_2),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).
tff(fact_479_int__gr__induct,axiom,
! [P_1: fun_int_bool,K_1: int,I_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_1))
=> ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(K_1),one_one_int)))
=> ( ! [I_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_2))
=> ( hBOOL(hAPP_int_bool(P_1,I_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(I_2),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).
tff(fact_480_dense__le,axiom,
! [Z_5: real,Y_8: real] :
( ! [X_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_8))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Z_5)) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_8),Z_5)) ) ).
tff(fact_481_dense__le__bounded,axiom,
! [Z_4: real,X_7: real,Y_7: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_7),Y_7))
=> ( ! [W_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_7),W_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,W_2),Y_7))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W_2),Z_4)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_7),Z_4)) ) ) ).
tff(fact_482_int__induct,axiom,
! [I_1: int,P_1: fun_int_bool,K_1: int] :
( hBOOL(hAPP_int_bool(P_1,K_1))
=> ( ! [I_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2))
=> ( hBOOL(hAPP_int_bool(P_1,I_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(I_2),one_one_int))) ) )
=> ( ! [I_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1))
=> ( hBOOL(hAPP_int_bool(P_1,I_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(I_2),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).
tff(fact_483_Euler_Oaux2,axiom,
! [B: int,A: int,C: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),C))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),C))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) ) ) ) ).
tff(fact_484_d22set__induct__old,axiom,
! [X_1: int,P_1: fun_int_bool] :
( ! [A_49: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_49))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(A_49),one_one_int))) )
=> hBOOL(hAPP_int_bool(P_1,A_49)) )
=> hBOOL(hAPP_int_bool(P_1,X_1)) ) ).
tff(fact_485_convex__bound__lt,axiom,
! [V_5: int,U_2: int,Y_6: int,X_6: int,A_48: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_6),A_48))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_6),A_48))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V_5))
=> ( ( hAPP_int_int(plus_plus_int(U_2),V_5) = one_one_int )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(U_2),X_6)),hAPP_int_int(times_times_int(V_5),Y_6))),A_48)) ) ) ) ) ) ).
tff(fact_486_convex__bound__lt,axiom,
! [V_5: real,U_2: real,Y_6: real,X_6: real,A_48: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_6),A_48))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_6),A_48))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),U_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),V_5))
=> ( ( hAPP_real_real(plus_plus_real(U_2),V_5) = one_one_real )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(U_2),X_6)),hAPP_real_real(times_times_real(V_5),Y_6))),A_48)) ) ) ) ) ) ).
tff(fact_487_number__of__le__ceiling,axiom,
! [V_1: int,X_1: real] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(V_1)),archim856651990g_real(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(minus_minus_real(number267125858f_real(V_1)),one_one_real)),X_1)) ) ).
tff(fact_488_ceiling__less__number__of,axiom,
! [X_1: real,V_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim856651990g_real(X_1)),number_number_of_int(V_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(minus_minus_real(number267125858f_real(V_1)),one_one_real))) ) ).
tff(fact_489_zero__less__ceiling,axiom,
! [X_1: real] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),archim856651990g_real(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) ) ).
tff(fact_490_zero__reorient,axiom,
! [X_1: int] :
( ( zero_zero_int = X_1 )
<=> ( X_1 = zero_zero_int ) ) ).
tff(fact_491_zero__reorient,axiom,
! [X_1: real] :
( ( zero_zero_real = X_1 )
<=> ( X_1 = zero_zero_real ) ) ).
tff(fact_492_zero__reorient,axiom,
! [X_1: nat] :
( ( zero_zero_nat = X_1 )
<=> ( X_1 = zero_zero_nat ) ) ).
tff(fact_493_ceiling__zero,axiom,
archim856651990g_real(zero_zero_real) = zero_zero_int ).
tff(fact_494_ceiling__le__zero,axiom,
! [X_1: real] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X_1)),zero_zero_int))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)) ) ).
tff(fact_495_ceiling__less__one,axiom,
! [X_1: real] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim856651990g_real(X_1)),one_one_int))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)) ) ).
tff(fact_496_one__le__ceiling,axiom,
! [X_1: real] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),archim856651990g_real(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) ) ).
tff(fact_497_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_47: nat] : hAPP_nat_nat(times_times_nat(zero_zero_nat),A_47) = zero_zero_nat ).
tff(fact_498_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_47: real] : hAPP_real_real(times_times_real(zero_zero_real),A_47) = zero_zero_real ).
tff(fact_499_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_47: int] : hAPP_int_int(times_times_int(zero_zero_int),A_47) = zero_zero_int ).
tff(fact_500_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_46: nat] : hAPP_nat_nat(times_times_nat(A_46),zero_zero_nat) = zero_zero_nat ).
tff(fact_501_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_46: real] : hAPP_real_real(times_times_real(A_46),zero_zero_real) = zero_zero_real ).
tff(fact_502_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_46: int] : hAPP_int_int(times_times_int(A_46),zero_zero_int) = zero_zero_int ).
tff(fact_503_mult__zero__left,axiom,
! [A_45: nat] : hAPP_nat_nat(times_times_nat(zero_zero_nat),A_45) = zero_zero_nat ).
tff(fact_504_mult__zero__left,axiom,
! [A_45: real] : hAPP_real_real(times_times_real(zero_zero_real),A_45) = zero_zero_real ).
tff(fact_505_mult__zero__left,axiom,
! [A_45: int] : hAPP_int_int(times_times_int(zero_zero_int),A_45) = zero_zero_int ).
tff(fact_506_mult_Ozero__left,axiom,
! [B_34: real] : hAPP_real_real(times_times_real(zero_zero_real),B_34) = zero_zero_real ).
tff(fact_507_mult__left_Ozero,axiom,
! [Y_5: real] : hAPP_real_real(times_times_real(zero_zero_real),Y_5) = zero_zero_real ).
tff(fact_508_mult__zero__right,axiom,
! [A_44: nat] : hAPP_nat_nat(times_times_nat(A_44),zero_zero_nat) = zero_zero_nat ).
tff(fact_509_mult__zero__right,axiom,
! [A_44: real] : hAPP_real_real(times_times_real(A_44),zero_zero_real) = zero_zero_real ).
tff(fact_510_mult__zero__right,axiom,
! [A_44: int] : hAPP_int_int(times_times_int(A_44),zero_zero_int) = zero_zero_int ).
tff(fact_511_mult_Ozero__right,axiom,
! [A_43: real] : hAPP_real_real(times_times_real(A_43),zero_zero_real) = zero_zero_real ).
tff(fact_512_mult__right_Ozero,axiom,
! [X_5: real] : hAPP_real_real(times_times_real(X_5),zero_zero_real) = zero_zero_real ).
tff(fact_513_mult__eq__0__iff,axiom,
! [Aa: real,Ba: real] :
( ( hAPP_real_real(times_times_real(Aa),Ba) = zero_zero_real )
<=> ( ( Aa = zero_zero_real )
| ( Ba = zero_zero_real ) ) ) ).
tff(fact_514_mult__eq__0__iff,axiom,
! [Aa: int,Ba: int] :
( ( hAPP_int_int(times_times_int(Aa),Ba) = zero_zero_int )
<=> ( ( Aa = zero_zero_int )
| ( Ba = zero_zero_int ) ) ) ).
tff(fact_515_no__zero__divisors,axiom,
! [B_33: nat,A_42: nat] :
( ( A_42 != zero_zero_nat )
=> ( ( B_33 != zero_zero_nat )
=> ( hAPP_nat_nat(times_times_nat(A_42),B_33) != zero_zero_nat ) ) ) ).
tff(fact_516_no__zero__divisors,axiom,
! [B_33: real,A_42: real] :
( ( A_42 != zero_zero_real )
=> ( ( B_33 != zero_zero_real )
=> ( hAPP_real_real(times_times_real(A_42),B_33) != zero_zero_real ) ) ) ).
tff(fact_517_no__zero__divisors,axiom,
! [B_33: int,A_42: int] :
( ( A_42 != zero_zero_int )
=> ( ( B_33 != zero_zero_int )
=> ( hAPP_int_int(times_times_int(A_42),B_33) != zero_zero_int ) ) ) ).
tff(fact_518_divisors__zero,axiom,
! [A_41: nat,B_32: nat] :
( ( hAPP_nat_nat(times_times_nat(A_41),B_32) = zero_zero_nat )
=> ( ( A_41 = zero_zero_nat )
| ( B_32 = zero_zero_nat ) ) ) ).
tff(fact_519_divisors__zero,axiom,
! [A_41: real,B_32: real] :
( ( hAPP_real_real(times_times_real(A_41),B_32) = zero_zero_real )
=> ( ( A_41 = zero_zero_real )
| ( B_32 = zero_zero_real ) ) ) ).
tff(fact_520_divisors__zero,axiom,
! [A_41: int,B_32: int] :
( ( hAPP_int_int(times_times_int(A_41),B_32) = zero_zero_int )
=> ( ( A_41 = zero_zero_int )
| ( B_32 = zero_zero_int ) ) ) ).
tff(fact_521_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_40: nat] : hAPP_nat_nat(plus_plus_nat(zero_zero_nat),A_40) = A_40 ).
tff(fact_522_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_40: real] : hAPP_real_real(plus_plus_real(zero_zero_real),A_40) = A_40 ).
tff(fact_523_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_40: int] : hAPP_int_int(plus_plus_int(zero_zero_int),A_40) = A_40 ).
tff(fact_524_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_39: nat] : hAPP_nat_nat(plus_plus_nat(A_39),zero_zero_nat) = A_39 ).
tff(fact_525_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_39: real] : hAPP_real_real(plus_plus_real(A_39),zero_zero_real) = A_39 ).
tff(fact_526_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_39: int] : hAPP_int_int(plus_plus_int(A_39),zero_zero_int) = A_39 ).
tff(fact_527_add__0__iff,axiom,
! [Ba: nat,Aa: nat] :
( ( Ba = hAPP_nat_nat(plus_plus_nat(Ba),Aa) )
<=> ( Aa = zero_zero_nat ) ) ).
tff(fact_528_add__0__iff,axiom,
! [Ba: real,Aa: real] :
( ( Ba = hAPP_real_real(plus_plus_real(Ba),Aa) )
<=> ( Aa = zero_zero_real ) ) ).
tff(fact_529_add__0__iff,axiom,
! [Ba: int,Aa: int] :
( ( Ba = hAPP_int_int(plus_plus_int(Ba),Aa) )
<=> ( Aa = zero_zero_int ) ) ).
tff(fact_530_add__0__left,axiom,
! [A_38: nat] : hAPP_nat_nat(plus_plus_nat(zero_zero_nat),A_38) = A_38 ).
tff(fact_531_add__0__left,axiom,
! [A_38: real] : hAPP_real_real(plus_plus_real(zero_zero_real),A_38) = A_38 ).
tff(fact_532_add__0__left,axiom,
! [A_38: int] : hAPP_int_int(plus_plus_int(zero_zero_int),A_38) = A_38 ).
tff(fact_533_add__0,axiom,
! [A_37: nat] : hAPP_nat_nat(plus_plus_nat(zero_zero_nat),A_37) = A_37 ).
tff(fact_534_add__0,axiom,
! [A_37: real] : hAPP_real_real(plus_plus_real(zero_zero_real),A_37) = A_37 ).
tff(fact_535_add__0,axiom,
! [A_37: int] : hAPP_int_int(plus_plus_int(zero_zero_int),A_37) = A_37 ).
tff(fact_536_double__zero__sym,axiom,
! [Aa: real] :
( ( zero_zero_real = hAPP_real_real(plus_plus_real(Aa),Aa) )
<=> ( Aa = zero_zero_real ) ) ).
tff(fact_537_double__zero__sym,axiom,
! [Aa: int] :
( ( zero_zero_int = hAPP_int_int(plus_plus_int(Aa),Aa) )
<=> ( Aa = zero_zero_int ) ) ).
tff(fact_538_add__0__right,axiom,
! [A_36: nat] : hAPP_nat_nat(plus_plus_nat(A_36),zero_zero_nat) = A_36 ).
tff(fact_539_add__0__right,axiom,
! [A_36: real] : hAPP_real_real(plus_plus_real(A_36),zero_zero_real) = A_36 ).
tff(fact_540_add__0__right,axiom,
! [A_36: int] : hAPP_int_int(plus_plus_int(A_36),zero_zero_int) = A_36 ).
tff(fact_541_add_Ocomm__neutral,axiom,
! [A_35: nat] : hAPP_nat_nat(plus_plus_nat(A_35),zero_zero_nat) = A_35 ).
tff(fact_542_add_Ocomm__neutral,axiom,
! [A_35: real] : hAPP_real_real(plus_plus_real(A_35),zero_zero_real) = A_35 ).
tff(fact_543_add_Ocomm__neutral,axiom,
! [A_35: int] : hAPP_int_int(plus_plus_int(A_35),zero_zero_int) = A_35 ).
tff(fact_544_double__eq__0__iff,axiom,
! [Aa: real] :
( ( hAPP_real_real(plus_plus_real(Aa),Aa) = zero_zero_real )
<=> ( Aa = zero_zero_real ) ) ).
tff(fact_545_double__eq__0__iff,axiom,
! [Aa: int] :
( ( hAPP_int_int(plus_plus_int(Aa),Aa) = zero_zero_int )
<=> ( Aa = zero_zero_int ) ) ).
tff(fact_546_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
tff(fact_547_one__neq__zero,axiom,
one_one_real != zero_zero_real ).
tff(fact_548_one__neq__zero,axiom,
one_one_int != zero_zero_int ).
tff(fact_549_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
tff(fact_550_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
tff(fact_551_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
tff(fact_552_diff__0__right,axiom,
! [A_34: real] : hAPP_real_real(minus_minus_real(A_34),zero_zero_real) = A_34 ).
tff(fact_553_diff__0__right,axiom,
! [A_34: int] : hAPP_int_int(minus_minus_int(A_34),zero_zero_int) = A_34 ).
tff(fact_554_diff__self,axiom,
! [A_33: real] : hAPP_real_real(minus_minus_real(A_33),A_33) = zero_zero_real ).
tff(fact_555_diff__self,axiom,
! [A_33: int] : hAPP_int_int(minus_minus_int(A_33),A_33) = zero_zero_int ).
tff(fact_556_eq__iff__diff__eq__0,axiom,
! [Aa: real,Ba: real] :
( ( Aa = Ba )
<=> ( hAPP_real_real(minus_minus_real(Aa),Ba) = zero_zero_real ) ) ).
tff(fact_557_eq__iff__diff__eq__0,axiom,
! [Aa: int,Ba: int] :
( ( Aa = Ba )
<=> ( hAPP_int_int(minus_minus_int(Aa),Ba) = zero_zero_int ) ) ).
tff(fact_558_right__minus__eq,axiom,
! [Aa: real,Ba: real] :
( ( hAPP_real_real(minus_minus_real(Aa),Ba) = zero_zero_real )
<=> ( Aa = Ba ) ) ).
tff(fact_559_right__minus__eq,axiom,
! [Aa: int,Ba: int] :
( ( hAPP_int_int(minus_minus_int(Aa),Ba) = zero_zero_int )
<=> ( Aa = Ba ) ) ).
tff(fact_560_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
tff(fact_561_zadd__0,axiom,
! [Z_1: int] : hAPP_int_int(plus_plus_int(zero_zero_int),Z_1) = Z_1 ).
tff(fact_562_zadd__0__right,axiom,
! [Z_1: int] : hAPP_int_int(plus_plus_int(Z_1),zero_zero_int) = Z_1 ).
tff(fact_563_ceiling__less__cancel,axiom,
! [X_4: real,Y_4: real] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim856651990g_real(X_4)),archim856651990g_real(Y_4)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_4),Y_4)) ) ).
tff(fact_564_ceiling__mono,axiom,
! [Y_3: real,X_3: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_3),X_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(Y_3)),archim856651990g_real(X_3))) ) ).
tff(fact_565_ceiling__one,axiom,
archim856651990g_real(one_one_real) = one_one_int ).
tff(fact_566_ceiling__number__of,axiom,
! [V_4: int] : archim856651990g_real(number267125858f_real(V_4)) = number_number_of_int(V_4) ).
tff(fact_567_zero__le__square,axiom,
! [A_32: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_32),A_32))) ).
tff(fact_568_zero__le__square,axiom,
! [A_32: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_32),A_32))) ).
tff(fact_569_zero__le__mult__iff,axiom,
! [Aa: real,Ba: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(Aa),Ba)))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Aa))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Ba)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ba),zero_zero_real)) ) ) ) ).
tff(fact_570_zero__le__mult__iff,axiom,
! [Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(Aa),Ba)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Aa))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ba)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),zero_zero_int)) ) ) ) ).
tff(fact_571_mult__le__0__iff,axiom,
! [Aa: real,Ba: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(Aa),Ba)),zero_zero_real))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Aa))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ba),zero_zero_real)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Ba)) ) ) ) ).
tff(fact_572_mult__le__0__iff,axiom,
! [Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(Aa),Ba)),zero_zero_int))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Aa))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),zero_zero_int)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ba)) ) ) ) ).
tff(fact_573_mult__nonneg__nonneg,axiom,
! [B_31: nat,A_31: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_31))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_31))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(A_31),B_31))) ) ) ).
tff(fact_574_mult__nonneg__nonneg,axiom,
! [B_31: real,A_31: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_31))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_31))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_31),B_31))) ) ) ).
tff(fact_575_mult__nonneg__nonneg,axiom,
! [B_31: int,A_31: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_31))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_31))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_31),B_31))) ) ) ).
tff(fact_576_mult__nonneg__nonpos,axiom,
! [B_30: nat,A_30: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_30))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_30),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_30),B_30)),zero_zero_nat)) ) ) ).
tff(fact_577_mult__nonneg__nonpos,axiom,
! [B_30: real,A_30: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_30))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_30),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_30),B_30)),zero_zero_real)) ) ) ).
tff(fact_578_mult__nonneg__nonpos,axiom,
! [B_30: int,A_30: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_30))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_30),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_30),B_30)),zero_zero_int)) ) ) ).
tff(fact_579_mult__nonneg__nonpos2,axiom,
! [B_29: nat,A_29: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_29))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_29),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(B_29),A_29)),zero_zero_nat)) ) ) ).
tff(fact_580_mult__nonneg__nonpos2,axiom,
! [B_29: real,A_29: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_29))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_29),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(B_29),A_29)),zero_zero_real)) ) ) ).
tff(fact_581_mult__nonneg__nonpos2,axiom,
! [B_29: int,A_29: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_29))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_29),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(B_29),A_29)),zero_zero_int)) ) ) ).
tff(fact_582_mult__nonpos__nonneg,axiom,
! [B_28: nat,A_28: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_28),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_28))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_28),B_28)),zero_zero_nat)) ) ) ).
tff(fact_583_mult__nonpos__nonneg,axiom,
! [B_28: real,A_28: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_28),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_28))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_28),B_28)),zero_zero_real)) ) ) ).
tff(fact_584_mult__nonpos__nonneg,axiom,
! [B_28: int,A_28: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_28),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_28))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_28),B_28)),zero_zero_int)) ) ) ).
tff(fact_585_mult__nonpos__nonpos,axiom,
! [B_27: real,A_27: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_27),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_27),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_27),B_27))) ) ) ).
tff(fact_586_mult__nonpos__nonpos,axiom,
! [B_27: int,A_27: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_27),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_27),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_27),B_27))) ) ) ).
tff(fact_587_mult__right__mono,axiom,
! [C_15: nat,A_26: nat,B_26: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_26),B_26))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_15))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_26),C_15)),hAPP_nat_nat(times_times_nat(B_26),C_15))) ) ) ).
tff(fact_588_mult__right__mono,axiom,
! [C_15: real,A_26: real,B_26: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_26),B_26))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_15))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_26),C_15)),hAPP_real_real(times_times_real(B_26),C_15))) ) ) ).
tff(fact_589_mult__right__mono,axiom,
! [C_15: int,A_26: int,B_26: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_26),B_26))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_15))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_26),C_15)),hAPP_int_int(times_times_int(B_26),C_15))) ) ) ).
tff(fact_590_mult__left__mono,axiom,
! [C_14: nat,A_25: nat,B_25: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_25),B_25))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_14))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(C_14),A_25)),hAPP_nat_nat(times_times_nat(C_14),B_25))) ) ) ).
tff(fact_591_mult__left__mono,axiom,
! [C_14: real,A_25: real,B_25: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_25),B_25))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_14))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(C_14),A_25)),hAPP_real_real(times_times_real(C_14),B_25))) ) ) ).
tff(fact_592_mult__left__mono,axiom,
! [C_14: int,A_25: int,B_25: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_25),B_25))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_14))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(C_14),A_25)),hAPP_int_int(times_times_int(C_14),B_25))) ) ) ).
tff(fact_593_comm__mult__left__mono,axiom,
! [C_13: nat,A_24: nat,B_24: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_24),B_24))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_13))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(C_13),A_24)),hAPP_nat_nat(times_times_nat(C_13),B_24))) ) ) ).
tff(fact_594_comm__mult__left__mono,axiom,
! [C_13: real,A_24: real,B_24: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_24),B_24))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_13))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(C_13),A_24)),hAPP_real_real(times_times_real(C_13),B_24))) ) ) ).
tff(fact_595_comm__mult__left__mono,axiom,
! [C_13: int,A_24: int,B_24: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_24),B_24))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(C_13),A_24)),hAPP_int_int(times_times_int(C_13),B_24))) ) ) ).
tff(fact_596_mult__right__mono__neg,axiom,
! [C_12: real,B_23: real,A_23: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_23),A_23))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_12),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_23),C_12)),hAPP_real_real(times_times_real(B_23),C_12))) ) ) ).
tff(fact_597_mult__right__mono__neg,axiom,
! [C_12: int,B_23: int,A_23: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_23),A_23))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_12),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_23),C_12)),hAPP_int_int(times_times_int(B_23),C_12))) ) ) ).
tff(fact_598_mult__left__mono__neg,axiom,
! [C_11: real,B_22: real,A_22: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_22),A_22))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_11),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(C_11),A_22)),hAPP_real_real(times_times_real(C_11),B_22))) ) ) ).
tff(fact_599_mult__left__mono__neg,axiom,
! [C_11: int,B_22: int,A_22: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_22),A_22))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_11),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(C_11),A_22)),hAPP_int_int(times_times_int(C_11),B_22))) ) ) ).
tff(fact_600_mult__mono_H,axiom,
! [C_10: nat,D_4: nat,A_21: nat,B_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_21),B_21))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_10),D_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_21))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_10))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_21),C_10)),hAPP_nat_nat(times_times_nat(B_21),D_4))) ) ) ) ) ).
tff(fact_601_mult__mono_H,axiom,
! [C_10: real,D_4: real,A_21: real,B_21: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_21),B_21))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_10),D_4))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_21))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_10))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_21),C_10)),hAPP_real_real(times_times_real(B_21),D_4))) ) ) ) ) ).
tff(fact_602_mult__mono_H,axiom,
! [C_10: int,D_4: int,A_21: int,B_21: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_21),B_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_10),D_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_10))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_21),C_10)),hAPP_int_int(times_times_int(B_21),D_4))) ) ) ) ) ).
tff(fact_603_mult__mono,axiom,
! [C_9: nat,D_3: nat,A_20: nat,B_20: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_20),B_20))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_9),D_3))
=> ( 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),C_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_20),C_9)),hAPP_nat_nat(times_times_nat(B_20),D_3))) ) ) ) ) ).
tff(fact_604_mult__mono,axiom,
! [C_9: real,D_3: real,A_20: real,B_20: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_20),B_20))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_9),D_3))
=> ( 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),C_9))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_20),C_9)),hAPP_real_real(times_times_real(B_20),D_3))) ) ) ) ) ).
tff(fact_605_mult__mono,axiom,
! [C_9: int,D_3: int,A_20: int,B_20: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_20),B_20))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_9),D_3))
=> ( 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),C_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_20),C_9)),hAPP_int_int(times_times_int(B_20),D_3))) ) ) ) ) ).
tff(fact_606_split__mult__pos__le,axiom,
! [B_19: real,A_19: real] :
( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_19))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_19)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_19),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_19),zero_zero_real)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_19),B_19))) ) ).
tff(fact_607_split__mult__pos__le,axiom,
! [B_19: int,A_19: int] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_19))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_19)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_19),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_19),zero_zero_int)) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_19),B_19))) ) ).
tff(fact_608_split__mult__neg__le,axiom,
! [B_18: nat,A_18: nat] :
( ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_18))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_18),zero_zero_nat)) )
| ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_18),zero_zero_nat))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_18)) ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_18),B_18)),zero_zero_nat)) ) ).
tff(fact_609_split__mult__neg__le,axiom,
! [B_18: real,A_18: real] :
( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_18))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_18),zero_zero_real)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_18),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_18)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_18),B_18)),zero_zero_real)) ) ).
tff(fact_610_split__mult__neg__le,axiom,
! [B_18: int,A_18: int] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_18))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_18),zero_zero_int)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_18),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_18)) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_18),B_18)),zero_zero_int)) ) ).
tff(fact_611_not__square__less__zero,axiom,
! [A_17: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_17),A_17)),zero_zero_real)) ).
tff(fact_612_not__square__less__zero,axiom,
! [A_17: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_17),A_17)),zero_zero_int)) ).
tff(fact_613_mult__less__cancel__right__disj,axiom,
! [Aa: real,C_1: real,Ba: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(Aa),C_1)),hAPP_real_real(times_times_real(Ba),C_1)))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_1),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ba),Aa)) ) ) ) ).
tff(fact_614_mult__less__cancel__right__disj,axiom,
! [Aa: int,C_1: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(Aa),C_1)),hAPP_int_int(times_times_int(Ba),C_1)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ba),Aa)) ) ) ) ).
tff(fact_615_mult__less__cancel__left__disj,axiom,
! [C_1: real,Aa: real,Ba: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_1),Aa)),hAPP_real_real(times_times_real(C_1),Ba)))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_1),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ba),Aa)) ) ) ) ).
tff(fact_616_mult__less__cancel__left__disj,axiom,
! [C_1: int,Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_1),Aa)),hAPP_int_int(times_times_int(C_1),Ba)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ba),Aa)) ) ) ) ).
tff(fact_617_mult__less__cancel__left__pos,axiom,
! [Aa: real,Ba: real,C_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_1),Aa)),hAPP_real_real(times_times_real(C_1),Ba)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba)) ) ) ).
tff(fact_618_mult__less__cancel__left__pos,axiom,
! [Aa: int,Ba: int,C_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_1),Aa)),hAPP_int_int(times_times_int(C_1),Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) ) ) ).
tff(fact_619_mult__pos__pos,axiom,
! [B_17: nat,A_16: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_16))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_17))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(A_16),B_17))) ) ) ).
tff(fact_620_mult__pos__pos,axiom,
! [B_17: real,A_16: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_16))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_17))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(A_16),B_17))) ) ) ).
tff(fact_621_mult__pos__pos,axiom,
! [B_17: int,A_16: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_16))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_17))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(A_16),B_17))) ) ) ).
tff(fact_622_mult__pos__neg,axiom,
! [B_16: nat,A_15: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_15))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_16),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_15),B_16)),zero_zero_nat)) ) ) ).
tff(fact_623_mult__pos__neg,axiom,
! [B_16: real,A_15: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_15))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_16),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_15),B_16)),zero_zero_real)) ) ) ).
tff(fact_624_mult__pos__neg,axiom,
! [B_16: int,A_15: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_15))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_16),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_15),B_16)),zero_zero_int)) ) ) ).
tff(fact_625_mult__pos__neg2,axiom,
! [B_15: 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_nat,B_15),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(B_15),A_14)),zero_zero_nat)) ) ) ).
tff(fact_626_mult__pos__neg2,axiom,
! [B_15: 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_real,B_15),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(B_15),A_14)),zero_zero_real)) ) ) ).
tff(fact_627_mult__pos__neg2,axiom,
! [B_15: 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_int,B_15),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(B_15),A_14)),zero_zero_int)) ) ) ).
tff(fact_628_zero__less__mult__pos,axiom,
! [A_13: nat,B_14: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(A_13),B_14)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_13))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_14)) ) ) ).
tff(fact_629_zero__less__mult__pos,axiom,
! [A_13: real,B_14: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(A_13),B_14)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_13))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_14)) ) ) ).
tff(fact_630_zero__less__mult__pos,axiom,
! [A_13: int,B_14: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(A_13),B_14)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_14)) ) ) ).
tff(fact_631_zero__less__mult__pos2,axiom,
! [B_13: nat,A_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(B_13),A_12)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_13)) ) ) ).
tff(fact_632_zero__less__mult__pos2,axiom,
! [B_13: real,A_12: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(B_13),A_12)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_12))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_13)) ) ) ).
tff(fact_633_zero__less__mult__pos2,axiom,
! [B_13: int,A_12: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(B_13),A_12)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_13)) ) ) ).
tff(fact_634_mult__less__cancel__left__neg,axiom,
! [Aa: real,Ba: real,C_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_1),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_1),Aa)),hAPP_real_real(times_times_real(C_1),Ba)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ba),Aa)) ) ) ).
tff(fact_635_mult__less__cancel__left__neg,axiom,
! [Aa: int,Ba: int,C_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_1),Aa)),hAPP_int_int(times_times_int(C_1),Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ba),Aa)) ) ) ).
tff(fact_636_mult__neg__pos,axiom,
! [B_12: nat,A_11: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_11),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_11),B_12)),zero_zero_nat)) ) ) ).
tff(fact_637_mult__neg__pos,axiom,
! [B_12: real,A_11: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_11),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_12))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_11),B_12)),zero_zero_real)) ) ) ).
tff(fact_638_mult__neg__pos,axiom,
! [B_12: int,A_11: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_11),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_11),B_12)),zero_zero_int)) ) ) ).
tff(fact_639_mult__neg__neg,axiom,
! [B_11: 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_real,B_11),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(A_10),B_11))) ) ) ).
tff(fact_640_mult__neg__neg,axiom,
! [B_11: 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_int,B_11),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(A_10),B_11))) ) ) ).
tff(fact_641_mult__strict__right__mono,axiom,
! [C_8: nat,A_9: nat,B_10: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_9),B_10))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_9),C_8)),hAPP_nat_nat(times_times_nat(B_10),C_8))) ) ) ).
tff(fact_642_mult__strict__right__mono,axiom,
! [C_8: real,A_9: real,B_10: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_9),B_10))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_8))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_9),C_8)),hAPP_real_real(times_times_real(B_10),C_8))) ) ) ).
tff(fact_643_mult__strict__right__mono,axiom,
! [C_8: int,A_9: int,B_10: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_9),B_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_9),C_8)),hAPP_int_int(times_times_int(B_10),C_8))) ) ) ).
tff(fact_644_mult__strict__left__mono,axiom,
! [C_7: nat,A_8: nat,B_9: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_8),B_9))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(C_7),A_8)),hAPP_nat_nat(times_times_nat(C_7),B_9))) ) ) ).
tff(fact_645_mult__strict__left__mono,axiom,
! [C_7: real,A_8: real,B_9: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_8),B_9))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_7))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_7),A_8)),hAPP_real_real(times_times_real(C_7),B_9))) ) ) ).
tff(fact_646_mult__strict__left__mono,axiom,
! [C_7: int,A_8: int,B_9: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_8),B_9))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_7),A_8)),hAPP_int_int(times_times_int(C_7),B_9))) ) ) ).
tff(fact_647_comm__mult__strict__left__mono,axiom,
! [C_6: nat,A_7: nat,B_8: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_7),B_8))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(C_6),A_7)),hAPP_nat_nat(times_times_nat(C_6),B_8))) ) ) ).
tff(fact_648_comm__mult__strict__left__mono,axiom,
! [C_6: real,A_7: real,B_8: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_7),B_8))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_6))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_6),A_7)),hAPP_real_real(times_times_real(C_6),B_8))) ) ) ).
tff(fact_649_comm__mult__strict__left__mono,axiom,
! [C_6: int,A_7: int,B_8: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_7),B_8))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_6))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_6),A_7)),hAPP_int_int(times_times_int(C_6),B_8))) ) ) ).
tff(fact_650_mult__strict__right__mono__neg,axiom,
! [C_5: real,B_7: real,A_6: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_7),A_6))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_5),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_6),C_5)),hAPP_real_real(times_times_real(B_7),C_5))) ) ) ).
tff(fact_651_mult__strict__right__mono__neg,axiom,
! [C_5: int,B_7: int,A_6: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_7),A_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_5),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_6),C_5)),hAPP_int_int(times_times_int(B_7),C_5))) ) ) ).
tff(fact_652_mult__strict__left__mono__neg,axiom,
! [C_4: real,B_6: real,A_5: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_6),A_5))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_4),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_4),A_5)),hAPP_real_real(times_times_real(C_4),B_6))) ) ) ).
tff(fact_653_mult__strict__left__mono__neg,axiom,
! [C_4: int,B_6: int,A_5: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_6),A_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_4),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_4),A_5)),hAPP_int_int(times_times_int(C_4),B_6))) ) ) ).
tff(fact_654_zero__le__double__add__iff__zero__le__single__add,axiom,
! [Aa: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(plus_plus_real(Aa),Aa)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Aa)) ) ).
tff(fact_655_zero__le__double__add__iff__zero__le__single__add,axiom,
! [Aa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(Aa),Aa)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Aa)) ) ).
tff(fact_656_double__add__le__zero__iff__single__add__le__zero,axiom,
! [Aa: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(Aa),Aa)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),zero_zero_real)) ) ).
tff(fact_657_double__add__le__zero__iff__single__add__le__zero,axiom,
! [Aa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(Aa),Aa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),zero_zero_int)) ) ).
tff(fact_658_add__nonneg__nonneg,axiom,
! [B_5: nat,A_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(plus_plus_nat(A_4),B_5))) ) ) ).
tff(fact_659_add__nonneg__nonneg,axiom,
! [B_5: real,A_4: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_4))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_5))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(plus_plus_real(A_4),B_5))) ) ) ).
tff(fact_660_add__nonneg__nonneg,axiom,
! [B_5: int,A_4: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(A_4),B_5))) ) ) ).
tff(fact_661_add__nonneg__eq__0__iff,axiom,
! [Y_1: nat,X_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_1))
=> ( ( hAPP_nat_nat(plus_plus_nat(X_1),Y_1) = zero_zero_nat )
<=> ( ( X_1 = zero_zero_nat )
& ( Y_1 = zero_zero_nat ) ) ) ) ) ).
tff(fact_662_add__nonneg__eq__0__iff,axiom,
! [Y_1: real,X_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1))
=> ( ( hAPP_real_real(plus_plus_real(X_1),Y_1) = zero_zero_real )
<=> ( ( X_1 = zero_zero_real )
& ( Y_1 = zero_zero_real ) ) ) ) ) ).
tff(fact_663_add__nonneg__eq__0__iff,axiom,
! [Y_1: int,X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1))
=> ( ( hAPP_int_int(plus_plus_int(X_1),Y_1) = zero_zero_int )
<=> ( ( X_1 = zero_zero_int )
& ( Y_1 = zero_zero_int ) ) ) ) ) ).
tff(fact_664_add__increasing,axiom,
! [B_4: nat,C_3: nat,A_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_4),C_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_4),hAPP_nat_nat(plus_plus_nat(A_3),C_3))) ) ) ).
tff(fact_665_add__increasing,axiom,
! [B_4: real,C_3: real,A_3: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_3))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),hAPP_real_real(plus_plus_real(A_3),C_3))) ) ) ).
tff(fact_666_add__increasing,axiom,
! [B_4: int,C_3: int,A_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_4),C_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_4),hAPP_int_int(plus_plus_int(A_3),C_3))) ) ) ).
tff(fact_667_add__increasing2,axiom,
! [B_3: nat,A_2: nat,C_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_3),A_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_3),hAPP_nat_nat(plus_plus_nat(A_2),C_2))) ) ) ).
tff(fact_668_add__increasing2,axiom,
! [B_3: real,A_2: real,C_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_3),A_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_3),hAPP_real_real(plus_plus_real(A_2),C_2))) ) ) ).
tff(fact_669_add__increasing2,axiom,
! [B_3: int,A_2: int,C_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_3),A_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_3),hAPP_int_int(plus_plus_int(A_2),C_2))) ) ) ).
tff(fact_670_add__nonpos__nonpos,axiom,
! [B_2: nat,A_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_2),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(A_1),B_2)),zero_zero_nat)) ) ) ).
tff(fact_671_add__nonpos__nonpos,axiom,
! [B_2: real,A_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_2),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_1),B_2)),zero_zero_real)) ) ) ).
tff(fact_672_add__nonpos__nonpos,axiom,
! [B_2: int,A_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_1),B_2)),zero_zero_int)) ) ) ).
tff(fact_673_zero__less__double__add__iff__zero__less__single__add,axiom,
! [Aa: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(plus_plus_real(Aa),Aa)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Aa)) ) ).
tff(fact_674_zero__less__double__add__iff__zero__less__single__add,axiom,
! [Aa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(Aa),Aa)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Aa)) ) ).
tff(fact_675_Euler_Oaux1,axiom,
! [A: int,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),A))
=> ( ( X != hAPP_int_int(minus_minus_int(A),one_one_int) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),hAPP_int_int(minus_minus_int(A),one_one_int))) ) ) ) ).
tff(fact_676_int__0__less__1,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
tff(fact_677_zmult__zless__mono2,axiom,
! [K: int,I: int,J: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(K),I)),hAPP_int_int(times_times_int(K),J))) ) ) ).
tff(fact_678_odd__nonzero,axiom,
! [Z_1: int] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),Z_1)),Z_1) != zero_zero_int ).
tff(fact_679_less__bin__lemma,axiom,
! [K_1: int,L: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(minus_minus_int(K_1),L)),zero_zero_int)) ) ).
tff(fact_680_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Z))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z)) ) ).
tff(fact_681_pos__zmult__eq__1__iff,axiom,
! [N_2: int,M_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M_1))
=> ( ( hAPP_int_int(times_times_int(M_1),N_2) = one_one_int )
<=> ( ( M_1 = one_one_int )
& ( N_2 = one_one_int ) ) ) ) ).
tff(fact_682_odd__less__0,axiom,
! [Z: 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)),Z)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),zero_zero_int)) ) ).
tff(fact_683_le__imp__0__less,axiom,
! [Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(one_one_int),Z_1))) ) ).
tff(fact_684_self__quotient__aux2,axiom,
! [R: int,Q: int,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( ( A = hAPP_int_int(plus_plus_int(R),hAPP_int_int(times_times_int(A),Q)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q),one_one_int)) ) ) ) ).
tff(fact_685_self__quotient__aux1,axiom,
! [R: int,Q: int,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( ( A = hAPP_int_int(plus_plus_int(R),hAPP_int_int(times_times_int(A),Q)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R),A))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Q)) ) ) ) ).
tff(fact_686_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q: int,R: int,B_1: int,Q_1: int,R_1: int] :
( ( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q)),R) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_1),Q_1)),R_1) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_1),Q_1)),R_1)),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q)) ) ) ) ) ) ) ).
tff(fact_687_unique__quotient__lemma__neg,axiom,
! [B: int,Q_1: int,R_1: int,Q: int,R: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q_1)),R_1)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q)),R)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),R))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),R_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q),Q_1)) ) ) ) ) ).
tff(fact_688_zdiv__mono2__lemma,axiom,
! [B: int,Q: int,R: int,B_1: int,Q_1: int,R_1: int] :
( ( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q)),R) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_1),Q_1)),R_1) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_1),Q_1)),R_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q),Q_1)) ) ) ) ) ) ) ).
tff(fact_689_unique__quotient__lemma,axiom,
! [B: int,Q_1: int,R_1: int,Q: int,R: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q_1)),R_1)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q)),R)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q)) ) ) ) ) ).
tff(fact_690_q__neg__lemma,axiom,
! [B_1: int,Q_1: int,R_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_1),Q_1)),R_1)),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),zero_zero_int)) ) ) ) ).
tff(fact_691_q__pos__lemma,axiom,
! [B_1: int,Q_1: int,R_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_1),Q_1)),R_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Q_1)) ) ) ) ).
tff(fact_692_xzgcd__linear,axiom,
! [M: int,R: int,S: int,T: int,N_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
=> ( ( xzgcd(M,N_1) = produc282740534nt_int(R,product_Pair_int_int(S,T)) )
=> ( R = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(S),M)),hAPP_int_int(times_times_int(T),N_1)) ) ) ) ).
tff(fact_693_xzgcda__linear,axiom,
! [M: int,N_1: int,R_1: int,S_1: int,S: int,T_1: int,T: int,Rn: int,Sn: int,Tn: int,R: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R))
=> ( ( xzgcda(M,N_1,R_1,R,S_1,S,T_1,T) = produc282740534nt_int(Rn,product_Pair_int_int(Sn,Tn)) )
=> ( ( R_1 = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(S_1),M)),hAPP_int_int(times_times_int(T_1),N_1)) )
=> ( ( R = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(S),M)),hAPP_int_int(times_times_int(T),N_1)) )
=> ( Rn = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Sn),M)),hAPP_int_int(times_times_int(Tn),N_1)) ) ) ) ) ) ).
tff(fact_694_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_695_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y: int,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(X),Y))) ) ) ).
tff(fact_696_pos__zmult__pos,axiom,
! [B: int,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(A),B)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) ) ) ).
tff(fact_697_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_698_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_699_xzgcd__def,axiom,
! [M: int,N_1: int] : xzgcd(M,N_1) = xzgcda(M,N_1,M,N_1,one_one_int,zero_zero_int,zero_zero_int,one_one_int) ).
tff(fact_700_decr__mult__lemma,axiom,
! [K_1: int,P_1: fun_int_bool,D_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_2))
=> ( ! [X_2: int] :
( hBOOL(hAPP_int_bool(P_1,X_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(X_2),D_2))) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1))
=> ! [X_2: int] :
( hBOOL(hAPP_int_bool(P_1,X_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(X_2),hAPP_int_int(times_times_int(K_1),D_2)))) ) ) ) ) ).
tff(fact_701_incr__mult__lemma,axiom,
! [K_1: int,P_1: fun_int_bool,D_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_2))
=> ( ! [X_2: int] :
( hBOOL(hAPP_int_bool(P_1,X_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(X_2),D_2))) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1))
=> ! [X_2: int] :
( hBOOL(hAPP_int_bool(P_1,X_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(X_2),hAPP_int_int(times_times_int(K_1),D_2)))) ) ) ) ) ).
tff(fact_702_divmod__int__relI,axiom,
! [A: int,B: int,Q: int,R: int] :
( ( A = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q)),R) )
=> ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R),B)) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),R))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R),zero_zero_int)) ) ) )
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,B),product_Pair_int_int(Q,R))) ) ) ).
tff(fact_703_tsub__def,axiom,
! [Y: int,X: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
=> ( nat_tsub(X,Y) = hAPP_int_int(minus_minus_int(X),Y) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
=> ( nat_tsub(X,Y) = zero_zero_int ) ) ) ).
tff(fact_704_divmod__int__rel__0,axiom,
! [B: int] :
( ( B != zero_zero_int )
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(zero_zero_int,B),product_Pair_int_int(zero_zero_int,zero_zero_int))) ) ).
tff(fact_705_self__remainder,axiom,
! [A: int,Q: int,R: int] :
( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,A),product_Pair_int_int(Q,R)))
=> ( ( A != zero_zero_int )
=> ( R = zero_zero_int ) ) ) ).
tff(fact_706_unique__quotient,axiom,
! [Q_1: int,R_1: int,A: int,B: int,Q: int,R: int] :
( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,B),product_Pair_int_int(Q,R)))
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,B),product_Pair_int_int(Q_1,R_1)))
=> ( ( B != zero_zero_int )
=> ( Q = Q_1 ) ) ) ) ).
tff(fact_707_unique__remainder,axiom,
! [Q_1: int,R_1: int,A: int,B: int,Q: int,R: int] :
( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,B),product_Pair_int_int(Q,R)))
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,B),product_Pair_int_int(Q_1,R_1)))
=> ( ( B != zero_zero_int )
=> ( R = R_1 ) ) ) ) ).
tff(fact_708_self__quotient,axiom,
! [A: int,Q: int,R: int] :
( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,A),product_Pair_int_int(Q,R)))
=> ( ( A != zero_zero_int )
=> ( Q = one_one_int ) ) ) ).
tff(fact_709_Nat__Transfer_Otransfer__nat__int__function__closures_I3_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),nat_tsub(X,Y))) ) ) ).
tff(fact_710_tsub__eq,axiom,
! [Y: int,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
=> ( nat_tsub(X,Y) = hAPP_int_int(minus_minus_int(X),Y) ) ) ).
tff(fact_711_conj__le__cong,axiom,
! [P_2: bool,P_1: bool,X_1: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(P_1)
<=> hBOOL(P_2) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
& hBOOL(P_1) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
& hBOOL(P_2) ) ) ) ).
tff(fact_712_imp__le__cong,axiom,
! [P_2: bool,P_1: bool,X_1: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(P_1)
<=> hBOOL(P_2) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hBOOL(P_1) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hBOOL(P_2) ) ) ) ).
tff(fact_713_minusinfinity,axiom,
! [P_1: fun_int_bool,P1: fun_int_bool,D_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_2))
=> ( ! [X_2: int,K_2: int] :
( hBOOL(hAPP_int_bool(P1,X_2))
<=> hBOOL(hAPP_int_bool(P1,hAPP_int_int(minus_minus_int(X_2),hAPP_int_int(times_times_int(K_2),D_2)))) )
=> ( ? [Z_3: int] :
! [X_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Z_3))
=> ( hBOOL(hAPP_int_bool(P_1,X_2))
<=> hBOOL(hAPP_int_bool(P1,X_2)) ) )
=> ( ? [X1: int] : hBOOL(hAPP_int_bool(P1,X1))
=> ? [X1: int] : hBOOL(hAPP_int_bool(P_1,X1)) ) ) ) ) ).
tff(fact_714_plusinfinity,axiom,
! [P_1: fun_int_bool,P_2: fun_int_bool,D_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_2))
=> ( ! [X_2: int,K_2: int] :
( hBOOL(hAPP_int_bool(P_2,X_2))
<=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(minus_minus_int(X_2),hAPP_int_int(times_times_int(K_2),D_2)))) )
=> ( ? [Z_3: int] :
! [X_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_3),X_2))
=> ( hBOOL(hAPP_int_bool(P_1,X_2))
<=> hBOOL(hAPP_int_bool(P_2,X_2)) ) )
=> ( ? [X1: int] : hBOOL(hAPP_int_bool(P_2,X1))
=> ? [X1: int] : hBOOL(hAPP_int_bool(P_1,X1)) ) ) ) ) ).
tff(fact_715_zfact_Osimps,axiom,
! [N_1: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),zero_zero_int))
=> ( zfact(N_1) = one_one_int ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),zero_zero_int))
=> ( zfact(N_1) = hAPP_int_int(times_times_int(N_1),zfact(hAPP_int_int(minus_minus_int(N_1),one_one_int))) ) ) ) ).
tff(fact_716_le__mult__floor,axiom,
! [B: real,A: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(archim1246769320r_real(A)),archim1246769320r_real(B))),archim1246769320r_real(hAPP_real_real(times_times_real(A),B)))) ) ) ).
tff(fact_717_posDivAlg__correct,axiom,
! [B: int,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,B),posDivAlg(A,B))) ) ) ).
tff(fact_718_incr__lemma,axiom,
! [Z_1: int,X: int,D_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),hAPP_int_int(plus_plus_int(X),hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(abs_abs_int(hAPP_int_int(minus_minus_int(X),Z_1))),one_one_int)),D_1)))) ) ).
tff(fact_719_decr__lemma,axiom,
! [X: int,Z_1: int,D_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(minus_minus_int(X),hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(abs_abs_int(hAPP_int_int(minus_minus_int(X),Z_1))),one_one_int)),D_1))),Z_1)) ) ).
tff(fact_720_real__le__refl,axiom,
! [W_1: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W_1),W_1)) ).
tff(fact_721_real__le__linear,axiom,
! [Z_1: real,W_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_1),W_1))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W_1),Z_1)) ) ).
tff(fact_722_real__add__left__mono,axiom,
! [Z_1: 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_1),X)),hAPP_real_real(plus_plus_real(Z_1),Y))) ) ).
tff(fact_723_real__le__trans,axiom,
! [K: real,I: real,J: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I),J))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,J),K))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I),K)) ) ) ).
tff(fact_724_real__le__antisym,axiom,
! [Z_1: real,W_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_1),W_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W_1),Z_1))
=> ( Z_1 = W_1 ) ) ) ).
tff(fact_725_real__add__mult__distrib,axiom,
! [Z1: real,Z2: real,W_1: real] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(Z1),Z2)),W_1) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Z1),W_1)),hAPP_real_real(times_times_real(Z2),W_1)) ).
tff(fact_726_real__mult__assoc,axiom,
! [Z1: real,Z2: real,Z3: real] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Z1),Z2)),Z3) = hAPP_real_real(times_times_real(Z1),hAPP_real_real(times_times_real(Z2),Z3)) ).
tff(fact_727_real__mult__commute,axiom,
! [Z_1: real,W_1: real] : hAPP_real_real(times_times_real(Z_1),W_1) = hAPP_real_real(times_times_real(W_1),Z_1) ).
tff(fact_728_real__mult__1,axiom,
! [Z_1: real] : hAPP_real_real(times_times_real(one_one_real),Z_1) = Z_1 ).
tff(fact_729_less__eq__real__def,axiom,
! [X_1: real,Y_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
| ( X_1 = Y_1 ) ) ) ).
tff(fact_730_real__less__def,axiom,
! [X_1: real,Y_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
& ( X_1 != Y_1 ) ) ) ).
tff(fact_731_real__le__eq__diff,axiom,
! [X_1: real,Y_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(minus_minus_real(X_1),Y_1)),zero_zero_real)) ) ).
tff(fact_732_real__mult__left__cancel,axiom,
! [Aa: real,Ba: real,C_1: real] :
( ( C_1 != zero_zero_real )
=> ( ( hAPP_real_real(times_times_real(C_1),Aa) = hAPP_real_real(times_times_real(C_1),Ba) )
<=> ( Aa = Ba ) ) ) ).
tff(fact_733_real__mult__right__cancel,axiom,
! [Aa: real,Ba: real,C_1: real] :
( ( C_1 != zero_zero_real )
=> ( ( hAPP_real_real(times_times_real(Aa),C_1) = hAPP_real_real(times_times_real(Ba),C_1) )
<=> ( Aa = Ba ) ) ) ).
tff(fact_734_real__two__squares__add__zero__iff,axiom,
! [X_1: real,Y_1: real] :
( ( hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_1),X_1)),hAPP_real_real(times_times_real(Y_1),Y_1)) = zero_zero_real )
<=> ( ( X_1 = zero_zero_real )
& ( Y_1 = zero_zero_real ) ) ) ).
tff(fact_735_real__zero__not__eq__one,axiom,
zero_zero_real != one_one_real ).
tff(fact_736_real__mult__less__iff1,axiom,
! [X_1: real,Y_1: real,Z: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(X_1),Z)),hAPP_real_real(times_times_real(Y_1),Z)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1)) ) ) ).
tff(fact_737_real__mult__order,axiom,
! [Y: real,X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(X),Y))) ) ) ).
tff(fact_738_real__mult__less__mono2,axiom,
! [X: real,Y: real,Z_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(Z_1),X)),hAPP_real_real(times_times_real(Z_1),Y))) ) ) ).
tff(fact_739_real__mult__le__cancel__iff2,axiom,
! [X_1: real,Y_1: real,Z: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(Z),X_1)),hAPP_real_real(times_times_real(Z),Y_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1)) ) ) ).
tff(fact_740_real__mult__le__cancel__iff1,axiom,
! [X_1: real,Y_1: real,Z: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(X_1),Z)),hAPP_real_real(times_times_real(Y_1),Z)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1)) ) ) ).
tff(fact_741_abs__zmult__eq__1,axiom,
! [M: int,N_1: int] :
( ( abs_abs_int(hAPP_int_int(times_times_int(M),N_1)) = one_one_int )
=> ( abs_abs_int(M) = one_one_int ) ) ).
tff(fact_742_zabs__less__one__iff,axiom,
! [Z: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,abs_abs_int(Z)),one_one_int))
<=> ( Z = zero_zero_int ) ) ).
tff(fact_743_posDivAlg__0,axiom,
! [B: int] : posDivAlg(zero_zero_int,B) = product_Pair_int_int(zero_zero_int,zero_zero_int) ).
tff(fact_744_pdivmod__posDivAlg,axiom,
! [K: int,L_1: int] :
( ( ( L_1 = zero_zero_int )
=> ( pdivmod(K,L_1) = product_Pair_int_int(zero_zero_int,abs_abs_int(K)) ) )
& ( ( L_1 != zero_zero_int )
=> ( pdivmod(K,L_1) = posDivAlg(abs_abs_int(K),abs_abs_int(L_1)) ) ) ) ).
tff(fact_745_nat__intermed__int__val,axiom,
! [K_1: int,F: fun_nat_int,N_2: nat,M_1: nat] :
( ! [I_2: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),I_2))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N_2)) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(hAPP_int_int(minus_minus_int(hAPP_nat_int(F,hAPP_nat_nat(plus_plus_nat(I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,M_1)),K_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N_2)))
=> ? [I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),I_2))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N_2))
& ( hAPP_nat_int(F,I_2) = K_1 ) ) ) ) ) ) ).
tff(fact_746_nat0__intermed__int__val,axiom,
! [K_1: int,F: fun_nat_int,N_2: nat] :
( ! [I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(hAPP_int_int(minus_minus_int(hAPP_nat_int(F,hAPP_nat_nat(plus_plus_nat(I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N_2)))
=> ? [I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N_2))
& ( hAPP_nat_int(F,I_2) = K_1 ) ) ) ) ) ).
tff(fact_747_int__val__lemma,axiom,
! [K_1: int,F: fun_nat_int,N_2: nat] :
( ! [I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(hAPP_int_int(minus_minus_int(hAPP_nat_int(F,hAPP_nat_nat(plus_plus_nat(I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N_2)))
=> ? [I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N_2))
& ( hAPP_nat_int(F,I_2) = K_1 ) ) ) ) ) ).
tff(fact_748_mult__eq__if,axiom,
! [N_1: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( hAPP_nat_nat(times_times_nat(M),N_1) = zero_zero_nat ) )
& ( ( M != zero_zero_nat )
=> ( hAPP_nat_nat(times_times_nat(M),N_1) = hAPP_nat_nat(plus_plus_nat(N_1),hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(M),one_one_nat)),N_1)) ) ) ) ).
tff(fact_749_abs__add__one__not__less__self,axiom,
! [X: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(abs_abs_real(X)),one_one_real)),X)) ).
tff(fact_750_abs__add__one__gt__zero,axiom,
! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(plus_plus_real(one_one_real),abs_abs_real(X)))) ).
tff(fact_751_le0,axiom,
! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1)) ).
tff(fact_752_less__zeroE,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).
tff(fact_753_sin__bound__lemma,axiom,
! [U_1: real,V: real,X: real,Y: real] :
( ( X = Y )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,abs_abs_real(U_1)),V))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,abs_abs_real(hAPP_real_real(minus_minus_real(hAPP_real_real(plus_plus_real(X),U_1)),Y))),V)) ) ) ).
tff(fact_754_nat__less__add__iff2,axiom,
! [U: nat,M_1: nat,N_2: nat,I_1: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_1),U)),M_1)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(J_2),I_1)),U)),N_2))) ) ) ).
tff(fact_755_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N_1: nat] : hAPP_nat_nat(times_times_nat(K),hAPP_nat_nat(minus_minus_nat(M),N_1)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(times_times_nat(K),M)),hAPP_nat_nat(times_times_nat(K),N_1)) ).
tff(fact_756_diff__mult__distrib,axiom,
! [M: nat,N_1: nat,K: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(M),N_1)),K) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(times_times_nat(M),K)),hAPP_nat_nat(times_times_nat(N_1),K)) ).
tff(fact_757_diff__commute,axiom,
! [I: nat,J: nat,K: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I),J)),K) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I),K)),J) ).
tff(fact_758_nat__mult__commute,axiom,
! [M: nat,N_1: nat] : hAPP_nat_nat(times_times_nat(M),N_1) = hAPP_nat_nat(times_times_nat(N_1),M) ).
tff(fact_759_nat__mult__assoc,axiom,
! [M: nat,N_1: nat,K: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(M),N_1)),K) = hAPP_nat_nat(times_times_nat(M),hAPP_nat_nat(times_times_nat(N_1),K)) ).
tff(fact_760_mult__0,axiom,
! [N_1: nat] : hAPP_nat_nat(times_times_nat(zero_zero_nat),N_1) = zero_zero_nat ).
tff(fact_761_mult__0__right,axiom,
! [M: nat] : hAPP_nat_nat(times_times_nat(M),zero_zero_nat) = zero_zero_nat ).
tff(fact_762_mult__is__0,axiom,
! [M_1: nat,N_2: nat] :
( ( hAPP_nat_nat(times_times_nat(M_1),N_2) = zero_zero_nat )
<=> ( ( M_1 = zero_zero_nat )
| ( N_2 = zero_zero_nat ) ) ) ).
tff(fact_763_nat__mult__eq__cancel__disj,axiom,
! [K_1: nat,M_1: nat,N_2: nat] :
( ( hAPP_nat_nat(times_times_nat(K_1),M_1) = hAPP_nat_nat(times_times_nat(K_1),N_2) )
<=> ( ( K_1 = zero_zero_nat )
| ( M_1 = N_2 ) ) ) ).
tff(fact_764_mult__cancel1,axiom,
! [K_1: nat,M_1: nat,N_2: nat] :
( ( hAPP_nat_nat(times_times_nat(K_1),M_1) = hAPP_nat_nat(times_times_nat(K_1),N_2) )
<=> ( ( M_1 = N_2 )
| ( K_1 = zero_zero_nat ) ) ) ).
tff(fact_765_mult__cancel2,axiom,
! [M_1: nat,K_1: nat,N_2: nat] :
( ( hAPP_nat_nat(times_times_nat(M_1),K_1) = hAPP_nat_nat(times_times_nat(N_2),K_1) )
<=> ( ( M_1 = N_2 )
| ( K_1 = zero_zero_nat ) ) ) ).
tff(fact_766_diffs0__imp__equal,axiom,
! [M: nat,N_1: nat] :
( ( hAPP_nat_nat(minus_minus_nat(M),N_1) = zero_zero_nat )
=> ( ( hAPP_nat_nat(minus_minus_nat(N_1),M) = zero_zero_nat )
=> ( M = N_1 ) ) ) ).
tff(fact_767_diff__self__eq__0,axiom,
! [M: nat] : hAPP_nat_nat(minus_minus_nat(M),M) = zero_zero_nat ).
tff(fact_768_minus__nat_Odiff__0,axiom,
! [M: nat] : hAPP_nat_nat(minus_minus_nat(M),zero_zero_nat) = M ).
tff(fact_769_diff__0__eq__0,axiom,
! [N_1: nat] : hAPP_nat_nat(minus_minus_nat(zero_zero_nat),N_1) = zero_zero_nat ).
tff(fact_770_nat__less__cases,axiom,
! [P_1: fun_nat_fun_nat_bool,M_1: nat,N_2: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_2),M_1)) )
=> ( ( ( M_1 = N_2 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_2),M_1)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),M_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_2),M_1)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_2),M_1)) ) ) ) ).
tff(fact_771_less__not__refl3,axiom,
! [S: nat,T: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S),T))
=> ( S != T ) ) ).
tff(fact_772_less__not__refl2,axiom,
! [N_1: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M))
=> ( M != N_1 ) ) ).
tff(fact_773_less__irrefl__nat,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1)) ).
tff(fact_774_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_775_nat__neq__iff,axiom,
! [M_1: nat,N_2: nat] :
( ( M_1 != N_2 )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),M_1)) ) ) ).
tff(fact_776_less__not__refl,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1)) ).
tff(fact_777_diff__less__mono2,axiom,
! [L_1: nat,M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),L_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(L_1),N_1)),hAPP_nat_nat(minus_minus_nat(L_1),M))) ) ) ).
tff(fact_778_less__imp__diff__less,axiom,
! [N_1: nat,J: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(J),N_1)),K)) ) ).
tff(fact_779_diff__add__inverse2,axiom,
! [M: nat,N_1: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(M),N_1)),N_1) = M ).
tff(fact_780_diff__add__inverse,axiom,
! [N_1: nat,M: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(N_1),M)),N_1) = M ).
tff(fact_781_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I),J)),K) = hAPP_nat_nat(minus_minus_nat(I),hAPP_nat_nat(plus_plus_nat(J),K)) ).
tff(fact_782_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N_1: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(K),M)),hAPP_nat_nat(plus_plus_nat(K),N_1)) = hAPP_nat_nat(minus_minus_nat(M),N_1) ).
tff(fact_783_diff__cancel2,axiom,
! [M: nat,K: nat,N_1: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(M),K)),hAPP_nat_nat(plus_plus_nat(N_1),K)) = hAPP_nat_nat(minus_minus_nat(M),N_1) ).
tff(fact_784_nat__add__right__cancel,axiom,
! [M_1: nat,K_1: nat,N_2: nat] :
( ( hAPP_nat_nat(plus_plus_nat(M_1),K_1) = hAPP_nat_nat(plus_plus_nat(N_2),K_1) )
<=> ( M_1 = N_2 ) ) ).
tff(fact_785_nat__add__left__cancel,axiom,
! [K_1: nat,M_1: nat,N_2: nat] :
( ( hAPP_nat_nat(plus_plus_nat(K_1),M_1) = hAPP_nat_nat(plus_plus_nat(K_1),N_2) )
<=> ( M_1 = N_2 ) ) ).
tff(fact_786_nat__add__assoc,axiom,
! [M: nat,N_1: nat,K: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(M),N_1)),K) = hAPP_nat_nat(plus_plus_nat(M),hAPP_nat_nat(plus_plus_nat(N_1),K)) ).
tff(fact_787_nat__add__left__commute,axiom,
! [X: nat,Y: nat,Z_1: nat] : hAPP_nat_nat(plus_plus_nat(X),hAPP_nat_nat(plus_plus_nat(Y),Z_1)) = hAPP_nat_nat(plus_plus_nat(Y),hAPP_nat_nat(plus_plus_nat(X),Z_1)) ).
tff(fact_788_nat__add__commute,axiom,
! [M: nat,N_1: nat] : hAPP_nat_nat(plus_plus_nat(M),N_1) = hAPP_nat_nat(plus_plus_nat(N_1),M) ).
tff(fact_789_add__mult__distrib2,axiom,
! [K: nat,M: nat,N_1: nat] : hAPP_nat_nat(times_times_nat(K),hAPP_nat_nat(plus_plus_nat(M),N_1)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(K),M)),hAPP_nat_nat(times_times_nat(K),N_1)) ).
tff(fact_790_add__mult__distrib,axiom,
! [M: nat,N_1: nat,K: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(M),N_1)),K) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(M),K)),hAPP_nat_nat(times_times_nat(N_1),K)) ).
tff(fact_791_left__add__mult__distrib,axiom,
! [I: nat,U_1: nat,J: nat,K: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I),U_1)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J),U_1)),K)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(I),J)),U_1)),K) ).
tff(fact_792_Nat_Odiff__le__self,axiom,
! [M: nat,N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(M),N_1)),M)) ).
tff(fact_793_diff__le__mono2,axiom,
! [L_1: nat,M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(L_1),N_1)),hAPP_nat_nat(minus_minus_nat(L_1),M))) ) ).
tff(fact_794_diff__le__mono,axiom,
! [L_1: nat,M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(M),L_1)),hAPP_nat_nat(minus_minus_nat(N_1),L_1))) ) ).
tff(fact_795_diff__diff__cancel,axiom,
! [I: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),N_1))
=> ( hAPP_nat_nat(minus_minus_nat(N_1),hAPP_nat_nat(minus_minus_nat(N_1),I)) = I ) ) ).
tff(fact_796_eq__diff__iff,axiom,
! [N_2: nat,K_1: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_2))
=> ( ( hAPP_nat_nat(minus_minus_nat(M_1),K_1) = hAPP_nat_nat(minus_minus_nat(N_2),K_1) )
<=> ( M_1 = N_2 ) ) ) ) ).
tff(fact_797_Nat_Odiff__diff__eq,axiom,
! [N_1: nat,K: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_1))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(M),K)),hAPP_nat_nat(minus_minus_nat(N_1),K)) = hAPP_nat_nat(minus_minus_nat(M),N_1) ) ) ) ).
tff(fact_798_le__diff__iff,axiom,
! [N_2: nat,K_1: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(M_1),K_1)),hAPP_nat_nat(minus_minus_nat(N_2),K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) ) ) ) ).
tff(fact_799_le__refl,axiom,
! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),N_1)) ).
tff(fact_800_nat__le__linear,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) ) ).
tff(fact_801_eq__imp__le,axiom,
! [M: nat,N_1: nat] :
( ( M = N_1 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).
tff(fact_802_le__trans,axiom,
! [K: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),K)) ) ) ).
tff(fact_803_le__antisym,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> ( M = N_1 ) ) ) ).
tff(fact_804_mult__le__mono,axiom,
! [K: nat,L_1: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),L_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(I),K)),hAPP_nat_nat(times_times_nat(J),L_1))) ) ) ).
tff(fact_805_mult__le__mono2,axiom,
! [K: nat,I: nat,J: nat] :
( 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(times_times_nat(K),I)),hAPP_nat_nat(times_times_nat(K),J))) ) ).
tff(fact_806_mult__le__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( 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(times_times_nat(I),K)),hAPP_nat_nat(times_times_nat(J),K))) ) ).
tff(fact_807_le__cube,axiom,
! [M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(times_times_nat(M),hAPP_nat_nat(times_times_nat(M),M)))) ).
tff(fact_808_le__square,axiom,
! [M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(times_times_nat(M),M))) ).
tff(fact_809_nat__mult__1,axiom,
! [N_1: nat] : hAPP_nat_nat(times_times_nat(one_one_nat),N_1) = N_1 ).
tff(fact_810_nat__1__eq__mult__iff,axiom,
! [M_1: nat,N_2: nat] :
( ( one_one_nat = hAPP_nat_nat(times_times_nat(M_1),N_2) )
<=> ( ( M_1 = one_one_nat )
& ( N_2 = one_one_nat ) ) ) ).
tff(fact_811_nat__mult__1__right,axiom,
! [N_1: nat] : hAPP_nat_nat(times_times_nat(N_1),one_one_nat) = N_1 ).
tff(fact_812_nat__mult__eq__1__iff,axiom,
! [M_1: nat,N_2: nat] :
( ( hAPP_nat_nat(times_times_nat(M_1),N_2) = one_one_nat )
<=> ( ( M_1 = one_one_nat )
& ( N_2 = one_one_nat ) ) ) ).
tff(fact_813_zero__less__diff,axiom,
! [N_2: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(minus_minus_nat(N_2),M_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) ) ).
tff(fact_814_diff__less,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(M),N_1)),M)) ) ) ).
tff(fact_815_not__less0,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).
tff(fact_816_neq0__conv,axiom,
! [N_2: nat] :
( ( N_2 != zero_zero_nat )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) ) ).
tff(fact_817_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_818_gr__implies__not0,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> ( N_1 != zero_zero_nat ) ) ).
tff(fact_819_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_820_mult__less__mono2,axiom,
! [K: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(K),I)),hAPP_nat_nat(times_times_nat(K),J))) ) ) ).
tff(fact_821_mult__less__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(I),K)),hAPP_nat_nat(times_times_nat(J),K))) ) ) ).
tff(fact_822_nat__mult__less__cancel1,axiom,
! [M_1: nat,N_2: nat,K_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(K_1),M_1)),hAPP_nat_nat(times_times_nat(K_1),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) ) ) ).
tff(fact_823_nat__mult__eq__cancel1,axiom,
! [M_1: nat,N_2: nat,K_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> ( ( hAPP_nat_nat(times_times_nat(K_1),M_1) = hAPP_nat_nat(times_times_nat(K_1),N_2) )
<=> ( M_1 = N_2 ) ) ) ).
tff(fact_824_mult__less__cancel2,axiom,
! [M_1: nat,K_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(M_1),K_1)),hAPP_nat_nat(times_times_nat(N_2),K_1)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) ) ) ).
tff(fact_825_mult__less__cancel1,axiom,
! [K_1: nat,M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(K_1),M_1)),hAPP_nat_nat(times_times_nat(K_1),N_2)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) ) ) ).
tff(fact_826_nat__0__less__mult__iff,axiom,
! [M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(M_1),N_2)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) ) ) ).
tff(fact_827_plus__nat_Oadd__0,axiom,
! [N_1: nat] : hAPP_nat_nat(plus_plus_nat(zero_zero_nat),N_1) = N_1 ).
tff(fact_828_Nat_Oadd__0__right,axiom,
! [M: nat] : hAPP_nat_nat(plus_plus_nat(M),zero_zero_nat) = M ).
tff(fact_829_add__is__0,axiom,
! [M_1: nat,N_2: nat] :
( ( hAPP_nat_nat(plus_plus_nat(M_1),N_2) = zero_zero_nat )
<=> ( ( M_1 = zero_zero_nat )
& ( N_2 = zero_zero_nat ) ) ) ).
tff(fact_830_add__eq__self__zero,axiom,
! [M: nat,N_1: nat] :
( ( hAPP_nat_nat(plus_plus_nat(M),N_1) = M )
=> ( N_1 = zero_zero_nat ) ) ).
tff(fact_831_diff__add__0,axiom,
! [N_1: nat,M: nat] : hAPP_nat_nat(minus_minus_nat(N_1),hAPP_nat_nat(plus_plus_nat(N_1),M)) = zero_zero_nat ).
tff(fact_832_diff__is__0__eq,axiom,
! [M_1: nat,N_2: nat] :
( ( hAPP_nat_nat(minus_minus_nat(M_1),N_2) = zero_zero_nat )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) ) ).
tff(fact_833_diff__is__0__eq_H,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> ( hAPP_nat_nat(minus_minus_nat(M),N_1) = zero_zero_nat ) ) ).
tff(fact_834_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_835_le__0__eq,axiom,
! [N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),zero_zero_nat))
<=> ( N_2 = zero_zero_nat ) ) ).
tff(fact_836_not__add__less1,axiom,
! [I: nat,J: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I),J)),I)) ).
tff(fact_837_not__add__less2,axiom,
! [J: nat,I: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(J),I)),I)) ).
tff(fact_838_nat__add__left__cancel__less,axiom,
! [K_1: nat,M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(K_1),M_1)),hAPP_nat_nat(plus_plus_nat(K_1),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) ) ).
tff(fact_839_trans__less__add1,axiom,
! [M: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(plus_plus_nat(J),M))) ) ).
tff(fact_840_trans__less__add2,axiom,
! [M: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(plus_plus_nat(M),J))) ) ).
tff(fact_841_add__less__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I),K)),hAPP_nat_nat(plus_plus_nat(J),K))) ) ).
tff(fact_842_add__less__mono,axiom,
! [K: nat,L_1: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),L_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I),K)),hAPP_nat_nat(plus_plus_nat(J),L_1))) ) ) ).
tff(fact_843_less__add__eq__less,axiom,
! [M: nat,N_1: nat,K: nat,L_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),L_1))
=> ( ( hAPP_nat_nat(plus_plus_nat(M),L_1) = hAPP_nat_nat(plus_plus_nat(K),N_1) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ) ).
tff(fact_844_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I),J)),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K)) ) ).
tff(fact_845_add__diff__inverse,axiom,
! [M: nat,N_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> ( hAPP_nat_nat(plus_plus_nat(N_1),hAPP_nat_nat(minus_minus_nat(M),N_1)) = M ) ) ).
tff(fact_846_less__diff__conv,axiom,
! [I_1: nat,J_2: nat,K_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(minus_minus_nat(J_2),K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_1),K_1)),J_2)) ) ).
tff(fact_847_mult__eq__self__implies__10,axiom,
! [M: nat,N_1: nat] :
( ( M = hAPP_nat_nat(times_times_nat(M),N_1) )
=> ( ( N_1 = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
tff(fact_848_less__or__eq__imp__le,axiom,
! [M: nat,N_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
| ( M = N_1 ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).
tff(fact_849_le__neq__implies__less,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> ( ( M != N_1 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ) ).
tff(fact_850_less__imp__le__nat,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).
tff(fact_851_le__eq__less__or__eq,axiom,
! [M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
| ( M_1 = N_2 ) ) ) ).
tff(fact_852_nat__less__le,axiom,
! [M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2))
& ( M_1 != N_2 ) ) ) ).
tff(fact_853_less__diff__iff,axiom,
! [N_2: nat,K_1: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(M_1),K_1)),hAPP_nat_nat(minus_minus_nat(N_2),K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) ) ) ) ).
tff(fact_854_diff__less__mono,axiom,
! [C: nat,A: nat,B: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C),A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(A),C)),hAPP_nat_nat(minus_minus_nat(B),C))) ) ) ).
tff(fact_855_diff__diff__right,axiom,
! [I: nat,K: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> ( hAPP_nat_nat(minus_minus_nat(I),hAPP_nat_nat(minus_minus_nat(J),K)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(I),K)),J) ) ) ).
tff(fact_856_le__diff__conv,axiom,
! [J_2: nat,K_1: nat,I_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(J_2),K_1)),I_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),hAPP_nat_nat(plus_plus_nat(I_1),K_1))) ) ).
tff(fact_857_le__add__diff,axiom,
! [M: nat,K: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(N_1),M)),K))) ) ).
tff(fact_858_le__add__diff__inverse,axiom,
! [N_1: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> ( hAPP_nat_nat(plus_plus_nat(N_1),hAPP_nat_nat(minus_minus_nat(M),N_1)) = M ) ) ).
tff(fact_859_add__diff__assoc,axiom,
! [I: nat,K: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> ( hAPP_nat_nat(plus_plus_nat(I),hAPP_nat_nat(minus_minus_nat(J),K)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(I),J)),K) ) ) ).
tff(fact_860_le__diff__conv2,axiom,
! [I_1: nat,K_1: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(minus_minus_nat(J_2),K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(I_1),K_1)),J_2)) ) ) ).
tff(fact_861_le__add__diff__inverse2,axiom,
! [N_1: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(minus_minus_nat(M),N_1)),N_1) = M ) ) ).
tff(fact_862_le__imp__diff__is__add,axiom,
! [K_1: nat,I_1: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_2))
=> ( ( hAPP_nat_nat(minus_minus_nat(J_2),I_1) = K_1 )
<=> ( J_2 = hAPP_nat_nat(plus_plus_nat(K_1),I_1) ) ) ) ).
tff(fact_863_diff__add__assoc,axiom,
! [I: nat,K: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(I),J)),K) = hAPP_nat_nat(plus_plus_nat(I),hAPP_nat_nat(minus_minus_nat(J),K)) ) ) ).
tff(fact_864_add__diff__assoc2,axiom,
! [I: nat,K: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(minus_minus_nat(J),K)),I) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(J),I)),K) ) ) ).
tff(fact_865_diff__add__assoc2,axiom,
! [I: nat,K: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(J),I)),K) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(minus_minus_nat(J),K)),I) ) ) ).
tff(fact_866_nat__eq__add__iff2,axiom,
! [U: nat,M_1: nat,N_2: nat,I_1: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_2))
=> ( ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_1),U)),M_1) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N_2) )
<=> ( M_1 = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(J_2),I_1)),U)),N_2) ) ) ) ).
tff(fact_867_nat__diff__add__eq2,axiom,
! [U_1: nat,M: nat,N_1: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I),U_1)),M)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J),U_1)),N_1)) = hAPP_nat_nat(minus_minus_nat(M),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(J),I)),U_1)),N_1)) ) ) ).
tff(fact_868_nat__le__add__iff2,axiom,
! [U: nat,M_1: nat,N_2: nat,I_1: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_1),U)),M_1)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(J_2),I_1)),U)),N_2))) ) ) ).
tff(fact_869_nat__eq__add__iff1,axiom,
! [U: nat,M_1: nat,N_2: nat,J_2: nat,I_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I_1))
=> ( ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_1),U)),M_1) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N_2) )
<=> ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(I_1),J_2)),U)),M_1) = N_2 ) ) ) ).
tff(fact_870_nat__diff__add__eq1,axiom,
! [U_1: nat,M: nat,N_1: nat,J: nat,I: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I),U_1)),M)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J),U_1)),N_1)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(I),J)),U_1)),M)),N_1) ) ) ).
tff(fact_871_nat__le__add__iff1,axiom,
! [U: nat,M_1: nat,N_2: nat,J_2: nat,I_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_1),U)),M_1)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(I_1),J_2)),U)),M_1)),N_2)) ) ) ).
tff(fact_872_add__leE,axiom,
! [M: nat,K: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M),K)),N_1))
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_1)) ) ) ).
tff(fact_873_add__leD1,axiom,
! [M: nat,K: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M),K)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).
tff(fact_874_add__leD2,axiom,
! [M: nat,K: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M),K)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_1)) ) ).
tff(fact_875_add__le__mono,axiom,
! [K: nat,L_1: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),L_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(I),K)),hAPP_nat_nat(plus_plus_nat(J),L_1))) ) ) ).
tff(fact_876_add__le__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( 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(plus_plus_nat(I),K)),hAPP_nat_nat(plus_plus_nat(J),K))) ) ).
tff(fact_877_trans__le__add2,axiom,
! [M: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(plus_plus_nat(M),J))) ) ).
tff(fact_878_trans__le__add1,axiom,
! [M: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(plus_plus_nat(J),M))) ) ).
tff(fact_879_nat__add__left__cancel__le,axiom,
! [K_1: nat,M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(K_1),M_1)),hAPP_nat_nat(plus_plus_nat(K_1),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) ) ).
tff(fact_880_le__iff__add,axiom,
! [M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2))
<=> ? [K_2: nat] : N_2 = hAPP_nat_nat(plus_plus_nat(M_1),K_2) ) ).
tff(fact_881_le__add1,axiom,
! [N_1: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(plus_plus_nat(N_1),M))) ).
tff(fact_882_le__add2,axiom,
! [N_1: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(plus_plus_nat(M),N_1))) ).
tff(fact_883_eq__diff__eq_H,axiom,
! [X_1: real,Y_1: real,Z: real] :
( ( X_1 = hAPP_real_real(minus_minus_real(Y_1),Z) )
<=> ( Y_1 = hAPP_real_real(plus_plus_real(X_1),Z) ) ) ).
tff(fact_884_nat__diff__split__asm,axiom,
! [P_1: fun_nat_bool,Aa: nat,Ba: nat] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(minus_minus_nat(Aa),Ba)))
<=> ~ ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba))
& ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
| ? [D: nat] :
( ( Aa = hAPP_nat_nat(plus_plus_nat(Ba),D) )
& ~ hBOOL(hAPP_nat_bool(P_1,D)) ) ) ) ).
tff(fact_885_nat__diff__split,axiom,
! [P_1: fun_nat_bool,Aa: nat,Ba: nat] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(minus_minus_nat(Aa),Ba)))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba))
=> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
& ! [D: nat] :
( ( Aa = hAPP_nat_nat(plus_plus_nat(Ba),D) )
=> hBOOL(hAPP_nat_bool(P_1,D)) ) ) ) ).
tff(fact_886_add__gr__0,axiom,
! [M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(plus_plus_nat(M_1),N_2)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) ) ) ).
tff(fact_887_mult__le__cancel1,axiom,
! [K_1: nat,M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(K_1),M_1)),hAPP_nat_nat(times_times_nat(K_1),N_2)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) ) ) ).
tff(fact_888_mult__le__cancel2,axiom,
! [M_1: nat,K_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(M_1),K_1)),hAPP_nat_nat(times_times_nat(N_2),K_1)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) ) ) ).
tff(fact_889_nat__mult__le__cancel1,axiom,
! [M_1: nat,N_2: nat,K_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(K_1),M_1)),hAPP_nat_nat(times_times_nat(K_1),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) ) ) ).
tff(fact_890_nat__less__add__iff1,axiom,
! [U: nat,M_1: nat,N_2: nat,J_2: nat,I_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_1),U)),M_1)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(I_1),J_2)),U)),M_1)),N_2)) ) ) ).
tff(fact_891_ex__least__nat__less,axiom,
! [N_2: nat,P_1: fun_nat_bool] :
( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(P_1,N_2))
=> ? [K_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),N_2))
& ! [I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),K_2))
=> ~ hBOOL(hAPP_nat_bool(P_1,I_2)) )
& hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(plus_plus_nat(K_2),one_one_nat))) ) ) ) ).
tff(fact_892_rabs__ratiotest__lemma,axiom,
! [X: real,Y: real,C: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,abs_abs_real(X)),hAPP_real_real(times_times_real(C),abs_abs_real(Y))))
=> ( X = zero_zero_real ) ) ) ).
tff(fact_893_mono__nat__linear__lb,axiom,
! [M_1: nat,K_1: nat,F: fun_nat_nat] :
( ! [M_2: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,M_2)),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,M_1)),K_1)),hAPP_nat_nat(F,hAPP_nat_nat(plus_plus_nat(M_1),K_1)))) ) ).
tff(fact_894_ex__least__nat__le,axiom,
! [N_2: nat,P_1: fun_nat_bool] :
( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(P_1,N_2))
=> ? [K_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_2))
& ! [I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),K_2))
=> ~ hBOOL(hAPP_nat_bool(P_1,I_2)) )
& hBOOL(hAPP_nat_bool(P_1,K_2)) ) ) ) ).
tff(fact_895_le__Suc__ex__iff,axiom,
! [K_1: nat,L: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),L))
<=> ? [N: nat] : L = hAPP_nat_nat(plus_plus_nat(K_1),N) ) ).
tff(fact_896_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ? [K_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
& ( hAPP_nat_nat(plus_plus_nat(I),K_2) = J ) ) ) ).
tff(fact_897_not__real__square__gt__zero,axiom,
! [X_1: real] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(X_1),X_1)))
<=> ( X_1 = zero_zero_real ) ) ).
tff(fact_898_mult__left__cancel,axiom,
! [N_1: nat,M: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( ( hAPP_nat_nat(times_times_nat(K),N_1) = hAPP_nat_nat(times_times_nat(K),M) )
=> ( N_1 = M ) ) ) ).
tff(fact_899_lemma__interval,axiom,
! [B: real,A: real,X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),B))
=> ? [D: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
& ! [Y_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,abs_abs_real(hAPP_real_real(minus_minus_real(X),Y_2))),D))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),Y_2))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),B)) ) ) ) ) ) ).
tff(fact_900_le__Suc__ex,axiom,
! [K: nat,L_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),L_1))
=> ? [N: nat] : L_1 = hAPP_nat_nat(plus_plus_nat(K),N) ) ).
tff(fact_901_less__mono__imp__le__mono,axiom,
! [I_1: nat,J_2: nat,F: fun_nat_nat] :
( ! [I_2: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,I_2)),hAPP_nat_nat(F,J_1))) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,I_1)),hAPP_nat_nat(F,J_2))) ) ) ).
tff(fact_902_lemma__interval__lt,axiom,
! [B: real,A: real,X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),B))
=> ? [D: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
& ! [Y_2: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,abs_abs_real(hAPP_real_real(minus_minus_real(X),Y_2))),D))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),Y_2))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),B)) ) ) ) ) ) ).
tff(fact_903_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_904_natceiling__add__one,axiom,
! [X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( natceiling(hAPP_real_real(plus_plus_real(X),one_one_real)) = hAPP_nat_nat(plus_plus_nat(natceiling(X)),one_one_nat) ) ) ).
tff(fact_905_natceiling__number__of__eq,axiom,
! [N_1: int] : natceiling(number267125858f_real(N_1)) = number_number_of_nat(N_1) ).
tff(fact_906_natceiling__zero,axiom,
natceiling(zero_zero_real) = zero_zero_nat ).
tff(fact_907_zero__le__natceiling,axiom,
! [X: real] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),natceiling(X))) ).
tff(fact_908_natceiling__mono,axiom,
! [X: real,Y: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X)),natceiling(Y))) ) ).
tff(fact_909_natceiling__one,axiom,
natceiling(one_one_real) = one_one_nat ).
tff(fact_910_natceiling__neg,axiom,
! [X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
=> ( natceiling(X) = zero_zero_nat ) ) ).
tff(fact_911_natceiling__le__eq__one,axiom,
! [X_1: real] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_1)),one_one_nat))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) ) ).
tff(fact_912_natfloor__add__one,axiom,
! [X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( natfloor(hAPP_real_real(plus_plus_real(X),one_one_real)) = hAPP_nat_nat(plus_plus_nat(natfloor(X)),one_one_nat) ) ) ).
tff(fact_913_nat__mult__eq__one,axiom,
! [N_2: nat,M_1: nat] :
( ( hAPP_nat_nat(times_times_nat(N_2),M_1) = one_one_nat )
<=> ( ( N_2 = one_one_nat )
& ( M_1 = one_one_nat ) ) ) ).
tff(fact_914_natceiling__eq,axiom,
! [N_1: nat,X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(plus_plus_real(hAPP_nat_real(real_nat,N_1)),one_one_real)))
=> ( natceiling(X) = hAPP_nat_nat(plus_plus_nat(N_1),one_one_nat) ) ) ) ).
tff(fact_915_real__of__nat__ge__zero,axiom,
! [N_1: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(real_nat,N_1))) ).
tff(fact_916_natceiling__real__of__nat,axiom,
! [N_1: nat] : natceiling(hAPP_nat_real(real_nat,N_1)) = N_1 ).
tff(fact_917_real__of__nat__less__iff,axiom,
! [N_2: nat,M_1: nat] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_2)),hAPP_nat_real(real_nat,M_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),M_1)) ) ).
tff(fact_918_real__of__nat__zero,axiom,
hAPP_nat_real(real_nat,zero_zero_nat) = zero_zero_real ).
tff(fact_919_real__of__nat__zero__iff,axiom,
! [N_2: nat] :
( ( hAPP_nat_real(real_nat,N_2) = zero_zero_real )
<=> ( N_2 = zero_zero_nat ) ) ).
tff(fact_920_abs__real__of__nat__cancel,axiom,
! [X: nat] : abs_abs_real(hAPP_nat_real(real_nat,X)) = hAPP_nat_real(real_nat,X) ).
tff(fact_921_natfloor__real__of__nat,axiom,
! [N_1: nat] : natfloor(hAPP_nat_real(real_nat,N_1)) = N_1 ).
tff(fact_922_Ints__real__of__nat,axiom,
! [N_2: nat] : hBOOL(member_real(hAPP_nat_real(real_nat,N_2),ring_1_Ints_real)) ).
tff(fact_923_natfloor__number__of__eq,axiom,
! [N_1: int] : natfloor(number267125858f_real(N_1)) = number_number_of_nat(N_1) ).
tff(fact_924_real__of__nat__inject,axiom,
! [N_2: nat,M_1: nat] :
( ( hAPP_nat_real(real_nat,N_2) = hAPP_nat_real(real_nat,M_1) )
<=> ( N_2 = M_1 ) ) ).
tff(fact_925_real__natfloor__le,axiom,
! [X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,natfloor(X))),X)) ) ).
tff(fact_926_le__natfloor,axiom,
! [X: nat,A: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,X)),A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),natfloor(A))) ) ).
tff(fact_927_real__natfloor__gt__diff__one,axiom,
! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(minus_minus_real(X),one_one_real)),hAPP_nat_real(real_nat,natfloor(X)))) ).
tff(fact_928_natfloor__subtract,axiom,
! [A: nat,X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A)),X))
=> ( natfloor(hAPP_real_real(minus_minus_real(X),hAPP_nat_real(real_nat,A))) = hAPP_nat_nat(minus_minus_nat(natfloor(X)),A) ) ) ).
tff(fact_929_real__natfloor__add__one__gt,axiom,
! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(plus_plus_real(hAPP_nat_real(real_nat,natfloor(X))),one_one_real))) ).
tff(fact_930_le__natfloor__eq,axiom,
! [Aa: nat,X_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Aa),natfloor(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,Aa)),X_1)) ) ) ).
tff(fact_931_not__real__of__nat__less__zero,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),zero_zero_real)) ).
tff(fact_932_real__of__nat__le__iff,axiom,
! [N_2: nat,M_1: nat] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_2)),hAPP_nat_real(real_nat,M_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),M_1)) ) ).
tff(fact_933_less__natfloor,axiom,
! [N_1: nat,X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,N_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,natfloor(X)),N_1)) ) ) ).
tff(fact_934_real__of__nat__mult,axiom,
! [M: nat,N_1: nat] : hAPP_nat_real(real_nat,hAPP_nat_nat(times_times_nat(M),N_1)) = hAPP_real_real(times_times_real(hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N_1)) ).
tff(fact_935_natfloor__add,axiom,
! [A: nat,X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( natfloor(hAPP_real_real(plus_plus_real(X),hAPP_nat_real(real_nat,A))) = hAPP_nat_nat(plus_plus_nat(natfloor(X)),A) ) ) ).
tff(fact_936_real__of__nat__add,axiom,
! [M: nat,N_1: nat] : hAPP_nat_real(real_nat,hAPP_nat_nat(plus_plus_nat(M),N_1)) = hAPP_real_real(plus_plus_real(hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N_1)) ).
tff(fact_937_ge__natfloor__plus__one__imp__gt,axiom,
! [Z_1: real,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(natfloor(Z_1)),one_one_nat)),N_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_1),hAPP_nat_real(real_nat,N_1))) ) ).
tff(fact_938_natfloor__eq,axiom,
! [N_1: nat,X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_1)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(plus_plus_real(hAPP_nat_real(real_nat,N_1)),one_one_real)))
=> ( natfloor(X) = N_1 ) ) ) ).
tff(fact_939_real__of__nat__1,axiom,
hAPP_nat_real(real_nat,one_one_nat) = one_one_real ).
tff(fact_940_real__natceiling__ge,axiom,
! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_nat_real(real_nat,natceiling(X)))) ).
tff(fact_941_natfloor__zero,axiom,
natfloor(zero_zero_real) = zero_zero_nat ).
tff(fact_942_zero__le__natfloor,axiom,
! [X: real] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),natfloor(X))) ).
tff(fact_943_natfloor__mono,axiom,
! [X: real,Y: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natfloor(X)),natfloor(Y))) ) ).
tff(fact_944_natfloor__one,axiom,
natfloor(one_one_real) = one_one_nat ).
tff(fact_945_real__of__nat__le__zero__cancel__iff,axiom,
! [N_2: nat] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_2)),zero_zero_real))
<=> ( N_2 = zero_zero_nat ) ) ).
tff(fact_946_real__of__nat__diff,axiom,
! [N_1: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> ( hAPP_nat_real(real_nat,hAPP_nat_nat(minus_minus_nat(M),N_1)) = hAPP_real_real(minus_minus_real(hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N_1)) ) ) ).
tff(fact_947_natceiling__le,axiom,
! [X: real,A: nat] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_nat_real(real_nat,A)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X)),A)) ) ).
tff(fact_948_natfloor__neg,axiom,
! [X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
=> ( natfloor(X) = zero_zero_nat ) ) ).
tff(fact_949_real__of__nat__gt__zero__cancel__iff,axiom,
! [N_2: nat] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) ) ).
tff(fact_950_nat__less__real__le,axiom,
! [N_2: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),M_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(hAPP_nat_real(real_nat,N_2)),one_one_real)),hAPP_nat_real(real_nat,M_1))) ) ).
tff(fact_951_nat__le__real__less,axiom,
! [N_2: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),M_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(plus_plus_real(hAPP_nat_real(real_nat,M_1)),one_one_real))) ) ).
tff(fact_952_natceiling__le__eq,axiom,
! [Aa: nat,X_1: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_1)),Aa))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_nat_real(real_nat,Aa))) ) ) ).
tff(fact_953_natceiling__subtract,axiom,
! [A: nat,X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A)),X))
=> ( natceiling(hAPP_real_real(minus_minus_real(X),hAPP_nat_real(real_nat,A))) = hAPP_nat_nat(minus_minus_nat(natceiling(X)),A) ) ) ).
tff(fact_954_le__natfloor__eq__one,axiom,
! [X_1: real] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),natfloor(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1)) ) ).
tff(fact_955_natceiling__add,axiom,
! [A: nat,X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( natceiling(hAPP_real_real(plus_plus_real(X),hAPP_nat_real(real_nat,A))) = hAPP_nat_nat(plus_plus_nat(natceiling(X)),A) ) ) ).
tff(fact_956_le__mult__natfloor,axiom,
! [B: real,A: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(natfloor(A)),natfloor(B))),natfloor(hAPP_real_real(times_times_real(A),B)))) ) ) ).
tff(fact_957_reals__Archimedean6,axiom,
! [R: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R))
=> ? [N: nat] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,hAPP_nat_nat(minus_minus_nat(N),one_one_nat))),R))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R),hAPP_nat_real(real_nat,N))) ) ) ).
tff(fact_958_reals__Archimedean3,axiom,
! [X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> ! [Y_2: real] :
? [N: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),hAPP_real_real(times_times_real(hAPP_nat_real(real_nat,N)),X))) ) ).
tff(fact_959_natceiling__add__number__of,axiom,
! [X: real,N_1: int] :
( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(N_1)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( natceiling(hAPP_real_real(plus_plus_real(X),number267125858f_real(N_1))) = hAPP_nat_nat(plus_plus_nat(natceiling(X)),number_number_of_nat(N_1)) ) ) ) ).
tff(fact_960_not__neg__0,axiom,
~ hBOOL(hAPP_int_bool(nat_neg,zero_zero_int)) ).
tff(fact_961_not__neg__1,axiom,
~ hBOOL(hAPP_int_bool(nat_neg,one_one_int)) ).
tff(fact_962_neg__def,axiom,
! [Z_2: int] :
( hBOOL(hAPP_int_bool(nat_neg,Z_2))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),zero_zero_int)) ) ).
tff(fact_963_not__neg__eq__ge__0,axiom,
! [X_1: int] :
( ~ hBOOL(hAPP_int_bool(nat_neg,X_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) ) ).
tff(fact_964_neg__imp__number__of__eq__0,axiom,
! [V: int] :
( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> ( number_number_of_nat(V) = zero_zero_nat ) ) ).
tff(fact_965_eq__nat__number__of,axiom,
! [V_1: int,V_3: int] :
( ( number_number_of_nat(V_1) = number_number_of_nat(V_3) )
<=> ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(V_3)),zero_zero_int)) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_1)))
=> ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_3)))
=> ( number_number_of_int(V_1) = zero_zero_int ) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_3)))
=> ( V_1 = V_3 ) ) ) ) ) ) ).
tff(fact_966_nat__number__of__add__left,axiom,
! [V_2: int,K: nat,V: int] :
( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_2)),K)) = hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_2)),K) ) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_2)))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_2)),K)) = hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),K) ) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_2)))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_2)),K)) = hAPP_nat_nat(plus_plus_nat(number_number_of_nat(hAPP_int_int(plus_plus_int(V),V_2))),K) ) ) ) ) ) ).
tff(fact_967_real__of__nat__number__of,axiom,
! [V: int] :
( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> ( hAPP_nat_real(real_nat,number_number_of_nat(V)) = zero_zero_real ) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> ( hAPP_nat_real(real_nat,number_number_of_nat(V)) = number267125858f_real(V) ) ) ) ).
tff(fact_968_le__natfloor__eq__number__of,axiom,
! [X_1: real,N_2: int] :
( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(N_2)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,number_number_of_nat(N_2)),natfloor(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(N_2)),X_1)) ) ) ) ).
tff(fact_969_natceiling__le__eq__number__of,axiom,
! [X_1: real,N_2: int] :
( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(N_2)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_1)),number_number_of_nat(N_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),number267125858f_real(N_2))) ) ) ) ).
tff(fact_970_natfloor__add__number__of,axiom,
! [X: real,N_1: int] :
( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(N_1)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( natfloor(hAPP_real_real(plus_plus_real(X),number267125858f_real(N_1))) = hAPP_nat_nat(plus_plus_nat(natfloor(X)),number_number_of_nat(N_1)) ) ) ) ).
tff(fact_971_reals__Archimedean2,axiom,
! [X: real] :
? [N: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,N))) ).
tff(fact_972_termination__basic__simps_I4_J,axiom,
! [Y: nat,X: nat,Z_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Z_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(plus_plus_nat(Y),Z_1))) ) ).
tff(fact_973_termination__basic__simps_I3_J,axiom,
! [Z_1: nat,X: nat,Y: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(plus_plus_nat(Y),Z_1))) ) ).
tff(fact_974_termination__basic__simps_I2_J,axiom,
! [Y: nat,X: nat,Z_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Z_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(plus_plus_nat(Y),Z_1))) ) ).
tff(fact_975_termination__basic__simps_I1_J,axiom,
! [Z_1: nat,X: nat,Y: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(plus_plus_nat(Y),Z_1))) ) ).
tff(fact_976_termination__basic__simps_I5_J,axiom,
! [X: nat,Y: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y)) ) ).
tff(fact_977_ceiling__eq2,axiom,
! [N_1: int,X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,N_1)),one_one_real)))
=> ( archim856651990g_real(X) = hAPP_int_int(plus_plus_int(N_1),one_one_int) ) ) ) ).
tff(fact_978_real__of__int__add,axiom,
! [X: int,Y: int] : hAPP_int_real(real_int,hAPP_int_int(plus_plus_int(X),Y)) = hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y)) ).
tff(fact_979_ceiling__add,axiom,
! [X: real,A: int] : archim856651990g_real(hAPP_real_real(plus_plus_real(X),hAPP_int_real(real_int,A))) = hAPP_int_int(plus_plus_int(archim856651990g_real(X)),A) ).
tff(fact_980_real__of__int__floor__cancel,axiom,
! [X_1: real] :
( ( hAPP_int_real(real_int,archim1246769320r_real(X_1)) = X_1 )
<=> ? [N: int] : X_1 = hAPP_int_real(real_int,N) ) ).
tff(fact_981_floor__real__of__int,axiom,
! [N_1: int] : archim1246769320r_real(hAPP_int_real(real_int,N_1)) = N_1 ).
tff(fact_982_real__of__int__ceiling__cancel,axiom,
! [X_1: real] :
( ( hAPP_int_real(real_int,archim856651990g_real(X_1)) = X_1 )
<=> ? [N: int] : X_1 = hAPP_int_real(real_int,N) ) ).
tff(fact_983_ceiling__real__of__int,axiom,
! [N_1: int] : archim856651990g_real(hAPP_int_real(real_int,N_1)) = N_1 ).
tff(fact_984_real__number__of,axiom,
! [V: int] : hAPP_int_real(real_int,number_number_of_int(V)) = number267125858f_real(V) ).
tff(fact_985_real__of__int__inject,axiom,
! [X_1: int,Y_1: int] :
( ( hAPP_int_real(real_int,X_1) = hAPP_int_real(real_int,Y_1) )
<=> ( X_1 = Y_1 ) ) ).
tff(fact_986_real__of__int__ceiling__ge,axiom,
! [R: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R),hAPP_int_real(real_int,archim856651990g_real(R)))) ).
tff(fact_987_real__of__one,axiom,
hAPP_int_real(real_int,one_one_int) = one_one_real ).
tff(fact_988_real__of__int__mult,axiom,
! [X: int,Y: int] : hAPP_int_real(real_int,hAPP_int_int(times_times_int(X),Y)) = hAPP_real_real(times_times_real(hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y)) ).
tff(fact_989_ceiling__le__eq,axiom,
! [X_1: real,Aa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X_1)),Aa))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_int_real(real_int,Aa))) ) ).
tff(fact_990_ceiling__le,axiom,
! [X: real,A: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,A)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X)),A)) ) ).
tff(fact_991_ceiling__le__real,axiom,
! [X: real,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X)),A))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,A))) ) ).
tff(fact_992_real__of__int__le__iff,axiom,
! [X_1: int,Y_1: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,X_1)),hAPP_int_real(real_int,Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1)) ) ).
tff(fact_993_real__of__int__less__iff,axiom,
! [X_1: int,Y_1: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,X_1)),hAPP_int_real(real_int,Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1)) ) ).
tff(fact_994_less__ceiling__eq,axiom,
! [Aa: int,X_1: real] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),archim856651990g_real(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,Aa)),X_1)) ) ).
tff(fact_995_real__of__int__zero__cancel,axiom,
! [X_1: int] :
( ( hAPP_int_real(real_int,X_1) = zero_zero_real )
<=> ( X_1 = zero_zero_int ) ) ).
tff(fact_996_real__of__int__zero,axiom,
hAPP_int_real(real_int,zero_zero_int) = zero_zero_real ).
tff(fact_997_ceiling__subtract,axiom,
! [X: real,A: int] : archim856651990g_real(hAPP_real_real(minus_minus_real(X),hAPP_int_real(real_int,A))) = hAPP_int_int(minus_minus_int(archim856651990g_real(X)),A) ).
tff(fact_998_real__of__int__diff,axiom,
! [X: int,Y: int] : hAPP_int_real(real_int,hAPP_int_int(minus_minus_int(X),Y)) = hAPP_real_real(minus_minus_real(hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y)) ).
tff(fact_999_real__of__int__floor__le,axiom,
! [R: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,archim1246769320r_real(R))),R)) ).
tff(fact_1000_real__of__int__abs,axiom,
! [X: int] : hAPP_int_real(real_int,abs_abs_int(X)) = abs_abs_real(hAPP_int_real(real_int,X)) ).
tff(fact_1001_Ints__real__of__int,axiom,
! [X_1: int] : hBOOL(member_real(hAPP_int_real(real_int,X_1),ring_1_Ints_real)) ).
tff(fact_1002_real__of__int__ceiling__le__add__one,axiom,
! [R: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,archim856651990g_real(R))),hAPP_real_real(plus_plus_real(R),one_one_real))) ).
tff(fact_1003_real__of__int__ceiling__diff__one__le,axiom,
! [R: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(minus_minus_real(hAPP_int_real(real_int,archim856651990g_real(R))),one_one_real)),R)) ).
tff(fact_1004_floor__less__eq,axiom,
! [X_1: real,Aa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim1246769320r_real(X_1)),Aa))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_int_real(real_int,Aa))) ) ).
tff(fact_1005_le__floor,axiom,
! [A: int,X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A)),X))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),archim1246769320r_real(X))) ) ).
tff(fact_1006_le__floor__eq,axiom,
! [Aa: int,X_1: real] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),archim1246769320r_real(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,Aa)),X_1)) ) ).
tff(fact_1007_real__le__floor,axiom,
! [A: int,X: real] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),archim1246769320r_real(X)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A)),X)) ) ).
tff(fact_1008_floor__add,axiom,
! [X: real,A: int] : archim1246769320r_real(hAPP_real_real(plus_plus_real(X),hAPP_int_real(real_int,A))) = hAPP_int_int(plus_plus_int(archim1246769320r_real(X)),A) ).
tff(fact_1009_floor__subtract,axiom,
! [X: real,A: int] : archim1246769320r_real(hAPP_real_real(minus_minus_real(X),hAPP_int_real(real_int,A))) = hAPP_int_int(minus_minus_int(archim1246769320r_real(X)),A) ).
tff(fact_1010_real__of__int__gt__zero__cancel__iff,axiom,
! [N_2: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_int_real(real_int,N_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_2)) ) ).
tff(fact_1011_real__of__int__lt__zero__cancel__iff,axiom,
! [N_2: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_2)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N_2),zero_zero_int)) ) ).
tff(fact_1012_real__of__int__ge__zero__cancel__iff,axiom,
! [N_2: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_int_real(real_int,N_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_2)) ) ).
tff(fact_1013_real__of__int__le__zero__cancel__iff,axiom,
! [N_2: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_2)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_2),zero_zero_int)) ) ).
tff(fact_1014_number__of__less__real__of__int__iff,axiom,
! [N_2: int,M_1: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(N_2)),hAPP_int_real(real_int,M_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(N_2)),M_1)) ) ).
tff(fact_1015_number__of__less__real__of__int__iff2,axiom,
! [M_1: int,N_2: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,M_1)),number267125858f_real(N_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M_1),number_number_of_int(N_2))) ) ).
tff(fact_1016_number__of__le__real__of__int__iff,axiom,
! [N_2: int,M_1: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(N_2)),hAPP_int_real(real_int,M_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(N_2)),M_1)) ) ).
tff(fact_1017_number__of__le__real__of__int__iff2,axiom,
! [M_1: int,N_2: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,M_1)),number267125858f_real(N_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M_1),number_number_of_int(N_2))) ) ).
tff(fact_1018_int__less__real__le,axiom,
! [N_2: int,M_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N_2),M_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,N_2)),one_one_real)),hAPP_int_real(real_int,M_1))) ) ).
tff(fact_1019_int__le__real__less,axiom,
! [N_2: int,M_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_2),M_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_2)),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,M_1)),one_one_real))) ) ).
tff(fact_1020_lemma__floor2,axiom,
! [N_1: int,X: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,X)),one_one_real)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),X)) ) ).
tff(fact_1021_ceiling__eq3,axiom,
! [N_1: int,X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(minus_minus_real(hAPP_int_real(real_int,N_1)),one_one_real)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,N_1)))
=> ( archim856651990g_real(X) = N_1 ) ) ) ).
tff(fact_1022_ceiling__less__eq,axiom,
! [X_1: real,Aa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim856651990g_real(X_1)),Aa))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(minus_minus_real(hAPP_int_real(real_int,Aa)),one_one_real))) ) ).
tff(fact_1023_le__ceiling__eq,axiom,
! [Aa: int,X_1: real] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),archim856651990g_real(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(minus_minus_real(hAPP_int_real(real_int,Aa)),one_one_real)),X_1)) ) ).
tff(fact_1024_real__of__int__floor__add__one__ge,axiom,
! [R: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,archim1246769320r_real(R))),one_one_real))) ).
tff(fact_1025_real__of__int__floor__add__one__gt,axiom,
! [R: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,archim1246769320r_real(R))),one_one_real))) ).
tff(fact_1026_floor__eq,axiom,
! [N_1: int,X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,N_1)),one_one_real)))
=> ( archim1246769320r_real(X) = N_1 ) ) ) ).
tff(fact_1027_real__of__int__floor__ge__diff__one,axiom,
! [R: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(minus_minus_real(R),one_one_real)),hAPP_int_real(real_int,archim1246769320r_real(R)))) ).
tff(fact_1028_real__of__int__floor__gt__diff__one,axiom,
! [R: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(minus_minus_real(R),one_one_real)),hAPP_int_real(real_int,archim1246769320r_real(R)))) ).
tff(fact_1029_lemma__floor,axiom,
! [N_1: int,M: int,R: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,M)),R))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,N_1)),one_one_real)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),N_1)) ) ) ).
tff(fact_1030_ceiling__eq,axiom,
! [N_1: int,X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,N_1)),one_one_real)))
=> ( archim856651990g_real(X) = hAPP_int_int(plus_plus_int(N_1),one_one_int) ) ) ) ).
tff(fact_1031_floor__eq2,axiom,
! [N_1: int,X: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_1)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,N_1)),one_one_real)))
=> ( archim1246769320r_real(X) = N_1 ) ) ) ).
tff(fact_1032_less__floor__eq,axiom,
! [Aa: int,X_1: real] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),archim1246769320r_real(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,Aa)),one_one_real)),X_1)) ) ).
tff(fact_1033_floor__le__eq,axiom,
! [X_1: real,Aa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim1246769320r_real(X_1)),Aa))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,Aa)),one_one_real))) ) ).
tff(fact_1034_reals__Archimedean__6b__int,axiom,
! [R: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R))
=> ? [N: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N)),R))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R),hAPP_int_real(real_int,hAPP_int_int(plus_plus_int(N),one_one_int)))) ) ) ).
tff(fact_1035_reals__Archimedean__6c__int,axiom,
! [R: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R),zero_zero_real))
=> ? [N: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N)),R))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R),hAPP_int_real(real_int,hAPP_int_int(plus_plus_int(N),one_one_int)))) ) ) ).
tff(fact_1036_real__lb__ub__int,axiom,
! [R: real] :
? [N: int] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N)),R))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R),hAPP_int_real(real_int,hAPP_int_int(plus_plus_int(N),one_one_int)))) ) ).
tff(fact_1037_ceiling__real__of__nat,axiom,
! [N_1: nat] : archim856651990g_real(hAPP_nat_real(real_nat,N_1)) = hAPP_nat_int(semiri1621563631at_int,N_1) ).
tff(fact_1038_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_1039_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_1040_transfer__int__nat__quantifiers_I1_J,axiom,
! [P_1: fun_int_bool] :
( ! [X_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> hBOOL(hAPP_int_bool(P_1,X_2)) )
<=> ! [X_2: nat] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_2))) ) ).
tff(fact_1041_transfer__int__nat__quantifiers_I2_J,axiom,
! [P_1: fun_int_bool] :
( ? [X_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
& hBOOL(hAPP_int_bool(P_1,X_2)) )
<=> ? [X_2: nat] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_2))) ) ).
tff(fact_1042_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z_1: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Z_1))) ).
tff(fact_1043_Nat__Transfer_Otransfer__int__nat__relations_I2_J,axiom,
! [X_1: nat,Y_1: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1)) ) ).
tff(fact_1044_zless__int,axiom,
! [M_1: nat,N_2: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) ) ).
tff(fact_1045_abs__int__eq,axiom,
! [M: nat] : abs_abs_int(hAPP_nat_int(semiri1621563631at_int,M)) = hAPP_nat_int(semiri1621563631at_int,M) ).
tff(fact_1046_int__eq__0__conv,axiom,
! [N_2: nat] :
( ( hAPP_nat_int(semiri1621563631at_int,N_2) = zero_zero_int )
<=> ( N_2 = zero_zero_nat ) ) ).
tff(fact_1047_int__0,axiom,
hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) = zero_zero_int ).
tff(fact_1048_transfer__int__nat__numerals_I1_J,axiom,
zero_zero_int = hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) ).
tff(fact_1049_transfer__int__nat__numerals_I2_J,axiom,
one_one_int = hAPP_nat_int(semiri1621563631at_int,one_one_nat) ).
tff(fact_1050_int__1,axiom,
hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int ).
tff(fact_1051_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_1052_zadd__int,axiom,
! [M: nat,N_1: nat] : hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(plus_plus_nat(M),N_1)) ).
tff(fact_1053_zadd__int__left,axiom,
! [M: nat,N_1: nat,Z_1: int] : hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M)),hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,N_1)),Z_1)) = hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(plus_plus_nat(M),N_1))),Z_1) ).
tff(fact_1054_zle__iff__zadd,axiom,
! [W: int,Z: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z))
<=> ? [N: nat] : Z = hAPP_int_int(plus_plus_int(W),hAPP_nat_int(semiri1621563631at_int,N)) ) ).
tff(fact_1055_zmult__int,axiom,
! [M: nat,N_1: nat] : hAPP_int_int(times_times_int(hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(times_times_nat(M),N_1)) ).
tff(fact_1056_int__mult,axiom,
! [M: nat,N_1: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(times_times_nat(M),N_1)) = hAPP_int_int(times_times_int(hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N_1)) ).
tff(fact_1057_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
! [X: nat,Y: nat] : hAPP_int_int(times_times_int(hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(times_times_nat(X),Y)) ).
tff(fact_1058_zle__int,axiom,
! [M_1: nat,N_2: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) ) ).
tff(fact_1059_Nat__Transfer_Otransfer__int__nat__relations_I3_J,axiom,
! [X_1: nat,Y_1: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1)) ) ).
tff(fact_1060_not__neg__int,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_nat_int(semiri1621563631at_int,N_1))) ).
tff(fact_1061_real__of__nat__def,axiom,
real_nat = semiri132038758t_real ).
tff(fact_1062_real__of__int__of__nat__eq,axiom,
! [N_1: nat] : hAPP_int_real(real_int,hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_real(real_nat,N_1) ).
tff(fact_1063_real__eq__of__nat,axiom,
real_nat = semiri132038758t_real ).
tff(fact_1064_floor__real__of__nat,axiom,
! [N_1: nat] : archim1246769320r_real(hAPP_nat_real(real_nat,N_1)) = hAPP_nat_int(semiri1621563631at_int,N_1) ).
tff(fact_1065_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
! [X: nat,Y: nat] : nat_tsub(hAPP_nat_int(semiri1621563631at_int,X),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(minus_minus_nat(X),Y)) ).
tff(fact_1066_int__le__0__conv,axiom,
! [N_2: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,N_2)),zero_zero_int))
<=> ( N_2 = zero_zero_nat ) ) ).
tff(fact_1067_zdiff__int,axiom,
! [N_1: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> ( hAPP_int_int(minus_minus_int(hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(minus_minus_nat(M),N_1)) ) ) ).
tff(fact_1068_zero__less__int__conv,axiom,
! [N_2: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) ) ).
tff(fact_1069_zmult__zless__mono2__lemma,axiom,
! [K: nat,I: int,J: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(hAPP_nat_int(semiri1621563631at_int,K)),I)),hAPP_int_int(times_times_int(hAPP_nat_int(semiri1621563631at_int,K)),J))) ) ) ).
tff(fact_1070_int__nat__number__of,axiom,
! [V: int] :
( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> ( hAPP_nat_int(semiri1621563631at_int,number_number_of_nat(V)) = zero_zero_int ) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> ( hAPP_nat_int(semiri1621563631at_int,number_number_of_nat(V)) = number_number_of_int(V) ) ) ) ).
tff(fact_1071_zdiff__int__split,axiom,
! [P_1: fun_int_bool,X_1: nat,Y_1: nat] :
( hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(minus_minus_nat(X_1),Y_1))))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)))) )
& ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
=> hBOOL(hAPP_int_bool(P_1,zero_zero_int)) ) ) ) ).
tff(fact_1072_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_1073_int__int__eq,axiom,
! [M_1: nat,N_2: nat] :
( ( hAPP_nat_int(semiri1621563631at_int,M_1) = hAPP_nat_int(semiri1621563631at_int,N_2) )
<=> ( M_1 = N_2 ) ) ).
tff(fact_1074_int__if__cong,axiom,
! [X_1: nat,Y_1: nat,P_1: bool] :
( ( hBOOL(P_1)
=> ( hAPP_nat_int(semiri1621563631at_int,X_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(if_nat(P_1,X_1),Y_1)) ) )
& ( ~ hBOOL(P_1)
=> ( hAPP_nat_int(semiri1621563631at_int,Y_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(if_nat(P_1,X_1),Y_1)) ) ) ) ).
tff(fact_1075_Nat__Transfer_Otransfer__int__nat__relations_I1_J,axiom,
! [X_1: nat,Y_1: nat] :
( ( hAPP_nat_int(semiri1621563631at_int,X_1) = hAPP_nat_int(semiri1621563631at_int,Y_1) )
<=> ( X_1 = Y_1 ) ) ).
tff(fact_1076_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_1077_nonneg__eq__int,axiom,
! [Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
=> ~ ! [M_2: nat] : Z_1 != hAPP_nat_int(semiri1621563631at_int,M_2) ) ).
tff(fact_1078_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_1079_int__diff__cases,axiom,
! [Z_1: int] :
~ ! [M_2: nat,N: nat] : Z_1 != hAPP_int_int(minus_minus_int(hAPP_nat_int(semiri1621563631at_int,M_2)),hAPP_nat_int(semiri1621563631at_int,N)) ).
tff(fact_1080_mult__nat__number__of,axiom,
! [V_2: int,V: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> ( hAPP_nat_nat(times_times_nat(number_number_of_nat(V)),number_number_of_nat(V_2)) = zero_zero_nat ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> ( hAPP_nat_nat(times_times_nat(number_number_of_nat(V)),number_number_of_nat(V_2)) = number_number_of_nat(hAPP_int_int(times_times_int(V),V_2)) ) ) ) ).
tff(fact_1081_nat__number__of__mult__left,axiom,
! [V_2: int,K: nat,V: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> ( hAPP_nat_nat(times_times_nat(number_number_of_nat(V)),hAPP_nat_nat(times_times_nat(number_number_of_nat(V_2)),K)) = zero_zero_nat ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> ( hAPP_nat_nat(times_times_nat(number_number_of_nat(V)),hAPP_nat_nat(times_times_nat(number_number_of_nat(V_2)),K)) = hAPP_nat_nat(times_times_nat(number_number_of_nat(hAPP_int_int(times_times_int(V),V_2))),K) ) ) ) ).
tff(fact_1082_diff__bin__simps_I1_J,axiom,
! [K: int] : hAPP_int_int(minus_minus_int(K),pls) = K ).
tff(fact_1083_mult__Pls,axiom,
! [W_1: int] : hAPP_int_int(times_times_int(pls),W_1) = pls ).
tff(fact_1084_rel__simps_I2_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),pls)) ).
tff(fact_1085_add__Pls,axiom,
! [K: int] : hAPP_int_int(plus_plus_int(pls),K) = K ).
tff(fact_1086_add__Pls__right,axiom,
! [K: int] : hAPP_int_int(plus_plus_int(K),pls) = K ).
tff(fact_1087_Pls__def,axiom,
pls = zero_zero_int ).
tff(fact_1088_rel__simps_I19_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),pls)) ).
tff(fact_1089_zero__is__num__zero,axiom,
zero_zero_int = number_number_of_int(pls) ).
tff(fact_1090_bin__less__0__simps_I1_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),zero_zero_int)) ).
tff(fact_1091_nat__number__of__Pls,axiom,
number_number_of_nat(pls) = zero_zero_nat ).
tff(fact_1092_semiring__norm_I113_J,axiom,
zero_zero_nat = number_number_of_nat(pls) ).
tff(fact_1093_not__neg__number__of__Pls,axiom,
~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(pls))) ).
tff(fact_1094_eq__0__number__of,axiom,
! [V_1: int] :
( ( zero_zero_nat = number_number_of_nat(V_1) )
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_1),pls)) ) ).
tff(fact_1095_eq__number__of__0,axiom,
! [V_1: int] :
( ( number_number_of_nat(V_1) = zero_zero_nat )
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_1),pls)) ) ).
tff(fact_1096_less__nat__number__of,axiom,
! [V_1: int,V_3: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,number_number_of_nat(V_1)),number_number_of_nat(V_3)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),V_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_3)) )
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),V_3)) ) ) ).
tff(fact_1097_number__of2,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(pls))) ).
tff(fact_1098_le__nat__number__of,axiom,
! [V_1: int,V_3: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,number_number_of_nat(V_1)),number_number_of_nat(V_3)))
<=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_1),V_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_1),pls)) ) ) ).
tff(fact_1099_less__0__number__of,axiom,
! [V_1: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),number_number_of_nat(V_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_1)) ) ).
tff(fact_1100_add__nat__number__of,axiom,
! [V_2: int,V: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),number_number_of_nat(V_2)) = number_number_of_nat(V_2) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),number_number_of_nat(V_2)) = number_number_of_nat(V) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),number_number_of_nat(V_2)) = number_number_of_nat(hAPP_int_int(plus_plus_int(V),V_2)) ) ) ) ) ) ).
tff(fact_1101_nat__1__add__number__of,axiom,
! [V: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> ( hAPP_nat_nat(plus_plus_nat(one_one_nat),number_number_of_nat(V)) = one_one_nat ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> ( hAPP_nat_nat(plus_plus_nat(one_one_nat),number_number_of_nat(V)) = number_number_of_nat(succ(V)) ) ) ) ).
tff(fact_1102_nat__number__of__add__1,axiom,
! [V: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),one_one_nat) = one_one_nat ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> ( hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),one_one_nat) = number_number_of_nat(succ(V)) ) ) ) ).
tff(fact_1103_succ__def,axiom,
! [K: int] : succ(K) = hAPP_int_int(plus_plus_int(K),one_one_int) ).
tff(fact_1104_nat__number__of__diff__1,axiom,
! [V: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),pls))
=> ( hAPP_nat_nat(minus_minus_nat(number_number_of_nat(V)),one_one_nat) = zero_zero_nat ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),pls))
=> ( hAPP_nat_nat(minus_minus_nat(number_number_of_nat(V)),one_one_nat) = number_number_of_nat(pred(V)) ) ) ) ).
tff(fact_1105_Suc__nat__number__of__add,axiom,
! [N_1: nat,V: int] :
( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> ( hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),N_1)) = hAPP_nat_nat(plus_plus_nat(one_one_nat),N_1) ) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> ( hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),N_1)) = hAPP_nat_nat(plus_plus_nat(number_number_of_nat(succ(V))),N_1) ) ) ) ).
tff(fact_1106_lessI,axiom,
! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,N_1))) ).
tff(fact_1107_Suc__mono,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N_1))) ) ).
tff(fact_1108_zero__less__Suc,axiom,
! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(suc,N_1))) ).
tff(fact_1109_succ__pred,axiom,
! [X: int] : succ(pred(X)) = X ).
tff(fact_1110_less__Suc__eq__0__disj,axiom,
! [M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N_2)))
<=> ( ( M_1 = zero_zero_nat )
| ? [J_1: nat] :
( ( M_1 = hAPP_nat_nat(suc,J_1) )
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),N_2)) ) ) ) ).
tff(fact_1111_less__Suc0,axiom,
! [N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),hAPP_nat_nat(suc,zero_zero_nat)))
<=> ( N_2 = zero_zero_nat ) ) ).
tff(fact_1112_gr0__conv__Suc,axiom,
! [N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2))
<=> ? [M_2: nat] : N_2 = hAPP_nat_nat(suc,M_2) ) ).
tff(fact_1113_add__is__1,axiom,
! [M_1: nat,N_2: nat] :
( ( hAPP_nat_nat(plus_plus_nat(M_1),N_2) = hAPP_nat_nat(suc,zero_zero_nat) )
<=> ( ( ( M_1 = hAPP_nat_nat(suc,zero_zero_nat) )
& ( N_2 = zero_zero_nat ) )
| ( ( M_1 = zero_zero_nat )
& ( N_2 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ) ).
tff(fact_1114_one__is__add,axiom,
! [M_1: nat,N_2: nat] :
( ( hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(plus_plus_nat(M_1),N_2) )
<=> ( ( ( M_1 = hAPP_nat_nat(suc,zero_zero_nat) )
& ( N_2 = zero_zero_nat ) )
| ( ( M_1 = zero_zero_nat )
& ( N_2 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ) ).
tff(fact_1115_mult__eq__1__iff,axiom,
! [M_1: nat,N_2: nat] :
( ( hAPP_nat_nat(times_times_nat(M_1),N_2) = hAPP_nat_nat(suc,zero_zero_nat) )
<=> ( ( M_1 = hAPP_nat_nat(suc,zero_zero_nat) )
& ( N_2 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).
tff(fact_1116_less__iff__Suc__add,axiom,
! [M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
<=> ? [K_2: nat] : N_2 = hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(M_1),K_2)) ) ).
tff(fact_1117_less__add__Suc2,axiom,
! [I: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(M),I)))) ).
tff(fact_1118_less__add__Suc1,axiom,
! [I: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(I),M)))) ).
tff(fact_1119_One__nat__def,axiom,
one_one_nat = hAPP_nat_nat(suc,zero_zero_nat) ).
tff(fact_1120_less__eq__Suc__le,axiom,
! [N_2: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),M_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_2)),M_1)) ) ).
tff(fact_1121_less__Suc__eq__le,axiom,
! [M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) ) ).
tff(fact_1122_Suc__le__eq,axiom,
! [M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_1)),N_2))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) ) ).
tff(fact_1123_le__imp__less__Suc,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N_1))) ) ).
tff(fact_1124_Suc__leI,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N_1)) ) ).
tff(fact_1125_le__less__Suc__eq,axiom,
! [M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),hAPP_nat_nat(suc,M_1)))
<=> ( N_2 = M_1 ) ) ) ).
tff(fact_1126_Suc__le__lessD,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ).
tff(fact_1127_Suc__mult__less__cancel1,axiom,
! [K_1: nat,M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(hAPP_nat_nat(suc,K_1)),M_1)),hAPP_nat_nat(times_times_nat(hAPP_nat_nat(suc,K_1)),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) ) ).
tff(fact_1128_diff__less__Suc,axiom,
! [M: nat,N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(M),N_1)),hAPP_nat_nat(suc,M))) ).
tff(fact_1129_mult__Suc,axiom,
! [M: nat,N_1: nat] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(suc,M)),N_1) = hAPP_nat_nat(plus_plus_nat(N_1),hAPP_nat_nat(times_times_nat(M),N_1)) ).
tff(fact_1130_mult__Suc__right,axiom,
! [M: nat,N_1: nat] : hAPP_nat_nat(times_times_nat(M),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(plus_plus_nat(M),hAPP_nat_nat(times_times_nat(M),N_1)) ).
tff(fact_1131_Suc__mult__le__cancel1,axiom,
! [K_1: nat,M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(hAPP_nat_nat(suc,K_1)),M_1)),hAPP_nat_nat(times_times_nat(hAPP_nat_nat(suc,K_1)),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) ) ).
tff(fact_1132_Suc__eq__plus1__left,axiom,
! [N_1: nat] : hAPP_nat_nat(suc,N_1) = hAPP_nat_nat(plus_plus_nat(one_one_nat),N_1) ).
tff(fact_1133_Suc__eq__plus1,axiom,
! [N_1: nat] : hAPP_nat_nat(suc,N_1) = hAPP_nat_nat(plus_plus_nat(N_1),one_one_nat) ).
tff(fact_1134_Suc__diff__le,axiom,
! [N_1: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,M)),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(minus_minus_nat(M),N_1)) ) ) ).
tff(fact_1135_diff__Suc__1,axiom,
! [N_1: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,N_1)),one_one_nat) = N_1 ).
tff(fact_1136_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N_1: nat] : hAPP_nat_nat(minus_minus_nat(M),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(M),one_one_nat)),N_1) ).
tff(fact_1137_Suc__leD,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).
tff(fact_1138_le__SucE,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N_1)))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> ( M = hAPP_nat_nat(suc,N_1) ) ) ) ).
tff(fact_1139_le__SucI,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N_1))) ) ).
tff(fact_1140_Suc__le__mono,axiom,
! [N_2: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_2)),hAPP_nat_nat(suc,M_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),M_1)) ) ).
tff(fact_1141_le__Suc__eq,axiom,
! [M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(suc,N_2)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2))
| ( M_1 = hAPP_nat_nat(suc,N_2) ) ) ) ).
tff(fact_1142_not__less__eq__eq,axiom,
! [M_1: nat,N_2: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_2)),M_1)) ) ).
tff(fact_1143_Suc__n__not__le__n,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),N_1)) ).
tff(fact_1144_Zero__not__Suc,axiom,
! [M: nat] : zero_zero_nat != hAPP_nat_nat(suc,M) ).
tff(fact_1145_nat_Osimps_I2_J,axiom,
! [Nat_3: nat] : zero_zero_nat != hAPP_nat_nat(suc,Nat_3) ).
tff(fact_1146_Suc__not__Zero,axiom,
! [M: nat] : hAPP_nat_nat(suc,M) != zero_zero_nat ).
tff(fact_1147_nat_Osimps_I3_J,axiom,
! [Nat_2: nat] : hAPP_nat_nat(suc,Nat_2) != zero_zero_nat ).
tff(fact_1148_Zero__neq__Suc,axiom,
! [M: nat] : zero_zero_nat != hAPP_nat_nat(suc,M) ).
tff(fact_1149_Suc__neq__Zero,axiom,
! [M: nat] : hAPP_nat_nat(suc,M) != zero_zero_nat ).
tff(fact_1150_not__less__eq,axiom,
! [M_1: nat,N_2: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),hAPP_nat_nat(suc,M_1))) ) ).
tff(fact_1151_less__Suc__eq,axiom,
! [M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N_2)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
| ( M_1 = N_2 ) ) ) ).
tff(fact_1152_Suc__less__eq,axiom,
! [M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) ) ).
tff(fact_1153_not__less__less__Suc__eq,axiom,
! [N_2: nat,M_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),hAPP_nat_nat(suc,M_1)))
<=> ( N_2 = M_1 ) ) ) ).
tff(fact_1154_less__antisym,axiom,
! [N_1: nat,M: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,M)))
=> ( M = N_1 ) ) ) ).
tff(fact_1155_less__SucI,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N_1))) ) ).
tff(fact_1156_Suc__lessI,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> ( ( hAPP_nat_nat(suc,M) != N_1 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),N_1)) ) ) ).
tff(fact_1157_less__trans__Suc,axiom,
! [K: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I)),K)) ) ) ).
tff(fact_1158_less__SucE,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N_1)))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> ( M = N_1 ) ) ) ).
tff(fact_1159_Suc__lessD,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ).
tff(fact_1160_Suc__less__SucD,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ).
tff(fact_1161_add__Suc__right,axiom,
! [M: nat,N_1: nat] : hAPP_nat_nat(plus_plus_nat(M),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(M),N_1)) ).
tff(fact_1162_add__Suc,axiom,
! [M: nat,N_1: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(suc,M)),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(M),N_1)) ).
tff(fact_1163_add__Suc__shift,axiom,
! [M: nat,N_1: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(suc,M)),N_1) = hAPP_nat_nat(plus_plus_nat(M),hAPP_nat_nat(suc,N_1)) ).
tff(fact_1164_Suc__diff__diff,axiom,
! [M: nat,N_1: nat,K: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,M)),N_1)),hAPP_nat_nat(suc,K)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(M),N_1)),K) ).
tff(fact_1165_diff__Suc__Suc,axiom,
! [M: nat,N_1: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(minus_minus_nat(M),N_1) ).
tff(fact_1166_n__not__Suc__n,axiom,
! [N_1: nat] : N_1 != hAPP_nat_nat(suc,N_1) ).
tff(fact_1167_Suc__n__not__n,axiom,
! [N_1: nat] : hAPP_nat_nat(suc,N_1) != N_1 ).
tff(fact_1168_nat_Oinject,axiom,
! [Nat_1: nat,Nat: nat] :
( ( hAPP_nat_nat(suc,Nat_1) = hAPP_nat_nat(suc,Nat) )
<=> ( Nat_1 = Nat ) ) ).
tff(fact_1169_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( hAPP_nat_nat(suc,X) = hAPP_nat_nat(suc,Y) )
=> ( X = Y ) ) ).
tff(fact_1170_Suc__mult__cancel1,axiom,
! [K_1: nat,M_1: nat,N_2: nat] :
( ( hAPP_nat_nat(times_times_nat(hAPP_nat_nat(suc,K_1)),M_1) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(suc,K_1)),N_2) )
<=> ( M_1 = N_2 ) ) ).
tff(fact_1171_pred__def,axiom,
! [K: int] : pred(K) = hAPP_int_int(minus_minus_int(K),one_one_int) ).
tff(fact_1172_le__iff__pred__less,axiom,
! [K_1: int,L: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pred(K_1)),L)) ) ).
tff(fact_1173_Suc__diff__number__of,axiom,
! [M: nat,V: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,M)),number_number_of_nat(V)) = hAPP_nat_nat(minus_minus_nat(M),number_number_of_nat(pred(V))) ) ) ).
tff(fact_1174_one__less__mult,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(times_times_nat(M),N_1))) ) ) ).
tff(fact_1175_n__less__n__mult__m,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(times_times_nat(N_1),M))) ) ) ).
tff(fact_1176_n__less__m__mult__n,axiom,
! [M: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(times_times_nat(M),N_1))) ) ) ).
tff(fact_1177_mn__eq__m__one,axiom,
! [N_1: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> ( ( hAPP_nat_nat(times_times_nat(M),N_1) = M )
=> ( N_1 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).
tff(fact_1178_prod__mn__less__k,axiom,
! [M: nat,K: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))
=> ( ( hAPP_nat_nat(times_times_nat(M),N_1) = K )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),K)) ) ) ) ) ).
tff(fact_1179_one__less__k,axiom,
! [M: nat,K: nat] :
( ( M != hAPP_nat_nat(times_times_nat(M),K) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(times_times_nat(M),K)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),K)) ) ) ).
tff(fact_1180_one__less__m,axiom,
! [M: nat,K: nat] :
( ( M != hAPP_nat_nat(times_times_nat(M),K) )
=> ( ( M != hAPP_nat_nat(suc,zero_zero_nat) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M)) ) ) ).
tff(fact_1181_Suc__pred,axiom,
! [N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hAPP_nat_nat(suc,hAPP_nat_nat(minus_minus_nat(N_1),hAPP_nat_nat(suc,zero_zero_nat))) = N_1 ) ) ).
tff(fact_1182_diff__Suc__less,axiom,
! [I: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(N_1),hAPP_nat_nat(suc,I))),N_1)) ) ).
tff(fact_1183_one__le__mult__iff,axiom,
! [M_1: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(times_times_nat(M_1),N_2)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_2)) ) ) ).
tff(fact_1184_real__of__nat__Suc__gt__zero,axiom,
! [N_1: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1)))) ).
tff(fact_1185_int__Suc0__eq__1,axiom,
hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,zero_zero_nat)) = one_one_int ).
tff(fact_1186_real__of__nat__one,axiom,
hAPP_nat_real(real_nat,hAPP_nat_nat(suc,zero_zero_nat)) = one_one_real ).
tff(fact_1187_diff__Suc__diff__eq1,axiom,
! [M: nat,K: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> ( hAPP_nat_nat(minus_minus_nat(M),hAPP_nat_nat(suc,hAPP_nat_nat(minus_minus_nat(J),K))) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(M),K)),hAPP_nat_nat(suc,J)) ) ) ).
tff(fact_1188_diff__Suc__diff__eq2,axiom,
! [M: nat,K: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,hAPP_nat_nat(minus_minus_nat(J),K))),M) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,J)),hAPP_nat_nat(plus_plus_nat(K),M)) ) ) ).
tff(fact_1189_zless__iff__Suc__zadd,axiom,
! [W: int,Z: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z))
<=> ? [N: nat] : Z = hAPP_int_int(plus_plus_int(W),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N))) ) ).
tff(fact_1190_int__Suc,axiom,
! [M: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,M)) = hAPP_int_int(plus_plus_int(one_one_int),hAPP_nat_int(semiri1621563631at_int,M)) ).
tff(fact_1191_real__of__nat__Suc,axiom,
! [N_1: nat] : hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1)) = hAPP_real_real(plus_plus_real(hAPP_nat_real(real_nat,N_1)),one_one_real) ).
tff(fact_1192_Suc__pred_H,axiom,
! [N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( N_1 = hAPP_nat_nat(suc,hAPP_nat_nat(minus_minus_nat(N_1),one_one_nat)) ) ) ).
tff(fact_1193_Suc__diff__1,axiom,
! [N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hAPP_nat_nat(suc,hAPP_nat_nat(minus_minus_nat(N_1),one_one_nat)) = N_1 ) ) ).
tff(fact_1194_add__eq__if,axiom,
! [N_1: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( hAPP_nat_nat(plus_plus_nat(M),N_1) = N_1 ) )
& ( ( M != zero_zero_nat )
=> ( hAPP_nat_nat(plus_plus_nat(M),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(minus_minus_nat(M),one_one_nat)),N_1)) ) ) ) ).
tff(fact_1195_expand__Suc,axiom,
! [V: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),number_number_of_nat(V)))
=> ( number_number_of_nat(V) = hAPP_nat_nat(suc,hAPP_nat_nat(minus_minus_nat(number_number_of_nat(V)),one_one_nat)) ) ) ).
tff(fact_1196_neg__number__of__pred__iff__0,axiom,
! [V_1: int] :
( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(pred(V_1))))
<=> ( number_number_of_nat(V_1) = zero_zero_nat ) ) ).
tff(fact_1197_Suc__nat__number__of,axiom,
! [V: int] :
( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> ( hAPP_nat_nat(suc,number_number_of_nat(V)) = one_one_nat ) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> ( hAPP_nat_nat(suc,number_number_of_nat(V)) = number_number_of_nat(succ(V)) ) ) ) ).
tff(fact_1198_reals__Archimedean6a,axiom,
! [R: real] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R))
=> ? [N: nat] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N)),R))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N)))) ) ) ).
%----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_int_int(times_times_int(twoSqu949963151sum2sq(product_Pair_int_int(a,b))),twoSqu949963151sum2sq(product_Pair_int_int(p,q))) = twoSqu949963151sum2sq(product_Pair_int_int(hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(a),p)),hAPP_int_int(times_times_int(b),q)),hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(a),q)),hAPP_int_int(times_times_int(b),p)))) ).
%------------------------------------------------------------------------------