TPTP Problem File: NUM926+4.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : NUM926+4 : TPTP v9.0.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 258, 5000 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : s2s_5000_fofmg_l258 [Bla11]
% Status : Theorem
% Rating : 0.73 v9.0.0, 0.75 v8.2.0, 0.83 v8.1.0, 0.72 v7.5.0, 0.78 v7.4.0, 0.77 v7.3.0, 0.72 v7.1.0, 0.70 v7.0.0, 0.90 v6.4.0, 0.88 v6.3.0, 0.83 v6.2.0, 0.92 v6.1.0, 0.97 v6.0.0, 0.91 v5.5.0, 0.96 v5.3.0
% Syntax : Number of formulae : 5480 (2351 unt; 0 def)
% Number of atoms : 12316 (4356 equ)
% Maximal formula atoms : 25 ( 2 avg)
% Number of connectives : 7762 ( 926 ~; 197 |; 877 &)
% ( 908 <=>;4854 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 4 avg)
% Maximal term depth : 21 ( 2 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 1-2 aty)
% Number of functors : 1073 (1073 usr; 426 con; 0-8 aty)
% Number of variables : 11702 (11425 !; 277 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 18:57:12
% : Encoded with monomorphized guards.
%------------------------------------------------------------------------------
%----Explicit typings (59)
fof(gsy_c_Archimedean__Field_Oceiling_000tc__RealDef__Oreal,axiom,
! [B_1_1] : is_int(archim856651990g_real(B_1_1)) ).
fof(gsy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_000tc__Rat__Orat,axiom,
! [B_1_1] : is_int(archim791455193or_rat(B_1_1)) ).
fof(gsy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_000tc__RealDef__Oreal,axiom,
! [B_1_1] : is_int(archim1246769320r_real(B_1_1)) ).
fof(gsy_c_Big__Operators_Ocomm__monoid__mult__class_Osetprod_000tc__Int__Oint_000tc_,axiom,
! [B_1_1,B_2_1] : is_int(big_co1548731110nt_int(B_1_1,B_2_1)) ).
fof(gsy_c_Code__Numeral_Oint__of,axiom,
! [B_1_1] : is_int(code_int_of(B_1_1)) ).
fof(gsy_c_Groups_Oone__class_Oone_000tc__Int__Oint,hypothesis,
is_int(one_one_int) ).
fof(gsy_c_Groups_Ozero__class_Ozero_000tc__Int__Oint,axiom,
is_int(zero_zero_int) ).
fof(gsy_c_HOL_OThe_000tc__Int__Oint,axiom,
! [B_1_1] : is_int(the_int(B_1_1)) ).
fof(gsy_c_HOL_Oundefined_000tc__Int__Oint,axiom,
is_int(undefined_int(int)) ).
fof(gsy_c_Hilbert__Choice_OEps_000tc__Int__Oint,axiom,
! [B_1_1] : is_int(hilbert_Eps_int(B_1_1)) ).
fof(gsy_c_Int_OMin,axiom,
is_int(min) ).
fof(gsy_c_Int_OPls,hypothesis,
is_int(pls) ).
fof(gsy_c_Limits_Otendsto_000tc__Complex__Ocomplex_000tc__Complex__Ocomplex,axiom,
! [B_1_1,B_2_1,B_3_1] : is_bool(tendst1507391555omplex(B_1_1,B_2_1,B_3_1)) ).
fof(gsy_c_Limits_Otendsto_000tc__Complex__Ocomplex_000tc__RealDef__Oreal,axiom,
! [B_1_1,B_2_1,B_3_1] : is_bool(tendsto_complex_real(B_1_1,B_2_1,B_3_1)) ).
fof(gsy_c_Limits_Otendsto_000tc__Nat__Onat_000tc__Complex__Ocomplex,axiom,
! [B_1_1,B_2_1,B_3_1] : is_bool(tendsto_nat_complex(B_1_1,B_2_1,B_3_1)) ).
fof(gsy_c_Limits_Otendsto_000tc__Nat__Onat_000tc__RealDef__Oreal,axiom,
! [B_1_1,B_2_1,B_3_1] : is_bool(tendsto_nat_real(B_1_1,B_2_1,B_3_1)) ).
fof(gsy_c_Limits_Otendsto_000tc__RealDef__Oreal_000tc__RealDef__Oreal,axiom,
! [B_1_1,B_2_1,B_3_1] : is_bool(tendsto_real_real(B_1_1,B_2_1,B_3_1)) ).
fof(gsy_c_Limits_Otrivial__limit_000tc__Nat__Onat,axiom,
! [B_1_1] : is_bool(trivial_limit_nat(B_1_1)) ).
fof(gsy_c_Quickcheck__Narrowing_Oint__of,axiom,
! [B_1_1] : is_int(quickcheck_int_of(B_1_1)) ).
fof(gsy_c_RealDef_Ovanishes,axiom,
! [B_1_1] : is_bool(vanishes(B_1_1)) ).
fof(gsy_c_SEQ_OBseq_000tc__RealDef__Oreal,axiom,
! [B_1_1] : is_bool(bseq_real(B_1_1)) ).
fof(gsy_c_SEQ_OCauchy_000tc__Complex__Ocomplex,axiom,
! [B_1_1] : is_bool(cauchy_complex(B_1_1)) ).
fof(gsy_c_SEQ_OCauchy_000tc__RealDef__Oreal,axiom,
! [B_1_1] : is_bool(cauchy_real(B_1_1)) ).
fof(gsy_c_SEQ_Omonoseq_000tc__RealDef__Oreal,axiom,
! [B_1_1] : is_bool(monoseq_real(B_1_1)) ).
fof(gsy_c_Series_Osummable_000tc__Complex__Ocomplex,axiom,
! [B_1_1] : is_bool(summable_complex(B_1_1)) ).
fof(gsy_c_Series_Osummable_000tc__RealDef__Oreal,axiom,
! [B_1_1] : is_bool(summable_real(B_1_1)) ).
fof(gsy_c_fFalse,axiom,
is_bool(fFalse) ).
fof(gsy_c_fTrue,axiom,
is_bool(fTrue) ).
fof(gsy_c_fdisj,axiom,
! [B_1_1,B_2_1] :
( ( is_bool(B_1_1)
& is_bool(B_2_1) )
=> is_bool(fdisj(B_1_1,B_2_1)) ) ).
fof(gsy_c_hAPP_000tc__Code____Numeral__Ocode____numeral_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_C24501650l_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__Complex__Ocomplex_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_complex_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__HOL__Obool_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] :
( is_bool(B_2_1)
=> is_bool(hAPP_bool_bool(B_1_1,B_2_1)) ) ).
fof(gsy_c_hAPP_000tc__Int__Oint_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] :
( is_int(B_2_1)
=> is_bool(hAPP_int_bool(B_1_1,B_2_1)) ) ).
fof(gsy_c_hAPP_000tc__Int__Oint_000tc__Int__Oint,hypothesis,
! [B_1_1,B_2_1] :
( is_int(B_2_1)
=> is_int(hAPP_int_int(B_1_1,B_2_1)) ) ).
fof(gsy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_nat_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__Nat__Onat_000tc__Int__Oint,hypothesis,
! [B_1_1,B_2_1] : is_int(hAPP_nat_int(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__Quickcheck____Narrowing__Ocode____int_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_Q2096512830t_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__Rat__Orat_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_rat_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__RealDef__Oreal_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_real_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_f448129468l_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__Int__Oint,axiom,
! [B_1_1,B_2_1] : is_int(hAPP_f1594865479ol_int(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_f54304608l_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Int__Oint,axiom,
! [B_1_1,B_2_1] : is_int(hAPP_f659380387ol_int(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_f1501066425l_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_f215623910l_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__fun_Itc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__f,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_f1469637905l_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__fun_Itc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__p,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_f526364711l_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__fun_Itc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__p,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_f115998247l_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_P603027463t_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oint,axiom,
! [B_1_1,B_2_1] : is_int(hAPP_P1175774780nt_int(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_P1555980039t_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__HOL__Obo,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_P1333854989l_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__I,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_P880235623l_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc__,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_P62333565t_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__prod_Itc__,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_P1463840893t_bool(B_1_1,B_2_1)) ).
fof(gsy_v_m,hypothesis,
is_int(m) ).
fof(gsy_v_s1____,axiom,
is_int(s1) ).
fof(gsy_v_s____,axiom,
is_int(s) ).
fof(gsy_v_t____,axiom,
is_int(t) ).
%----Relevant facts (5196)
fof(fact_0_tpos,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),t)) ).
fof(fact_1__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06,axiom,
( t = one_one_int
=> ? [X,Y] :
( is_int(X)
& is_int(Y)
& hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int) ) ) ).
fof(fact_2__0961_A_060_At_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06,axiom,
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),t))
=> ? [X,Y] :
( is_int(X)
& is_int(Y)
& hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int) ) ) ).
fof(fact_3_t__l__p,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,t),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) ).
fof(fact_4_p,axiom,
hBOOL(hAPP_int_bool(zprime,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) ).
fof(fact_5_t,axiom,
hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),t) ).
fof(fact_6_qf1pt,axiom,
hBOOL(hAPP_int_bool(twoSqu1770640450sum2sq,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),t))) ).
fof(fact_7_zadd__power2,axiom,
! [A_3,B_2] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)),B_2))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_8_zadd__power3,axiom,
! [A_3,B_2] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),A_3)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) ).
fof(fact_9_power2__sum,axiom,
! [X_2,Y_2] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Y_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2)),Y_2)) ).
fof(fact_10_power2__sum,axiom,
! [X_2,Y_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Y_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2)),Y_2)) ).
fof(fact_11_power2__sum,axiom,
! [X_2,Y_2] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Y_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2)),Y_2)) ).
fof(fact_12_power2__sum,axiom,
! [X_2,Y_2] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_2),Y_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2)),Y_2)) ).
fof(fact_13_power2__sum,axiom,
! [X_2,Y_2] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Y_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2)),Y_2)) ).
fof(fact_14_power2__eq__square__number__of,axiom,
! [W_17] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_i769753017umeral(number1443263063umeral,W_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,W_17)),hAPP_i769753017umeral(number1443263063umeral,W_17)) ).
fof(fact_15_power2__eq__square__number__of,axiom,
! [W_17] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_i1732201573de_int(number1226105091de_int,W_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,W_17)),hAPP_i1732201573de_int(number1226105091de_int,W_17)) ).
fof(fact_16_power2__eq__square__number__of,axiom,
! [W_17] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_int_rat(number_number_of_rat,W_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,W_17)),hAPP_int_rat(number_number_of_rat,W_17)) ).
fof(fact_17_power2__eq__square__number__of,axiom,
! [W_17] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,W_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,W_17)),hAPP_int_int(number_number_of_int,W_17)) ).
fof(fact_18_power2__eq__square__number__of,axiom,
! [W_17] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,W_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,W_17)),hAPP_int_real(number267125858f_real,W_17)) ).
fof(fact_19_power2__eq__square__number__of,axiom,
! [W_17] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,W_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,W_17)),hAPP_int_nat(number_number_of_nat,W_17)) ).
fof(fact_20_power2__eq__square__number__of,axiom,
! [W_17] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_int_complex(number528085621omplex,W_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,W_17)),hAPP_int_complex(number528085621omplex,W_17)) ).
fof(fact_21_cube__square,axiom,
! [A_3] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) ).
fof(fact_22_one__power2,axiom,
hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,one_one_int),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_int ).
fof(fact_23_one__power2,axiom,
hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,one_one_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_nat ).
fof(fact_24_one__power2,axiom,
hAPP_nat_real(hAPP_r474017924t_real(power_power_real,one_one_real),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_real ).
fof(fact_25_one__power2,axiom,
hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,one_on1645066479umeral),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_on1645066479umeral ).
fof(fact_26_one__power2,axiom,
hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,one_one_complex),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_complex ).
fof(fact_27_one__power2,axiom,
hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,one_on1684967323de_int),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_on1684967323de_int ).
fof(fact_28_one__power2,axiom,
hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,one_one_rat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_rat ).
fof(fact_29_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_49] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_49),X_49) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_49),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_30_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_49] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_49),X_49) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_49),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_31_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_49] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_49),X_49) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_49),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_32_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_49] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_49),X_49) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_49),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_33_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_49] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_49),X_49) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_49),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_34_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_49] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_49),X_49) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_49),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_35_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_49] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_49),X_49) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_49),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_36_power2__eq__square,axiom,
! [A_239] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_239),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_239),A_239) ).
fof(fact_37_power2__eq__square,axiom,
! [A_239] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_239),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_239),A_239) ).
fof(fact_38_power2__eq__square,axiom,
! [A_239] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_239),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_239),A_239) ).
fof(fact_39_power2__eq__square,axiom,
! [A_239] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_239),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_239),A_239) ).
fof(fact_40_power2__eq__square,axiom,
! [A_239] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_239),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_239),A_239) ).
fof(fact_41_power2__eq__square,axiom,
! [A_239] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_239),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_239),A_239) ).
fof(fact_42_power2__eq__square,axiom,
! [A_239] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_239),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_239),A_239) ).
fof(fact_43_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_48,N_69] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_48),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_48),N_69)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_48),N_69)) ).
fof(fact_44_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_48,N_69] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_48),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_48),N_69)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_48),N_69)) ).
fof(fact_45_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_48,N_69] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_48),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_48),N_69)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_48),N_69)) ).
fof(fact_46_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_48,N_69] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_48),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_48),N_69)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_48),N_69)) ).
fof(fact_47_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_48,N_69] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_48),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_48),N_69)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_48),N_69)) ).
fof(fact_48_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_48,N_69] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_48),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_48),N_69)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_48),N_69)) ).
fof(fact_49_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_48,N_69] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_48),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_48),N_69)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_48),N_69)) ).
fof(fact_50_add__special_I2_J,axiom,
! [W_16] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W_16)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_16)) ).
fof(fact_51_add__special_I2_J,axiom,
! [W_16] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_int_real(number267125858f_real,W_16)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_16)) ).
fof(fact_52_add__special_I2_J,axiom,
! [W_16] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),hAPP_int_complex(number528085621omplex,W_16)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_16)) ).
fof(fact_53_add__special_I2_J,axiom,
! [W_16] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,W_16)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_16)) ).
fof(fact_54_add__special_I3_J,axiom,
! [V_23] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_23)),one_one_int) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),hAPP_int_int(bit1,pls))) ).
fof(fact_55_add__special_I3_J,axiom,
! [V_23] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_23)),one_one_real) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),hAPP_int_int(bit1,pls))) ).
fof(fact_56_add__special_I3_J,axiom,
! [V_23] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_23)),one_one_complex) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),hAPP_int_int(bit1,pls))) ).
fof(fact_57_add__special_I3_J,axiom,
! [V_23] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_23)),one_one_rat) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),hAPP_int_int(bit1,pls))) ).
fof(fact_58_one__add__one__is__two,axiom,
hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int) = hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).
fof(fact_59_one__add__one__is__two,axiom,
hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),one_one_real) = hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).
fof(fact_60_one__add__one__is__two,axiom,
hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),one_one_complex) = hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).
fof(fact_61_one__add__one__is__two,axiom,
hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).
fof(fact_62__096_B_Bthesis_O_A_I_B_Bt_O_As_A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_L_A1_,axiom,
~ ! [T] :
( is_int(T)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int) != hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),T) ) ).
fof(fact_63_zle__refl,axiom,
! [W] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),W)) ).
fof(fact_64_zle__linear,axiom,
! [Z_1,W] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_1)) ) ).
fof(fact_65_zless__le,axiom,
! [Z_2,W_1] :
( ( is_int(Z_2)
& is_int(W_1) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),W_1))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),W_1))
& Z_2 != W_1 ) ) ) ).
fof(fact_66_zless__linear,axiom,
! [X_1,Y_1] :
( ( is_int(X_1)
& is_int(Y_1) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
| X_1 = Y_1
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),X_1)) ) ) ).
fof(fact_67_zle__trans,axiom,
! [K_2,I,J_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_2),K_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),K_2)) ) ) ).
fof(fact_68_zle__antisym,axiom,
! [Z_1,W] :
( ( is_int(Z_1)
& is_int(W) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_1))
=> Z_1 = W ) ) ) ).
fof(fact_69_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_47,P_8,Q_12] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_47),P_8)),Q_12) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_47),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) ).
fof(fact_70_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_47,P_8,Q_12] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_47),P_8)),Q_12) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_47),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) ).
fof(fact_71_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_47,P_8,Q_12] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_47),P_8)),Q_12) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_47),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) ).
fof(fact_72_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_47,P_8,Q_12] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_47),P_8)),Q_12) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_47),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) ).
fof(fact_73_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_47,P_8,Q_12] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_47),P_8)),Q_12) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_47),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) ).
fof(fact_74_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_47,P_8,Q_12] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_47),P_8)),Q_12) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_47),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) ).
fof(fact_75_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_47,P_8,Q_12] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_47),P_8)),Q_12) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_47),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) ).
fof(fact_76_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_46] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_46),one_one_nat) = X_46 ).
fof(fact_77_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_46] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_46),one_one_nat) = X_46 ).
fof(fact_78_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_46] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_46),one_one_nat) = X_46 ).
fof(fact_79_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_46] :
( is_int(X_46)
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_46),one_one_nat) = X_46 ) ).
fof(fact_80_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_46] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_46),one_one_nat) = X_46 ).
fof(fact_81_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_46] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_46),one_one_nat) = X_46 ).
fof(fact_82_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_46] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_46),one_one_nat) = X_46 ).
fof(fact_83_zpower__zpower,axiom,
! [X_1,Y_1,Z_1] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),Y_1)),Z_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Y_1),Z_1)) ).
fof(fact_84_le__number__of__eq__not__less,axiom,
! [V,W_1] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_i769753017umeral(number1443263063umeral,V)),hAPP_i769753017umeral(number1443263063umeral,W_1)))
<=> ~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_i769753017umeral(number1443263063umeral,W_1)),hAPP_i769753017umeral(number1443263063umeral,V))) ) ).
fof(fact_85_le__number__of__eq__not__less,axiom,
! [V,W_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,V)),hAPP_int_int(number_number_of_int,W_1)))
<=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W_1)),hAPP_int_int(number_number_of_int,V))) ) ).
fof(fact_86_le__number__of__eq__not__less,axiom,
! [V,W_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,W_1)))
<=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,W_1)),hAPP_int_nat(number_number_of_nat,V))) ) ).
fof(fact_87_le__number__of__eq__not__less,axiom,
! [V,W_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,V)),hAPP_int_real(number267125858f_real,W_1)))
<=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,W_1)),hAPP_int_real(number267125858f_real,V))) ) ).
fof(fact_88_le__number__of__eq__not__less,axiom,
! [V,W_1] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_i1732201573de_int(number1226105091de_int,V)),hAPP_i1732201573de_int(number1226105091de_int,W_1)))
<=> ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_i1732201573de_int(number1226105091de_int,W_1)),hAPP_i1732201573de_int(number1226105091de_int,V))) ) ).
fof(fact_89_le__number__of__eq__not__less,axiom,
! [V,W_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,V)),hAPP_int_rat(number_number_of_rat,W_1)))
<=> ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,W_1)),hAPP_int_rat(number_number_of_rat,V))) ) ).
fof(fact_90_less__number__of,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,X_2)),hAPP_int_int(number_number_of_int,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Y_2)) ) ).
fof(fact_91_less__number__of,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,X_2)),hAPP_int_real(number267125858f_real,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Y_2)) ) ).
fof(fact_92_less__number__of,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,X_2)),hAPP_int_rat(number_number_of_rat,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Y_2)) ) ).
fof(fact_93_le__number__of,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,X_2)),hAPP_int_int(number_number_of_int,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_2)) ) ).
fof(fact_94_le__number__of,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,X_2)),hAPP_int_real(number267125858f_real,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_2)) ) ).
fof(fact_95_le__number__of,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,X_2)),hAPP_int_rat(number_number_of_rat,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_2)) ) ).
fof(fact_96_zadd__zless__mono,axiom,
! [Z_4,Z_1,W_15,W] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_15),W))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_4),Z_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_15),Z_4)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),Z_1))) ) ) ).
fof(fact_97_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_45,P_7,Q_11] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_45),P_7)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_45),Q_11)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_45),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_11)) ).
fof(fact_98_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_45,P_7,Q_11] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_45),P_7)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_45),Q_11)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_45),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_11)) ).
fof(fact_99_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_45,P_7,Q_11] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_45),P_7)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_45),Q_11)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_45),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_11)) ).
fof(fact_100_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_45,P_7,Q_11] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_45),P_7)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_45),Q_11)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_45),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_11)) ).
fof(fact_101_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_45,P_7,Q_11] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_45),P_7)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_45),Q_11)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_45),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_11)) ).
fof(fact_102_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_45,P_7,Q_11] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_45),P_7)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_45),Q_11)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_45),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_11)) ).
fof(fact_103_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_45,P_7,Q_11] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_45),P_7)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_45),Q_11)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_45),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_11)) ).
fof(fact_104_zpower__zadd__distrib,axiom,
! [X_1,Y_1,Z_1] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_1),Z_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),Y_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),Z_1)) ).
fof(fact_105_nat__mult__2,axiom,
! [Z_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_1),Z_1) ).
fof(fact_106_nat__mult__2__right,axiom,
! [Z_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_1),Z_1) ).
fof(fact_107_nat__1__add__1,axiom,
hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).
fof(fact_108_less__int__code_I16_J,axiom,
! [K1,K2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit1,K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).
fof(fact_109_rel__simps_I17_J,axiom,
! [K_1,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit1,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) ) ).
fof(fact_110_less__eq__int__code_I16_J,axiom,
! [K1,K2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit1,K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).
fof(fact_111_rel__simps_I34_J,axiom,
! [K_1,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit1,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) ) ).
fof(fact_112_rel__simps_I2_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),pls)) ).
fof(fact_113_less__int__code_I13_J,axiom,
! [K1,K2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit0,K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).
fof(fact_114_rel__simps_I14_J,axiom,
! [K_1,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit0,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) ) ).
fof(fact_115_rel__simps_I19_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),pls)) ).
fof(fact_116_less__eq__int__code_I13_J,axiom,
! [K1,K2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit0,K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).
fof(fact_117_rel__simps_I31_J,axiom,
! [K_1,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit0,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) ) ).
fof(fact_118_less__number__of__int__code,axiom,
! [K_1,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,K_1)),hAPP_int_int(number_number_of_int,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) ) ).
fof(fact_119_less__eq__number__of__int__code,axiom,
! [K_1,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,K_1)),hAPP_int_int(number_number_of_int,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) ) ).
fof(fact_120_zadd__strict__right__mono,axiom,
! [K_2,I,J_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I),K_2)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,J_2),K_2))) ) ).
fof(fact_121_zadd__left__mono,axiom,
! [K_2,I,J_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),I)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),J_2))) ) ).
fof(fact_122_add__nat__number__of,axiom,
! [V_3,V_2] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,V_3) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_3),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,V_2) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_3),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),V_3)) ) ) ) ) ).
fof(fact_123_nat__numeral__1__eq__1,axiom,
hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)) = one_one_nat ).
fof(fact_124_Numeral1__eq1__nat,axiom,
one_one_nat = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)) ).
fof(fact_125_rel__simps_I29_J,axiom,
! [K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),pls)) ) ).
fof(fact_126_rel__simps_I5_J,axiom,
! [K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),hAPP_int_int(bit1,K_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_1)) ) ).
fof(fact_127_less__eq__int__code_I15_J,axiom,
! [K1,K2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit0,K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).
fof(fact_128_rel__simps_I33_J,axiom,
! [K_1,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit0,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) ) ).
fof(fact_129_less__int__code_I14_J,axiom,
! [K1,K2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit1,K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).
fof(fact_130_rel__simps_I15_J,axiom,
! [K_1,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit1,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) ) ).
fof(fact_131_zless__imp__add1__zle,axiom,
! [W,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),one_one_int)),Z_1)) ) ).
fof(fact_132_add1__zle__eq,axiom,
! [W_1,Z_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),one_one_int)),Z_2))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) ) ).
fof(fact_133_zle__add1__eq__le,axiom,
! [W_1,Z_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_2),one_one_int)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_2)) ) ).
fof(fact_134_zprime__2,axiom,
hBOOL(hAPP_int_bool(zprime,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_135_is__mult__sum2sq,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(twoSqu1770640450sum2sq,X_1))
=> ( hBOOL(hAPP_int_bool(twoSqu1770640450sum2sq,Y_1))
=> hBOOL(hAPP_int_bool(twoSqu1770640450sum2sq,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1))) ) ) ).
fof(fact_136_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6,Ly_4,Rx_6,Ry_4] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_6),Ly_4)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_6),Ry_4)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_6),Rx_6)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Ly_4),Ry_4)) ).
fof(fact_137_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6,Ly_4,Rx_6,Ry_4] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),Ly_4)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_6),Ry_4)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),Rx_6)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly_4),Ry_4)) ).
fof(fact_138_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6,Ly_4,Rx_6,Ry_4] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),Ly_4)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_6),Ry_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),Rx_6)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly_4),Ry_4)) ).
fof(fact_139_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6,Ly_4,Rx_6,Ry_4] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_6),Ly_4)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_6),Ry_4)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_6),Rx_6)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly_4),Ry_4)) ).
fof(fact_140_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6,Ly_4,Rx_6,Ry_4] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_6),Ly_4)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_6),Ry_4)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_6),Rx_6)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly_4),Ry_4)) ).
fof(fact_141_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6,Ly_4,Rx_6,Ry_4] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_6),Ly_4)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_6),Ry_4)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_6),Rx_6)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly_4),Ry_4)) ).
fof(fact_142_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6,Ly_4,Rx_6,Ry_4] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_6),Ly_4)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_6),Ry_4)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_6),Rx_6)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ly_4),Ry_4)) ).
fof(fact_143_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5,Ly_3,Rx_5,Ry_3] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_5),Ly_3)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_5),Ry_3)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_5),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_5),Ly_3)),Ry_3)) ).
fof(fact_144_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5,Ly_3,Rx_5,Ry_3] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_5),Ly_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_5),Ry_3)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_5),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_5),Ly_3)),Ry_3)) ).
fof(fact_145_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5,Ly_3,Rx_5,Ry_3] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),Ly_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_5),Ry_3)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),Ly_3)),Ry_3)) ).
fof(fact_146_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5,Ly_3,Rx_5,Ry_3] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_5),Ly_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_5),Ry_3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_5),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_5),Ly_3)),Ry_3)) ).
fof(fact_147_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5,Ly_3,Rx_5,Ry_3] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_5),Ly_3)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_5),Ry_3)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_5),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_5),Ly_3)),Ry_3)) ).
fof(fact_148_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5,Ly_3,Rx_5,Ry_3] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_5),Ly_3)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_5),Ry_3)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_5),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_5),Ly_3)),Ry_3)) ).
fof(fact_149_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5,Ly_3,Rx_5,Ry_3] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_5),Ly_3)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_5),Ry_3)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_5),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_5),Ly_3)),Ry_3)) ).
fof(fact_150_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4,Ly_2,Rx_4,Ry_2] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_4),Ly_2)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_4),Ry_2)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_4),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Ly_2),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_4),Ry_2))) ).
fof(fact_151_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4,Ly_2,Rx_4,Ry_2] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_4),Ly_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_4),Ry_2)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_4),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_4),Ry_2))) ).
fof(fact_152_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4,Ly_2,Rx_4,Ry_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_4),Ly_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_4),Ry_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_4),Ry_2))) ).
fof(fact_153_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4,Ly_2,Rx_4,Ry_2] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_4),Ly_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_4),Ry_2)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_4),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_4),Ry_2))) ).
fof(fact_154_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4,Ly_2,Rx_4,Ry_2] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_4),Ly_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_4),Ry_2)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_4),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly_2),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_4),Ry_2))) ).
fof(fact_155_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4,Ly_2,Rx_4,Ry_2] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_4),Ly_2)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_4),Ry_2)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_4),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly_2),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_4),Ry_2))) ).
fof(fact_156_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4,Ly_2,Rx_4,Ry_2] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_4),Ly_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_4),Ry_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_4),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ly_2),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_4),Ry_2))) ).
fof(fact_157_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_1,Rx_3] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_3),Ly_1)),Rx_3) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_3),Rx_3)),Ly_1) ).
fof(fact_158_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_1,Rx_3] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_3),Ly_1)),Rx_3) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_3),Rx_3)),Ly_1) ).
fof(fact_159_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_1,Rx_3] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_3),Ly_1)),Rx_3) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_3),Rx_3)),Ly_1) ).
fof(fact_160_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_1,Rx_3] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_3),Ly_1)),Rx_3) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_3),Rx_3)),Ly_1) ).
fof(fact_161_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_1,Rx_3] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_3),Ly_1)),Rx_3) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_3),Rx_3)),Ly_1) ).
fof(fact_162_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_1,Rx_3] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_3),Ly_1)),Rx_3) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_3),Rx_3)),Ly_1) ).
fof(fact_163_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_1,Rx_3] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_3),Ly_1)),Rx_3) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_3),Rx_3)),Ly_1) ).
fof(fact_164_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2,Ly,Rx_2] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_2),Ly)),Rx_2) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_2),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Ly),Rx_2)) ).
fof(fact_165_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2,Ly,Rx_2] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_2),Ly)),Rx_2) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly),Rx_2)) ).
fof(fact_166_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2,Ly,Rx_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_2),Ly)),Rx_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly),Rx_2)) ).
fof(fact_167_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2,Ly,Rx_2] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_2),Ly)),Rx_2) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly),Rx_2)) ).
fof(fact_168_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2,Ly,Rx_2] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_2),Ly)),Rx_2) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_2),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly),Rx_2)) ).
fof(fact_169_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2,Ly,Rx_2] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_2),Ly)),Rx_2) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_2),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly),Rx_2)) ).
fof(fact_170_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2,Ly,Rx_2] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_2),Ly)),Rx_2) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_2),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ly),Rx_2)) ).
fof(fact_171_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1,Rx_1,Ry_1] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_1),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_1),Ry_1)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_1),Rx_1)),Ry_1) ).
fof(fact_172_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1,Rx_1,Ry_1] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_1),Ry_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_1),Rx_1)),Ry_1) ).
fof(fact_173_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1,Rx_1,Ry_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_1),Ry_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_1),Rx_1)),Ry_1) ).
fof(fact_174_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1,Rx_1,Ry_1] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_1),Ry_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_1),Rx_1)),Ry_1) ).
fof(fact_175_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1,Rx_1,Ry_1] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_1),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_1),Ry_1)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_1),Rx_1)),Ry_1) ).
fof(fact_176_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1,Rx_1,Ry_1] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_1),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_1),Ry_1)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_1),Rx_1)),Ry_1) ).
fof(fact_177_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1,Rx_1,Ry_1] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_1),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_1),Ry_1)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_1),Rx_1)),Ry_1) ).
fof(fact_178_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx,Rx,Ry] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx),Ry)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx),Ry)) ).
fof(fact_179_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx,Rx,Ry] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx),Ry)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),Ry)) ).
fof(fact_180_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx,Rx,Ry] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx),Ry)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),Ry)) ).
fof(fact_181_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx,Rx,Ry] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx),Ry)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx),Ry)) ).
fof(fact_182_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx,Rx,Ry] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx),Ry)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx),Ry)) ).
fof(fact_183_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx,Rx,Ry] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx),Ry)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx),Ry)) ).
fof(fact_184_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx,Rx,Ry] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx),Ry)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx),Ry)) ).
fof(fact_185_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_238,B_167] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_238),B_167) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_167),A_238) ).
fof(fact_186_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_238,B_167] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_238),B_167) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_167),A_238) ).
fof(fact_187_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_238,B_167] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_238),B_167) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_167),A_238) ).
fof(fact_188_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_238,B_167] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_238),B_167) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_167),A_238) ).
fof(fact_189_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_238,B_167] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_238),B_167) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_167),A_238) ).
fof(fact_190_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_238,B_167] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_238),B_167) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_167),A_238) ).
fof(fact_191_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_238,B_167] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_238),B_167) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_167),A_238) ).
fof(fact_192_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_237,B_166,C_97,D_20] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_237),B_166)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_97),D_20)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_237),C_97)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_166),D_20)) ).
fof(fact_193_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_237,B_166,C_97,D_20] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_237),B_166)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_97),D_20)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_237),C_97)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_166),D_20)) ).
fof(fact_194_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_237,B_166,C_97,D_20] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_237),B_166)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_97),D_20)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_237),C_97)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_166),D_20)) ).
fof(fact_195_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_237,B_166,C_97,D_20] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_237),B_166)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_97),D_20)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_237),C_97)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_166),D_20)) ).
fof(fact_196_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_237,B_166,C_97,D_20] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_237),B_166)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_97),D_20)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_237),C_97)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_166),D_20)) ).
fof(fact_197_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_237,B_166,C_97,D_20] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_237),B_166)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_97),D_20)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_237),C_97)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_166),D_20)) ).
fof(fact_198_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_237,B_166,C_97,D_20] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_237),B_166)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_97),D_20)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_237),C_97)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_166),D_20)) ).
fof(fact_199_mem__def,axiom,
! [X_2,A_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A_1))
<=> hBOOL(hAPP_int_bool(A_1,X_2)) ) ).
fof(fact_200_mem__def,axiom,
! [X_2,A_1] :
( hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,X_2),A_1))
<=> hBOOL(hAPP_P1463840893t_bool(A_1,X_2)) ) ).
fof(fact_201_mem__def,axiom,
! [X_2,A_1] :
( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_2),A_1))
<=> hBOOL(hAPP_f448129468l_bool(A_1,X_2)) ) ).
fof(fact_202_mem__def,axiom,
! [X_2,A_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_1))
<=> hBOOL(hAPP_nat_bool(A_1,X_2)) ) ).
fof(fact_203_mem__def,axiom,
! [X_2,A_1] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),A_1))
<=> hBOOL(hAPP_real_bool(A_1,X_2)) ) ).
fof(fact_204_mem__def,axiom,
! [X_2,A_1] :
( hBOOL(hAPP_f526364711l_bool(hAPP_P1062890741l_bool(member2143287562nt_int,X_2),A_1))
<=> hBOOL(hAPP_P62333565t_bool(A_1,X_2)) ) ).
fof(fact_205_mem__def,axiom,
! [X_2,A_1] :
( hBOOL(hAPP_f1469637905l_bool(hAPP_P1681069257l_bool(member308914164t_bool,X_2),A_1))
<=> hBOOL(hAPP_P880235623l_bool(A_1,X_2)) ) ).
fof(fact_206_Collect__def,axiom,
! [P_1] : collect_int(P_1) = P_1 ).
fof(fact_207_Collect__def,axiom,
! [P_1] : collect_nat(P_1) = P_1 ).
fof(fact_208_Collect__def,axiom,
! [P_1] : collec50511176nt_int(P_1) = P_1 ).
fof(fact_209_Collect__def,axiom,
! [P_1] : collec1347809874nt_int(P_1) = P_1 ).
fof(fact_210_Collect__def,axiom,
! [P_1] : collect_real(P_1) = P_1 ).
fof(fact_211_Collect__def,axiom,
! [P_1] : collec1979865426at_nat(P_1) = P_1 ).
fof(fact_212_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_236,B_165,C_96] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_236),B_165)),C_96) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_236),C_96)),B_165) ).
fof(fact_213_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_236,B_165,C_96] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_236),B_165)),C_96) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_236),C_96)),B_165) ).
fof(fact_214_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_236,B_165,C_96] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_236),B_165)),C_96) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_236),C_96)),B_165) ).
fof(fact_215_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_236,B_165,C_96] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_236),B_165)),C_96) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_236),C_96)),B_165) ).
fof(fact_216_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_236,B_165,C_96] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_236),B_165)),C_96) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_236),C_96)),B_165) ).
fof(fact_217_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_236,B_165,C_96] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_236),B_165)),C_96) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_236),C_96)),B_165) ).
fof(fact_218_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_236,B_165,C_96] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_236),B_165)),C_96) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_236),C_96)),B_165) ).
fof(fact_219_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_235,B_164,C_95] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_235),B_164)),C_95) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_235),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_164),C_95)) ).
fof(fact_220_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_235,B_164,C_95] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_235),B_164)),C_95) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_235),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_164),C_95)) ).
fof(fact_221_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_235,B_164,C_95] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_235),B_164)),C_95) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_235),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_164),C_95)) ).
fof(fact_222_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_235,B_164,C_95] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_235),B_164)),C_95) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_235),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_164),C_95)) ).
fof(fact_223_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_235,B_164,C_95] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_235),B_164)),C_95) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_235),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_164),C_95)) ).
fof(fact_224_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_235,B_164,C_95] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_235),B_164)),C_95) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_235),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_164),C_95)) ).
fof(fact_225_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_235,B_164,C_95] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_235),B_164)),C_95) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_235),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_164),C_95)) ).
fof(fact_226_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_234,C_94,D_19] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_234),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_94),D_19)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_234),C_94)),D_19) ).
fof(fact_227_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_234,C_94,D_19] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_234),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_94),D_19)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_234),C_94)),D_19) ).
fof(fact_228_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_234,C_94,D_19] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_234),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_94),D_19)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_234),C_94)),D_19) ).
fof(fact_229_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_234,C_94,D_19] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_234),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_94),D_19)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_234),C_94)),D_19) ).
fof(fact_230_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_234,C_94,D_19] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_234),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_94),D_19)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_234),C_94)),D_19) ).
fof(fact_231_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_234,C_94,D_19] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_234),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_94),D_19)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_234),C_94)),D_19) ).
fof(fact_232_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_234,C_94,D_19] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_234),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_94),D_19)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_234),C_94)),D_19) ).
fof(fact_233_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_233,C_93,D_18] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_233),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_93),D_18)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_93),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_233),D_18)) ).
fof(fact_234_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_233,C_93,D_18] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_233),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_93),D_18)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_93),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_233),D_18)) ).
fof(fact_235_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_233,C_93,D_18] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_233),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_93),D_18)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_93),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_233),D_18)) ).
fof(fact_236_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_233,C_93,D_18] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_233),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_93),D_18)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_93),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_233),D_18)) ).
fof(fact_237_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_233,C_93,D_18] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_233),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_93),D_18)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_93),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_233),D_18)) ).
fof(fact_238_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_233,C_93,D_18] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_233),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_93),D_18)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_93),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_233),D_18)) ).
fof(fact_239_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_233,C_93,D_18] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_233),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_93),D_18)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_93),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_233),D_18)) ).
fof(fact_240_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_232,C_92] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_232),C_92) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_92),A_232) ).
fof(fact_241_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_232,C_92] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_232),C_92) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_92),A_232) ).
fof(fact_242_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_232,C_92] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_232),C_92) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_92),A_232) ).
fof(fact_243_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_232,C_92] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_232),C_92) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_92),A_232) ).
fof(fact_244_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_232,C_92] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_232),C_92) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_92),A_232) ).
fof(fact_245_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_232,C_92] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_232),C_92) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_92),A_232) ).
fof(fact_246_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_232,C_92] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_232),C_92) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_92),A_232) ).
fof(fact_247_eq__number__of,axiom,
! [X_2,Y_2] :
( ( is_int(X_2)
& is_int(Y_2) )
=> ( hAPP_int_int(number_number_of_int,X_2) = hAPP_int_int(number_number_of_int,Y_2)
<=> X_2 = Y_2 ) ) ).
fof(fact_248_eq__number__of,axiom,
! [X_2,Y_2] :
( ( is_int(X_2)
& is_int(Y_2) )
=> ( hAPP_int_real(number267125858f_real,X_2) = hAPP_int_real(number267125858f_real,Y_2)
<=> X_2 = Y_2 ) ) ).
fof(fact_249_eq__number__of,axiom,
! [X_2,Y_2] :
( ( is_int(X_2)
& is_int(Y_2) )
=> ( hAPP_int_complex(number528085621omplex,X_2) = hAPP_int_complex(number528085621omplex,Y_2)
<=> X_2 = Y_2 ) ) ).
fof(fact_250_eq__number__of,axiom,
! [X_2,Y_2] :
( ( is_int(X_2)
& is_int(Y_2) )
=> ( hAPP_int_rat(number_number_of_rat,X_2) = hAPP_int_rat(number_number_of_rat,Y_2)
<=> X_2 = Y_2 ) ) ).
fof(fact_251_number__of__reorient,axiom,
! [W_1,X_2] :
( hAPP_int_nat(number_number_of_nat,W_1) = X_2
<=> X_2 = hAPP_int_nat(number_number_of_nat,W_1) ) ).
fof(fact_252_number__of__reorient,axiom,
! [W_1,X_2] :
( is_int(X_2)
=> ( hAPP_int_int(number_number_of_int,W_1) = X_2
<=> X_2 = hAPP_int_int(number_number_of_int,W_1) ) ) ).
fof(fact_253_number__of__reorient,axiom,
! [W_1,X_2] :
( hAPP_int_real(number267125858f_real,W_1) = X_2
<=> X_2 = hAPP_int_real(number267125858f_real,W_1) ) ).
fof(fact_254_number__of__reorient,axiom,
! [W_1,X_2] :
( hAPP_i769753017umeral(number1443263063umeral,W_1) = X_2
<=> X_2 = hAPP_i769753017umeral(number1443263063umeral,W_1) ) ).
fof(fact_255_number__of__reorient,axiom,
! [W_1,X_2] :
( hAPP_int_complex(number528085621omplex,W_1) = X_2
<=> X_2 = hAPP_int_complex(number528085621omplex,W_1) ) ).
fof(fact_256_number__of__reorient,axiom,
! [W_1,X_2] :
( hAPP_i1732201573de_int(number1226105091de_int,W_1) = X_2
<=> X_2 = hAPP_i1732201573de_int(number1226105091de_int,W_1) ) ).
fof(fact_257_number__of__reorient,axiom,
! [W_1,X_2] :
( hAPP_int_rat(number_number_of_rat,W_1) = X_2
<=> X_2 = hAPP_int_rat(number_number_of_rat,W_1) ) ).
fof(fact_258_rel__simps_I51_J,axiom,
! [K_1,L] :
( ( is_int(K_1)
& is_int(L) )
=> ( hAPP_int_int(bit1,K_1) = hAPP_int_int(bit1,L)
<=> K_1 = L ) ) ).
fof(fact_259_rel__simps_I48_J,axiom,
! [K_1,L] :
( ( is_int(K_1)
& is_int(L) )
=> ( hAPP_int_int(bit0,K_1) = hAPP_int_int(bit0,L)
<=> K_1 = L ) ) ).
fof(fact_260_zmult__assoc,axiom,
! [Z1,Z2,Z3] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),Z2)),Z3) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),Z3)) ).
fof(fact_261_zmult__commute,axiom,
! [Z_1,W] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),W) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z_1) ).
fof(fact_262_number__of__is__id,axiom,
! [K_2] :
( is_int(K_2)
=> hAPP_int_int(number_number_of_int,K_2) = K_2 ) ).
fof(fact_263_zadd__assoc,axiom,
! [Z1,Z2,Z3] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)),Z3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z2),Z3)) ).
fof(fact_264_zadd__left__commute,axiom,
! [X_1,Y_1,Z_1] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y_1),Z_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Z_1)) ).
fof(fact_265_zadd__commute,axiom,
! [Z_1,W] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),Z_1) ).
fof(fact_266_rel__simps_I12_J,axiom,
! [K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),pls)) ) ).
fof(fact_267_less__int__code_I15_J,axiom,
! [K1,K2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit0,K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).
fof(fact_268_rel__simps_I16_J,axiom,
! [K_1,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit0,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) ) ).
fof(fact_269_rel__simps_I10_J,axiom,
! [K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),pls)) ) ).
fof(fact_270_rel__simps_I4_J,axiom,
! [K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),hAPP_int_int(bit0,K_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),K_1)) ) ).
fof(fact_271_rel__simps_I22_J,axiom,
! [K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),hAPP_int_int(bit1,K_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_1)) ) ).
fof(fact_272_less__eq__int__code_I14_J,axiom,
! [K1,K2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit1,K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).
fof(fact_273_rel__simps_I32_J,axiom,
! [K_1,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit1,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) ) ).
fof(fact_274_rel__simps_I27_J,axiom,
! [K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),pls)) ) ).
fof(fact_275_rel__simps_I21_J,axiom,
! [K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),hAPP_int_int(bit0,K_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_1)) ) ).
fof(fact_276_zless__add1__eq,axiom,
! [W_1,Z_2] :
( ( is_int(W_1)
& is_int(Z_2) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_2),one_one_int)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2))
| W_1 = Z_2 ) ) ) ).
fof(fact_277_power__even__eq,axiom,
! [A_231,N_68] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_231),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_231),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_278_power__even__eq,axiom,
! [A_231,N_68] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_231),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_231),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_279_power__even__eq,axiom,
! [A_231,N_68] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_231),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_231),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_280_power__even__eq,axiom,
! [A_231,N_68] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_231),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_231),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_281_power__even__eq,axiom,
! [A_231,N_68] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_231),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_231),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_282_power__even__eq,axiom,
! [A_231,N_68] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_231),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_231),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_283_power__even__eq,axiom,
! [A_231,N_68] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_231),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_231),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_284_less__special_I4_J,axiom,
! [X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,X_2)),one_one_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),hAPP_int_int(bit1,pls))) ) ).
fof(fact_285_less__special_I4_J,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,X_2)),one_one_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),hAPP_int_int(bit1,pls))) ) ).
fof(fact_286_less__special_I4_J,axiom,
! [X_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,X_2)),one_one_rat))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),hAPP_int_int(bit1,pls))) ) ).
fof(fact_287_less__special_I2_J,axiom,
! [Y_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(number_number_of_int,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,pls)),Y_2)) ) ).
fof(fact_288_less__special_I2_J,axiom,
! [Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_int_real(number267125858f_real,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,pls)),Y_2)) ) ).
fof(fact_289_less__special_I2_J,axiom,
! [Y_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,pls)),Y_2)) ) ).
fof(fact_290_le__special_I4_J,axiom,
! [X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,X_2)),one_one_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),hAPP_int_int(bit1,pls))) ) ).
fof(fact_291_le__special_I4_J,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,X_2)),one_one_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),hAPP_int_int(bit1,pls))) ) ).
fof(fact_292_le__special_I4_J,axiom,
! [X_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,X_2)),one_one_rat))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),hAPP_int_int(bit1,pls))) ) ).
fof(fact_293_le__special_I2_J,axiom,
! [Y_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_int_int(number_number_of_int,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,pls)),Y_2)) ) ).
fof(fact_294_le__special_I2_J,axiom,
! [Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_int_real(number267125858f_real,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,pls)),Y_2)) ) ).
fof(fact_295_le__special_I2_J,axiom,
! [Y_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,pls)),Y_2)) ) ).
fof(fact_296_crossproduct__eq,axiom,
! [W_1,Y_2,X_2,Z_2] :
( ( is_int(W_1)
& is_int(Y_2)
& is_int(X_2)
& is_int(Z_2) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Y_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Z_2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Z_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Y_2))
<=> ( W_1 = X_2
| Y_2 = Z_2 ) ) ) ).
fof(fact_297_crossproduct__eq,axiom,
! [W_1,Y_2,X_2,Z_2] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,W_1),Y_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_2),Z_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,W_1),Z_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_2),Y_2))
<=> ( W_1 = X_2
| Y_2 = Z_2 ) ) ).
fof(fact_298_crossproduct__eq,axiom,
! [W_1,Y_2,X_2,Z_2] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,W_1),Y_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_2),Z_2)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,W_1),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_2),Y_2))
<=> ( W_1 = X_2
| Y_2 = Z_2 ) ) ).
fof(fact_299_crossproduct__eq,axiom,
! [W_1,Y_2,X_2,Z_2] :
( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,W_1),Y_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_2),Z_2)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,W_1),Z_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_2),Y_2))
<=> ( W_1 = X_2
| Y_2 = Z_2 ) ) ).
fof(fact_300_crossproduct__eq,axiom,
! [W_1,Y_2,X_2,Z_2] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,W_1),Y_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_2),Z_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,W_1),Z_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_2),Y_2))
<=> ( W_1 = X_2
| Y_2 = Z_2 ) ) ).
fof(fact_301_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_230,M_29,B_163] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_230),M_29)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_163),M_29)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_230),B_163)),M_29) ).
fof(fact_302_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_230,M_29,B_163] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_230),M_29)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_163),M_29)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_230),B_163)),M_29) ).
fof(fact_303_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_230,M_29,B_163] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_230),M_29)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_163),M_29)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_230),B_163)),M_29) ).
fof(fact_304_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_230,M_29,B_163] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_230),M_29)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_163),M_29)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_230),B_163)),M_29) ).
fof(fact_305_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_230,M_29,B_163] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_230),M_29)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_163),M_29)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_230),B_163)),M_29) ).
fof(fact_306_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_230,M_29,B_163] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_230),M_29)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_163),M_29)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_230),B_163)),M_29) ).
fof(fact_307_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_230,M_29,B_163] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_230),M_29)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_163),M_29)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_230),B_163)),M_29) ).
fof(fact_308_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_229,B_162,C_91] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_229),B_162)),C_91) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_229),C_91)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_162),C_91)) ).
fof(fact_309_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_229,B_162,C_91] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_229),B_162)),C_91) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_229),C_91)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_162),C_91)) ).
fof(fact_310_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_229,B_162,C_91] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_229),B_162)),C_91) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_229),C_91)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_162),C_91)) ).
fof(fact_311_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_229,B_162,C_91] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_229),B_162)),C_91) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_229),C_91)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_162),C_91)) ).
fof(fact_312_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_229,B_162,C_91] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_229),B_162)),C_91) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_229),C_91)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_162),C_91)) ).
fof(fact_313_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_229,B_162,C_91] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_229),B_162)),C_91) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_229),C_91)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_162),C_91)) ).
fof(fact_314_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_229,B_162,C_91] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_229),B_162)),C_91) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_229),C_91)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_162),C_91)) ).
fof(fact_315_crossproduct__noteq,axiom,
! [C,D,A,B] :
( ( is_int(C)
& is_int(D)
& is_int(A)
& is_int(B) )
=> ( ( A != B
& C != D )
<=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D)) != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),C)) ) ) ).
fof(fact_316_crossproduct__noteq,axiom,
! [C,D,A,B] :
( ( A != B
& C != D )
<=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),D)) != hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),D)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),C)) ) ).
fof(fact_317_crossproduct__noteq,axiom,
! [C,D,A,B] :
( ( A != B
& C != D )
<=> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),D)) != hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),D)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),C)) ) ).
fof(fact_318_crossproduct__noteq,axiom,
! [C,D,A,B] :
( ( A != B
& C != D )
<=> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),C)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),D)) != hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),D)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),C)) ) ).
fof(fact_319_crossproduct__noteq,axiom,
! [C,D,A,B] :
( ( A != B
& C != D )
<=> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),D)) != hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),D)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),C)) ) ).
fof(fact_320_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_44,Y_36,Z_15] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_44),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,Y_36),Z_15)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_44),Y_36)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_44),Z_15)) ).
fof(fact_321_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_44,Y_36,Z_15] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_44),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y_36),Z_15)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_44),Y_36)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_44),Z_15)) ).
fof(fact_322_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_44,Y_36,Z_15] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_44),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_36),Z_15)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_44),Y_36)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_44),Z_15)) ).
fof(fact_323_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_44,Y_36,Z_15] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_44),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y_36),Z_15)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_44),Y_36)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_44),Z_15)) ).
fof(fact_324_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_44,Y_36,Z_15] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_44),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Y_36),Z_15)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_44),Y_36)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_44),Z_15)) ).
fof(fact_325_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_44,Y_36,Z_15] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_44),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,Y_36),Z_15)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_44),Y_36)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_44),Z_15)) ).
fof(fact_326_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_44,Y_36,Z_15] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_44),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Y_36),Z_15)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_44),Y_36)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_44),Z_15)) ).
fof(fact_327_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_228] :
( is_int(A_228)
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_228),one_one_int) = A_228 ) ).
fof(fact_328_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_228] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_228),one_one_nat) = A_228 ).
fof(fact_329_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_228] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_228),one_one_real) = A_228 ).
fof(fact_330_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_228] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_228),one_on1645066479umeral) = A_228 ).
fof(fact_331_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_228] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_228),one_one_complex) = A_228 ).
fof(fact_332_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_228] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_228),one_on1684967323de_int) = A_228 ).
fof(fact_333_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_228] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_228),one_one_rat) = A_228 ).
fof(fact_334_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_227] :
( is_int(A_227)
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_227) = A_227 ) ).
fof(fact_335_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_227] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_227) = A_227 ).
fof(fact_336_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_227] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_227) = A_227 ).
fof(fact_337_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_227] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_227) = A_227 ).
fof(fact_338_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_227] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_227) = A_227 ).
fof(fact_339_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_227] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_227) = A_227 ).
fof(fact_340_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_227] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_227) = A_227 ).
fof(fact_341_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_43,Y_35,Q_10] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_43),Y_35)),Q_10) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_43),Q_10)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_35),Q_10)) ).
fof(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_43,Y_35,Q_10] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_43),Y_35)),Q_10) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_43),Q_10)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_35),Q_10)) ).
fof(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_43,Y_35,Q_10] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_43),Y_35)),Q_10) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_43),Q_10)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_35),Q_10)) ).
fof(fact_344_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_43,Y_35,Q_10] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_43),Y_35)),Q_10) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_43),Q_10)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_35),Q_10)) ).
fof(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_43,Y_35,Q_10] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_43),Y_35)),Q_10) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_43),Q_10)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_35),Q_10)) ).
fof(fact_346_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_43,Y_35,Q_10] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_43),Y_35)),Q_10) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_43),Q_10)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_35),Q_10)) ).
fof(fact_347_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_43,Y_35,Q_10] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_43),Y_35)),Q_10) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_43),Q_10)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_35),Q_10)) ).
fof(fact_348_rel__simps_I46_J,axiom,
! [K_2] : hAPP_int_int(bit1,K_2) != pls ).
fof(fact_349_rel__simps_I39_J,axiom,
! [L_2] : pls != hAPP_int_int(bit1,L_2) ).
fof(fact_350_rel__simps_I50_J,axiom,
! [K_2,L_2] : hAPP_int_int(bit1,K_2) != hAPP_int_int(bit0,L_2) ).
fof(fact_351_rel__simps_I49_J,axiom,
! [K_2,L_2] : hAPP_int_int(bit0,K_2) != hAPP_int_int(bit1,L_2) ).
fof(fact_352_rel__simps_I44_J,axiom,
! [K_1] :
( is_int(K_1)
=> ( hAPP_int_int(bit0,K_1) = pls
<=> K_1 = pls ) ) ).
fof(fact_353_rel__simps_I38_J,axiom,
! [L] :
( is_int(L)
=> ( pls = hAPP_int_int(bit0,L)
<=> pls = L ) ) ).
fof(fact_354_Bit0__Pls,axiom,
hAPP_int_int(bit0,pls) = pls ).
fof(fact_355_mult__Pls,axiom,
! [W] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,pls),W) = pls ).
fof(fact_356_mult__Bit0,axiom,
! [K_2,L_2] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(bit0,K_2)),L_2) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),L_2)) ).
fof(fact_357_add__Pls__right,axiom,
! [K_2] :
( is_int(K_2)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),pls) = K_2 ) ).
fof(fact_358_add__Pls,axiom,
! [K_2] :
( is_int(K_2)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,pls),K_2) = K_2 ) ).
fof(fact_359_add__Bit0__Bit0,axiom,
! [K_2,L_2] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),L_2)) ).
fof(fact_360_Bit0__def,axiom,
! [K_2] : hAPP_int_int(bit0,K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),K_2) ).
fof(fact_361_zmult__1__right,axiom,
! [Z_1] :
( is_int(Z_1)
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),one_one_int) = Z_1 ) ).
fof(fact_362_zmult__1,axiom,
! [Z_1] :
( is_int(Z_1)
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),Z_1) = Z_1 ) ).
fof(fact_363_times__numeral__code_I5_J,axiom,
! [V_2,W] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_2),W)) ).
fof(fact_364_zadd__zmult__distrib,axiom,
! [Z1,Z2,W] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),W)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),W)) ).
fof(fact_365_zadd__zmult__distrib2,axiom,
! [W,Z1,Z2] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z2)) ).
fof(fact_366_plus__numeral__code_I9_J,axiom,
! [V_2,W] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),W)) ).
fof(fact_367_semiring__mult__number__of,axiom,
! [V_22,V_21] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22))
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_21)),hAPP_int_int(number_number_of_int,V_22)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22)) ) ) ).
fof(fact_368_semiring__mult__number__of,axiom,
! [V_22,V_21] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_21)),hAPP_int_nat(number_number_of_nat,V_22)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22)) ) ) ).
fof(fact_369_semiring__mult__number__of,axiom,
! [V_22,V_21] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22))
=> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_21)),hAPP_int_real(number267125858f_real,V_22)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22)) ) ) ).
fof(fact_370_semiring__mult__number__of,axiom,
! [V_22,V_21] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22))
=> hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_21)),hAPP_int_complex(number528085621omplex,V_22)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22)) ) ) ).
fof(fact_371_semiring__mult__number__of,axiom,
! [V_22,V_21] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22))
=> hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_21)),hAPP_int_rat(number_number_of_rat,V_22)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22)) ) ) ).
fof(fact_372_semiring__add__number__of,axiom,
! [V_20,V_19] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20))
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_19)),hAPP_int_int(number_number_of_int,V_20)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_19),V_20)) ) ) ).
fof(fact_373_semiring__add__number__of,axiom,
! [V_20,V_19] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_19)),hAPP_int_nat(number_number_of_nat,V_20)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_19),V_20)) ) ) ).
fof(fact_374_semiring__add__number__of,axiom,
! [V_20,V_19] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20))
=> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_19)),hAPP_int_real(number267125858f_real,V_20)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_19),V_20)) ) ) ).
fof(fact_375_semiring__add__number__of,axiom,
! [V_20,V_19] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20))
=> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_19)),hAPP_int_complex(number528085621omplex,V_20)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_19),V_20)) ) ) ).
fof(fact_376_semiring__add__number__of,axiom,
! [V_20,V_19] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20))
=> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_19)),hAPP_int_rat(number_number_of_rat,V_20)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_19),V_20)) ) ) ).
fof(fact_377_power2__ge__self,axiom,
! [X_1] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_378_left__distrib__number__of,axiom,
! [A_226,B_161,V_18] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_226),B_161)),hAPP_i769753017umeral(number1443263063umeral,V_18)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_226),hAPP_i769753017umeral(number1443263063umeral,V_18))),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_161),hAPP_i769753017umeral(number1443263063umeral,V_18))) ).
fof(fact_379_left__distrib__number__of,axiom,
! [A_226,B_161,V_18] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_226),B_161)),hAPP_int_int(number_number_of_int,V_18)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_226),hAPP_int_int(number_number_of_int,V_18))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_161),hAPP_int_int(number_number_of_int,V_18))) ).
fof(fact_380_left__distrib__number__of,axiom,
! [A_226,B_161,V_18] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_226),B_161)),hAPP_int_nat(number_number_of_nat,V_18)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_226),hAPP_int_nat(number_number_of_nat,V_18))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_161),hAPP_int_nat(number_number_of_nat,V_18))) ).
fof(fact_381_left__distrib__number__of,axiom,
! [A_226,B_161,V_18] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_226),B_161)),hAPP_int_real(number267125858f_real,V_18)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_226),hAPP_int_real(number267125858f_real,V_18))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_161),hAPP_int_real(number267125858f_real,V_18))) ).
fof(fact_382_left__distrib__number__of,axiom,
! [A_226,B_161,V_18] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_226),B_161)),hAPP_int_complex(number528085621omplex,V_18)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_226),hAPP_int_complex(number528085621omplex,V_18))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_161),hAPP_int_complex(number528085621omplex,V_18))) ).
fof(fact_383_left__distrib__number__of,axiom,
! [A_226,B_161,V_18] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_226),B_161)),hAPP_i1732201573de_int(number1226105091de_int,V_18)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_226),hAPP_i1732201573de_int(number1226105091de_int,V_18))),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_161),hAPP_i1732201573de_int(number1226105091de_int,V_18))) ).
fof(fact_384_left__distrib__number__of,axiom,
! [A_226,B_161,V_18] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_226),B_161)),hAPP_int_rat(number_number_of_rat,V_18)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_226),hAPP_int_rat(number_number_of_rat,V_18))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_161),hAPP_int_rat(number_number_of_rat,V_18))) ).
fof(fact_385_right__distrib__number__of,axiom,
! [V_17,B_160,C_90] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_17)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_160),C_90)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_17)),B_160)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_17)),C_90)) ).
fof(fact_386_right__distrib__number__of,axiom,
! [V_17,B_160,C_90] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_17)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_160),C_90)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_17)),B_160)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_17)),C_90)) ).
fof(fact_387_right__distrib__number__of,axiom,
! [V_17,B_160,C_90] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_17)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_160),C_90)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_17)),B_160)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_17)),C_90)) ).
fof(fact_388_right__distrib__number__of,axiom,
! [V_17,B_160,C_90] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_17)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_160),C_90)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_17)),B_160)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_17)),C_90)) ).
fof(fact_389_right__distrib__number__of,axiom,
! [V_17,B_160,C_90] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_17)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_160),C_90)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_17)),B_160)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_17)),C_90)) ).
fof(fact_390_right__distrib__number__of,axiom,
! [V_17,B_160,C_90] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_17)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_160),C_90)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_17)),B_160)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_17)),C_90)) ).
fof(fact_391_right__distrib__number__of,axiom,
! [V_17,B_160,C_90] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_17)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_160),C_90)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_17)),B_160)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_17)),C_90)) ).
fof(fact_392_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_225,M_28] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_225),M_28)),M_28) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_225),one_one_int)),M_28) ).
fof(fact_393_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_225,M_28] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_225),M_28)),M_28) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_225),one_one_nat)),M_28) ).
fof(fact_394_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_225,M_28] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_225),M_28)),M_28) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_225),one_one_real)),M_28) ).
fof(fact_395_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_225,M_28] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_225),M_28)),M_28) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_225),one_on1645066479umeral)),M_28) ).
fof(fact_396_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_225,M_28] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_225),M_28)),M_28) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_225),one_one_complex)),M_28) ).
fof(fact_397_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_225,M_28] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_225),M_28)),M_28) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_225),one_on1684967323de_int)),M_28) ).
fof(fact_398_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_225,M_28] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_225),M_28)),M_28) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_225),one_one_rat)),M_28) ).
fof(fact_399_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_27,A_224] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_27),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_224),M_27)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_224),one_one_int)),M_27) ).
fof(fact_400_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_27,A_224] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_27),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_224),M_27)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_224),one_one_nat)),M_27) ).
fof(fact_401_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_27,A_224] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,M_27),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_224),M_27)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_224),one_one_real)),M_27) ).
fof(fact_402_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_27,A_224] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,M_27),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_224),M_27)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_224),one_on1645066479umeral)),M_27) ).
fof(fact_403_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_27,A_224] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,M_27),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_224),M_27)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_224),one_one_complex)),M_27) ).
fof(fact_404_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_27,A_224] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,M_27),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_224),M_27)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_224),one_on1684967323de_int)),M_27) ).
fof(fact_405_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_27,A_224] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,M_27),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_224),M_27)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_224),one_one_rat)),M_27) ).
fof(fact_406_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_26] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_26),M_26) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)),M_26) ).
fof(fact_407_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_26] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_26),M_26) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)),M_26) ).
fof(fact_408_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_26] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,M_26),M_26) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),one_one_real)),M_26) ).
fof(fact_409_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_26] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,M_26),M_26) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,one_on1645066479umeral),one_on1645066479umeral)),M_26) ).
fof(fact_410_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_26] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,M_26),M_26) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),one_one_complex)),M_26) ).
fof(fact_411_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_26] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,M_26),M_26) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,one_on1684967323de_int),one_on1684967323de_int)),M_26) ).
fof(fact_412_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_26] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,M_26),M_26) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)),M_26) ).
fof(fact_413_add__numeral__0,axiom,
! [A_223] :
( is_int(A_223)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,pls)),A_223) = A_223 ) ).
fof(fact_414_add__numeral__0,axiom,
! [A_223] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,pls)),A_223) = A_223 ).
fof(fact_415_add__numeral__0,axiom,
! [A_223] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,pls)),A_223) = A_223 ).
fof(fact_416_add__numeral__0,axiom,
! [A_223] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,pls)),A_223) = A_223 ).
fof(fact_417_add__numeral__0__right,axiom,
! [A_222] :
( is_int(A_222)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_222),hAPP_int_int(number_number_of_int,pls)) = A_222 ) ).
fof(fact_418_add__numeral__0__right,axiom,
! [A_222] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_222),hAPP_int_real(number267125858f_real,pls)) = A_222 ).
fof(fact_419_add__numeral__0__right,axiom,
! [A_222] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_222),hAPP_int_complex(number528085621omplex,pls)) = A_222 ).
fof(fact_420_add__numeral__0__right,axiom,
! [A_222] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_222),hAPP_int_rat(number_number_of_rat,pls)) = A_222 ).
fof(fact_421_mult__number__of__left,axiom,
! [V_16,W_14,Z_14] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_16)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,W_14)),Z_14)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_16),W_14))),Z_14) ).
fof(fact_422_mult__number__of__left,axiom,
! [V_16,W_14,Z_14] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_16)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,W_14)),Z_14)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_16),W_14))),Z_14) ).
fof(fact_423_mult__number__of__left,axiom,
! [V_16,W_14,Z_14] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_16)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,W_14)),Z_14)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_16),W_14))),Z_14) ).
fof(fact_424_mult__number__of__left,axiom,
! [V_16,W_14,Z_14] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_16)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,W_14)),Z_14)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_16),W_14))),Z_14) ).
fof(fact_425_arith__simps_I32_J,axiom,
! [V_15,W_13] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_15)),hAPP_int_int(number_number_of_int,W_13)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_15),W_13)) ).
fof(fact_426_arith__simps_I32_J,axiom,
! [V_15,W_13] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_15)),hAPP_int_real(number267125858f_real,W_13)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_15),W_13)) ).
fof(fact_427_arith__simps_I32_J,axiom,
! [V_15,W_13] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_15)),hAPP_int_complex(number528085621omplex,W_13)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_15),W_13)) ).
fof(fact_428_arith__simps_I32_J,axiom,
! [V_15,W_13] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_15)),hAPP_int_rat(number_number_of_rat,W_13)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_15),W_13)) ).
fof(fact_429_number__of__mult,axiom,
! [V_14,W_12] : hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_14),W_12)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_14)),hAPP_int_int(number_number_of_int,W_12)) ).
fof(fact_430_number__of__mult,axiom,
! [V_14,W_12] : hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_14),W_12)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_14)),hAPP_int_real(number267125858f_real,W_12)) ).
fof(fact_431_number__of__mult,axiom,
! [V_14,W_12] : hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_14),W_12)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_14)),hAPP_int_complex(number528085621omplex,W_12)) ).
fof(fact_432_number__of__mult,axiom,
! [V_14,W_12] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_14),W_12)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_14)),hAPP_int_rat(number_number_of_rat,W_12)) ).
fof(fact_433_add__number__of__left,axiom,
! [V_13,W_11,Z_13] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_13)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W_11)),Z_13)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_13),W_11))),Z_13) ).
fof(fact_434_add__number__of__left,axiom,
! [V_13,W_11,Z_13] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_13)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,W_11)),Z_13)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_13),W_11))),Z_13) ).
fof(fact_435_add__number__of__left,axiom,
! [V_13,W_11,Z_13] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_13)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,W_11)),Z_13)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_13),W_11))),Z_13) ).
fof(fact_436_add__number__of__left,axiom,
! [V_13,W_11,Z_13] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_13)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,W_11)),Z_13)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_13),W_11))),Z_13) ).
fof(fact_437_add__number__of__eq,axiom,
! [V_12,W_10] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_12)),hAPP_int_int(number_number_of_int,W_10)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_12),W_10)) ).
fof(fact_438_add__number__of__eq,axiom,
! [V_12,W_10] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_12)),hAPP_int_real(number267125858f_real,W_10)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_12),W_10)) ).
fof(fact_439_add__number__of__eq,axiom,
! [V_12,W_10] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_12)),hAPP_int_complex(number528085621omplex,W_10)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_12),W_10)) ).
fof(fact_440_add__number__of__eq,axiom,
! [V_12,W_10] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_12)),hAPP_int_rat(number_number_of_rat,W_10)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_12),W_10)) ).
fof(fact_441_number__of__add,axiom,
! [V_11,W_9] : hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_11),W_9)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_11)),hAPP_int_int(number_number_of_int,W_9)) ).
fof(fact_442_number__of__add,axiom,
! [V_11,W_9] : hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_11),W_9)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_11)),hAPP_int_real(number267125858f_real,W_9)) ).
fof(fact_443_number__of__add,axiom,
! [V_11,W_9] : hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_11),W_9)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_11)),hAPP_int_complex(number528085621omplex,W_9)) ).
fof(fact_444_number__of__add,axiom,
! [V_11,W_9] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_11),W_9)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_11)),hAPP_int_rat(number_number_of_rat,W_9)) ).
fof(fact_445_add__Bit1__Bit0,axiom,
! [K_2,L_2] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),L_2)) ).
fof(fact_446_add__Bit0__Bit1,axiom,
! [K_2,L_2] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),L_2)) ).
fof(fact_447_Bit1__def,axiom,
! [K_2] : hAPP_int_int(bit1,K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),K_2)),K_2) ).
fof(fact_448_number__of__Bit1,axiom,
! [W_8] : hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,W_8)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W_8))),hAPP_int_int(number_number_of_int,W_8)) ).
fof(fact_449_number__of__Bit1,axiom,
! [W_8] : hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,W_8)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_int_real(number267125858f_real,W_8))),hAPP_int_real(number267125858f_real,W_8)) ).
fof(fact_450_number__of__Bit1,axiom,
! [W_8] : hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,W_8)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),hAPP_int_complex(number528085621omplex,W_8))),hAPP_int_complex(number528085621omplex,W_8)) ).
fof(fact_451_number__of__Bit1,axiom,
! [W_8] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,W_8)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,W_8))),hAPP_int_rat(number_number_of_rat,W_8)) ).
fof(fact_452_mult__numeral__1,axiom,
! [A_221] :
( is_int(A_221)
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls))),A_221) = A_221 ) ).
fof(fact_453_mult__numeral__1,axiom,
! [A_221] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls))),A_221) = A_221 ).
fof(fact_454_mult__numeral__1,axiom,
! [A_221] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls))),A_221) = A_221 ).
fof(fact_455_mult__numeral__1,axiom,
! [A_221] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls))),A_221) = A_221 ).
fof(fact_456_mult__numeral__1__right,axiom,
! [A_220] :
( is_int(A_220)
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_220),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls))) = A_220 ) ).
fof(fact_457_mult__numeral__1__right,axiom,
! [A_220] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_220),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls))) = A_220 ).
fof(fact_458_mult__numeral__1__right,axiom,
! [A_220] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_220),hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls))) = A_220 ).
fof(fact_459_mult__numeral__1__right,axiom,
! [A_220] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_220),hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls))) = A_220 ).
fof(fact_460_semiring__numeral__1__eq__1,axiom,
hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls)) = one_one_int ).
fof(fact_461_semiring__numeral__1__eq__1,axiom,
hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)) = one_one_nat ).
fof(fact_462_semiring__numeral__1__eq__1,axiom,
hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls)) = one_one_real ).
fof(fact_463_semiring__numeral__1__eq__1,axiom,
hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls)) = one_one_complex ).
fof(fact_464_semiring__numeral__1__eq__1,axiom,
hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls)) = one_one_rat ).
fof(fact_465_numeral__1__eq__1,axiom,
hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls)) = one_one_int ).
fof(fact_466_numeral__1__eq__1,axiom,
hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls)) = one_one_real ).
fof(fact_467_numeral__1__eq__1,axiom,
hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls)) = one_one_complex ).
fof(fact_468_numeral__1__eq__1,axiom,
hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls)) = one_one_rat ).
fof(fact_469_semiring__norm_I110_J,axiom,
one_one_int = hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls)) ).
fof(fact_470_semiring__norm_I110_J,axiom,
one_one_real = hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls)) ).
fof(fact_471_semiring__norm_I110_J,axiom,
one_one_complex = hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls)) ).
fof(fact_472_semiring__norm_I110_J,axiom,
one_one_rat = hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls)) ).
fof(fact_473_one__is__num__one,axiom,
one_one_int = hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls)) ).
fof(fact_474_mult__Bit1,axiom,
! [K_2,L_2] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(bit1,K_2)),L_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),L_2))),L_2) ).
fof(fact_475_double__number__of__Bit0,axiom,
! [W_7] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)),hAPP_int_int(number_number_of_int,W_7)) = hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W_7)) ).
fof(fact_476_double__number__of__Bit0,axiom,
! [W_7] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),one_one_real)),hAPP_int_real(number267125858f_real,W_7)) = hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,W_7)) ).
fof(fact_477_double__number__of__Bit0,axiom,
! [W_7] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),one_one_complex)),hAPP_int_complex(number528085621omplex,W_7)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,W_7)) ).
fof(fact_478_double__number__of__Bit0,axiom,
! [W_7] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)),hAPP_int_rat(number_number_of_rat,W_7)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,W_7)) ).
fof(fact_479_power3__eq__cube,axiom,
! [A_219] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_219),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_219),A_219)),A_219) ).
fof(fact_480_power3__eq__cube,axiom,
! [A_219] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_219),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_219),A_219)),A_219) ).
fof(fact_481_power3__eq__cube,axiom,
! [A_219] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_219),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_219),A_219)),A_219) ).
fof(fact_482_power3__eq__cube,axiom,
! [A_219] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_219),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_219),A_219)),A_219) ).
fof(fact_483_power3__eq__cube,axiom,
! [A_219] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_219),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_219),A_219)),A_219) ).
fof(fact_484_power3__eq__cube,axiom,
! [A_219] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_219),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_219),A_219)),A_219) ).
fof(fact_485_power3__eq__cube,axiom,
! [A_219] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_219),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_219),A_219)),A_219) ).
fof(fact_486_quartic__square__square,axiom,
! [X_1] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_487_semiring__mult__2,axiom,
! [Z_12] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_12) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_12),Z_12) ).
fof(fact_488_semiring__mult__2,axiom,
! [Z_12] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_12) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_12),Z_12) ).
fof(fact_489_semiring__mult__2,axiom,
! [Z_12] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_12) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_12),Z_12) ).
fof(fact_490_semiring__mult__2,axiom,
! [Z_12] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_12) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_12),Z_12) ).
fof(fact_491_semiring__mult__2,axiom,
! [Z_12] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_12) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_12),Z_12) ).
fof(fact_492_mult__2,axiom,
! [Z_11] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_11) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_11),Z_11) ).
fof(fact_493_mult__2,axiom,
! [Z_11] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_11) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_11),Z_11) ).
fof(fact_494_mult__2,axiom,
! [Z_11] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_11) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_11),Z_11) ).
fof(fact_495_mult__2,axiom,
! [Z_11] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_11) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_11),Z_11) ).
fof(fact_496_semiring__mult__2__right,axiom,
! [Z_10] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_10),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_10),Z_10) ).
fof(fact_497_semiring__mult__2__right,axiom,
! [Z_10] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Z_10),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_10),Z_10) ).
fof(fact_498_semiring__mult__2__right,axiom,
! [Z_10] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_10),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_10),Z_10) ).
fof(fact_499_semiring__mult__2__right,axiom,
! [Z_10] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_10),hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_10),Z_10) ).
fof(fact_500_semiring__mult__2__right,axiom,
! [Z_10] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Z_10),hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_10),Z_10) ).
fof(fact_501_mult__2__right,axiom,
! [Z_9] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_9),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_9),Z_9) ).
fof(fact_502_mult__2__right,axiom,
! [Z_9] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_9),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_9),Z_9) ).
fof(fact_503_mult__2__right,axiom,
! [Z_9] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_9),hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_9),Z_9) ).
fof(fact_504_mult__2__right,axiom,
! [Z_9] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Z_9),hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_9),Z_9) ).
fof(fact_505_semiring__one__add__one__is__two,axiom,
hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int) = hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).
fof(fact_506_semiring__one__add__one__is__two,axiom,
hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).
fof(fact_507_semiring__one__add__one__is__two,axiom,
hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),one_one_real) = hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).
fof(fact_508_semiring__one__add__one__is__two,axiom,
hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),one_one_complex) = hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).
fof(fact_509_semiring__one__add__one__is__two,axiom,
hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).
fof(fact_510_p0,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) ).
fof(fact_511__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_L_A1_096,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int))) ).
fof(fact_512_prime__g__5,axiom,
! [P_3] :
( is_int(P_3)
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( P_3 != hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))
=> ( P_3 != hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_3)) ) ) ) ) ).
fof(fact_513__096sum2sq_A_Is_M_A1_J_A_061_A_I4_A_K_Am_A_L_A1_J_A_K_At_096,axiom,
hAPP_P1175774780nt_int(twoSqu1907779896sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,s),one_one_int)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),t) ).
fof(fact_514_real__sum__squared__expand,axiom,
! [X_1,Y_1] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1)),Y_1)) ).
fof(fact_515_four__x__squared,axiom,
! [X_1] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_516_power__less__power__Suc,axiom,
! [N_67,A_218] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_218))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_218),N_67)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_218),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_218),N_67)))) ) ).
fof(fact_517_power__less__power__Suc,axiom,
! [N_67,A_218] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_218))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_218),N_67)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_218),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_218),N_67)))) ) ).
fof(fact_518_power__less__power__Suc,axiom,
! [N_67,A_218] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_218))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_218),N_67)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_218),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_218),N_67)))) ) ).
fof(fact_519_power__less__power__Suc,axiom,
! [N_67,A_218] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_218))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_218),N_67)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_218),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_218),N_67)))) ) ).
fof(fact_520_power__less__power__Suc,axiom,
! [N_67,A_218] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_218))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_218),N_67)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_218),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_218),N_67)))) ) ).
fof(fact_521_power__less__power__Suc,axiom,
! [N_67,A_218] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_218))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_218),N_67)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_218),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_218),N_67)))) ) ).
fof(fact_522_power__gt1__lemma,axiom,
! [N_66,A_217] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_217))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_217),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_217),N_66)))) ) ).
fof(fact_523_power__gt1__lemma,axiom,
! [N_66,A_217] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_217))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_217),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_217),N_66)))) ) ).
fof(fact_524_power__gt1__lemma,axiom,
! [N_66,A_217] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_217))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_217),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_217),N_66)))) ) ).
fof(fact_525_power__gt1__lemma,axiom,
! [N_66,A_217] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_217))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_217),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_217),N_66)))) ) ).
fof(fact_526_power__gt1__lemma,axiom,
! [N_66,A_217] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_217))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_217),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_217),N_66)))) ) ).
fof(fact_527_power__gt1__lemma,axiom,
! [N_66,A_217] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_217))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_217),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_217),N_66)))) ) ).
fof(fact_528_power__le__imp__le__exp,axiom,
! [M_25,N_65,A_216] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_216))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_216),M_25)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_216),N_65)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_65)) ) ) ).
fof(fact_529_power__le__imp__le__exp,axiom,
! [M_25,N_65,A_216] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_216))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_216),M_25)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_216),N_65)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_65)) ) ) ).
fof(fact_530_power__le__imp__le__exp,axiom,
! [M_25,N_65,A_216] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_216))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_216),M_25)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_216),N_65)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_65)) ) ) ).
fof(fact_531_power__le__imp__le__exp,axiom,
! [M_25,N_65,A_216] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_216))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_216),M_25)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_216),N_65)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_65)) ) ) ).
fof(fact_532_power__le__imp__le__exp,axiom,
! [M_25,N_65,A_216] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_216))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_216),M_25)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_216),N_65)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_65)) ) ) ).
fof(fact_533_power__le__imp__le__exp,axiom,
! [M_25,N_65,A_216] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_216))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_216),M_25)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_216),N_65)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_65)) ) ) ).
fof(fact_534_power__increasing__iff,axiom,
! [X_2,Y_2,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),X_2)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2)) ) ) ).
fof(fact_535_power__increasing__iff,axiom,
! [X_2,Y_2,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2)) ) ) ).
fof(fact_536_power__increasing__iff,axiom,
! [X_2,Y_2,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B),X_2)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B),Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2)) ) ) ).
fof(fact_537_power__increasing__iff,axiom,
! [X_2,Y_2,B] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),B))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B),X_2)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B),Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2)) ) ) ).
fof(fact_538_power__increasing__iff,axiom,
! [X_2,Y_2,B] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),B))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B),X_2)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B),Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2)) ) ) ).
fof(fact_539_power__increasing__iff,axiom,
! [X_2,Y_2,B] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),B))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B),X_2)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B),Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2)) ) ) ).
fof(fact_540__096_091s_A_094_A2_A_061_As1_A_094_A2_093_A_Imod_A4_A_K_Am_A_L_A1_J_096,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) ).
fof(fact_541_int__ge__induct,axiom,
! [P_1,K_1,I_1] :
( 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] :
( is_int(I_2)
=> ( 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(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))) ) ) )
=> hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).
fof(fact_542_s0p,axiom,
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),s))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,s),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(s1),s),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) ) ).
fof(fact_543__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s1,axiom,
? [X] :
( is_int(X)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(s1),X),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)))
& ! [Y] :
( is_int(Y)
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(s1),Y),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) )
=> Y = X ) ) ) ).
fof(fact_544__096_B_Bthesis_O_A_I_B_Bs_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_,axiom,
~ ! [S_2] :
( is_int(S_2)
=> ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),S_2))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,S_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(s1),S_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) ) ) ).
fof(fact_545_s1,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) ).
fof(fact_546_power__eq__0__iff,axiom,
! [A,N] :
( is_int(A)
=> ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),N) = zero_zero_int
<=> ( A = zero_zero_int
& N != zero_zero_nat ) ) ) ).
fof(fact_547_power__eq__0__iff,axiom,
! [A,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),N) = zero_zero_nat
<=> ( A = zero_zero_nat
& N != zero_zero_nat ) ) ).
fof(fact_548_power__eq__0__iff,axiom,
! [A,N] :
( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),N) = zero_zero_real
<=> ( A = zero_zero_real
& N != zero_zero_nat ) ) ).
fof(fact_549_power__eq__0__iff,axiom,
! [A,N] :
( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A),N) = zero_z126310315umeral
<=> ( A = zero_z126310315umeral
& N != zero_zero_nat ) ) ).
fof(fact_550_power__eq__0__iff,axiom,
! [A,N] :
( hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A),N) = zero_zero_complex
<=> ( A = zero_zero_complex
& N != zero_zero_nat ) ) ).
fof(fact_551_power__eq__0__iff,axiom,
! [A,N] :
( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A),N) = zero_z891286103de_int
<=> ( A = zero_z891286103de_int
& N != zero_zero_nat ) ) ).
fof(fact_552_power__eq__0__iff,axiom,
! [A,N] :
( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),N) = zero_zero_rat
<=> ( A = zero_zero_rat
& N != zero_zero_nat ) ) ).
fof(fact_553_le__imp__power__dvd,axiom,
! [A_215,M_24,N_64] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_24),N_64))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_215),M_24)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_215),N_64))) ) ).
fof(fact_554_le__imp__power__dvd,axiom,
! [A_215,M_24,N_64] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_24),N_64))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_215),M_24)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_215),N_64))) ) ).
fof(fact_555_le__imp__power__dvd,axiom,
! [A_215,M_24,N_64] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_24),N_64))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_215),M_24)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_215),N_64))) ) ).
fof(fact_556_le__imp__power__dvd,axiom,
! [A_215,M_24,N_64] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_24),N_64))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_215),M_24)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_215),N_64))) ) ).
fof(fact_557_le__imp__power__dvd,axiom,
! [A_215,M_24,N_64] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_24),N_64))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_215),M_24)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_215),N_64))) ) ).
fof(fact_558_le__imp__power__dvd,axiom,
! [A_215,M_24,N_64] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_24),N_64))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_215),M_24)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_215),N_64))) ) ).
fof(fact_559_le__imp__power__dvd,axiom,
! [A_215,M_24,N_64] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_24),N_64))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_215),M_24)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_215),N_64))) ) ).
fof(fact_560_dvd__power__le,axiom,
! [N_63,M_23,X_42,Y_34] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,X_42),Y_34))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_63),M_23))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_42),N_63)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_34),M_23))) ) ) ).
fof(fact_561_dvd__power__le,axiom,
! [N_63,M_23,X_42,Y_34] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,X_42),Y_34))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_63),M_23))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_42),N_63)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_34),M_23))) ) ) ).
fof(fact_562_dvd__power__le,axiom,
! [N_63,M_23,X_42,Y_34] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_42),Y_34))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_63),M_23))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_42),N_63)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_34),M_23))) ) ) ).
fof(fact_563_dvd__power__le,axiom,
! [N_63,M_23,X_42,Y_34] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_42),Y_34))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_63),M_23))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_42),N_63)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_34),M_23))) ) ) ).
fof(fact_564_dvd__power__le,axiom,
! [N_63,M_23,X_42,Y_34] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_42),Y_34))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_63),M_23))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_42),N_63)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_34),M_23))) ) ) ).
fof(fact_565_dvd__power__le,axiom,
! [N_63,M_23,X_42,Y_34] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_42),Y_34))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_63),M_23))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_42),N_63)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_34),M_23))) ) ) ).
fof(fact_566_dvd__power__le,axiom,
! [N_63,M_23,X_42,Y_34] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_42),Y_34))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_63),M_23))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_42),N_63)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_34),M_23))) ) ) ).
fof(fact_567_power__le__dvd,axiom,
! [M_22,A_214,N_62,B_159] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_214),N_62)),B_159))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_22),N_62))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_214),M_22)),B_159)) ) ) ).
fof(fact_568_power__le__dvd,axiom,
! [M_22,A_214,N_62,B_159] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_214),N_62)),B_159))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_22),N_62))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_214),M_22)),B_159)) ) ) ).
fof(fact_569_power__le__dvd,axiom,
! [M_22,A_214,N_62,B_159] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_214),N_62)),B_159))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_22),N_62))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_214),M_22)),B_159)) ) ) ).
fof(fact_570_power__le__dvd,axiom,
! [M_22,A_214,N_62,B_159] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_214),N_62)),B_159))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_22),N_62))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_214),M_22)),B_159)) ) ) ).
fof(fact_571_power__le__dvd,axiom,
! [M_22,A_214,N_62,B_159] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_214),N_62)),B_159))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_22),N_62))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_214),M_22)),B_159)) ) ) ).
fof(fact_572_power__le__dvd,axiom,
! [M_22,A_214,N_62,B_159] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_214),N_62)),B_159))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_22),N_62))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_214),M_22)),B_159)) ) ) ).
fof(fact_573_power__le__dvd,axiom,
! [M_22,A_214,N_62,B_159] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_214),N_62)),B_159))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_22),N_62))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_214),M_22)),B_159)) ) ) ).
fof(fact_574_power__eq__imp__eq__base,axiom,
! [A_213,N_61,B_158] :
( ( is_int(A_213)
& is_int(B_158) )
=> ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_213),N_61) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_158),N_61)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_213))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_158))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_61))
=> A_213 = B_158 ) ) ) ) ) ).
fof(fact_575_power__eq__imp__eq__base,axiom,
! [A_213,N_61,B_158] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_213),N_61) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_158),N_61)
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_213))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_158))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_61))
=> A_213 = B_158 ) ) ) ) ).
fof(fact_576_power__eq__imp__eq__base,axiom,
! [A_213,N_61,B_158] :
( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_213),N_61) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_158),N_61)
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_213))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_158))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_61))
=> A_213 = B_158 ) ) ) ) ).
fof(fact_577_power__eq__imp__eq__base,axiom,
! [A_213,N_61,B_158] :
( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_213),N_61) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_158),N_61)
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_213))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_158))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_61))
=> A_213 = B_158 ) ) ) ) ).
fof(fact_578_power__eq__imp__eq__base,axiom,
! [A_213,N_61,B_158] :
( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_213),N_61) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_158),N_61)
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_213))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_158))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_61))
=> A_213 = B_158 ) ) ) ) ).
fof(fact_579_power__eq__imp__eq__base,axiom,
! [A_213,N_61,B_158] :
( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_213),N_61) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_158),N_61)
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_213))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_158))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_61))
=> A_213 = B_158 ) ) ) ) ).
fof(fact_580_zdvd__not__zless,axiom,
! [N_1,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M),N_1))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),M)) ) ) ).
fof(fact_581_zdvd__antisym__nonneg,axiom,
! [N_1,M] :
( ( is_int(N_1)
& is_int(M) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),M))
=> M = N_1 ) ) ) ) ) ).
fof(fact_582_zdvd__mult__cancel,axiom,
! [K_2,M,N_1] :
( is_int(K_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1)))
=> ( K_2 != zero_zero_int
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N_1)) ) ) ) ).
fof(fact_583_dvd__power__same,axiom,
! [N_60,X_41,Y_33] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,X_41),Y_33))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_41),N_60)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_33),N_60))) ) ).
fof(fact_584_dvd__power__same,axiom,
! [N_60,X_41,Y_33] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,X_41),Y_33))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_41),N_60)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_33),N_60))) ) ).
fof(fact_585_dvd__power__same,axiom,
! [N_60,X_41,Y_33] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_41),Y_33))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_41),N_60)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_33),N_60))) ) ).
fof(fact_586_dvd__power__same,axiom,
! [N_60,X_41,Y_33] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_41),Y_33))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_41),N_60)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_33),N_60))) ) ).
fof(fact_587_dvd__power__same,axiom,
! [N_60,X_41,Y_33] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_41),Y_33))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_41),N_60)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_33),N_60))) ) ).
fof(fact_588_dvd__power__same,axiom,
! [N_60,X_41,Y_33] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_41),Y_33))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_41),N_60)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_33),N_60))) ) ).
fof(fact_589_dvd__power__same,axiom,
! [N_60,X_41,Y_33] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_41),Y_33))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_41),N_60)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_33),N_60))) ) ).
fof(fact_590_field__power__not__zero,axiom,
! [N_59,A_212] :
( is_int(A_212)
=> ( A_212 != zero_zero_int
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_212),N_59) != zero_zero_int ) ) ).
fof(fact_591_field__power__not__zero,axiom,
! [N_59,A_212] :
( A_212 != zero_zero_real
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_212),N_59) != zero_zero_real ) ).
fof(fact_592_field__power__not__zero,axiom,
! [N_59,A_212] :
( A_212 != zero_zero_complex
=> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_212),N_59) != zero_zero_complex ) ).
fof(fact_593_field__power__not__zero,axiom,
! [N_59,A_212] :
( A_212 != zero_zero_rat
=> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_212),N_59) != zero_zero_rat ) ).
fof(fact_594_power__0__left,axiom,
! [N_58] :
( ( N_58 = zero_zero_nat
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),N_58) = one_one_int )
& ( N_58 != zero_zero_nat
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),N_58) = zero_zero_int ) ) ).
fof(fact_595_power__0__left,axiom,
! [N_58] :
( ( N_58 = zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N_58) = one_one_nat )
& ( N_58 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N_58) = zero_zero_nat ) ) ).
fof(fact_596_power__0__left,axiom,
! [N_58] :
( ( N_58 = zero_zero_nat
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N_58) = one_one_real )
& ( N_58 != zero_zero_nat
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N_58) = zero_zero_real ) ) ).
fof(fact_597_power__0__left,axiom,
! [N_58] :
( ( N_58 = zero_zero_nat
=> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),N_58) = one_on1645066479umeral )
& ( N_58 != zero_zero_nat
=> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),N_58) = zero_z126310315umeral ) ) ).
fof(fact_598_power__0__left,axiom,
! [N_58] :
( ( N_58 = zero_zero_nat
=> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),N_58) = one_one_complex )
& ( N_58 != zero_zero_nat
=> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),N_58) = zero_zero_complex ) ) ).
fof(fact_599_power__0__left,axiom,
! [N_58] :
( ( N_58 = zero_zero_nat
=> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),N_58) = one_on1684967323de_int )
& ( N_58 != zero_zero_nat
=> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),N_58) = zero_z891286103de_int ) ) ).
fof(fact_600_power__0__left,axiom,
! [N_58] :
( ( N_58 = zero_zero_nat
=> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),N_58) = one_one_rat )
& ( N_58 != zero_zero_nat
=> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),N_58) = zero_zero_rat ) ) ).
fof(fact_601_zdvd__imp__le,axiom,
! [Z_1,N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_1),N_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),N_1)) ) ) ).
fof(fact_602_power__strict__mono,axiom,
! [N_57,A_211,B_157] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_211),B_157))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_211))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_57))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_211),N_57)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_157),N_57))) ) ) ) ).
fof(fact_603_power__strict__mono,axiom,
! [N_57,A_211,B_157] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_211),B_157))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_211))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_57))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_211),N_57)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_157),N_57))) ) ) ) ).
fof(fact_604_power__strict__mono,axiom,
! [N_57,A_211,B_157] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_211),B_157))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_211))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_57))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_211),N_57)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_157),N_57))) ) ) ) ).
fof(fact_605_power__strict__mono,axiom,
! [N_57,A_211,B_157] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_211),B_157))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_211))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_57))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_211),N_57)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_157),N_57))) ) ) ) ).
fof(fact_606_power__strict__mono,axiom,
! [N_57,A_211,B_157] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_211),B_157))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_211))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_57))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_211),N_57)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_157),N_57))) ) ) ) ).
fof(fact_607_power__strict__mono,axiom,
! [N_57,A_211,B_157] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_211),B_157))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_211))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_57))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_211),N_57)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_157),N_57))) ) ) ) ).
fof(fact_608_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_210] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,zero_zero_int),A_210) = zero_zero_int ).
fof(fact_609_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_210] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A_210) = zero_zero_nat ).
fof(fact_610_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_210] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),A_210) = zero_zero_real ).
fof(fact_611_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_210] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,zero_z126310315umeral),A_210) = zero_z126310315umeral ).
fof(fact_612_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_210] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),A_210) = zero_zero_complex ).
fof(fact_613_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_210] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,zero_z891286103de_int),A_210) = zero_z891286103de_int ).
fof(fact_614_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_210] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A_210) = zero_zero_rat ).
fof(fact_615_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_209] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_209),zero_zero_int) = zero_zero_int ).
fof(fact_616_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_209] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_209),zero_zero_nat) = zero_zero_nat ).
fof(fact_617_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_209] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_209),zero_zero_real) = zero_zero_real ).
fof(fact_618_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_209] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_209),zero_z126310315umeral) = zero_z126310315umeral ).
fof(fact_619_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_209] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_209),zero_zero_complex) = zero_zero_complex ).
fof(fact_620_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_209] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_209),zero_z891286103de_int) = zero_z891286103de_int ).
fof(fact_621_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_209] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_209),zero_zero_rat) = zero_zero_rat ).
fof(fact_622_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_208] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_208) = A_208 ).
fof(fact_623_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_208] :
( is_int(A_208)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_208) = A_208 ) ).
fof(fact_624_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_208] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_208) = A_208 ).
fof(fact_625_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_208] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_208) = A_208 ).
fof(fact_626_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_208] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_208) = A_208 ).
fof(fact_627_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_208] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_208) = A_208 ).
fof(fact_628_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_208] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_208) = A_208 ).
fof(fact_629_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_207] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_207),zero_z126310315umeral) = A_207 ).
fof(fact_630_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_207] :
( is_int(A_207)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_207),zero_zero_int) = A_207 ) ).
fof(fact_631_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_207] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_207),zero_zero_nat) = A_207 ).
fof(fact_632_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_207] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_207),zero_zero_real) = A_207 ).
fof(fact_633_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_207] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_207),zero_zero_complex) = A_207 ).
fof(fact_634_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_207] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_207),zero_z891286103de_int) = A_207 ).
fof(fact_635_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_207] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_207),zero_zero_rat) = A_207 ).
fof(fact_636_add__0__iff,axiom,
! [B,A] :
( ( is_int(B)
& is_int(A) )
=> ( B = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B),A)
<=> A = zero_zero_int ) ) ).
fof(fact_637_add__0__iff,axiom,
! [B,A] :
( B = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),A)
<=> A = zero_zero_nat ) ).
fof(fact_638_add__0__iff,axiom,
! [B,A] :
( B = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B),A)
<=> A = zero_zero_real ) ).
fof(fact_639_add__0__iff,axiom,
! [B,A] :
( B = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B),A)
<=> A = zero_zero_complex ) ).
fof(fact_640_add__0__iff,axiom,
! [B,A] :
( B = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B),A)
<=> A = zero_zero_rat ) ).
fof(fact_641_double__eq__0__iff,axiom,
! [A] :
( is_int(A)
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int
<=> A = zero_zero_int ) ) ).
fof(fact_642_double__eq__0__iff,axiom,
! [A] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A) = zero_zero_real
<=> A = zero_zero_real ) ).
fof(fact_643_double__eq__0__iff,axiom,
! [A] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat
<=> A = zero_zero_rat ) ).
fof(fact_644_Pls__def,axiom,
pls = zero_zero_int ).
fof(fact_645_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
fof(fact_646_zadd__0,axiom,
! [Z_1] :
( is_int(Z_1)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),Z_1) = Z_1 ) ).
fof(fact_647_zadd__0__right,axiom,
! [Z_1] :
( is_int(Z_1)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),zero_zero_int) = Z_1 ) ).
fof(fact_648_zero__le__power,axiom,
! [N_56,A_206] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_206))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_206),N_56))) ) ).
fof(fact_649_zero__le__power,axiom,
! [N_56,A_206] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_206))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_206),N_56))) ) ).
fof(fact_650_zero__le__power,axiom,
! [N_56,A_206] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_206))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_206),N_56))) ) ).
fof(fact_651_zero__le__power,axiom,
! [N_56,A_206] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_206))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_206),N_56))) ) ).
fof(fact_652_zero__le__power,axiom,
! [N_56,A_206] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_206))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_206),N_56))) ) ).
fof(fact_653_zero__le__power,axiom,
! [N_56,A_206] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_206))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_206),N_56))) ) ).
fof(fact_654_power__mono,axiom,
! [N_55,A_205,B_156] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_205),B_156))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_205))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_205),N_55)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_156),N_55))) ) ) ).
fof(fact_655_power__mono,axiom,
! [N_55,A_205,B_156] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_205),B_156))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_205))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_205),N_55)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_156),N_55))) ) ) ).
fof(fact_656_power__mono,axiom,
! [N_55,A_205,B_156] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_205),B_156))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_205))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_205),N_55)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_156),N_55))) ) ) ).
fof(fact_657_power__mono,axiom,
! [N_55,A_205,B_156] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_205),B_156))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_205))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_205),N_55)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_156),N_55))) ) ) ).
fof(fact_658_power__mono,axiom,
! [N_55,A_205,B_156] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_205),B_156))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_205))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_205),N_55)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_156),N_55))) ) ) ).
fof(fact_659_power__mono,axiom,
! [N_55,A_205,B_156] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_205),B_156))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_205))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_205),N_55)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_156),N_55))) ) ) ).
fof(fact_660_zero__less__power,axiom,
! [N_54,A_204] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_204))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_204),N_54))) ) ).
fof(fact_661_zero__less__power,axiom,
! [N_54,A_204] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_204))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_204),N_54))) ) ).
fof(fact_662_zero__less__power,axiom,
! [N_54,A_204] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_204))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_204),N_54))) ) ).
fof(fact_663_zero__less__power,axiom,
! [N_54,A_204] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_204))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_204),N_54))) ) ).
fof(fact_664_zero__less__power,axiom,
! [N_54,A_204] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_204))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_204),N_54))) ) ).
fof(fact_665_zero__less__power,axiom,
! [N_54,A_204] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_204))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_204),N_54))) ) ).
fof(fact_666_zcong__zpower__zmult,axiom,
! [Z_1,X_1,Y_1,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),Y_1)),one_one_int),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Y_1),Z_1))),one_one_int),P_3)) ) ).
fof(fact_667_zdvd__reduce,axiom,
! [K_1,N,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),Ma))))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N)) ) ).
fof(fact_668_zdvd__period,axiom,
! [C,X_2,Ta,A,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),D))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Ta)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),D))),Ta))) ) ) ).
fof(fact_669_power__less__imp__less__base,axiom,
! [A_203,N_53,B_155] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_203),N_53)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_155),N_53)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_155))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_203),B_155)) ) ) ).
fof(fact_670_power__less__imp__less__base,axiom,
! [A_203,N_53,B_155] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_203),N_53)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_155),N_53)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_155))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_203),B_155)) ) ) ).
fof(fact_671_power__less__imp__less__base,axiom,
! [A_203,N_53,B_155] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_203),N_53)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_155),N_53)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_155))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_203),B_155)) ) ) ).
fof(fact_672_power__less__imp__less__base,axiom,
! [A_203,N_53,B_155] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_203),N_53)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_155),N_53)))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_155))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_203),B_155)) ) ) ).
fof(fact_673_power__less__imp__less__base,axiom,
! [A_203,N_53,B_155] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_203),N_53)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_155),N_53)))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_155))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_203),B_155)) ) ) ).
fof(fact_674_power__less__imp__less__base,axiom,
! [A_203,N_53,B_155] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_203),N_53)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_155),N_53)))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_155))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_203),B_155)) ) ) ).
fof(fact_675_power__decreasing,axiom,
! [A_202,N_52,N_51] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_52),N_51))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_202))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_202),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_202),N_51)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_202),N_52))) ) ) ) ).
fof(fact_676_power__decreasing,axiom,
! [A_202,N_52,N_51] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_52),N_51))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_202))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_202),one_one_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_202),N_51)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_202),N_52))) ) ) ) ).
fof(fact_677_power__decreasing,axiom,
! [A_202,N_52,N_51] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_52),N_51))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_202))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_202),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_202),N_51)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_202),N_52))) ) ) ) ).
fof(fact_678_power__decreasing,axiom,
! [A_202,N_52,N_51] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_52),N_51))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_202))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_202),one_on1645066479umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_202),N_51)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_202),N_52))) ) ) ) ).
fof(fact_679_power__decreasing,axiom,
! [A_202,N_52,N_51] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_52),N_51))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_202))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_202),one_on1684967323de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_202),N_51)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_202),N_52))) ) ) ) ).
fof(fact_680_power__decreasing,axiom,
! [A_202,N_52,N_51] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_52),N_51))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_202))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_202),one_one_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_202),N_51)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_202),N_52))) ) ) ) ).
fof(fact_681_power__strict__decreasing,axiom,
! [A_201,N_50,N_49] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_50),N_49))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_201))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_201),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_201),N_49)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_201),N_50))) ) ) ) ).
fof(fact_682_power__strict__decreasing,axiom,
! [A_201,N_50,N_49] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_50),N_49))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_201))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_201),one_one_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_201),N_49)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_201),N_50))) ) ) ) ).
fof(fact_683_power__strict__decreasing,axiom,
! [A_201,N_50,N_49] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_50),N_49))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_201))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_201),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_201),N_49)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_201),N_50))) ) ) ) ).
fof(fact_684_power__strict__decreasing,axiom,
! [A_201,N_50,N_49] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_50),N_49))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_201))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_201),one_on1645066479umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_201),N_49)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_201),N_50))) ) ) ) ).
fof(fact_685_power__strict__decreasing,axiom,
! [A_201,N_50,N_49] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_50),N_49))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_201))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_201),one_on1684967323de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_201),N_49)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_201),N_50))) ) ) ) ).
fof(fact_686_power__strict__decreasing,axiom,
! [A_201,N_50,N_49] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_50),N_49))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_201))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_201),one_one_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_201),N_49)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_201),N_50))) ) ) ) ).
fof(fact_687_even__less__0__iff,axiom,
! [A] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),zero_zero_int)) ) ).
fof(fact_688_even__less__0__iff,axiom,
! [A] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),zero_zero_real)) ) ).
fof(fact_689_even__less__0__iff,axiom,
! [A] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),zero_zero_rat)) ) ).
fof(fact_690_sum__squares__eq__zero__iff,axiom,
! [X_2,Y_2] :
( ( is_int(X_2)
& is_int(Y_2) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),X_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_2),Y_2)) = zero_zero_int
<=> ( X_2 = zero_zero_int
& Y_2 = zero_zero_int ) ) ) ).
fof(fact_691_sum__squares__eq__zero__iff,axiom,
! [X_2,Y_2] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_2),X_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2)) = zero_zero_real
<=> ( X_2 = zero_zero_real
& Y_2 = zero_zero_real ) ) ).
fof(fact_692_sum__squares__eq__zero__iff,axiom,
! [X_2,Y_2] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_2),X_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_2),Y_2)) = zero_zero_rat
<=> ( X_2 = zero_zero_rat
& Y_2 = zero_zero_rat ) ) ).
fof(fact_693_add__scale__eq__noteq,axiom,
! [C_89,D_17,A_200,B_154,R_8] :
( ( is_int(C_89)
& is_int(D_17)
& is_int(R_8) )
=> ( R_8 != zero_zero_int
=> ( ( A_200 = B_154
& C_89 != D_17 )
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_200),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_8),C_89)) != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_154),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_8),D_17)) ) ) ) ).
fof(fact_694_add__scale__eq__noteq,axiom,
! [C_89,D_17,A_200,B_154,R_8] :
( R_8 != zero_zero_nat
=> ( ( A_200 = B_154
& C_89 != D_17 )
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_200),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_8),C_89)) != hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_154),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_8),D_17)) ) ) ).
fof(fact_695_add__scale__eq__noteq,axiom,
! [C_89,D_17,A_200,B_154,R_8] :
( R_8 != zero_zero_real
=> ( ( A_200 = B_154
& C_89 != D_17 )
=> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_200),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_8),C_89)) != hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_154),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_8),D_17)) ) ) ).
fof(fact_696_add__scale__eq__noteq,axiom,
! [C_89,D_17,A_200,B_154,R_8] :
( R_8 != zero_zero_complex
=> ( ( A_200 = B_154
& C_89 != D_17 )
=> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_200),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,R_8),C_89)) != hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_154),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,R_8),D_17)) ) ) ).
fof(fact_697_add__scale__eq__noteq,axiom,
! [C_89,D_17,A_200,B_154,R_8] :
( R_8 != zero_zero_rat
=> ( ( A_200 = B_154
& C_89 != D_17 )
=> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_200),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,R_8),C_89)) != hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_154),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,R_8),D_17)) ) ) ).
fof(fact_698_zprime__zdvd__power,axiom,
! [A_3,N_1,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),A_3)) ) ) ).
fof(fact_699_semiring__norm_I112_J,axiom,
zero_zero_int = hAPP_int_int(number_number_of_int,pls) ).
fof(fact_700_semiring__norm_I112_J,axiom,
zero_zero_real = hAPP_int_real(number267125858f_real,pls) ).
fof(fact_701_semiring__norm_I112_J,axiom,
zero_zero_complex = hAPP_int_complex(number528085621omplex,pls) ).
fof(fact_702_semiring__norm_I112_J,axiom,
zero_zero_rat = hAPP_int_rat(number_number_of_rat,pls) ).
fof(fact_703_number__of__Pls,axiom,
hAPP_int_int(number_number_of_int,pls) = zero_zero_int ).
fof(fact_704_number__of__Pls,axiom,
hAPP_int_real(number267125858f_real,pls) = zero_zero_real ).
fof(fact_705_number__of__Pls,axiom,
hAPP_int_complex(number528085621omplex,pls) = zero_zero_complex ).
fof(fact_706_number__of__Pls,axiom,
hAPP_int_rat(number_number_of_rat,pls) = zero_zero_rat ).
fof(fact_707_semiring__numeral__0__eq__0,axiom,
hAPP_int_int(number_number_of_int,pls) = zero_zero_int ).
fof(fact_708_semiring__numeral__0__eq__0,axiom,
hAPP_int_nat(number_number_of_nat,pls) = zero_zero_nat ).
fof(fact_709_semiring__numeral__0__eq__0,axiom,
hAPP_int_real(number267125858f_real,pls) = zero_zero_real ).
fof(fact_710_semiring__numeral__0__eq__0,axiom,
hAPP_int_complex(number528085621omplex,pls) = zero_zero_complex ).
fof(fact_711_semiring__numeral__0__eq__0,axiom,
hAPP_int_rat(number_number_of_rat,pls) = zero_zero_rat ).
fof(fact_712_bin__less__0__simps_I4_J,axiom,
! [W_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,W_1)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),zero_zero_int)) ) ).
fof(fact_713_bin__less__0__simps_I1_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),zero_zero_int)) ).
fof(fact_714_bin__less__0__simps_I3_J,axiom,
! [W_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,W_1)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),zero_zero_int)) ) ).
fof(fact_715_zero__is__num__zero,axiom,
zero_zero_int = hAPP_int_int(number_number_of_int,pls) ).
fof(fact_716_int__0__less__1,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
fof(fact_717_pos__zmult__pos,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) ) ) ).
fof(fact_718_zmult__zless__mono2,axiom,
! [K_2,I,J_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),I)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),J_2))) ) ) ).
fof(fact_719_odd__nonzero,axiom,
! [Z_1] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_1)),Z_1) != zero_zero_int ).
fof(fact_720_power__Suc__less,axiom,
! [N_48,A_199] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_199))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_199),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_199),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_199),N_48))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_199),N_48))) ) ) ).
fof(fact_721_power__Suc__less,axiom,
! [N_48,A_199] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_199))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_199),one_one_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_199),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_199),N_48))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_199),N_48))) ) ) ).
fof(fact_722_power__Suc__less,axiom,
! [N_48,A_199] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_199))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_199),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_199),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_199),N_48))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_199),N_48))) ) ) ).
fof(fact_723_power__Suc__less,axiom,
! [N_48,A_199] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_199))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_199),one_on1645066479umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_199),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_199),N_48))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_199),N_48))) ) ) ).
fof(fact_724_power__Suc__less,axiom,
! [N_48,A_199] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_199))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_199),one_on1684967323de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_199),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_199),N_48))),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_199),N_48))) ) ) ).
fof(fact_725_power__Suc__less,axiom,
! [N_48,A_199] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_199))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_199),one_one_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_199),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_199),N_48))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_199),N_48))) ) ) ).
fof(fact_726_zprime__power__zdvd__cancel__left,axiom,
! [N_1,B_2,A_3,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N_1)),B_2)) ) ) ) ).
fof(fact_727_zprime__power__zdvd__cancel__right,axiom,
! [N_1,A_3,B_2,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N_1)),A_3)) ) ) ) ).
fof(fact_728_sum__squares__ge__zero,axiom,
! [X_40,Y_32] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_40),X_40)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_32),Y_32)))) ).
fof(fact_729_sum__squares__ge__zero,axiom,
! [X_40,Y_32] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_40),X_40)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_32),Y_32)))) ).
fof(fact_730_sum__squares__ge__zero,axiom,
! [X_40,Y_32] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_40),X_40)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_32),Y_32)))) ).
fof(fact_731_sum__squares__le__zero__iff,axiom,
! [X_2,Y_2] :
( ( is_int(X_2)
& is_int(Y_2) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),X_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_2),Y_2))),zero_zero_int))
<=> ( X_2 = zero_zero_int
& Y_2 = zero_zero_int ) ) ) ).
fof(fact_732_sum__squares__le__zero__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_2),X_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2))),zero_zero_real))
<=> ( X_2 = zero_zero_real
& Y_2 = zero_zero_real ) ) ).
fof(fact_733_sum__squares__le__zero__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_2),X_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_2),Y_2))),zero_zero_rat))
<=> ( X_2 = zero_zero_rat
& Y_2 = zero_zero_rat ) ) ).
fof(fact_734_less__nat__number__of,axiom,
! [V,V_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),V_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_1)) )
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),V_1)) ) ) ).
fof(fact_735_not__sum__squares__lt__zero,axiom,
! [X_39,Y_31] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_39),X_39)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_31),Y_31))),zero_zero_int)) ).
fof(fact_736_not__sum__squares__lt__zero,axiom,
! [X_39,Y_31] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_39),X_39)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_31),Y_31))),zero_zero_real)) ).
fof(fact_737_not__sum__squares__lt__zero,axiom,
! [X_39,Y_31] : ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_39),X_39)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_31),Y_31))),zero_zero_rat)) ).
fof(fact_738_sum__squares__gt__zero__iff,axiom,
! [X_2,Y_2] :
( ( is_int(X_2)
& is_int(Y_2) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),X_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_2),Y_2))))
<=> ( X_2 != zero_zero_int
| Y_2 != zero_zero_int ) ) ) ).
fof(fact_739_sum__squares__gt__zero__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_2),X_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2))))
<=> ( X_2 != zero_zero_real
| Y_2 != zero_zero_real ) ) ).
fof(fact_740_sum__squares__gt__zero__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_2),X_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_2),Y_2))))
<=> ( X_2 != zero_zero_rat
| Y_2 != zero_zero_rat ) ) ).
fof(fact_741_le__nat__number__of,axiom,
! [V,V_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1)))
<=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),V_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),pls)) ) ) ).
fof(fact_742_number__of__Bit0,axiom,
! [W_6] : hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W_6)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),hAPP_int_int(number_number_of_int,W_6))),hAPP_int_int(number_number_of_int,W_6)) ).
fof(fact_743_number__of__Bit0,axiom,
! [W_6] : hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,W_6)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),hAPP_int_real(number267125858f_real,W_6))),hAPP_int_real(number267125858f_real,W_6)) ).
fof(fact_744_number__of__Bit0,axiom,
! [W_6] : hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,W_6)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),hAPP_int_complex(number528085621omplex,W_6))),hAPP_int_complex(number528085621omplex,W_6)) ).
fof(fact_745_number__of__Bit0,axiom,
! [W_6] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,W_6)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),hAPP_int_rat(number_number_of_rat,W_6))),hAPP_int_rat(number_number_of_rat,W_6)) ).
fof(fact_746_power__one__right,axiom,
! [A_198] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_198),one_one_nat) = A_198 ).
fof(fact_747_power__one__right,axiom,
! [A_198] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_198),one_one_nat) = A_198 ).
fof(fact_748_power__one__right,axiom,
! [A_198] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_198),one_one_nat) = A_198 ).
fof(fact_749_power__one__right,axiom,
! [A_198] :
( is_int(A_198)
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_198),one_one_nat) = A_198 ) ).
fof(fact_750_power__one__right,axiom,
! [A_198] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_198),one_one_nat) = A_198 ).
fof(fact_751_power__one__right,axiom,
! [A_198] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_198),one_one_nat) = A_198 ).
fof(fact_752_power__one__right,axiom,
! [A_198] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_198),one_one_nat) = A_198 ).
fof(fact_753_int__one__le__iff__zero__less,axiom,
! [Z_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Z_2))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2)) ) ).
fof(fact_754_pos__zmult__eq__1__iff,axiom,
! [N,Ma] :
( ( is_int(N)
& is_int(Ma) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),N) = one_one_int
<=> ( Ma = one_one_int
& N = one_one_int ) ) ) ) ).
fof(fact_755_odd__less__0,axiom,
! [Z_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_2)),Z_2)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),zero_zero_int)) ) ).
fof(fact_756_less__special_I1_J,axiom,
! [Y_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Y_2)) ) ).
fof(fact_757_less__special_I1_J,axiom,
! [Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_int_real(number267125858f_real,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Y_2)) ) ).
fof(fact_758_less__special_I1_J,axiom,
! [Y_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_int_rat(number_number_of_rat,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Y_2)) ) ).
fof(fact_759_less__special_I3_J,axiom,
! [X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,X_2)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),pls)) ) ).
fof(fact_760_less__special_I3_J,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,X_2)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),pls)) ) ).
fof(fact_761_less__special_I3_J,axiom,
! [X_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,X_2)),zero_zero_rat))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),pls)) ) ).
fof(fact_762_le__special_I1_J,axiom,
! [Y_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Y_2)) ) ).
fof(fact_763_le__special_I1_J,axiom,
! [Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_int_real(number267125858f_real,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Y_2)) ) ).
fof(fact_764_le__special_I1_J,axiom,
! [Y_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_int_rat(number_number_of_rat,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Y_2)) ) ).
fof(fact_765_le__special_I3_J,axiom,
! [X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,X_2)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),pls)) ) ).
fof(fact_766_le__special_I3_J,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,X_2)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),pls)) ) ).
fof(fact_767_le__special_I3_J,axiom,
! [X_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,X_2)),zero_zero_rat))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),pls)) ) ).
fof(fact_768_le__imp__0__less,axiom,
! [Z_1] :
( 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(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_1))) ) ).
fof(fact_769_zero__power2,axiom,
hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_int ).
fof(fact_770_zero__power2,axiom,
hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_nat ).
fof(fact_771_zero__power2,axiom,
hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_real ).
fof(fact_772_zero__power2,axiom,
hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_z126310315umeral ).
fof(fact_773_zero__power2,axiom,
hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_complex ).
fof(fact_774_zero__power2,axiom,
hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_z891286103de_int ).
fof(fact_775_zero__power2,axiom,
hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_rat ).
fof(fact_776_zero__eq__power2,axiom,
! [A] :
( is_int(A)
=> ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_int
<=> A = zero_zero_int ) ) ).
fof(fact_777_zero__eq__power2,axiom,
! [A] :
( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_real
<=> A = zero_zero_real ) ).
fof(fact_778_zero__eq__power2,axiom,
! [A] :
( hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_complex
<=> A = zero_zero_complex ) ).
fof(fact_779_zero__eq__power2,axiom,
! [A] :
( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_rat
<=> A = zero_zero_rat ) ).
fof(fact_780_zero__le__power2,axiom,
! [A_197] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_197),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_781_zero__le__power2,axiom,
! [A_197] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_197),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_782_zero__le__power2,axiom,
! [A_197] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_197),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_783_power2__le__imp__le,axiom,
! [X_38,Y_30] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_38),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_30),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_30))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_38),Y_30)) ) ) ).
fof(fact_784_power2__le__imp__le,axiom,
! [X_38,Y_30] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_38),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_30),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_30))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_38),Y_30)) ) ) ).
fof(fact_785_power2__le__imp__le,axiom,
! [X_38,Y_30] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_38),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_30),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_30))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_38),Y_30)) ) ) ).
fof(fact_786_power2__le__imp__le,axiom,
! [X_38,Y_30] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_38),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_30),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Y_30))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,X_38),Y_30)) ) ) ).
fof(fact_787_power2__le__imp__le,axiom,
! [X_38,Y_30] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_38),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_30),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Y_30))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_38),Y_30)) ) ) ).
fof(fact_788_power2__le__imp__le,axiom,
! [X_38,Y_30] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_38),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_30),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_30))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_38),Y_30)) ) ) ).
fof(fact_789_power2__eq__imp__eq,axiom,
! [X_37,Y_29] :
( ( is_int(X_37)
& is_int(Y_29) )
=> ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_37),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_37))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_29))
=> X_37 = Y_29 ) ) ) ) ).
fof(fact_790_power2__eq__imp__eq,axiom,
! [X_37,Y_29] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_37),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),X_37))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_29))
=> X_37 = Y_29 ) ) ) ).
fof(fact_791_power2__eq__imp__eq,axiom,
! [X_37,Y_29] :
( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_37),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_37))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_29))
=> X_37 = Y_29 ) ) ) ).
fof(fact_792_power2__eq__imp__eq,axiom,
! [X_37,Y_29] :
( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_37),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),X_37))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Y_29))
=> X_37 = Y_29 ) ) ) ).
fof(fact_793_power2__eq__imp__eq,axiom,
! [X_37,Y_29] :
( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_37),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),X_37))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Y_29))
=> X_37 = Y_29 ) ) ) ).
fof(fact_794_power2__eq__imp__eq,axiom,
! [X_37,Y_29] :
( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_37),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_37))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_29))
=> X_37 = Y_29 ) ) ) ).
fof(fact_795_power2__less__0,axiom,
! [A_196] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_196),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_int)) ).
fof(fact_796_power2__less__0,axiom,
! [A_196] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_196),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_real)) ).
fof(fact_797_power2__less__0,axiom,
! [A_196] : ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_196),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_rat)) ).
fof(fact_798_zero__less__power2,axiom,
! [A] :
( is_int(A)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
<=> A != zero_zero_int ) ) ).
fof(fact_799_zero__less__power2,axiom,
! [A] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
<=> A != zero_zero_real ) ).
fof(fact_800_zero__less__power2,axiom,
! [A] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
<=> A != zero_zero_rat ) ).
fof(fact_801_sum__power2__eq__zero__iff,axiom,
! [X_2,Y_2] :
( ( is_int(X_2)
& is_int(Y_2) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_int
<=> ( X_2 = zero_zero_int
& Y_2 = zero_zero_int ) ) ) ).
fof(fact_802_sum__power2__eq__zero__iff,axiom,
! [X_2,Y_2] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real
<=> ( X_2 = zero_zero_real
& Y_2 = zero_zero_real ) ) ).
fof(fact_803_sum__power2__eq__zero__iff,axiom,
! [X_2,Y_2] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_rat
<=> ( X_2 = zero_zero_rat
& Y_2 = zero_zero_rat ) ) ).
fof(fact_804_power__commutes,axiom,
! [A_195,N_47] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_195),N_47)),A_195) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_195),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_195),N_47)) ).
fof(fact_805_power__commutes,axiom,
! [A_195,N_47] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_195),N_47)),A_195) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_195),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_195),N_47)) ).
fof(fact_806_power__commutes,axiom,
! [A_195,N_47] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_195),N_47)),A_195) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_195),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_195),N_47)) ).
fof(fact_807_power__commutes,axiom,
! [A_195,N_47] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_195),N_47)),A_195) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_195),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_195),N_47)) ).
fof(fact_808_power__commutes,axiom,
! [A_195,N_47] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_195),N_47)),A_195) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_195),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_195),N_47)) ).
fof(fact_809_power__commutes,axiom,
! [A_195,N_47] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_195),N_47)),A_195) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_195),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_195),N_47)) ).
fof(fact_810_power__commutes,axiom,
! [A_195,N_47] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_195),N_47)),A_195) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_195),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_195),N_47)) ).
fof(fact_811_power__mult__distrib,axiom,
! [A_194,B_153,N_46] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_194),B_153)),N_46) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_194),N_46)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_153),N_46)) ).
fof(fact_812_power__mult__distrib,axiom,
! [A_194,B_153,N_46] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_194),B_153)),N_46) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_194),N_46)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_153),N_46)) ).
fof(fact_813_power__mult__distrib,axiom,
! [A_194,B_153,N_46] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_194),B_153)),N_46) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_194),N_46)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_153),N_46)) ).
fof(fact_814_power__mult__distrib,axiom,
! [A_194,B_153,N_46] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_194),B_153)),N_46) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_194),N_46)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_153),N_46)) ).
fof(fact_815_power__mult__distrib,axiom,
! [A_194,B_153,N_46] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_194),B_153)),N_46) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_194),N_46)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_153),N_46)) ).
fof(fact_816_power__mult__distrib,axiom,
! [A_194,B_153,N_46] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_194),B_153)),N_46) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_194),N_46)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_153),N_46)) ).
fof(fact_817_power__mult__distrib,axiom,
! [A_194,B_153,N_46] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_194),B_153)),N_46) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_194),N_46)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,B_153),N_46)) ).
fof(fact_818_power__add,axiom,
! [A_193,M_21,N_45] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_193),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_21),N_45)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_193),M_21)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_193),N_45)) ).
fof(fact_819_power__add,axiom,
! [A_193,M_21,N_45] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_193),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_21),N_45)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_193),M_21)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_193),N_45)) ).
fof(fact_820_power__add,axiom,
! [A_193,M_21,N_45] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_193),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_21),N_45)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_193),M_21)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_193),N_45)) ).
fof(fact_821_power__add,axiom,
! [A_193,M_21,N_45] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_193),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_21),N_45)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_193),M_21)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_193),N_45)) ).
fof(fact_822_power__add,axiom,
! [A_193,M_21,N_45] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_193),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_21),N_45)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_193),M_21)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_193),N_45)) ).
fof(fact_823_power__add,axiom,
! [A_193,M_21,N_45] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_193),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_21),N_45)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_193),M_21)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_193),N_45)) ).
fof(fact_824_power__add,axiom,
! [A_193,M_21,N_45] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_193),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_21),N_45)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_193),M_21)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_193),N_45)) ).
fof(fact_825_power__one,axiom,
! [N_44] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,one_one_int),N_44) = one_one_int ).
fof(fact_826_power__one,axiom,
! [N_44] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,one_one_nat),N_44) = one_one_nat ).
fof(fact_827_power__one,axiom,
! [N_44] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,one_one_real),N_44) = one_one_real ).
fof(fact_828_power__one,axiom,
! [N_44] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,one_on1645066479umeral),N_44) = one_on1645066479umeral ).
fof(fact_829_power__one,axiom,
! [N_44] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,one_one_complex),N_44) = one_one_complex ).
fof(fact_830_power__one,axiom,
! [N_44] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,one_on1684967323de_int),N_44) = one_on1684967323de_int ).
fof(fact_831_power__one,axiom,
! [N_44] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,one_one_rat),N_44) = one_one_rat ).
fof(fact_832_power__mult,axiom,
! [A_192,M_20,N_43] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_192),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_20),N_43)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_192),M_20)),N_43) ).
fof(fact_833_power__mult,axiom,
! [A_192,M_20,N_43] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_192),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_20),N_43)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_192),M_20)),N_43) ).
fof(fact_834_power__mult,axiom,
! [A_192,M_20,N_43] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_192),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_20),N_43)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_192),M_20)),N_43) ).
fof(fact_835_power__mult,axiom,
! [A_192,M_20,N_43] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_192),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_20),N_43)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_192),M_20)),N_43) ).
fof(fact_836_power__mult,axiom,
! [A_192,M_20,N_43] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_192),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_20),N_43)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_192),M_20)),N_43) ).
fof(fact_837_power__mult,axiom,
! [A_192,M_20,N_43] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_192),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_20),N_43)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_192),M_20)),N_43) ).
fof(fact_838_power__mult,axiom,
! [A_192,M_20,N_43] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_192),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_20),N_43)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_192),M_20)),N_43) ).
fof(fact_839_power2__less__imp__less,axiom,
! [X_36,Y_28] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_36),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_28),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_28))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_36),Y_28)) ) ) ).
fof(fact_840_power2__less__imp__less,axiom,
! [X_36,Y_28] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_36),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_28),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_28))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_36),Y_28)) ) ) ).
fof(fact_841_power2__less__imp__less,axiom,
! [X_36,Y_28] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_36),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_28),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_28))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_36),Y_28)) ) ) ).
fof(fact_842_power2__less__imp__less,axiom,
! [X_36,Y_28] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_36),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_28),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Y_28))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,X_36),Y_28)) ) ) ).
fof(fact_843_power2__less__imp__less,axiom,
! [X_36,Y_28] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_36),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_28),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Y_28))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_36),Y_28)) ) ) ).
fof(fact_844_power2__less__imp__less,axiom,
! [X_36,Y_28] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_36),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_28),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_28))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_36),Y_28)) ) ) ).
fof(fact_845_sum__power2__ge__zero,axiom,
! [X_35,Y_27] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_35),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_27),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).
fof(fact_846_sum__power2__ge__zero,axiom,
! [X_35,Y_27] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_35),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_27),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).
fof(fact_847_sum__power2__ge__zero,axiom,
! [X_35,Y_27] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_35),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_27),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).
fof(fact_848_sum__power2__le__zero__iff,axiom,
! [X_2,Y_2] :
( ( is_int(X_2)
& is_int(Y_2) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_int))
<=> ( X_2 = zero_zero_int
& Y_2 = zero_zero_int ) ) ) ).
fof(fact_849_sum__power2__le__zero__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_real))
<=> ( X_2 = zero_zero_real
& Y_2 = zero_zero_real ) ) ).
fof(fact_850_sum__power2__le__zero__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_rat))
<=> ( X_2 = zero_zero_rat
& Y_2 = zero_zero_rat ) ) ).
fof(fact_851_not__sum__power2__lt__zero,axiom,
! [X_34,Y_26] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_34),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_26),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_int)) ).
fof(fact_852_not__sum__power2__lt__zero,axiom,
! [X_34,Y_26] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_34),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_26),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_real)) ).
fof(fact_853_not__sum__power2__lt__zero,axiom,
! [X_34,Y_26] : ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_34),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_26),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_rat)) ).
fof(fact_854_sum__power2__gt__zero__iff,axiom,
! [X_2,Y_2] :
( ( is_int(X_2)
& is_int(Y_2) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))
<=> ( X_2 != zero_zero_int
| Y_2 != zero_zero_int ) ) ) ).
fof(fact_855_sum__power2__gt__zero__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))
<=> ( X_2 != zero_zero_real
| Y_2 != zero_zero_real ) ) ).
fof(fact_856_sum__power2__gt__zero__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))
<=> ( X_2 != zero_zero_rat
| Y_2 != zero_zero_rat ) ) ).
fof(fact_857_zero__le__even__power_H,axiom,
! [A_191,N_42] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_191),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_42)))) ).
fof(fact_858_zero__le__even__power_H,axiom,
! [A_191,N_42] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_191),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_42)))) ).
fof(fact_859_zero__le__even__power_H,axiom,
! [A_191,N_42] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_191),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_42)))) ).
fof(fact_860_one__le__power,axiom,
! [N_41,A_190] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_190))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_190),N_41))) ) ).
fof(fact_861_one__le__power,axiom,
! [N_41,A_190] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_190))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_190),N_41))) ) ).
fof(fact_862_one__le__power,axiom,
! [N_41,A_190] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_190))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_190),N_41))) ) ).
fof(fact_863_one__le__power,axiom,
! [N_41,A_190] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),A_190))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_190),N_41))) ) ).
fof(fact_864_one__le__power,axiom,
! [N_41,A_190] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),A_190))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_190),N_41))) ) ).
fof(fact_865_one__le__power,axiom,
! [N_41,A_190] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),A_190))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_190),N_41))) ) ).
fof(fact_866_power__increasing,axiom,
! [A_189,N_40,N_39] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_40),N_39))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_189))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_189),N_40)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_189),N_39))) ) ) ).
fof(fact_867_power__increasing,axiom,
! [A_189,N_40,N_39] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_40),N_39))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_189))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_189),N_40)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_189),N_39))) ) ) ).
fof(fact_868_power__increasing,axiom,
! [A_189,N_40,N_39] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_40),N_39))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_189))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_189),N_40)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_189),N_39))) ) ) ).
fof(fact_869_power__increasing,axiom,
! [A_189,N_40,N_39] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_40),N_39))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),A_189))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_189),N_40)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_189),N_39))) ) ) ).
fof(fact_870_power__increasing,axiom,
! [A_189,N_40,N_39] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_40),N_39))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),A_189))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_189),N_40)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_189),N_39))) ) ) ).
fof(fact_871_power__increasing,axiom,
! [A_189,N_40,N_39] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_40),N_39))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),A_189))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_189),N_40)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_189),N_39))) ) ) ).
fof(fact_872_power__inject__exp,axiom,
! [Ma,N,A] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A))
=> ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),Ma) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),N)
<=> Ma = N ) ) ).
fof(fact_873_power__inject__exp,axiom,
! [Ma,N,A] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),N)
<=> Ma = N ) ) ).
fof(fact_874_power__inject__exp,axiom,
! [Ma,N,A] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A))
=> ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),Ma) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),N)
<=> Ma = N ) ) ).
fof(fact_875_power__inject__exp,axiom,
! [Ma,N,A] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A))
=> ( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A),Ma) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A),N)
<=> Ma = N ) ) ).
fof(fact_876_power__inject__exp,axiom,
! [Ma,N,A] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A))
=> ( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A),Ma) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A),N)
<=> Ma = N ) ) ).
fof(fact_877_power__inject__exp,axiom,
! [Ma,N,A] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A))
=> ( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),Ma) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),N)
<=> Ma = N ) ) ).
fof(fact_878_power__strict__increasing__iff,axiom,
! [X_2,Y_2,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),X_2)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) ) ) ).
fof(fact_879_power__strict__increasing__iff,axiom,
! [X_2,Y_2,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) ) ) ).
fof(fact_880_power__strict__increasing__iff,axiom,
! [X_2,Y_2,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B),X_2)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B),Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) ) ) ).
fof(fact_881_power__strict__increasing__iff,axiom,
! [X_2,Y_2,B] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),B))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B),X_2)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B),Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) ) ) ).
fof(fact_882_power__strict__increasing__iff,axiom,
! [X_2,Y_2,B] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),B))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B),X_2)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B),Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) ) ) ).
fof(fact_883_power__strict__increasing__iff,axiom,
! [X_2,Y_2,B] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),B))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B),X_2)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B),Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) ) ) ).
fof(fact_884_power__less__imp__less__exp,axiom,
! [M_19,N_38,A_188] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_188))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_188),M_19)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_188),N_38)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_19),N_38)) ) ) ).
fof(fact_885_power__less__imp__less__exp,axiom,
! [M_19,N_38,A_188] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_188))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_188),M_19)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_188),N_38)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_19),N_38)) ) ) ).
fof(fact_886_power__less__imp__less__exp,axiom,
! [M_19,N_38,A_188] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_188))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_188),M_19)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_188),N_38)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_19),N_38)) ) ) ).
fof(fact_887_power__less__imp__less__exp,axiom,
! [M_19,N_38,A_188] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_188))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_188),M_19)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_188),N_38)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_19),N_38)) ) ) ).
fof(fact_888_power__less__imp__less__exp,axiom,
! [M_19,N_38,A_188] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_188))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_188),M_19)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_188),N_38)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_19),N_38)) ) ) ).
fof(fact_889_power__less__imp__less__exp,axiom,
! [M_19,N_38,A_188] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_188))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_188),M_19)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_188),N_38)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_19),N_38)) ) ) ).
fof(fact_890_power__strict__increasing,axiom,
! [A_187,N_37,N_36] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_37),N_36))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_187))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_187),N_37)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_187),N_36))) ) ) ).
fof(fact_891_power__strict__increasing,axiom,
! [A_187,N_37,N_36] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_37),N_36))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_187))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_187),N_37)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_187),N_36))) ) ) ).
fof(fact_892_power__strict__increasing,axiom,
! [A_187,N_37,N_36] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_37),N_36))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_187))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_187),N_37)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_187),N_36))) ) ) ).
fof(fact_893_power__strict__increasing,axiom,
! [A_187,N_37,N_36] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_37),N_36))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_187))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_187),N_37)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_187),N_36))) ) ) ).
fof(fact_894_power__strict__increasing,axiom,
! [A_187,N_37,N_36] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_37),N_36))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_187))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_187),N_37)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_187),N_36))) ) ) ).
fof(fact_895_power__strict__increasing,axiom,
! [A_187,N_37,N_36] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_37),N_36))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_187))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_187),N_37)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_187),N_36))) ) ) ).
fof(fact_896_s,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) ).
fof(fact_897_Euler_Oaux____1,axiom,
! [Y_1,X_1,P_3] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_1),P_3))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y_1)) ) ) ).
fof(fact_898_int__pos__lt__two__imp__zero__or__one,axiom,
! [X_1] :
( is_int(X_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
=> ( X_1 = zero_zero_int
| X_1 = one_one_int ) ) ) ) ).
fof(fact_899_even__power__le__0__imp__0,axiom,
! [A_186,K_8] :
( is_int(A_186)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_186),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_8))),zero_zero_int))
=> A_186 = zero_zero_int ) ) ).
fof(fact_900_even__power__le__0__imp__0,axiom,
! [A_186,K_8] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_186),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_8))),zero_zero_real))
=> A_186 = zero_zero_real ) ).
fof(fact_901_even__power__le__0__imp__0,axiom,
! [A_186,K_8] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_186),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_8))),zero_zero_rat))
=> A_186 = zero_zero_rat ) ).
fof(fact_902_zprime__def,axiom,
! [P_5] :
( is_int(P_5)
=> ( hBOOL(hAPP_int_bool(zprime,P_5))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),P_5))
& ! [M_1] :
( is_int(M_1)
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M_1),P_5)) )
=> ( M_1 = one_one_int
| M_1 = P_5 ) ) ) ) ) ) ).
fof(fact_903_self__quotient__aux2,axiom,
! [R_3,Q_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
=> ( A_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,R_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),one_one_int)) ) ) ) ).
fof(fact_904_self__quotient__aux1,axiom,
! [R_3,Q_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
=> ( A_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,R_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),A_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Q_2)) ) ) ) ).
fof(fact_905_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_906_convex__bound__lt,axiom,
! [V_10,U_3,Y_25,X_33,A_185] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_33),A_185))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_25),A_185))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V_10))
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U_3),V_10) = one_one_int
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,U_3),X_33)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_10),Y_25))),A_185)) ) ) ) ) ) ).
fof(fact_907_convex__bound__lt,axiom,
! [V_10,U_3,Y_25,X_33,A_185] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_33),A_185))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_25),A_185))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),U_3))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),V_10))
=> ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,U_3),V_10) = one_one_real
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_3),X_33)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,V_10),Y_25))),A_185)) ) ) ) ) ) ).
fof(fact_908_convex__bound__lt,axiom,
! [V_10,U_3,Y_25,X_33,A_185] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_33),A_185))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_25),A_185))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),U_3))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),V_10))
=> ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,U_3),V_10) = one_one_rat
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,U_3),X_33)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,V_10),Y_25))),A_185)) ) ) ) ) ) ).
fof(fact_909_zprime__zdvd__zmult,axiom,
! [N_1,P_3,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),M))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),N_1)) ) ) ) ) ).
fof(fact_910_zcong__zprime__prod__zero,axiom,
! [B_2,A_3,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),zero_zero_int),P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B_2),zero_zero_int),P_3)) ) ) ) ) ).
fof(fact_911_dvd__0__right,axiom,
! [A_184] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_184),zero_zero_real)) ).
fof(fact_912_dvd__0__right,axiom,
! [A_184] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_184),zero_z126310315umeral)) ).
fof(fact_913_dvd__0__right,axiom,
! [A_184] : hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_184),zero_zero_complex)) ).
fof(fact_914_dvd__0__right,axiom,
! [A_184] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_184),zero_z891286103de_int)) ).
fof(fact_915_dvd__0__right,axiom,
! [A_184] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_184),zero_zero_rat)) ).
fof(fact_916_dvd__0__right,axiom,
! [A_184] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_184),zero_zero_int)) ).
fof(fact_917_dvd__0__right,axiom,
! [A_184] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_184),zero_zero_nat)) ).
fof(fact_918__096_B_Bthesis_O_A_I_B_Bs1_O_A_091s1_A_094_A2_A_061_A_N1_093_A_Imod_A4_,axiom,
~ ! [S1] :
( is_int(S1)
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,S1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) ) ).
fof(fact_919__096Legendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_061_A1_096,axiom,
hAPP_int_int(legendre(hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)) = one_one_int ).
fof(fact_920_nat__zero__less__power__iff,axiom,
! [X_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_2))
| N = zero_zero_nat ) ) ).
fof(fact_921_zero__less__power__nat__eq,axiom,
! [X_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),N)))
<=> ( N = zero_zero_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_2)) ) ) ).
fof(fact_922_nat__power__less__imp__less,axiom,
! [M,N_1,I] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),I))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ) ).
fof(fact_923_zero__less__power__nat__eq__number__of,axiom,
! [X_2,W_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),hAPP_int_nat(number_number_of_nat,W_1))))
<=> ( hAPP_int_nat(number_number_of_nat,W_1) = zero_zero_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_2)) ) ) ).
fof(fact_924_rel__simps_I47_J,axiom,
! [K_1] :
( is_int(K_1)
=> ( hAPP_int_int(bit1,K_1) = min
<=> K_1 = min ) ) ).
fof(fact_925_rel__simps_I43_J,axiom,
! [L] :
( is_int(L)
=> ( min = hAPP_int_int(bit1,L)
<=> min = L ) ) ).
fof(fact_926_Bit1__Min,axiom,
hAPP_int_int(bit1,min) = min ).
fof(fact_927_rel__simps_I37_J,axiom,
pls != min ).
fof(fact_928_rel__simps_I40_J,axiom,
min != pls ).
fof(fact_929_rel__simps_I45_J,axiom,
! [K_2] : hAPP_int_int(bit0,K_2) != min ).
fof(fact_930_rel__simps_I42_J,axiom,
! [L_2] : min != hAPP_int_int(bit0,L_2) ).
fof(fact_931_rel__simps_I7_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),min)) ).
fof(fact_932_rel__simps_I24_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),min)) ).
fof(fact_933_not__real__square__gt__zero,axiom,
! [X_2] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_2),X_2)))
<=> X_2 = zero_zero_real ) ).
fof(fact_934_zpower__zdvd__prop1,axiom,
! [P_3,Y_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_1),N_1))) ) ) ).
fof(fact_935_rel__simps_I13_J,axiom,
! [K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),min))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),min)) ) ).
fof(fact_936_rel__simps_I9_J,axiom,
! [K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),hAPP_int_int(bit1,K_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K_1)) ) ).
fof(fact_937_rel__simps_I3_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),min)) ).
fof(fact_938_rel__simps_I6_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),pls)) ).
fof(fact_939_rel__simps_I8_J,axiom,
! [K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),hAPP_int_int(bit0,K_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K_1)) ) ).
fof(fact_940_bin__less__0__simps_I2_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),zero_zero_int)) ).
fof(fact_941_rel__simps_I30_J,axiom,
! [K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),min))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),min)) ) ).
fof(fact_942_rel__simps_I26_J,axiom,
! [K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),hAPP_int_int(bit1,K_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),K_1)) ) ).
fof(fact_943_rel__simps_I20_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),min)) ).
fof(fact_944_rel__simps_I23_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),pls)) ).
fof(fact_945_rel__simps_I28_J,axiom,
! [K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),min))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),min)) ) ).
fof(fact_946_eq__number__of__Pls__Min,axiom,
hAPP_int_int(number_number_of_int,pls) != hAPP_int_int(number_number_of_int,min) ).
fof(fact_947_power__dvd__imp__le,axiom,
! [I,M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N_1)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),I))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ) ).
fof(fact_948_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_32] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_32),zero_zero_nat) = one_one_int ).
fof(fact_949_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_32] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_32),zero_zero_nat) = one_one_nat ).
fof(fact_950_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_32] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_32),zero_zero_nat) = one_one_real ).
fof(fact_951_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_32] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_32),zero_zero_nat) = one_on1645066479umeral ).
fof(fact_952_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_32] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_32),zero_zero_nat) = one_one_complex ).
fof(fact_953_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_32] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_32),zero_zero_nat) = one_on1684967323de_int ).
fof(fact_954_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_32] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_32),zero_zero_nat) = one_one_rat ).
fof(fact_955_power__0,axiom,
! [A_183] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_183),zero_zero_nat) = one_one_int ).
fof(fact_956_power__0,axiom,
! [A_183] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_183),zero_zero_nat) = one_one_nat ).
fof(fact_957_power__0,axiom,
! [A_183] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_183),zero_zero_nat) = one_one_real ).
fof(fact_958_power__0,axiom,
! [A_183] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_183),zero_zero_nat) = one_on1645066479umeral ).
fof(fact_959_power__0,axiom,
! [A_183] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_183),zero_zero_nat) = one_one_complex ).
fof(fact_960_power__0,axiom,
! [A_183] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_183),zero_zero_nat) = one_on1684967323de_int ).
fof(fact_961_power__0,axiom,
! [A_183] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_183),zero_zero_nat) = one_one_rat ).
fof(fact_962_nat__number__of__Pls,axiom,
hAPP_int_nat(number_number_of_nat,pls) = zero_zero_nat ).
fof(fact_963_semiring__norm_I113_J,axiom,
zero_zero_nat = hAPP_int_nat(number_number_of_nat,pls) ).
fof(fact_964_zpower__zdvd__prop2,axiom,
! [Y_1,N_1,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_1),N_1)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y_1)) ) ) ) ).
fof(fact_965_rel__simps_I25_J,axiom,
! [K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),hAPP_int_int(bit0,K_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K_1)) ) ).
fof(fact_966_rel__simps_I11_J,axiom,
! [K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),min))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),min)) ) ).
fof(fact_967_pos__zmult__eq__1__iff__lemma,axiom,
! [M,N_1] :
( is_int(M)
=> ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N_1) = one_one_int
=> ( M = one_one_int
| M = hAPP_int_int(number_number_of_int,min) ) ) ) ).
fof(fact_968_zmult__eq__1__iff,axiom,
! [Ma,N] :
( ( is_int(Ma)
& is_int(N) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),N) = one_one_int
<=> ( ( Ma = one_one_int
& N = one_one_int )
| ( Ma = hAPP_int_int(number_number_of_int,min)
& N = hAPP_int_int(number_number_of_int,min) ) ) ) ) ).
fof(fact_969_linorder__neqE__linordered__idom,axiom,
! [X_31,Y_24] :
( ( is_int(X_31)
& is_int(Y_24) )
=> ( X_31 != Y_24
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_31),Y_24))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_24),X_31)) ) ) ) ).
fof(fact_970_linorder__neqE__linordered__idom,axiom,
! [X_31,Y_24] :
( X_31 != Y_24
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_31),Y_24))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_24),X_31)) ) ) ).
fof(fact_971_linorder__neqE__linordered__idom,axiom,
! [X_31,Y_24] :
( X_31 != Y_24
=> ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_31),Y_24))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_24),X_31)) ) ) ).
fof(fact_972_one__less__power,axiom,
! [N_35,A_182] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_182))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_35))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_182),N_35))) ) ) ).
fof(fact_973_one__less__power,axiom,
! [N_35,A_182] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_182))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_35))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_182),N_35))) ) ) ).
fof(fact_974_one__less__power,axiom,
! [N_35,A_182] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_182))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_35))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_182),N_35))) ) ) ).
fof(fact_975_one__less__power,axiom,
! [N_35,A_182] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_182))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_35))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_182),N_35))) ) ) ).
fof(fact_976_one__less__power,axiom,
! [N_35,A_182] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_182))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_35))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_182),N_35))) ) ) ).
fof(fact_977_one__less__power,axiom,
! [N_35,A_182] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_182))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_35))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_182),N_35))) ) ) ).
fof(fact_978_dvd__refl,axiom,
! [A_181] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_181),A_181)) ).
fof(fact_979_dvd__refl,axiom,
! [A_181] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_181),A_181)) ).
fof(fact_980_dvd__refl,axiom,
! [A_181] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_181),A_181)) ).
fof(fact_981_dvd__refl,axiom,
! [A_181] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_181),A_181)) ).
fof(fact_982_dvd__refl,axiom,
! [A_181] : hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_181),A_181)) ).
fof(fact_983_dvd__refl,axiom,
! [A_181] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_181),A_181)) ).
fof(fact_984_dvd__refl,axiom,
! [A_181] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_181),A_181)) ).
fof(fact_985_dvd__trans,axiom,
! [C_88,A_180,B_152] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_180),B_152))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B_152),C_88))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_180),C_88)) ) ) ).
fof(fact_986_dvd__trans,axiom,
! [C_88,A_180,B_152] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_180),B_152))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,B_152),C_88))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_180),C_88)) ) ) ).
fof(fact_987_dvd__trans,axiom,
! [C_88,A_180,B_152] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_180),B_152))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,B_152),C_88))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_180),C_88)) ) ) ).
fof(fact_988_dvd__trans,axiom,
! [C_88,A_180,B_152] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_180),B_152))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_152),C_88))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_180),C_88)) ) ) ).
fof(fact_989_dvd__trans,axiom,
! [C_88,A_180,B_152] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_180),B_152))
=> ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,B_152),C_88))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_180),C_88)) ) ) ).
fof(fact_990_dvd__trans,axiom,
! [C_88,A_180,B_152] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_180),B_152))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_152),C_88))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_180),C_88)) ) ) ).
fof(fact_991_dvd__trans,axiom,
! [C_88,A_180,B_152] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_180),B_152))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_152),C_88))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_180),C_88)) ) ) ).
fof(fact_992_dvd__power,axiom,
! [X_30,N_34] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34))
| X_30 = one_one_real )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_30),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_30),N_34))) ) ).
fof(fact_993_dvd__power,axiom,
! [X_30,N_34] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34))
| X_30 = one_on1645066479umeral )
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,X_30),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_30),N_34))) ) ).
fof(fact_994_dvd__power,axiom,
! [X_30,N_34] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34))
| X_30 = one_one_complex )
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_30),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_30),N_34))) ) ).
fof(fact_995_dvd__power,axiom,
! [X_30,N_34] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34))
| X_30 = one_on1684967323de_int )
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,X_30),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_30),N_34))) ) ).
fof(fact_996_dvd__power,axiom,
! [X_30,N_34] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34))
| X_30 = one_one_rat )
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_30),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_30),N_34))) ) ).
fof(fact_997_dvd__power,axiom,
! [X_30,N_34] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34))
| X_30 = one_one_int )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_30),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_30),N_34))) ) ).
fof(fact_998_dvd__power,axiom,
! [X_30,N_34] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34))
| X_30 = one_one_nat )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_30),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_30),N_34))) ) ).
fof(fact_999_less__0__number__of,axiom,
! [V] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,V)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V)) ) ).
fof(fact_1000_eq__number__of__0,axiom,
! [V] :
( hAPP_int_nat(number_number_of_nat,V) = zero_zero_nat
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),pls)) ) ).
fof(fact_1001_eq__0__number__of,axiom,
! [V] :
( zero_zero_nat = hAPP_int_nat(number_number_of_nat,V)
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),pls)) ) ).
fof(fact_1002_zcong__sym,axiom,
! [A,B,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B),A),Ma)) ) ).
fof(fact_1003_zcong__refl,axiom,
! [K_2,M] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),K_2),M)) ).
fof(fact_1004_zcong__eq__trans,axiom,
! [D_3,C_1,A_3,B_2,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
=> ( B_2 = C_1
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),D_3),M)) ) ) ) ).
fof(fact_1005_zcong__trans,axiom,
! [C_1,A_3,B_2,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B_2),C_1),M))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),C_1),M)) ) ) ).
fof(fact_1006_pos2,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_1007_zcong__neg__1__impl__ne__1,axiom,
! [X_1,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),hAPP_int_int(number_number_of_int,min)),P_3))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),one_one_int),P_3)) ) ) ).
fof(fact_1008_nat__number__of__mult__left,axiom,
! [V_3,K_2,V_2] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)) = zero_zero_nat )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_2),V_3))),K_2) ) ) ).
fof(fact_1009_mult__nat__number__of,axiom,
! [V_3,V_2] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = zero_zero_nat )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_2),V_3)) ) ) ).
fof(fact_1010_order__le__neq__implies__less,axiom,
! [X_29,Y_23] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,X_29),Y_23))
=> ( X_29 != Y_23
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,X_29),Y_23)) ) ) ).
fof(fact_1011_order__le__neq__implies__less,axiom,
! [X_29,Y_23] :
( ( is_int(X_29)
& is_int(Y_23) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_29),Y_23))
=> ( X_29 != Y_23
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_29),Y_23)) ) ) ) ).
fof(fact_1012_order__le__neq__implies__less,axiom,
! [X_29,Y_23] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_29),Y_23))
=> ( X_29 != Y_23
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_29),Y_23)) ) ) ).
fof(fact_1013_order__le__neq__implies__less,axiom,
! [X_29,Y_23] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_29),Y_23))
=> ( X_29 != Y_23
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_29),Y_23)) ) ) ).
fof(fact_1014_order__le__neq__implies__less,axiom,
! [X_29,Y_23] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_29),Y_23))
=> ( X_29 != Y_23
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_29),Y_23)) ) ) ).
fof(fact_1015_order__le__neq__implies__less,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_2),Y_2))
=> ( X_2 != Y_2
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_2),Y_2)) ) ) ).
fof(fact_1016_order__le__neq__implies__less,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_2),Y_2))
=> ( X_2 != Y_2
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_2),Y_2)) ) ) ).
fof(fact_1017_order__le__neq__implies__less,axiom,
! [X_29,Y_23] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_29),Y_23))
=> ( X_29 != Y_23
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_29),Y_23)) ) ) ).
fof(fact_1018_divisors__zero,axiom,
! [A_179,B_151] :
( ( is_int(A_179)
& is_int(B_151) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_179),B_151) = zero_zero_int
=> ( A_179 = zero_zero_int
| B_151 = zero_zero_int ) ) ) ).
fof(fact_1019_divisors__zero,axiom,
! [A_179,B_151] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_179),B_151) = zero_zero_nat
=> ( A_179 = zero_zero_nat
| B_151 = zero_zero_nat ) ) ).
fof(fact_1020_divisors__zero,axiom,
! [A_179,B_151] :
( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_179),B_151) = zero_zero_real
=> ( A_179 = zero_zero_real
| B_151 = zero_zero_real ) ) ).
fof(fact_1021_divisors__zero,axiom,
! [A_179,B_151] :
( hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_179),B_151) = zero_z126310315umeral
=> ( A_179 = zero_z126310315umeral
| B_151 = zero_z126310315umeral ) ) ).
fof(fact_1022_divisors__zero,axiom,
! [A_179,B_151] :
( hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_179),B_151) = zero_zero_complex
=> ( A_179 = zero_zero_complex
| B_151 = zero_zero_complex ) ) ).
fof(fact_1023_divisors__zero,axiom,
! [A_179,B_151] :
( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_179),B_151) = zero_z891286103de_int
=> ( A_179 = zero_z891286103de_int
| B_151 = zero_z891286103de_int ) ) ).
fof(fact_1024_divisors__zero,axiom,
! [A_179,B_151] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_179),B_151) = zero_zero_rat
=> ( A_179 = zero_zero_rat
| B_151 = zero_zero_rat ) ) ).
fof(fact_1025_no__zero__divisors,axiom,
! [B_150,A_178] :
( ( is_int(B_150)
& is_int(A_178) )
=> ( A_178 != zero_zero_int
=> ( B_150 != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_178),B_150) != zero_zero_int ) ) ) ).
fof(fact_1026_no__zero__divisors,axiom,
! [B_150,A_178] :
( A_178 != zero_zero_nat
=> ( B_150 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_178),B_150) != zero_zero_nat ) ) ).
fof(fact_1027_no__zero__divisors,axiom,
! [B_150,A_178] :
( A_178 != zero_zero_real
=> ( B_150 != zero_zero_real
=> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_178),B_150) != zero_zero_real ) ) ).
fof(fact_1028_no__zero__divisors,axiom,
! [B_150,A_178] :
( A_178 != zero_z126310315umeral
=> ( B_150 != zero_z126310315umeral
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_178),B_150) != zero_z126310315umeral ) ) ).
fof(fact_1029_no__zero__divisors,axiom,
! [B_150,A_178] :
( A_178 != zero_zero_complex
=> ( B_150 != zero_zero_complex
=> hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_178),B_150) != zero_zero_complex ) ) ).
fof(fact_1030_no__zero__divisors,axiom,
! [B_150,A_178] :
( A_178 != zero_z891286103de_int
=> ( B_150 != zero_z891286103de_int
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_178),B_150) != zero_z891286103de_int ) ) ).
fof(fact_1031_no__zero__divisors,axiom,
! [B_150,A_178] :
( A_178 != zero_zero_rat
=> ( B_150 != zero_zero_rat
=> hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_178),B_150) != zero_zero_rat ) ) ).
fof(fact_1032_mult__eq__0__iff,axiom,
! [A,B] :
( ( is_int(A)
& is_int(B) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B) = zero_zero_int
<=> ( A = zero_zero_int
| B = zero_zero_int ) ) ) ).
fof(fact_1033_mult__eq__0__iff,axiom,
! [A,B] :
( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),B) = zero_zero_real
<=> ( A = zero_zero_real
| B = zero_zero_real ) ) ).
fof(fact_1034_mult__eq__0__iff,axiom,
! [A,B] :
( hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),B) = zero_zero_complex
<=> ( A = zero_zero_complex
| B = zero_zero_complex ) ) ).
fof(fact_1035_mult__eq__0__iff,axiom,
! [A,B] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat
<=> ( A = zero_zero_rat
| B = zero_zero_rat ) ) ).
fof(fact_1036_mult__zero__right,axiom,
! [A_177] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_177),zero_zero_int) = zero_zero_int ).
fof(fact_1037_mult__zero__right,axiom,
! [A_177] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_177),zero_zero_nat) = zero_zero_nat ).
fof(fact_1038_mult__zero__right,axiom,
! [A_177] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_177),zero_zero_real) = zero_zero_real ).
fof(fact_1039_mult__zero__right,axiom,
! [A_177] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_177),zero_z126310315umeral) = zero_z126310315umeral ).
fof(fact_1040_mult__zero__right,axiom,
! [A_177] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_177),zero_zero_complex) = zero_zero_complex ).
fof(fact_1041_mult__zero__right,axiom,
! [A_177] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_177),zero_z891286103de_int) = zero_z891286103de_int ).
fof(fact_1042_mult__zero__right,axiom,
! [A_177] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_177),zero_zero_rat) = zero_zero_rat ).
fof(fact_1043_mult__zero__left,axiom,
! [A_176] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,zero_zero_int),A_176) = zero_zero_int ).
fof(fact_1044_mult__zero__left,axiom,
! [A_176] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A_176) = zero_zero_nat ).
fof(fact_1045_mult__zero__left,axiom,
! [A_176] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),A_176) = zero_zero_real ).
fof(fact_1046_mult__zero__left,axiom,
! [A_176] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,zero_z126310315umeral),A_176) = zero_z126310315umeral ).
fof(fact_1047_mult__zero__left,axiom,
! [A_176] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),A_176) = zero_zero_complex ).
fof(fact_1048_mult__zero__left,axiom,
! [A_176] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,zero_z891286103de_int),A_176) = zero_z891286103de_int ).
fof(fact_1049_mult__zero__left,axiom,
! [A_176] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A_176) = zero_zero_rat ).
fof(fact_1050_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
fof(fact_1051_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
fof(fact_1052_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
fof(fact_1053_zero__neq__one,axiom,
zero_z126310315umeral != one_on1645066479umeral ).
fof(fact_1054_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
fof(fact_1055_zero__neq__one,axiom,
zero_z891286103de_int != one_on1684967323de_int ).
fof(fact_1056_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
fof(fact_1057_one__neq__zero,axiom,
one_one_int != zero_zero_int ).
fof(fact_1058_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
fof(fact_1059_one__neq__zero,axiom,
one_one_real != zero_zero_real ).
fof(fact_1060_one__neq__zero,axiom,
one_on1645066479umeral != zero_z126310315umeral ).
fof(fact_1061_one__neq__zero,axiom,
one_one_complex != zero_zero_complex ).
fof(fact_1062_one__neq__zero,axiom,
one_on1684967323de_int != zero_z891286103de_int ).
fof(fact_1063_one__neq__zero,axiom,
one_one_rat != zero_zero_rat ).
fof(fact_1064_combine__common__factor,axiom,
! [A_175,E_3,B_149,C_87] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_175),E_3)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_149),E_3)),C_87)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_175),B_149)),E_3)),C_87) ).
fof(fact_1065_combine__common__factor,axiom,
! [A_175,E_3,B_149,C_87] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_175),E_3)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_149),E_3)),C_87)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_175),B_149)),E_3)),C_87) ).
fof(fact_1066_combine__common__factor,axiom,
! [A_175,E_3,B_149,C_87] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_175),E_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_149),E_3)),C_87)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_175),B_149)),E_3)),C_87) ).
fof(fact_1067_combine__common__factor,axiom,
! [A_175,E_3,B_149,C_87] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_175),E_3)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_149),E_3)),C_87)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_175),B_149)),E_3)),C_87) ).
fof(fact_1068_combine__common__factor,axiom,
! [A_175,E_3,B_149,C_87] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_175),E_3)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_149),E_3)),C_87)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_175),B_149)),E_3)),C_87) ).
fof(fact_1069_combine__common__factor,axiom,
! [A_175,E_3,B_149,C_87] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_175),E_3)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_149),E_3)),C_87)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_175),B_149)),E_3)),C_87) ).
fof(fact_1070_combine__common__factor,axiom,
! [A_175,E_3,B_149,C_87] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_175),E_3)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_149),E_3)),C_87)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_175),B_149)),E_3)),C_87) ).
fof(fact_1071_comm__semiring__class_Odistrib,axiom,
! [A_174,B_148,C_86] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_174),B_148)),C_86) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_174),C_86)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_148),C_86)) ).
fof(fact_1072_comm__semiring__class_Odistrib,axiom,
! [A_174,B_148,C_86] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_174),B_148)),C_86) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_174),C_86)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_148),C_86)) ).
fof(fact_1073_comm__semiring__class_Odistrib,axiom,
! [A_174,B_148,C_86] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_174),B_148)),C_86) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_174),C_86)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_148),C_86)) ).
fof(fact_1074_comm__semiring__class_Odistrib,axiom,
! [A_174,B_148,C_86] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_174),B_148)),C_86) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_174),C_86)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_148),C_86)) ).
fof(fact_1075_comm__semiring__class_Odistrib,axiom,
! [A_174,B_148,C_86] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_174),B_148)),C_86) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_174),C_86)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_148),C_86)) ).
fof(fact_1076_comm__semiring__class_Odistrib,axiom,
! [A_174,B_148,C_86] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_174),B_148)),C_86) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_174),C_86)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_148),C_86)) ).
fof(fact_1077_comm__semiring__class_Odistrib,axiom,
! [A_174,B_148,C_86] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_174),B_148)),C_86) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_174),C_86)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_148),C_86)) ).
fof(fact_1078_dvd__0__left,axiom,
! [A_173] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,zero_zero_real),A_173))
=> A_173 = zero_zero_real ) ).
fof(fact_1079_dvd__0__left,axiom,
! [A_173] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,zero_z126310315umeral),A_173))
=> A_173 = zero_z126310315umeral ) ).
fof(fact_1080_dvd__0__left,axiom,
! [A_173] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,zero_zero_complex),A_173))
=> A_173 = zero_zero_complex ) ).
fof(fact_1081_dvd__0__left,axiom,
! [A_173] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,zero_z891286103de_int),A_173))
=> A_173 = zero_z891286103de_int ) ).
fof(fact_1082_dvd__0__left,axiom,
! [A_173] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,zero_zero_rat),A_173))
=> A_173 = zero_zero_rat ) ).
fof(fact_1083_dvd__0__left,axiom,
! [A_173] :
( is_int(A_173)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,zero_zero_int),A_173))
=> A_173 = zero_zero_int ) ) ).
fof(fact_1084_dvd__0__left,axiom,
! [A_173] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_173))
=> A_173 = zero_zero_nat ) ).
fof(fact_1085_dvd__mult__right,axiom,
! [A_172,B_147,C_85] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_172),B_147)),C_85))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,B_147),C_85)) ) ).
fof(fact_1086_dvd__mult__right,axiom,
! [A_172,B_147,C_85] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_172),B_147)),C_85))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_147),C_85)) ) ).
fof(fact_1087_dvd__mult__right,axiom,
! [A_172,B_147,C_85] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_172),B_147)),C_85))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,B_147),C_85)) ) ).
fof(fact_1088_dvd__mult__right,axiom,
! [A_172,B_147,C_85] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_172),B_147)),C_85))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B_147),C_85)) ) ).
fof(fact_1089_dvd__mult__right,axiom,
! [A_172,B_147,C_85] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_172),B_147)),C_85))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,B_147),C_85)) ) ).
fof(fact_1090_dvd__mult__right,axiom,
! [A_172,B_147,C_85] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_172),B_147)),C_85))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_147),C_85)) ) ).
fof(fact_1091_dvd__mult__right,axiom,
! [A_172,B_147,C_85] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_172),B_147)),C_85))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_147),C_85)) ) ).
fof(fact_1092_dvd__mult__left,axiom,
! [A_171,B_146,C_84] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_171),B_146)),C_84))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_171),C_84)) ) ).
fof(fact_1093_dvd__mult__left,axiom,
! [A_171,B_146,C_84] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_171),B_146)),C_84))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_171),C_84)) ) ).
fof(fact_1094_dvd__mult__left,axiom,
! [A_171,B_146,C_84] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_171),B_146)),C_84))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_171),C_84)) ) ).
fof(fact_1095_dvd__mult__left,axiom,
! [A_171,B_146,C_84] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_171),B_146)),C_84))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_171),C_84)) ) ).
fof(fact_1096_dvd__mult__left,axiom,
! [A_171,B_146,C_84] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_171),B_146)),C_84))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_171),C_84)) ) ).
fof(fact_1097_dvd__mult__left,axiom,
! [A_171,B_146,C_84] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_171),B_146)),C_84))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_171),C_84)) ) ).
fof(fact_1098_dvd__mult__left,axiom,
! [A_171,B_146,C_84] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_171),B_146)),C_84))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_171),C_84)) ) ).
fof(fact_1099_dvdI,axiom,
! [A_170,B_145,K_7] :
( A_170 = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_145),K_7)
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,B_145),A_170)) ) ).
fof(fact_1100_dvdI,axiom,
! [A_170,B_145,K_7] :
( A_170 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_145),K_7)
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_145),A_170)) ) ).
fof(fact_1101_dvdI,axiom,
! [A_170,B_145,K_7] :
( A_170 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_145),K_7)
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,B_145),A_170)) ) ).
fof(fact_1102_dvdI,axiom,
! [A_170,B_145,K_7] :
( A_170 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_145),K_7)
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B_145),A_170)) ) ).
fof(fact_1103_dvdI,axiom,
! [A_170,B_145,K_7] :
( A_170 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_145),K_7)
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,B_145),A_170)) ) ).
fof(fact_1104_dvdI,axiom,
! [A_170,B_145,K_7] :
( A_170 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_145),K_7)
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_145),A_170)) ) ).
fof(fact_1105_dvdI,axiom,
! [A_170,B_145,K_7] :
( A_170 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_145),K_7)
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_145),A_170)) ) ).
fof(fact_1106_mult__dvd__mono,axiom,
! [C_83,D_16,A_169,B_144] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_169),B_144))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,C_83),D_16))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_169),C_83)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_144),D_16))) ) ) ).
fof(fact_1107_mult__dvd__mono,axiom,
! [C_83,D_16,A_169,B_144] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_169),B_144))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,C_83),D_16))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_169),C_83)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_144),D_16))) ) ) ).
fof(fact_1108_mult__dvd__mono,axiom,
! [C_83,D_16,A_169,B_144] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_169),B_144))
=> ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,C_83),D_16))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_169),C_83)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_144),D_16))) ) ) ).
fof(fact_1109_mult__dvd__mono,axiom,
! [C_83,D_16,A_169,B_144] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_169),B_144))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,C_83),D_16))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_169),C_83)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_144),D_16))) ) ) ).
fof(fact_1110_mult__dvd__mono,axiom,
! [C_83,D_16,A_169,B_144] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_169),B_144))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,C_83),D_16))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_169),C_83)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_144),D_16))) ) ) ).
fof(fact_1111_mult__dvd__mono,axiom,
! [C_83,D_16,A_169,B_144] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_169),B_144))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C_83),D_16))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_169),C_83)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_144),D_16))) ) ) ).
fof(fact_1112_mult__dvd__mono,axiom,
! [C_83,D_16,A_169,B_144] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_169),B_144))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_83),D_16))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_169),C_83)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_144),D_16))) ) ) ).
fof(fact_1113_dvd__mult,axiom,
! [B_143,A_168,C_82] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_168),C_82))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_168),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_143),C_82))) ) ).
fof(fact_1114_dvd__mult,axiom,
! [B_143,A_168,C_82] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_168),C_82))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_168),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_143),C_82))) ) ).
fof(fact_1115_dvd__mult,axiom,
! [B_143,A_168,C_82] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_168),C_82))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_168),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_143),C_82))) ) ).
fof(fact_1116_dvd__mult,axiom,
! [B_143,A_168,C_82] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_168),C_82))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_168),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_143),C_82))) ) ).
fof(fact_1117_dvd__mult,axiom,
! [B_143,A_168,C_82] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_168),C_82))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_168),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_143),C_82))) ) ).
fof(fact_1118_dvd__mult,axiom,
! [B_143,A_168,C_82] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_168),C_82))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_168),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_143),C_82))) ) ).
fof(fact_1119_dvd__mult,axiom,
! [B_143,A_168,C_82] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_168),C_82))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_168),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_143),C_82))) ) ).
fof(fact_1120_dvd__mult2,axiom,
! [C_81,A_167,B_142] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_167),B_142))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_167),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_142),C_81))) ) ).
fof(fact_1121_dvd__mult2,axiom,
! [C_81,A_167,B_142] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_167),B_142))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_167),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_142),C_81))) ) ).
fof(fact_1122_dvd__mult2,axiom,
! [C_81,A_167,B_142] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_167),B_142))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_167),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_142),C_81))) ) ).
fof(fact_1123_dvd__mult2,axiom,
! [C_81,A_167,B_142] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_167),B_142))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_167),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_142),C_81))) ) ).
fof(fact_1124_dvd__mult2,axiom,
! [C_81,A_167,B_142] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_167),B_142))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_167),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_142),C_81))) ) ).
fof(fact_1125_dvd__mult2,axiom,
! [C_81,A_167,B_142] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_167),B_142))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_167),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_142),C_81))) ) ).
fof(fact_1126_dvd__mult2,axiom,
! [C_81,A_167,B_142] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_167),B_142))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_167),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_142),C_81))) ) ).
fof(fact_1127_dvd__triv__right,axiom,
! [A_166,B_141] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_166),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_141),A_166))) ).
fof(fact_1128_dvd__triv__right,axiom,
! [A_166,B_141] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_166),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_141),A_166))) ).
fof(fact_1129_dvd__triv__right,axiom,
! [A_166,B_141] : hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_166),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_141),A_166))) ).
fof(fact_1130_dvd__triv__right,axiom,
! [A_166,B_141] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_166),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_141),A_166))) ).
fof(fact_1131_dvd__triv__right,axiom,
! [A_166,B_141] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_166),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_141),A_166))) ).
fof(fact_1132_dvd__triv__right,axiom,
! [A_166,B_141] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_166),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_141),A_166))) ).
fof(fact_1133_dvd__triv__right,axiom,
! [A_166,B_141] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_166),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_141),A_166))) ).
fof(fact_1134_dvd__triv__left,axiom,
! [A_165,B_140] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_165),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_165),B_140))) ).
fof(fact_1135_dvd__triv__left,axiom,
! [A_165,B_140] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_165),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_165),B_140))) ).
fof(fact_1136_dvd__triv__left,axiom,
! [A_165,B_140] : hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_165),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_165),B_140))) ).
fof(fact_1137_dvd__triv__left,axiom,
! [A_165,B_140] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_165),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_165),B_140))) ).
fof(fact_1138_dvd__triv__left,axiom,
! [A_165,B_140] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_165),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_165),B_140))) ).
fof(fact_1139_dvd__triv__left,axiom,
! [A_165,B_140] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_165),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_165),B_140))) ).
fof(fact_1140_dvd__triv__left,axiom,
! [A_165,B_140] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_165),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_165),B_140))) ).
fof(fact_1141_dvd__add,axiom,
! [C_80,A_164,B_139] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_164),B_139))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_164),C_80))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_164),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_139),C_80))) ) ) ).
fof(fact_1142_dvd__add,axiom,
! [C_80,A_164,B_139] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_164),B_139))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_164),C_80))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_164),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_139),C_80))) ) ) ).
fof(fact_1143_dvd__add,axiom,
! [C_80,A_164,B_139] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_164),B_139))
=> ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_164),C_80))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_164),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_139),C_80))) ) ) ).
fof(fact_1144_dvd__add,axiom,
! [C_80,A_164,B_139] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_164),B_139))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_164),C_80))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_164),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_139),C_80))) ) ) ).
fof(fact_1145_dvd__add,axiom,
! [C_80,A_164,B_139] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_164),B_139))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_164),C_80))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_164),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_139),C_80))) ) ) ).
fof(fact_1146_dvd__add,axiom,
! [C_80,A_164,B_139] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_164),B_139))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_164),C_80))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_164),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_139),C_80))) ) ) ).
fof(fact_1147_dvd__add,axiom,
! [C_80,A_164,B_139] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_164),B_139))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_164),C_80))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_164),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_139),C_80))) ) ) ).
fof(fact_1148_one__dvd,axiom,
! [A_163] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,one_one_real),A_163)) ).
fof(fact_1149_one__dvd,axiom,
! [A_163] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,one_on1645066479umeral),A_163)) ).
fof(fact_1150_one__dvd,axiom,
! [A_163] : hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,one_one_complex),A_163)) ).
fof(fact_1151_one__dvd,axiom,
! [A_163] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,one_on1684967323de_int),A_163)) ).
fof(fact_1152_one__dvd,axiom,
! [A_163] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,one_one_rat),A_163)) ).
fof(fact_1153_one__dvd,axiom,
! [A_163] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,one_one_int),A_163)) ).
fof(fact_1154_one__dvd,axiom,
! [A_163] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_163)) ).
fof(fact_1155_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)) ).
fof(fact_1156_Euler_Oaux2,axiom,
! [B_2,A_3,C_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),C_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),C_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),B_2))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),A_3)) ) ) ) ).
fof(fact_1157_IntPrimes_Ozcong__zero,axiom,
! [A,B] :
( ( is_int(A)
& is_int(B) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),zero_zero_int))
<=> A = B ) ) ).
fof(fact_1158_zcong__id,axiom,
! [M] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(M),zero_zero_int),M)) ).
fof(fact_1159_zcong__1,axiom,
! [A_3,B_2] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),one_one_int)) ).
fof(fact_1160_zcong__zmult,axiom,
! [C_1,D_3,A_3,B_2,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),D_3)),M)) ) ) ).
fof(fact_1161_zcong__scalar2,axiom,
! [K_2,A_3,B_2,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),A_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),B_2)),M)) ) ).
fof(fact_1162_zcong__scalar,axiom,
! [K_2,A_3,B_2,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),K_2)),M)) ) ).
fof(fact_1163_zcong__zmult__prop2,axiom,
! [C,D,A,B,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D),A)),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D),B)),Ma)) ) ) ).
fof(fact_1164_zcong__zmult__prop1,axiom,
! [C,D,A,B,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D)),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D)),Ma)) ) ) ).
fof(fact_1165_zcong__zmult__self,axiom,
! [A_3,M,B_2] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),M)),M)) ).
fof(fact_1166_zcong__zadd,axiom,
! [C_1,D_3,A_3,B_2,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),D_3)),M)) ) ) ).
fof(fact_1167_zcong__shift,axiom,
! [C_1,A_3,B_2,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),C_1)),M)) ) ).
fof(fact_1168_zcong__zpower,axiom,
! [Z_1,X_1,Y_1,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),Y_1),M))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),Z_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_1),Z_1)),M)) ) ).
fof(fact_1169_power__m1__even,axiom,
! [N_33] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_33)) = one_one_int ).
fof(fact_1170_power__m1__even,axiom,
! [N_33] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_33)) = one_one_real ).
fof(fact_1171_power__m1__even,axiom,
! [N_33] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_int_complex(number528085621omplex,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_33)) = one_one_complex ).
fof(fact_1172_power__m1__even,axiom,
! [N_33] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_int_rat(number_number_of_rat,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_33)) = one_one_rat ).
fof(fact_1173_split__mult__neg__le,axiom,
! [B_138,A_162] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_162))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_138),zero_zero_int)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_162),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_138)) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_162),B_138)),zero_zero_int)) ) ).
fof(fact_1174_split__mult__neg__le,axiom,
! [B_138,A_162] :
( ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_162))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_138),zero_zero_nat)) )
| ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_162),zero_zero_nat))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_138)) ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_162),B_138)),zero_zero_nat)) ) ).
fof(fact_1175_split__mult__neg__le,axiom,
! [B_138,A_162] :
( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_162))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_138),zero_zero_real)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_162),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_138)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_162),B_138)),zero_zero_real)) ) ).
fof(fact_1176_split__mult__neg__le,axiom,
! [B_138,A_162] :
( ( ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_162))
& hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_138),zero_z126310315umeral)) )
| ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_162),zero_z126310315umeral))
& hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_138)) ) )
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_162),B_138)),zero_z126310315umeral)) ) ).
fof(fact_1177_split__mult__neg__le,axiom,
! [B_138,A_162] :
( ( ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_162))
& hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_138),zero_z891286103de_int)) )
| ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_162),zero_z891286103de_int))
& hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_138)) ) )
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_162),B_138)),zero_z891286103de_int)) ) ).
fof(fact_1178_split__mult__neg__le,axiom,
! [B_138,A_162] :
( ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_162))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_138),zero_zero_rat)) )
| ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_162),zero_zero_rat))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_138)) ) )
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_162),B_138)),zero_zero_rat)) ) ).
fof(fact_1179_split__mult__pos__le,axiom,
! [B_137,A_161] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_161))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_137)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_161),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_137),zero_zero_int)) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_161),B_137))) ) ).
fof(fact_1180_split__mult__pos__le,axiom,
! [B_137,A_161] :
( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_161))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_137)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_161),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_137),zero_zero_real)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_161),B_137))) ) ).
fof(fact_1181_split__mult__pos__le,axiom,
! [B_137,A_161] :
( ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_161))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_137)) )
| ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_161),zero_zero_rat))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_137),zero_zero_rat)) ) )
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_161),B_137))) ) ).
fof(fact_1182_mult__mono,axiom,
! [C_79,D_15,A_160,B_136] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_160),B_136))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_79),D_15))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_136))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_79))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_160),C_79)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_136),D_15))) ) ) ) ) ).
fof(fact_1183_mult__mono,axiom,
! [C_79,D_15,A_160,B_136] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_160),B_136))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_79),D_15))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_136))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_79))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_160),C_79)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_136),D_15))) ) ) ) ) ).
fof(fact_1184_mult__mono,axiom,
! [C_79,D_15,A_160,B_136] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_160),B_136))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_79),D_15))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_136))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_79))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_160),C_79)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_136),D_15))) ) ) ) ) ).
fof(fact_1185_mult__mono,axiom,
! [C_79,D_15,A_160,B_136] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_160),B_136))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_79),D_15))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_136))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_79))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_160),C_79)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_136),D_15))) ) ) ) ) ).
fof(fact_1186_mult__mono,axiom,
! [C_79,D_15,A_160,B_136] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_160),B_136))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_79),D_15))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_136))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_79))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_160),C_79)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_136),D_15))) ) ) ) ) ).
fof(fact_1187_mult__mono,axiom,
! [C_79,D_15,A_160,B_136] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_160),B_136))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_79),D_15))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_136))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_79))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_160),C_79)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_136),D_15))) ) ) ) ) ).
fof(fact_1188_mult__mono_H,axiom,
! [C_78,D_14,A_159,B_135] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_159),B_135))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_78),D_14))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_159))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_78))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_159),C_78)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_135),D_14))) ) ) ) ) ).
fof(fact_1189_mult__mono_H,axiom,
! [C_78,D_14,A_159,B_135] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_159),B_135))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_78),D_14))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_159))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_78))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_159),C_78)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_135),D_14))) ) ) ) ) ).
fof(fact_1190_mult__mono_H,axiom,
! [C_78,D_14,A_159,B_135] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_159),B_135))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_78),D_14))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_159))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_78))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_159),C_78)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_135),D_14))) ) ) ) ) ).
fof(fact_1191_mult__mono_H,axiom,
! [C_78,D_14,A_159,B_135] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_159),B_135))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_78),D_14))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_159))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_78))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_159),C_78)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_135),D_14))) ) ) ) ) ).
fof(fact_1192_mult__mono_H,axiom,
! [C_78,D_14,A_159,B_135] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_159),B_135))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_78),D_14))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_159))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_78))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_159),C_78)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_135),D_14))) ) ) ) ) ).
fof(fact_1193_mult__mono_H,axiom,
! [C_78,D_14,A_159,B_135] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_159),B_135))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_78),D_14))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_159))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_78))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_159),C_78)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_135),D_14))) ) ) ) ) ).
fof(fact_1194_mult__left__mono__neg,axiom,
! [C_77,B_134,A_158] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_134),A_158))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_77),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_77),A_158)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_77),B_134))) ) ) ).
fof(fact_1195_mult__left__mono__neg,axiom,
! [C_77,B_134,A_158] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_134),A_158))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_77),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_77),A_158)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_77),B_134))) ) ) ).
fof(fact_1196_mult__left__mono__neg,axiom,
! [C_77,B_134,A_158] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_134),A_158))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_77),zero_zero_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_77),A_158)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_77),B_134))) ) ) ).
fof(fact_1197_mult__right__mono__neg,axiom,
! [C_76,B_133,A_157] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_133),A_157))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_76),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_157),C_76)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_133),C_76))) ) ) ).
fof(fact_1198_mult__right__mono__neg,axiom,
! [C_76,B_133,A_157] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_133),A_157))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_76),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_157),C_76)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_133),C_76))) ) ) ).
fof(fact_1199_mult__right__mono__neg,axiom,
! [C_76,B_133,A_157] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_133),A_157))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_76),zero_zero_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_157),C_76)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_133),C_76))) ) ) ).
fof(fact_1200_comm__mult__left__mono,axiom,
! [C_75,A_156,B_132] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_156),B_132))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_75))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_75),A_156)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_75),B_132))) ) ) ).
fof(fact_1201_comm__mult__left__mono,axiom,
! [C_75,A_156,B_132] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_156),B_132))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_75))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_75),A_156)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_75),B_132))) ) ) ).
fof(fact_1202_comm__mult__left__mono,axiom,
! [C_75,A_156,B_132] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_156),B_132))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_75))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_75),A_156)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_75),B_132))) ) ) ).
fof(fact_1203_comm__mult__left__mono,axiom,
! [C_75,A_156,B_132] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_156),B_132))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_75))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_75),A_156)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_75),B_132))) ) ) ).
fof(fact_1204_comm__mult__left__mono,axiom,
! [C_75,A_156,B_132] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_156),B_132))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_75))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_75),A_156)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_75),B_132))) ) ) ).
fof(fact_1205_comm__mult__left__mono,axiom,
! [C_75,A_156,B_132] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_156),B_132))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_75))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_75),A_156)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_75),B_132))) ) ) ).
fof(fact_1206_mult__left__mono,axiom,
! [C_74,A_155,B_131] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_155),B_131))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_74))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_74),A_155)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_74),B_131))) ) ) ).
fof(fact_1207_mult__left__mono,axiom,
! [C_74,A_155,B_131] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_155),B_131))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_74))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_74),A_155)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_74),B_131))) ) ) ).
fof(fact_1208_mult__left__mono,axiom,
! [C_74,A_155,B_131] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_155),B_131))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_74))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_74),A_155)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_74),B_131))) ) ) ).
fof(fact_1209_mult__left__mono,axiom,
! [C_74,A_155,B_131] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_155),B_131))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_74))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_74),A_155)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_74),B_131))) ) ) ).
fof(fact_1210_mult__left__mono,axiom,
! [C_74,A_155,B_131] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_155),B_131))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_74))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_74),A_155)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_74),B_131))) ) ) ).
fof(fact_1211_mult__left__mono,axiom,
! [C_74,A_155,B_131] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_155),B_131))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_74))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_74),A_155)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_74),B_131))) ) ) ).
fof(fact_1212_mult__right__mono,axiom,
! [C_73,A_154,B_130] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_154),B_130))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_73))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_154),C_73)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_130),C_73))) ) ) ).
fof(fact_1213_mult__right__mono,axiom,
! [C_73,A_154,B_130] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_154),B_130))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_73))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_154),C_73)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_130),C_73))) ) ) ).
fof(fact_1214_mult__right__mono,axiom,
! [C_73,A_154,B_130] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_154),B_130))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_73))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_154),C_73)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_130),C_73))) ) ) ).
fof(fact_1215_mult__right__mono,axiom,
! [C_73,A_154,B_130] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_154),B_130))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_73))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_154),C_73)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_130),C_73))) ) ) ).
fof(fact_1216_mult__right__mono,axiom,
! [C_73,A_154,B_130] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_154),B_130))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_73))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_154),C_73)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_130),C_73))) ) ) ).
fof(fact_1217_mult__right__mono,axiom,
! [C_73,A_154,B_130] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_154),B_130))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_73))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_154),C_73)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_130),C_73))) ) ) ).
fof(fact_1218_mult__nonpos__nonpos,axiom,
! [B_129,A_153] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_153),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_129),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_153),B_129))) ) ) ).
fof(fact_1219_mult__nonpos__nonpos,axiom,
! [B_129,A_153] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_153),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_129),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_153),B_129))) ) ) ).
fof(fact_1220_mult__nonpos__nonpos,axiom,
! [B_129,A_153] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_153),zero_zero_rat))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_129),zero_zero_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_153),B_129))) ) ) ).
fof(fact_1221_mult__nonpos__nonneg,axiom,
! [B_128,A_152] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_152),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_128))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_152),B_128)),zero_zero_int)) ) ) ).
fof(fact_1222_mult__nonpos__nonneg,axiom,
! [B_128,A_152] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_152),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_128))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_152),B_128)),zero_zero_nat)) ) ) ).
fof(fact_1223_mult__nonpos__nonneg,axiom,
! [B_128,A_152] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_152),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_128))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_152),B_128)),zero_zero_real)) ) ) ).
fof(fact_1224_mult__nonpos__nonneg,axiom,
! [B_128,A_152] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_152),zero_z126310315umeral))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_128))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_152),B_128)),zero_z126310315umeral)) ) ) ).
fof(fact_1225_mult__nonpos__nonneg,axiom,
! [B_128,A_152] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_152),zero_z891286103de_int))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_128))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_152),B_128)),zero_z891286103de_int)) ) ) ).
fof(fact_1226_mult__nonpos__nonneg,axiom,
! [B_128,A_152] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_152),zero_zero_rat))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_128))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_152),B_128)),zero_zero_rat)) ) ) ).
fof(fact_1227_mult__nonneg__nonpos2,axiom,
! [B_127,A_151] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_151))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_127),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_127),A_151)),zero_zero_int)) ) ) ).
fof(fact_1228_mult__nonneg__nonpos2,axiom,
! [B_127,A_151] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_151))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_127),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_127),A_151)),zero_zero_nat)) ) ) ).
fof(fact_1229_mult__nonneg__nonpos2,axiom,
! [B_127,A_151] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_151))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_127),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_127),A_151)),zero_zero_real)) ) ) ).
fof(fact_1230_mult__nonneg__nonpos2,axiom,
! [B_127,A_151] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_151))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_127),zero_z126310315umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_127),A_151)),zero_z126310315umeral)) ) ) ).
fof(fact_1231_mult__nonneg__nonpos2,axiom,
! [B_127,A_151] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_151))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_127),zero_z891286103de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_127),A_151)),zero_z891286103de_int)) ) ) ).
fof(fact_1232_mult__nonneg__nonpos2,axiom,
! [B_127,A_151] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_151))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_127),zero_zero_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_127),A_151)),zero_zero_rat)) ) ) ).
fof(fact_1233_mult__nonneg__nonpos,axiom,
! [B_126,A_150] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_150))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_126),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_150),B_126)),zero_zero_int)) ) ) ).
fof(fact_1234_mult__nonneg__nonpos,axiom,
! [B_126,A_150] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_150))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_126),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_150),B_126)),zero_zero_nat)) ) ) ).
fof(fact_1235_mult__nonneg__nonpos,axiom,
! [B_126,A_150] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_150))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_126),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_150),B_126)),zero_zero_real)) ) ) ).
fof(fact_1236_mult__nonneg__nonpos,axiom,
! [B_126,A_150] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_150))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_126),zero_z126310315umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_150),B_126)),zero_z126310315umeral)) ) ) ).
fof(fact_1237_mult__nonneg__nonpos,axiom,
! [B_126,A_150] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_150))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_126),zero_z891286103de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_150),B_126)),zero_z891286103de_int)) ) ) ).
fof(fact_1238_mult__nonneg__nonpos,axiom,
! [B_126,A_150] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_150))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_126),zero_zero_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_150),B_126)),zero_zero_rat)) ) ) ).
fof(fact_1239_mult__nonneg__nonneg,axiom,
! [B_125,A_149] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_149))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_125))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_149),B_125))) ) ) ).
fof(fact_1240_mult__nonneg__nonneg,axiom,
! [B_125,A_149] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_149))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_125))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_149),B_125))) ) ) ).
fof(fact_1241_mult__nonneg__nonneg,axiom,
! [B_125,A_149] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_149))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_125))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_149),B_125))) ) ) ).
fof(fact_1242_mult__nonneg__nonneg,axiom,
! [B_125,A_149] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_149))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_125))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_149),B_125))) ) ) ).
fof(fact_1243_mult__nonneg__nonneg,axiom,
! [B_125,A_149] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_149))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_125))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_149),B_125))) ) ) ).
fof(fact_1244_mult__nonneg__nonneg,axiom,
! [B_125,A_149] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_149))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_125))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_149),B_125))) ) ) ).
fof(fact_1245_mult__le__0__iff,axiom,
! [A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B)),zero_zero_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_eq_int,B),zero_zero_int)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B)) ) ) ) ).
fof(fact_1246_mult__le__0__iff,axiom,
! [A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),B)),zero_zero_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,B),zero_zero_real)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B)) ) ) ) ).
fof(fact_1247_mult__le__0__iff,axiom,
! [A,B] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat))
<=> ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B),zero_zero_rat)) )
| ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),zero_zero_rat))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ) ) ).
fof(fact_1248_zero__le__mult__iff,axiom,
! [A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) ) ) ) ).
fof(fact_1249_zero__le__mult__iff,axiom,
! [A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),B)))
<=> ( ( 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_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),zero_zero_real)) ) ) ) ).
fof(fact_1250_zero__le__mult__iff,axiom,
! [A,B] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),B)))
<=> ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B)) )
| ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),zero_zero_rat))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B),zero_zero_rat)) ) ) ) ).
fof(fact_1251_zero__le__square,axiom,
! [A_148] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_148),A_148))) ).
fof(fact_1252_zero__le__square,axiom,
! [A_148] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_148),A_148))) ).
fof(fact_1253_zero__le__square,axiom,
! [A_148] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_148),A_148))) ).
fof(fact_1254_mult__strict__left__mono__neg,axiom,
! [C_72,B_124,A_147] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_124),A_147))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_72),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_72),A_147)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_72),B_124))) ) ) ).
fof(fact_1255_mult__strict__left__mono__neg,axiom,
! [C_72,B_124,A_147] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_124),A_147))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_72),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_72),A_147)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_72),B_124))) ) ) ).
fof(fact_1256_mult__strict__left__mono__neg,axiom,
! [C_72,B_124,A_147] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_124),A_147))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_72),zero_zero_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_72),A_147)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_72),B_124))) ) ) ).
fof(fact_1257_mult__strict__right__mono__neg,axiom,
! [C_71,B_123,A_146] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_123),A_146))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_71),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_146),C_71)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_123),C_71))) ) ) ).
fof(fact_1258_mult__strict__right__mono__neg,axiom,
! [C_71,B_123,A_146] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_123),A_146))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_71),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_146),C_71)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_123),C_71))) ) ) ).
fof(fact_1259_mult__strict__right__mono__neg,axiom,
! [C_71,B_123,A_146] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_123),A_146))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_71),zero_zero_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_146),C_71)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_123),C_71))) ) ) ).
fof(fact_1260_comm__mult__strict__left__mono,axiom,
! [C_70,A_145,B_122] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_145),B_122))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_70))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_70),A_145)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_70),B_122))) ) ) ).
fof(fact_1261_comm__mult__strict__left__mono,axiom,
! [C_70,A_145,B_122] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_145),B_122))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_70))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_70),A_145)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_70),B_122))) ) ) ).
fof(fact_1262_comm__mult__strict__left__mono,axiom,
! [C_70,A_145,B_122] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_145),B_122))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_70))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_70),A_145)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_70),B_122))) ) ) ).
fof(fact_1263_comm__mult__strict__left__mono,axiom,
! [C_70,A_145,B_122] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_145),B_122))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_70))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_70),A_145)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_70),B_122))) ) ) ).
fof(fact_1264_comm__mult__strict__left__mono,axiom,
! [C_70,A_145,B_122] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_145),B_122))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_70))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_70),A_145)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_70),B_122))) ) ) ).
fof(fact_1265_comm__mult__strict__left__mono,axiom,
! [C_70,A_145,B_122] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_145),B_122))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_70))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_70),A_145)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_70),B_122))) ) ) ).
fof(fact_1266_mult__strict__left__mono,axiom,
! [C_69,A_144,B_121] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_144),B_121))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_69))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_69),A_144)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_69),B_121))) ) ) ).
fof(fact_1267_mult__strict__left__mono,axiom,
! [C_69,A_144,B_121] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_144),B_121))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_69))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_69),A_144)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_69),B_121))) ) ) ).
fof(fact_1268_mult__strict__left__mono,axiom,
! [C_69,A_144,B_121] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_144),B_121))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_69))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_69),A_144)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_69),B_121))) ) ) ).
fof(fact_1269_mult__strict__left__mono,axiom,
! [C_69,A_144,B_121] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_144),B_121))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_69))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_69),A_144)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_69),B_121))) ) ) ).
fof(fact_1270_mult__strict__left__mono,axiom,
! [C_69,A_144,B_121] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_144),B_121))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_69))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_69),A_144)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_69),B_121))) ) ) ).
fof(fact_1271_mult__strict__left__mono,axiom,
! [C_69,A_144,B_121] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_144),B_121))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_69))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_69),A_144)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_69),B_121))) ) ) ).
fof(fact_1272_mult__strict__right__mono,axiom,
! [C_68,A_143,B_120] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_143),B_120))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_68))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_143),C_68)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_120),C_68))) ) ) ).
fof(fact_1273_mult__strict__right__mono,axiom,
! [C_68,A_143,B_120] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_143),B_120))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_68))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_143),C_68)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_120),C_68))) ) ) ).
fof(fact_1274_mult__strict__right__mono,axiom,
! [C_68,A_143,B_120] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_143),B_120))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_68))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_143),C_68)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_120),C_68))) ) ) ).
fof(fact_1275_mult__strict__right__mono,axiom,
! [C_68,A_143,B_120] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_143),B_120))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_68))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_143),C_68)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_120),C_68))) ) ) ).
fof(fact_1276_mult__strict__right__mono,axiom,
! [C_68,A_143,B_120] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_143),B_120))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_68))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_143),C_68)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_120),C_68))) ) ) ).
fof(fact_1277_mult__strict__right__mono,axiom,
! [C_68,A_143,B_120] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_143),B_120))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_68))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_143),C_68)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_120),C_68))) ) ) ).
fof(fact_1278_mult__neg__neg,axiom,
! [B_119,A_142] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_142),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_119),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_142),B_119))) ) ) ).
fof(fact_1279_mult__neg__neg,axiom,
! [B_119,A_142] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_142),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_119),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_142),B_119))) ) ) ).
fof(fact_1280_mult__neg__neg,axiom,
! [B_119,A_142] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_142),zero_zero_rat))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_119),zero_zero_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_142),B_119))) ) ) ).
fof(fact_1281_mult__neg__pos,axiom,
! [B_118,A_141] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_141),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_118))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_141),B_118)),zero_zero_int)) ) ) ).
fof(fact_1282_mult__neg__pos,axiom,
! [B_118,A_141] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_141),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_118))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_141),B_118)),zero_zero_nat)) ) ) ).
fof(fact_1283_mult__neg__pos,axiom,
! [B_118,A_141] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_141),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_118))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_141),B_118)),zero_zero_real)) ) ) ).
fof(fact_1284_mult__neg__pos,axiom,
! [B_118,A_141] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_141),zero_z126310315umeral))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_118))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_141),B_118)),zero_z126310315umeral)) ) ) ).
fof(fact_1285_mult__neg__pos,axiom,
! [B_118,A_141] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_141),zero_z891286103de_int))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_118))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_141),B_118)),zero_z891286103de_int)) ) ) ).
fof(fact_1286_mult__neg__pos,axiom,
! [B_118,A_141] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_141),zero_zero_rat))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_118))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_141),B_118)),zero_zero_rat)) ) ) ).
fof(fact_1287_mult__less__cancel__left__neg,axiom,
! [A,B,C] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A)) ) ) ).
fof(fact_1288_mult__less__cancel__left__neg,axiom,
! [A,B,C] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B),A)) ) ) ).
fof(fact_1289_mult__less__cancel__left__neg,axiom,
! [A,B,C] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B),A)) ) ) ).
fof(fact_1290_zero__less__mult__pos2,axiom,
! [B_117,A_140] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_117),A_140)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_140))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_117)) ) ) ).
fof(fact_1291_zero__less__mult__pos2,axiom,
! [B_117,A_140] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_117),A_140)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_140))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_117)) ) ) ).
fof(fact_1292_zero__less__mult__pos2,axiom,
! [B_117,A_140] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_117),A_140)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_140))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_117)) ) ) ).
fof(fact_1293_zero__less__mult__pos2,axiom,
! [B_117,A_140] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_117),A_140)))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_140))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_117)) ) ) ).
fof(fact_1294_zero__less__mult__pos2,axiom,
! [B_117,A_140] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_117),A_140)))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_140))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_117)) ) ) ).
fof(fact_1295_zero__less__mult__pos2,axiom,
! [B_117,A_140] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_117),A_140)))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_140))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_117)) ) ) ).
fof(fact_1296_zero__less__mult__pos,axiom,
! [A_139,B_116] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_139),B_116)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_139))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_116)) ) ) ).
fof(fact_1297_zero__less__mult__pos,axiom,
! [A_139,B_116] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_139),B_116)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_139))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_116)) ) ) ).
fof(fact_1298_zero__less__mult__pos,axiom,
! [A_139,B_116] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_139),B_116)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_139))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_116)) ) ) ).
fof(fact_1299_zero__less__mult__pos,axiom,
! [A_139,B_116] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_139),B_116)))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_139))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_116)) ) ) ).
fof(fact_1300_zero__less__mult__pos,axiom,
! [A_139,B_116] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_139),B_116)))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_139))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_116)) ) ) ).
fof(fact_1301_zero__less__mult__pos,axiom,
! [A_139,B_116] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_139),B_116)))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_139))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_116)) ) ) ).
fof(fact_1302_mult__pos__neg2,axiom,
! [B_115,A_138] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_138))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_115),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_115),A_138)),zero_zero_int)) ) ) ).
fof(fact_1303_mult__pos__neg2,axiom,
! [B_115,A_138] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_138))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_115),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_115),A_138)),zero_zero_nat)) ) ) ).
fof(fact_1304_mult__pos__neg2,axiom,
! [B_115,A_138] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_138))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_115),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_115),A_138)),zero_zero_real)) ) ) ).
fof(fact_1305_mult__pos__neg2,axiom,
! [B_115,A_138] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_138))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_115),zero_z126310315umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_115),A_138)),zero_z126310315umeral)) ) ) ).
fof(fact_1306_mult__pos__neg2,axiom,
! [B_115,A_138] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_138))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_115),zero_z891286103de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_115),A_138)),zero_z891286103de_int)) ) ) ).
fof(fact_1307_mult__pos__neg2,axiom,
! [B_115,A_138] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_138))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_115),zero_zero_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_115),A_138)),zero_zero_rat)) ) ) ).
fof(fact_1308_mult__pos__neg,axiom,
! [B_114,A_137] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_137))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_114),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_137),B_114)),zero_zero_int)) ) ) ).
fof(fact_1309_mult__pos__neg,axiom,
! [B_114,A_137] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_137))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_114),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_137),B_114)),zero_zero_nat)) ) ) ).
fof(fact_1310_mult__pos__neg,axiom,
! [B_114,A_137] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_137))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_114),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_137),B_114)),zero_zero_real)) ) ) ).
fof(fact_1311_mult__pos__neg,axiom,
! [B_114,A_137] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_137))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_114),zero_z126310315umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_137),B_114)),zero_z126310315umeral)) ) ) ).
fof(fact_1312_mult__pos__neg,axiom,
! [B_114,A_137] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_137))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_114),zero_z891286103de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_137),B_114)),zero_z891286103de_int)) ) ) ).
fof(fact_1313_mult__pos__neg,axiom,
! [B_114,A_137] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_137))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_114),zero_zero_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_137),B_114)),zero_zero_rat)) ) ) ).
fof(fact_1314_mult__pos__pos,axiom,
! [B_113,A_136] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_136))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_113))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_136),B_113))) ) ) ).
fof(fact_1315_mult__pos__pos,axiom,
! [B_113,A_136] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_136))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_113))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_136),B_113))) ) ) ).
fof(fact_1316_mult__pos__pos,axiom,
! [B_113,A_136] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_136))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_113))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_136),B_113))) ) ) ).
fof(fact_1317_mult__pos__pos,axiom,
! [B_113,A_136] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_136))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_113))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_136),B_113))) ) ) ).
fof(fact_1318_mult__pos__pos,axiom,
! [B_113,A_136] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_136))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_113))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_136),B_113))) ) ) ).
fof(fact_1319_mult__pos__pos,axiom,
! [B_113,A_136] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_136))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_113))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_136),B_113))) ) ) ).
fof(fact_1320_mult__less__cancel__left__pos,axiom,
! [A,B,C] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) ) ) ).
fof(fact_1321_mult__less__cancel__left__pos,axiom,
! [A,B,C] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) ) ) ).
fof(fact_1322_mult__less__cancel__left__pos,axiom,
! [A,B,C] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)) ) ) ).
fof(fact_1323_mult__less__cancel__left__disj,axiom,
! [C,A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A)) ) ) ) ).
fof(fact_1324_mult__less__cancel__left__disj,axiom,
! [C,A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B),A)) ) ) ) ).
fof(fact_1325_mult__less__cancel__left__disj,axiom,
! [C,A,B] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B)))
<=> ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)) )
| ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B),A)) ) ) ) ).
fof(fact_1326_mult__less__cancel__right__disj,axiom,
! [A,C,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),C)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A)) ) ) ) ).
fof(fact_1327_mult__less__cancel__right__disj,axiom,
! [A,C,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),C)))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B),A)) ) ) ) ).
fof(fact_1328_mult__less__cancel__right__disj,axiom,
! [A,C,B] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),C)))
<=> ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)) )
| ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B),A)) ) ) ) ).
fof(fact_1329_not__square__less__zero,axiom,
! [A_135] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_135),A_135)),zero_zero_int)) ).
fof(fact_1330_not__square__less__zero,axiom,
! [A_135] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_135),A_135)),zero_zero_real)) ).
fof(fact_1331_not__square__less__zero,axiom,
! [A_135] : ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_135),A_135)),zero_zero_rat)) ).
fof(fact_1332_pos__add__strict,axiom,
! [B_112,C_67,A_134] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_134))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_112),C_67))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_112),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_134),C_67))) ) ) ).
fof(fact_1333_pos__add__strict,axiom,
! [B_112,C_67,A_134] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_134))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_112),C_67))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_112),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_134),C_67))) ) ) ).
fof(fact_1334_pos__add__strict,axiom,
! [B_112,C_67,A_134] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_134))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_112),C_67))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_112),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_134),C_67))) ) ) ).
fof(fact_1335_pos__add__strict,axiom,
! [B_112,C_67,A_134] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_134))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_112),C_67))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_112),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_134),C_67))) ) ) ).
fof(fact_1336_pos__add__strict,axiom,
! [B_112,C_67,A_134] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_134))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_112),C_67))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_112),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_134),C_67))) ) ) ).
fof(fact_1337_pos__add__strict,axiom,
! [B_112,C_67,A_134] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_134))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_112),C_67))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_112),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_134),C_67))) ) ) ).
fof(fact_1338_zero__le__one,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).
fof(fact_1339_zero__le__one,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).
fof(fact_1340_zero__le__one,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).
fof(fact_1341_zero__le__one,axiom,
hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),one_on1645066479umeral)) ).
fof(fact_1342_zero__le__one,axiom,
hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),one_on1684967323de_int)) ).
fof(fact_1343_zero__le__one,axiom,
hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).
fof(fact_1344_not__one__le__zero,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),zero_zero_int)) ).
fof(fact_1345_not__one__le__zero,axiom,
~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),zero_zero_nat)) ).
fof(fact_1346_not__one__le__zero,axiom,
~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),zero_zero_real)) ).
fof(fact_1347_not__one__le__zero,axiom,
~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),zero_z126310315umeral)) ).
fof(fact_1348_not__one__le__zero,axiom,
~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),zero_z891286103de_int)) ).
fof(fact_1349_not__one__le__zero,axiom,
~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),zero_zero_rat)) ).
fof(fact_1350_zero__less__one,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
fof(fact_1351_zero__less__one,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).
fof(fact_1352_zero__less__one,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),one_one_real)) ).
fof(fact_1353_zero__less__one,axiom,
hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),one_on1645066479umeral)) ).
fof(fact_1354_zero__less__one,axiom,
hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),one_on1684967323de_int)) ).
fof(fact_1355_zero__less__one,axiom,
hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).
fof(fact_1356_not__one__less__zero,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),zero_zero_int)) ).
fof(fact_1357_not__one__less__zero,axiom,
~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).
fof(fact_1358_not__one__less__zero,axiom,
~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),zero_zero_real)) ).
fof(fact_1359_not__one__less__zero,axiom,
~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),zero_z126310315umeral)) ).
fof(fact_1360_not__one__less__zero,axiom,
~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),zero_z891286103de_int)) ).
fof(fact_1361_not__one__less__zero,axiom,
~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),zero_zero_rat)) ).
fof(fact_1362_less__1__mult,axiom,
! [N_32,M_18] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M_18))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),N_32))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_18),N_32))) ) ) ).
fof(fact_1363_less__1__mult,axiom,
! [N_32,M_18] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),M_18))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_32))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_18),N_32))) ) ) ).
fof(fact_1364_less__1__mult,axiom,
! [N_32,M_18] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),M_18))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),N_32))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,M_18),N_32))) ) ) ).
fof(fact_1365_less__1__mult,axiom,
! [N_32,M_18] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),M_18))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),N_32))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,M_18),N_32))) ) ) ).
fof(fact_1366_less__1__mult,axiom,
! [N_32,M_18] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),M_18))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),N_32))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,M_18),N_32))) ) ) ).
fof(fact_1367_less__1__mult,axiom,
! [N_32,M_18] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),M_18))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),N_32))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,M_18),N_32))) ) ) ).
fof(fact_1368_less__add__one,axiom,
! [A_133] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_133),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_133),one_one_int))) ).
fof(fact_1369_less__add__one,axiom,
! [A_133] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_133),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_133),one_one_nat))) ).
fof(fact_1370_less__add__one,axiom,
! [A_133] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_133),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_133),one_one_real))) ).
fof(fact_1371_less__add__one,axiom,
! [A_133] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_133),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_133),one_on1645066479umeral))) ).
fof(fact_1372_less__add__one,axiom,
! [A_133] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_133),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_133),one_on1684967323de_int))) ).
fof(fact_1373_less__add__one,axiom,
! [A_133] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_133),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_133),one_one_rat))) ).
fof(fact_1374_dvd__mult__cancel__left,axiom,
! [C,A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)))
<=> ( C = zero_zero_real
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A),B)) ) ) ).
fof(fact_1375_dvd__mult__cancel__left,axiom,
! [C,A,B] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,C),A)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,C),B)))
<=> ( C = zero_zero_complex
| hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A),B)) ) ) ).
fof(fact_1376_dvd__mult__cancel__left,axiom,
! [C,A,B] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B)))
<=> ( C = zero_zero_rat
| hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A),B)) ) ) ).
fof(fact_1377_dvd__mult__cancel__left,axiom,
! [C,A,B] :
( is_int(C)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)))
<=> ( C = zero_zero_int
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B)) ) ) ) ).
fof(fact_1378_dvd__mult__cancel__right,axiom,
! [A,C,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),C)))
<=> ( C = zero_zero_real
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A),B)) ) ) ).
fof(fact_1379_dvd__mult__cancel__right,axiom,
! [A,C,B] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),C)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),C)))
<=> ( C = zero_zero_complex
| hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A),B)) ) ) ).
fof(fact_1380_dvd__mult__cancel__right,axiom,
! [A,C,B] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),C)))
<=> ( C = zero_zero_rat
| hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A),B)) ) ) ).
fof(fact_1381_dvd__mult__cancel__right,axiom,
! [A,C,B] :
( is_int(C)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),C)))
<=> ( C = zero_zero_int
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B)) ) ) ) ).
fof(fact_1382_power__eq__0__iff__number__of,axiom,
! [A,W_1] :
( is_int(A)
=> ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(number_number_of_nat,W_1)) = zero_zero_int
<=> ( A = zero_zero_int
& hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat ) ) ) ).
fof(fact_1383_power__eq__0__iff__number__of,axiom,
! [A,W_1] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),hAPP_int_nat(number_number_of_nat,W_1)) = zero_zero_nat
<=> ( A = zero_zero_nat
& hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat ) ) ).
fof(fact_1384_power__eq__0__iff__number__of,axiom,
! [A,W_1] :
( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),hAPP_int_nat(number_number_of_nat,W_1)) = zero_zero_real
<=> ( A = zero_zero_real
& hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat ) ) ).
fof(fact_1385_power__eq__0__iff__number__of,axiom,
! [A,W_1] :
( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A),hAPP_int_nat(number_number_of_nat,W_1)) = zero_z126310315umeral
<=> ( A = zero_z126310315umeral
& hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat ) ) ).
fof(fact_1386_power__eq__0__iff__number__of,axiom,
! [A,W_1] :
( hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A),hAPP_int_nat(number_number_of_nat,W_1)) = zero_zero_complex
<=> ( A = zero_zero_complex
& hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat ) ) ).
fof(fact_1387_power__eq__0__iff__number__of,axiom,
! [A,W_1] :
( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A),hAPP_int_nat(number_number_of_nat,W_1)) = zero_z891286103de_int
<=> ( A = zero_z891286103de_int
& hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat ) ) ).
fof(fact_1388_power__eq__0__iff__number__of,axiom,
! [A,W_1] :
( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),hAPP_int_nat(number_number_of_nat,W_1)) = zero_zero_rat
<=> ( A = zero_zero_rat
& hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat ) ) ).
fof(fact_1389_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)) ).
fof(fact_1390_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1))) ) ) ).
fof(fact_1391_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_1))) ) ) ).
fof(fact_1392_zcong__not,axiom,
! [B_2,M,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_3))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) ) ) ) ) ).
fof(fact_1393_zcong__not__zero,axiom,
! [M,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),M))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),M)) ) ) ).
fof(fact_1394_zcong__less__eq,axiom,
! [M,Y_1,X_1] :
( ( is_int(Y_1)
& is_int(X_1) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),Y_1),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),M))
=> X_1 = Y_1 ) ) ) ) ) ) ) ).
fof(fact_1395_zdvd__bounds,axiom,
! [N_1,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),zero_zero_int))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),M)) ) ) ).
fof(fact_1396_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [N_1,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N_1))) ) ).
fof(fact_1397_zcong__iff__lin,axiom,
! [A,B,Ma] :
( is_int(B)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma))
<=> ? [K] :
( is_int(K)
& B = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),K)) ) ) ) ).
fof(fact_1398_zcong__zero__equiv__div,axiom,
! [A,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),A)) ) ).
fof(fact_1399_zcong__eq__zdvd__prop,axiom,
! [X_2,P_5] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_2),zero_zero_int),P_5))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_5),X_2)) ) ).
fof(fact_1400_zprime__zdvd__zmult__better,axiom,
! [M,N_1,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),M))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),N_1)) ) ) ) ).
fof(fact_1401_mult__le__cancel__left__pos,axiom,
! [A,B,C] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B)) ) ) ).
fof(fact_1402_mult__le__cancel__left__pos,axiom,
! [A,B,C] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) ) ) ).
fof(fact_1403_mult__le__cancel__left__pos,axiom,
! [A,B,C] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B)) ) ) ).
fof(fact_1404_mult__le__cancel__left__neg,axiom,
! [A,B,C] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) ) ) ).
fof(fact_1405_mult__le__cancel__left__neg,axiom,
! [A,B,C] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),A)) ) ) ).
fof(fact_1406_mult__le__cancel__left__neg,axiom,
! [A,B,C] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B),A)) ) ) ).
fof(fact_1407_mult__strict__mono,axiom,
! [C_66,D_13,A_132,B_111] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_132),B_111))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_66),D_13))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_111))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_66))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_132),C_66)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_111),D_13))) ) ) ) ) ).
fof(fact_1408_mult__strict__mono,axiom,
! [C_66,D_13,A_132,B_111] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_132),B_111))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_66),D_13))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_111))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_66))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_132),C_66)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_111),D_13))) ) ) ) ) ).
fof(fact_1409_mult__strict__mono,axiom,
! [C_66,D_13,A_132,B_111] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_132),B_111))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_66),D_13))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_111))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_66))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_132),C_66)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_111),D_13))) ) ) ) ) ).
fof(fact_1410_mult__strict__mono,axiom,
! [C_66,D_13,A_132,B_111] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_132),B_111))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_66),D_13))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_111))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_66))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_132),C_66)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_111),D_13))) ) ) ) ) ).
fof(fact_1411_mult__strict__mono,axiom,
! [C_66,D_13,A_132,B_111] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_132),B_111))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_66),D_13))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_111))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_66))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_132),C_66)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_111),D_13))) ) ) ) ) ).
fof(fact_1412_mult__strict__mono,axiom,
! [C_66,D_13,A_132,B_111] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_132),B_111))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_66),D_13))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_111))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_66))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_132),C_66)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_111),D_13))) ) ) ) ) ).
fof(fact_1413_mult__strict__mono_H,axiom,
! [C_65,D_12,A_131,B_110] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_131),B_110))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_65),D_12))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_131))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_65))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_131),C_65)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_110),D_12))) ) ) ) ) ).
fof(fact_1414_mult__strict__mono_H,axiom,
! [C_65,D_12,A_131,B_110] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_131),B_110))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_65),D_12))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_131))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_65))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_131),C_65)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_110),D_12))) ) ) ) ) ).
fof(fact_1415_mult__strict__mono_H,axiom,
! [C_65,D_12,A_131,B_110] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_131),B_110))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_65),D_12))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_131))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_65))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_131),C_65)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_110),D_12))) ) ) ) ) ).
fof(fact_1416_mult__strict__mono_H,axiom,
! [C_65,D_12,A_131,B_110] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_131),B_110))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_65),D_12))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_131))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_65))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_131),C_65)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_110),D_12))) ) ) ) ) ).
fof(fact_1417_mult__strict__mono_H,axiom,
! [C_65,D_12,A_131,B_110] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_131),B_110))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_65),D_12))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_131))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_65))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_131),C_65)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_110),D_12))) ) ) ) ) ).
fof(fact_1418_mult__strict__mono_H,axiom,
! [C_65,D_12,A_131,B_110] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_131),B_110))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_65),D_12))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_131))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_65))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_131),C_65)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_110),D_12))) ) ) ) ) ).
fof(fact_1419_mult__less__le__imp__less,axiom,
! [C_64,D_11,A_130,B_109] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_130),B_109))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_64),D_11))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_130))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_64))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_130),C_64)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_109),D_11))) ) ) ) ) ).
fof(fact_1420_mult__less__le__imp__less,axiom,
! [C_64,D_11,A_130,B_109] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_130),B_109))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_64),D_11))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_130))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_64))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_130),C_64)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_109),D_11))) ) ) ) ) ).
fof(fact_1421_mult__less__le__imp__less,axiom,
! [C_64,D_11,A_130,B_109] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_130),B_109))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_64),D_11))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_130))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_64))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_130),C_64)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_109),D_11))) ) ) ) ) ).
fof(fact_1422_mult__less__le__imp__less,axiom,
! [C_64,D_11,A_130,B_109] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_130),B_109))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_64),D_11))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_130))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_64))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_130),C_64)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_109),D_11))) ) ) ) ) ).
fof(fact_1423_mult__less__le__imp__less,axiom,
! [C_64,D_11,A_130,B_109] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_130),B_109))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_64),D_11))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_130))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_64))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_130),C_64)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_109),D_11))) ) ) ) ) ).
fof(fact_1424_mult__less__le__imp__less,axiom,
! [C_64,D_11,A_130,B_109] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_130),B_109))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_64),D_11))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_130))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_64))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_130),C_64)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_109),D_11))) ) ) ) ) ).
fof(fact_1425_mult__le__less__imp__less,axiom,
! [C_63,D_10,A_129,B_108] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_129),B_108))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_63),D_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_129))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_63))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_129),C_63)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_108),D_10))) ) ) ) ) ).
fof(fact_1426_mult__le__less__imp__less,axiom,
! [C_63,D_10,A_129,B_108] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_129),B_108))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_63),D_10))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_129))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_63))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_129),C_63)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_108),D_10))) ) ) ) ) ).
fof(fact_1427_mult__le__less__imp__less,axiom,
! [C_63,D_10,A_129,B_108] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_129),B_108))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_63),D_10))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_129))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_63))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_129),C_63)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_108),D_10))) ) ) ) ) ).
fof(fact_1428_mult__le__less__imp__less,axiom,
! [C_63,D_10,A_129,B_108] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_129),B_108))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_63),D_10))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_129))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_63))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_129),C_63)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_108),D_10))) ) ) ) ) ).
fof(fact_1429_mult__le__less__imp__less,axiom,
! [C_63,D_10,A_129,B_108] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_129),B_108))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_63),D_10))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_129))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_63))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_129),C_63)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_108),D_10))) ) ) ) ) ).
fof(fact_1430_mult__le__less__imp__less,axiom,
! [C_63,D_10,A_129,B_108] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_129),B_108))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_63),D_10))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_129))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_63))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_129),C_63)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_108),D_10))) ) ) ) ) ).
fof(fact_1431_mult__right__less__imp__less,axiom,
! [A_128,C_62,B_107] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_128),C_62)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_107),C_62)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_62))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_128),B_107)) ) ) ).
fof(fact_1432_mult__right__less__imp__less,axiom,
! [A_128,C_62,B_107] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_128),C_62)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_107),C_62)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_62))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_128),B_107)) ) ) ).
fof(fact_1433_mult__right__less__imp__less,axiom,
! [A_128,C_62,B_107] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_128),C_62)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_107),C_62)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_62))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_128),B_107)) ) ) ).
fof(fact_1434_mult__right__less__imp__less,axiom,
! [A_128,C_62,B_107] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_128),C_62)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_107),C_62)))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_62))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_128),B_107)) ) ) ).
fof(fact_1435_mult__right__less__imp__less,axiom,
! [A_128,C_62,B_107] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_128),C_62)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_107),C_62)))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_62))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_128),B_107)) ) ) ).
fof(fact_1436_mult__right__less__imp__less,axiom,
! [A_128,C_62,B_107] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_128),C_62)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_107),C_62)))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_62))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_128),B_107)) ) ) ).
fof(fact_1437_mult__less__imp__less__right,axiom,
! [A_127,C_61,B_106] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_127),C_61)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_106),C_61)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_61))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_127),B_106)) ) ) ).
fof(fact_1438_mult__less__imp__less__right,axiom,
! [A_127,C_61,B_106] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_127),C_61)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_106),C_61)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_61))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_127),B_106)) ) ) ).
fof(fact_1439_mult__less__imp__less__right,axiom,
! [A_127,C_61,B_106] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_127),C_61)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_106),C_61)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_61))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_127),B_106)) ) ) ).
fof(fact_1440_mult__less__imp__less__right,axiom,
! [A_127,C_61,B_106] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_127),C_61)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_106),C_61)))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_61))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_127),B_106)) ) ) ).
fof(fact_1441_mult__less__imp__less__right,axiom,
! [A_127,C_61,B_106] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_127),C_61)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_106),C_61)))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_61))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_127),B_106)) ) ) ).
fof(fact_1442_mult__less__imp__less__right,axiom,
! [A_127,C_61,B_106] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_127),C_61)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_106),C_61)))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_61))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_127),B_106)) ) ) ).
fof(fact_1443_mult__left__less__imp__less,axiom,
! [C_60,A_126,B_105] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_60),A_126)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_60),B_105)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_60))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_126),B_105)) ) ) ).
fof(fact_1444_mult__left__less__imp__less,axiom,
! [C_60,A_126,B_105] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_60),A_126)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_60),B_105)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_60))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_126),B_105)) ) ) ).
fof(fact_1445_mult__left__less__imp__less,axiom,
! [C_60,A_126,B_105] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_60),A_126)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_60),B_105)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_60))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_126),B_105)) ) ) ).
fof(fact_1446_mult__left__less__imp__less,axiom,
! [C_60,A_126,B_105] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_60),A_126)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_60),B_105)))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_60))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_126),B_105)) ) ) ).
fof(fact_1447_mult__left__less__imp__less,axiom,
! [C_60,A_126,B_105] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_60),A_126)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_60),B_105)))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_60))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_126),B_105)) ) ) ).
fof(fact_1448_mult__left__less__imp__less,axiom,
! [C_60,A_126,B_105] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_60),A_126)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_60),B_105)))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_60))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_126),B_105)) ) ) ).
fof(fact_1449_mult__less__imp__less__left,axiom,
! [C_59,A_125,B_104] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_59),A_125)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_59),B_104)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_59))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_125),B_104)) ) ) ).
fof(fact_1450_mult__less__imp__less__left,axiom,
! [C_59,A_125,B_104] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_59),A_125)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_59),B_104)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_59))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_125),B_104)) ) ) ).
fof(fact_1451_mult__less__imp__less__left,axiom,
! [C_59,A_125,B_104] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_59),A_125)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_59),B_104)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_59))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_125),B_104)) ) ) ).
fof(fact_1452_mult__less__imp__less__left,axiom,
! [C_59,A_125,B_104] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_59),A_125)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_59),B_104)))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_59))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_125),B_104)) ) ) ).
fof(fact_1453_mult__less__imp__less__left,axiom,
! [C_59,A_125,B_104] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_59),A_125)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_59),B_104)))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_59))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_125),B_104)) ) ) ).
fof(fact_1454_mult__less__imp__less__left,axiom,
! [C_59,A_125,B_104] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_59),A_125)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_59),B_104)))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_59))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_125),B_104)) ) ) ).
fof(fact_1455_mult__right__le__imp__le,axiom,
! [A_124,C_58,B_103] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_124),C_58)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_103),C_58)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_58))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_124),B_103)) ) ) ).
fof(fact_1456_mult__right__le__imp__le,axiom,
! [A_124,C_58,B_103] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_124),C_58)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_103),C_58)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_58))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_124),B_103)) ) ) ).
fof(fact_1457_mult__right__le__imp__le,axiom,
! [A_124,C_58,B_103] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_124),C_58)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_103),C_58)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_58))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_124),B_103)) ) ) ).
fof(fact_1458_mult__right__le__imp__le,axiom,
! [A_124,C_58,B_103] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_124),C_58)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_103),C_58)))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_58))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_124),B_103)) ) ) ).
fof(fact_1459_mult__right__le__imp__le,axiom,
! [A_124,C_58,B_103] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_124),C_58)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_103),C_58)))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_58))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_124),B_103)) ) ) ).
fof(fact_1460_mult__right__le__imp__le,axiom,
! [A_124,C_58,B_103] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_124),C_58)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_103),C_58)))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_58))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_124),B_103)) ) ) ).
fof(fact_1461_mult__left__le__imp__le,axiom,
! [C_57,A_123,B_102] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_57),A_123)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_57),B_102)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_57))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_123),B_102)) ) ) ).
fof(fact_1462_mult__left__le__imp__le,axiom,
! [C_57,A_123,B_102] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_57),A_123)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_57),B_102)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_57))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_123),B_102)) ) ) ).
fof(fact_1463_mult__left__le__imp__le,axiom,
! [C_57,A_123,B_102] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_57),A_123)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_57),B_102)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_57))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_123),B_102)) ) ) ).
fof(fact_1464_mult__left__le__imp__le,axiom,
! [C_57,A_123,B_102] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_57),A_123)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_57),B_102)))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_57))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_123),B_102)) ) ) ).
fof(fact_1465_mult__left__le__imp__le,axiom,
! [C_57,A_123,B_102] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_57),A_123)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_57),B_102)))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_57))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_123),B_102)) ) ) ).
fof(fact_1466_mult__left__le__imp__le,axiom,
! [C_57,A_123,B_102] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_57),A_123)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_57),B_102)))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_57))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_123),B_102)) ) ) ).
fof(fact_1467_mult__left__le__one__le,axiom,
! [Y_22,X_28] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_28))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_22))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_22),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_22),X_28)),X_28)) ) ) ) ).
fof(fact_1468_mult__left__le__one__le,axiom,
! [Y_22,X_28] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_28))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_22))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_22),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_22),X_28)),X_28)) ) ) ) ).
fof(fact_1469_mult__left__le__one__le,axiom,
! [Y_22,X_28] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_28))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_22))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_22),one_one_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_22),X_28)),X_28)) ) ) ) ).
fof(fact_1470_mult__right__le__one__le,axiom,
! [Y_21,X_27] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_27))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_21),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_27),Y_21)),X_27)) ) ) ) ).
fof(fact_1471_mult__right__le__one__le,axiom,
! [Y_21,X_27] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_27))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_21))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_21),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_27),Y_21)),X_27)) ) ) ) ).
fof(fact_1472_mult__right__le__one__le,axiom,
! [Y_21,X_27] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_27))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_21))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_21),one_one_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_27),Y_21)),X_27)) ) ) ) ).
fof(fact_1473_zero__less__two,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int))) ).
fof(fact_1474_zero__less__two,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).
fof(fact_1475_zero__less__two,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),one_one_real))) ).
fof(fact_1476_zero__less__two,axiom,
hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,one_on1645066479umeral),one_on1645066479umeral))) ).
fof(fact_1477_zero__less__two,axiom,
hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,one_on1684967323de_int),one_on1684967323de_int))) ).
fof(fact_1478_zero__less__two,axiom,
hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))) ).
fof(fact_1479_power__0__left__number__of,axiom,
! [W_5] :
( ( hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),hAPP_int_nat(number_number_of_nat,W_5)) = one_one_int )
& ( hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),hAPP_int_nat(number_number_of_nat,W_5)) = zero_zero_int ) ) ).
fof(fact_1480_power__0__left__number__of,axiom,
! [W_5] :
( ( hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,W_5)) = one_one_nat )
& ( hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,W_5)) = zero_zero_nat ) ) ).
fof(fact_1481_power__0__left__number__of,axiom,
! [W_5] :
( ( hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),hAPP_int_nat(number_number_of_nat,W_5)) = one_one_real )
& ( hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),hAPP_int_nat(number_number_of_nat,W_5)) = zero_zero_real ) ) ).
fof(fact_1482_power__0__left__number__of,axiom,
! [W_5] :
( ( hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat
=> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_int_nat(number_number_of_nat,W_5)) = one_on1645066479umeral )
& ( hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat
=> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_int_nat(number_number_of_nat,W_5)) = zero_z126310315umeral ) ) ).
fof(fact_1483_power__0__left__number__of,axiom,
! [W_5] :
( ( hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat
=> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_int_nat(number_number_of_nat,W_5)) = one_one_complex )
& ( hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat
=> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_int_nat(number_number_of_nat,W_5)) = zero_zero_complex ) ) ).
fof(fact_1484_power__0__left__number__of,axiom,
! [W_5] :
( ( hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat
=> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),hAPP_int_nat(number_number_of_nat,W_5)) = one_on1684967323de_int )
& ( hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat
=> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),hAPP_int_nat(number_number_of_nat,W_5)) = zero_z891286103de_int ) ) ).
fof(fact_1485_power__0__left__number__of,axiom,
! [W_5] :
( ( hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat
=> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),hAPP_int_nat(number_number_of_nat,W_5)) = one_one_rat )
& ( hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat
=> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),hAPP_int_nat(number_number_of_nat,W_5)) = zero_zero_rat ) ) ).
fof(fact_1486_zcong__zless__imp__eq,axiom,
! [B_2,M,A_3] :
( ( is_int(B_2)
& is_int(A_3) )
=> ( 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_int,A_3),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
=> A_3 = B_2 ) ) ) ) ) ) ).
fof(fact_1487_zcong__zless__0,axiom,
! [M,A_3] :
( is_int(A_3)
=> ( 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_int,A_3),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),M))
=> A_3 = zero_zero_int ) ) ) ) ).
fof(fact_1488_Int2_Ozcong__zero,axiom,
! [M,X_1] :
( is_int(X_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),M))
=> X_1 = zero_zero_int ) ) ) ) ).
fof(fact_1489_zcong__zmult__prop3,axiom,
! [Y_1,X_1,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Y_1),zero_zero_int),P_3))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1)),zero_zero_int),P_3)) ) ) ) ).
fof(fact_1490_convex__bound__le,axiom,
! [V_9,U_2,Y_20,X_26,A_122] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_26),A_122))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_20),A_122))
=> ( 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_9))
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U_2),V_9) = one_one_int
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,U_2),X_26)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_9),Y_20))),A_122)) ) ) ) ) ) ).
fof(fact_1491_convex__bound__le,axiom,
! [V_9,U_2,Y_20,X_26,A_122] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_26),A_122))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_20),A_122))
=> ( 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_9))
=> ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,U_2),V_9) = one_one_real
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_2),X_26)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,V_9),Y_20))),A_122)) ) ) ) ) ) ).
fof(fact_1492_convex__bound__le,axiom,
! [V_9,U_2,Y_20,X_26,A_122] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_26),A_122))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_20),A_122))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),U_2))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),V_9))
=> ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,U_2),V_9) = one_one_rat
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,U_2),X_26)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,V_9),Y_20))),A_122)) ) ) ) ) ) ).
fof(fact_1493_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) ).
fof(fact_1494_zdiv__mono2__neg__lemma,axiom,
! [B_2,Q_2,R_3,B_5,Q_8,R_5] :
( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5)),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_8),Q_2)) ) ) ) ) ) ) ).
fof(fact_1495_unique__quotient__lemma__neg,axiom,
! [B_2,Q_8,R_5,Q_2,R_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_8)),R_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),Q_8)) ) ) ) ) ).
fof(fact_1496_zdiv__mono2__lemma,axiom,
! [B_2,Q_2,R_3,B_5,Q_8,R_5] :
( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_5),B_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),Q_8)) ) ) ) ) ) ) ).
fof(fact_1497_unique__quotient__lemma,axiom,
! [B_2,Q_8,R_5,Q_2,R_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_8)),R_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_5),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_8),Q_2)) ) ) ) ) ).
fof(fact_1498_q__neg__lemma,axiom,
! [B_5,Q_8,R_5] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5)),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_8),zero_zero_int)) ) ) ) ).
fof(fact_1499_q__pos__lemma,axiom,
! [B_5,Q_8,R_5] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_5),B_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Q_8)) ) ) ) ).
fof(fact_1500_zcong__zprime__prod__zero__contra,axiom,
! [B_2,A_3,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
=> ( ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3))
& ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B_2),zero_zero_int),P_3)) )
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),zero_zero_int),P_3)) ) ) ) ).
fof(fact_1501__096QuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_096,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),hAPP_int_int(number_number_of_int,min))) ).
fof(fact_1502__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_N_A_N1_096,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(number_number_of_int,min)))) ).
fof(fact_1503_neg__one__power__eq__mod__m,axiom,
! [J_2,K_2,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),J_2)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),K_2)),M))
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),J_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),K_2) ) ) ).
fof(fact_1504_one__not__neg__one__mod__m,axiom,
! [M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(one_one_int),hAPP_int_int(number_number_of_int,min)),M)) ) ).
fof(fact_1505__096s_A_094_A2_A_N_A_N1_A_061_As_A_094_A2_A_L_A1_096,axiom,
hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(number_number_of_int,min)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int) ).
fof(fact_1506_divides__cases,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),M))
=> ( M = zero_zero_nat
| M = N_1
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)),M)) ) ) ).
fof(fact_1507__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_,axiom,
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),hAPP_int_int(number_number_of_int,min)))
=> hAPP_int_int(legendre(hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)) != one_one_int ) ).
fof(fact_1508_Legendre__1mod4,axiom,
! [M] :
( hBOOL(hAPP_int_bool(zprime,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),M)),one_one_int)))
=> hAPP_int_int(legendre(hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),M)),one_one_int)) = one_one_int ) ).
fof(fact_1509_realpow__two__sum__zero__iff,axiom,
! [X_2,Y_2] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real
<=> ( X_2 = zero_zero_real
& Y_2 = zero_zero_real ) ) ).
fof(fact_1510_two__realpow__ge__one,axiom,
! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1))) ).
fof(fact_1511_real__zero__not__eq__one,axiom,
zero_zero_real != one_one_real ).
fof(fact_1512_real__less__def,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2))
& X_2 != Y_2 ) ) ).
fof(fact_1513_less__eq__real__def,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2))
| X_2 = Y_2 ) ) ).
fof(fact_1514_divides__exp,axiom,
! [N_1,X_1,Y_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_1),N_1))) ) ).
fof(fact_1515_divides__antisym,axiom,
! [X_2,Y_2] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) )
<=> X_2 = Y_2 ) ).
fof(fact_1516_number__of__diff,axiom,
! [V_8,W_4] : hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_8),W_4)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,V_8)),hAPP_int_int(number_number_of_int,W_4)) ).
fof(fact_1517_number__of__diff,axiom,
! [V_8,W_4] : hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_8),W_4)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,V_8)),hAPP_int_real(number267125858f_real,W_4)) ).
fof(fact_1518_number__of__diff,axiom,
! [V_8,W_4] : hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_8),W_4)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,V_8)),hAPP_int_complex(number528085621omplex,W_4)) ).
fof(fact_1519_number__of__diff,axiom,
! [V_8,W_4] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_8),W_4)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,V_8)),hAPP_int_rat(number_number_of_rat,W_4)) ).
fof(fact_1520_real__le__antisym,axiom,
! [Z_1,W] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_1),W))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z_1))
=> Z_1 = W ) ) ).
fof(fact_1521_real__le__trans,axiom,
! [K_2,I,J_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I),J_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,J_2),K_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I),K_2)) ) ) ).
fof(fact_1522_real__le__linear,axiom,
! [Z_1,W] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_1),W))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z_1)) ) ).
fof(fact_1523_real__le__refl,axiom,
! [W] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),W)) ).
fof(fact_1524_add__diff__add,axiom,
! [A_121,C_56,B_101,D_9] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_121),C_56)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_101),D_9)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_121),B_101)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C_56),D_9)) ).
fof(fact_1525_add__diff__add,axiom,
! [A_121,C_56,B_101,D_9] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_121),C_56)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_101),D_9)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_121),B_101)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C_56),D_9)) ).
fof(fact_1526_add__diff__add,axiom,
! [A_121,C_56,B_101,D_9] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_121),C_56)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_101),D_9)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_121),B_101)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,C_56),D_9)) ).
fof(fact_1527_add__diff__add,axiom,
! [A_121,C_56,B_101,D_9] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_121),C_56)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_101),D_9)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_121),B_101)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C_56),D_9)) ).
fof(fact_1528_dvd__diff,axiom,
! [Z_8,X_25,Y_19] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_25),Y_19))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_25),Z_8))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_25),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_19),Z_8))) ) ) ).
fof(fact_1529_dvd__diff,axiom,
! [Z_8,X_25,Y_19] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_25),Y_19))
=> ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_25),Z_8))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_25),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_19),Z_8))) ) ) ).
fof(fact_1530_dvd__diff,axiom,
! [Z_8,X_25,Y_19] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_25),Y_19))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_25),Z_8))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_25),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Y_19),Z_8))) ) ) ).
fof(fact_1531_dvd__diff,axiom,
! [Z_8,X_25,Y_19] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_25),Y_19))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_25),Z_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_25),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_19),Z_8))) ) ) ).
fof(fact_1532_diff__bin__simps_I1_J,axiom,
! [K_2] :
( is_int(K_2)
=> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),pls) = K_2 ) ).
fof(fact_1533_diff__bin__simps_I7_J,axiom,
! [K_2,L_2] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),L_2)) ).
fof(fact_1534_zdiff__zmult__distrib2,axiom,
! [W,Z1,Z2] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z2)) ).
fof(fact_1535_zdiff__zmult__distrib,axiom,
! [Z1,Z2,W] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2)),W) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),W)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),W)) ).
fof(fact_1536_Int2_Oaux1,axiom,
! [A_3,B_2,C_1] :
( is_int(A_3)
=> ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2) = C_1
=> A_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_1),B_2) ) ) ).
fof(fact_1537_zcong__zdiff,axiom,
! [C_1,D_3,A_3,B_2,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),D_3)),M)) ) ) ).
fof(fact_1538_zdvd__zdiffD,axiom,
! [K_2,M,N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),N_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) ) ) ).
fof(fact_1539_mult__diff__mult,axiom,
! [X_24,Y_18,A_120,B_100] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_24),Y_18)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_120),B_100)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_24),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_18),B_100))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_24),A_120)),B_100)) ).
fof(fact_1540_mult__diff__mult,axiom,
! [X_24,Y_18,A_120,B_100] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_24),Y_18)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_120),B_100)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_24),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_18),B_100))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_24),A_120)),B_100)) ).
fof(fact_1541_mult__diff__mult,axiom,
! [X_24,Y_18,A_120,B_100] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_24),Y_18)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_120),B_100)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_24),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_18),B_100))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_24),A_120)),B_100)) ).
fof(fact_1542_mult__diff__mult,axiom,
! [X_24,Y_18,A_120,B_100] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_24),Y_18)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_120),B_100)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_24),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Y_18),B_100))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_24),A_120)),B_100)) ).
fof(fact_1543_real__squared__diff__one__factored,axiom,
! [X_23] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_23),X_23)),one_one_int) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_23),one_one_int)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_23),one_one_int)) ).
fof(fact_1544_real__squared__diff__one__factored,axiom,
! [X_23] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_23),X_23)),one_one_real) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_23),one_one_real)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_23),one_one_real)) ).
fof(fact_1545_real__squared__diff__one__factored,axiom,
! [X_23] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_23),X_23)),one_one_complex) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_23),one_one_complex)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_23),one_one_complex)) ).
fof(fact_1546_real__squared__diff__one__factored,axiom,
! [X_23] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_23),X_23)),one_one_rat) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_23),one_one_rat)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_23),one_one_rat)) ).
fof(fact_1547_eq__add__iff2,axiom,
! [A,E_2,C,B,D] :
( is_int(C)
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D)
<=> C = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B),A)),E_2)),D) ) ) ).
fof(fact_1548_eq__add__iff2,axiom,
! [A,E_2,C,B,D] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D)
<=> C = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),E_2)),D) ) ).
fof(fact_1549_eq__add__iff2,axiom,
! [A,E_2,C,B,D] :
( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),E_2)),C) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),E_2)),D)
<=> C = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,B),A)),E_2)),D) ) ).
fof(fact_1550_eq__add__iff2,axiom,
! [A,E_2,C,B,D] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D)
<=> C = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B),A)),E_2)),D) ) ).
fof(fact_1551_eq__add__iff1,axiom,
! [A,E_2,C,B,D] :
( is_int(D)
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D)
<=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),E_2)),C) = D ) ) ).
fof(fact_1552_eq__add__iff1,axiom,
! [A,E_2,C,B,D] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D)
<=> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),E_2)),C) = D ) ).
fof(fact_1553_eq__add__iff1,axiom,
! [A,E_2,C,B,D] :
( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),E_2)),C) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),E_2)),D)
<=> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A),B)),E_2)),C) = D ) ).
fof(fact_1554_eq__add__iff1,axiom,
! [A,E_2,C,B,D] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D)
<=> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),E_2)),C) = D ) ).
fof(fact_1555_right__diff__distrib__number__of,axiom,
! [V_7,B_99,C_55] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_7)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_99),C_55)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_7)),B_99)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_7)),C_55)) ).
fof(fact_1556_right__diff__distrib__number__of,axiom,
! [V_7,B_99,C_55] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_7)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_99),C_55)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_7)),B_99)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_7)),C_55)) ).
fof(fact_1557_right__diff__distrib__number__of,axiom,
! [V_7,B_99,C_55] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_7)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,B_99),C_55)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_7)),B_99)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_7)),C_55)) ).
fof(fact_1558_right__diff__distrib__number__of,axiom,
! [V_7,B_99,C_55] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_7)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B_99),C_55)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_7)),B_99)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_7)),C_55)) ).
fof(fact_1559_left__diff__distrib__number__of,axiom,
! [A_119,B_98,V_6] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_119),B_98)),hAPP_int_int(number_number_of_int,V_6)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_119),hAPP_int_int(number_number_of_int,V_6))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_98),hAPP_int_int(number_number_of_int,V_6))) ).
fof(fact_1560_left__diff__distrib__number__of,axiom,
! [A_119,B_98,V_6] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_119),B_98)),hAPP_int_real(number267125858f_real,V_6)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_119),hAPP_int_real(number267125858f_real,V_6))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_98),hAPP_int_real(number267125858f_real,V_6))) ).
fof(fact_1561_left__diff__distrib__number__of,axiom,
! [A_119,B_98,V_6] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_119),B_98)),hAPP_int_complex(number528085621omplex,V_6)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_119),hAPP_int_complex(number528085621omplex,V_6))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_98),hAPP_int_complex(number528085621omplex,V_6))) ).
fof(fact_1562_left__diff__distrib__number__of,axiom,
! [A_119,B_98,V_6] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_119),B_98)),hAPP_int_rat(number_number_of_rat,V_6)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_119),hAPP_int_rat(number_number_of_rat,V_6))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_98),hAPP_int_rat(number_number_of_rat,V_6))) ).
fof(fact_1563_diff__bin__simps_I9_J,axiom,
! [K_2,L_2] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),L_2)) ).
fof(fact_1564_diff__bin__simps_I10_J,axiom,
! [K_2,L_2] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),L_2)) ).
fof(fact_1565_diff__bin__simps_I3_J,axiom,
! [L_2] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,pls),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,pls),L_2)) ).
fof(fact_1566_less__bin__lemma,axiom,
! [K_1,L] :
( 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(hAPP_int_fun_int_int(minus_minus_int,K_1),L)),zero_zero_int)) ) ).
fof(fact_1567_xzgcda__linear__aux1,axiom,
! [A_3,R_3,B_2,M,C_1,D_3,N_1] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),B_2))),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),D_3))),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),N_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D_3),N_1)))) ).
fof(fact_1568_zcong__def,axiom,
! [A,B,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B))) ) ).
fof(fact_1569_le__add__iff2,axiom,
! [A,E_2,C,B,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B),A)),E_2)),D))) ) ).
fof(fact_1570_le__add__iff2,axiom,
! [A,E_2,C,B,D] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),E_2)),D))) ) ).
fof(fact_1571_le__add__iff2,axiom,
! [A,E_2,C,B,D] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B),A)),E_2)),D))) ) ).
fof(fact_1572_le__add__iff1,axiom,
! [A,E_2,C,B,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),E_2)),C)),D)) ) ).
fof(fact_1573_le__add__iff1,axiom,
! [A,E_2,C,B,D] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),E_2)),C)),D)) ) ).
fof(fact_1574_le__add__iff1,axiom,
! [A,E_2,C,B,D] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),E_2)),C)),D)) ) ).
fof(fact_1575_less__add__iff2,axiom,
! [A,E_2,C,B,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B),A)),E_2)),D))) ) ).
fof(fact_1576_less__add__iff2,axiom,
! [A,E_2,C,B,D] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),E_2)),D))) ) ).
fof(fact_1577_less__add__iff2,axiom,
! [A,E_2,C,B,D] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B),A)),E_2)),D))) ) ).
fof(fact_1578_less__add__iff1,axiom,
! [A,E_2,C,B,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),E_2)),C)),D)) ) ).
fof(fact_1579_less__add__iff1,axiom,
! [A,E_2,C,B,D] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),E_2)),C)),D)) ) ).
fof(fact_1580_less__add__iff1,axiom,
! [A,E_2,C,B,D] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),E_2)),C)),D)) ) ).
fof(fact_1581_add__number__of__diff1,axiom,
! [V_5,W_3,C_54] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_5)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,W_3)),C_54)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),W_3))),C_54) ).
fof(fact_1582_add__number__of__diff1,axiom,
! [V_5,W_3,C_54] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_5)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,W_3)),C_54)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),W_3))),C_54) ).
fof(fact_1583_add__number__of__diff1,axiom,
! [V_5,W_3,C_54] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_5)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,W_3)),C_54)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),W_3))),C_54) ).
fof(fact_1584_add__number__of__diff1,axiom,
! [V_5,W_3,C_54] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_5)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,W_3)),C_54)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),W_3))),C_54) ).
fof(fact_1585_Euler_Oaux1,axiom,
! [A_3,X_1] :
( is_int(X_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),A_3))
=> ( X_1 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),one_one_int)
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),one_one_int))) ) ) ) ) ).
fof(fact_1586_zle__diff1__eq,axiom,
! [W_1,Z_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),one_one_int)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) ) ).
fof(fact_1587_diff__bin__simps_I4_J,axiom,
! [L_2] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,pls),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2)) ).
fof(fact_1588_diff__bin__simps_I5_J,axiom,
! [L_2] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2)) ).
fof(fact_1589_diff__bin__simps_I6_J,axiom,
! [L_2] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2)) ).
fof(fact_1590_inv__not__p__minus__1__aux,axiom,
! [A,P_5] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))),one_one_int),P_5))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),P_5)) ) ).
fof(fact_1591_divides__rev,axiom,
! [A_3,N_1,B_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),N_1)))
=> ( N_1 != zero_zero_nat
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) ) ) ).
fof(fact_1592_divides__exp2,axiom,
! [X_1,Y_1,N_1] :
( N_1 != zero_zero_nat
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1)),Y_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) ) ) ).
fof(fact_1593_nat__power__eq__0__iff,axiom,
! [Ma,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Ma),N) = zero_zero_nat
<=> ( N != zero_zero_nat
& Ma = zero_zero_nat ) ) ).
fof(fact_1594_divides__add__revr,axiom,
! [B_2,D_3,A_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),A_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_3),B_2)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),B_2)) ) ) ).
fof(fact_1595_real__mult__1,axiom,
! [Z_1] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),Z_1) = Z_1 ).
fof(fact_1596_real__mult__commute,axiom,
! [Z_1,W] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_1),W) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,W),Z_1) ).
fof(fact_1597_real__mult__assoc,axiom,
! [Z1,Z2,Z3] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z1),Z2)),Z3) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z2),Z3)) ).
fof(fact_1598_real__mult__right__cancel,axiom,
! [A,B,C] :
( C != zero_zero_real
=> ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),C) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),C)
<=> A = B ) ) ).
fof(fact_1599_real__mult__left__cancel,axiom,
! [A,B,C] :
( C != zero_zero_real
=> ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)
<=> A = B ) ) ).
fof(fact_1600_real__mult__less__iff1,axiom,
! [X_2,Y_2,Z_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_2),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Z_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) ) ) ).
fof(fact_1601_real__mult__le__cancel__iff1,axiom,
! [X_2,Y_2,Z_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_2),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Z_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)) ) ) ).
fof(fact_1602_real__mult__le__cancel__iff2,axiom,
! [X_2,Y_2,Z_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_2),X_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_2),Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)) ) ) ).
fof(fact_1603_real__mult__order,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_1))) ) ) ).
fof(fact_1604_real__mult__less__mono2,axiom,
! [X_1,Y_1,Z_1] :
( 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_1),Y_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_1),Y_1))) ) ) ).
fof(fact_1605_divides__mul__l,axiom,
! [C_1,A_3,B_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_1),A_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_1),B_2))) ) ).
fof(fact_1606_divides__mul__r,axiom,
! [C_1,A_3,B_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1))) ) ).
fof(fact_1607_real__add__left__mono,axiom,
! [Z_1,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,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_1),Y_1))) ) ).
fof(fact_1608_mult__sum2sq,axiom,
! [A_3,B_2,P_3,Q_2] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(twoSqu1907779896sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_3),B_2))),hAPP_P1175774780nt_int(twoSqu1907779896sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2))) = hAPP_P1175774780nt_int(twoSqu1907779896sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),P_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),P_3)))) ).
fof(fact_1609_zcong__square,axiom,
! [A_3,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),A_3)),one_one_int),P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),one_one_int),P_3))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),P_3)) ) ) ) ) ).
fof(fact_1610_zcong__square__zless,axiom,
! [A_3,P_3] :
( is_int(A_3)
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),A_3)),one_one_int),P_3))
=> ( A_3 = one_one_int
| A_3 = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int) ) ) ) ) ) ) ).
fof(fact_1611_divides__ge,axiom,
! [A_3,B_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
=> ( B_2 = zero_zero_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_3),B_2)) ) ) ).
fof(fact_1612_nat__mult__dvd__cancel__disj_H,axiom,
! [Ma,K_1,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_1)))
<=> ( K_1 = zero_zero_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),N)) ) ) ).
fof(fact_1613_exp__eq__1,axiom,
! [X_2,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),N) = one_one_nat
<=> ( X_2 = one_one_nat
| N = zero_zero_nat ) ) ).
fof(fact_1614_zspecial__product,axiom,
! [A_3,B_2] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_1615_nat__mult__eq__one,axiom,
! [N,Ma] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Ma) = one_one_nat
<=> ( N = one_one_nat
& Ma = one_one_nat ) ) ).
fof(fact_1616_real__add__mult__distrib,axiom,
! [Z1,Z2,W] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z1),Z2)),W) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z1),W)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z2),W)) ).
fof(fact_1617_real__two__squares__add__zero__iff,axiom,
! [X_2,Y_2] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_2),X_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2)) = zero_zero_real
<=> ( X_2 = zero_zero_real
& Y_2 = zero_zero_real ) ) ).
fof(fact_1618_power2__diff,axiom,
! [X_22,Y_17] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_22),Y_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_22),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_17),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_22)),Y_17)) ).
fof(fact_1619_power2__diff,axiom,
! [X_22,Y_17] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_22),Y_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_22),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_17),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_22)),Y_17)) ).
fof(fact_1620_power2__diff,axiom,
! [X_22,Y_17] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_22),Y_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_22),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_17),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_22)),Y_17)) ).
fof(fact_1621_power2__diff,axiom,
! [X_22,Y_17] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_22),Y_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_22),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_17),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_22)),Y_17)) ).
fof(fact_1622_zdiff__power2,axiom,
! [A_3,B_2] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)),B_2))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_1623_zdiff__power3,axiom,
! [A_3,B_2] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),A_3)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) ).
fof(fact_1624_divides__div__not,axiom,
! [X_1,Q_2,N_1,R_3] :
( X_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Q_2),N_1)),R_3)
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),R_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_3),N_1))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),X_1)) ) ) ) ).
fof(fact_1625_neg__one__power,axiom,
! [N_1] :
( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),N_1) = one_one_int
| hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),N_1) = hAPP_int_int(number_number_of_int,min) ) ).
fof(fact_1626_Legendre__def,axiom,
! [A_3,P_3] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3))
=> hAPP_int_int(legendre(A_3),P_3) = zero_zero_int )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_3))
=> hAPP_int_int(legendre(A_3),P_3) = one_one_int )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_3))
=> hAPP_int_int(legendre(A_3),P_3) = hAPP_int_int(number_number_of_int,min) ) ) ) ) ).
fof(fact_1627_QuadRes__def,axiom,
! [Ma,X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,Ma),X_2))
<=> ? [Y] :
( is_int(Y)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2),Ma)) ) ) ).
fof(fact_1628_realpow__pos__nth,axiom,
! [A_3,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
=> ? [R_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))
& hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_1),N_1) = A_3 ) ) ) ).
fof(fact_1629_realpow__pos__nth__unique,axiom,
! [A_3,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
=> ? [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
& hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N_1) = A_3
& ! [Y] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
& hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),N_1) = A_3 )
=> Y = X ) ) ) ) ).
fof(fact_1630_dvd__mult__cancel1,axiom,
! [N,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N)),Ma))
<=> N = one_one_nat ) ) ).
fof(fact_1631_dvd__mult__cancel2,axiom,
! [N,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Ma)),Ma))
<=> N = one_one_nat ) ) ).
fof(fact_1632_le0,axiom,
! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1)) ).
fof(fact_1633_number__of1,axiom,
! [N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,N_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,N_1))))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,N_1)))) ) ) ).
fof(fact_1634_less__zeroE,axiom,
! [N_1] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).
fof(fact_1635_is__sum2sq__def,axiom,
! [X_2] :
( is_int(X_2)
=> ( hBOOL(hAPP_int_bool(twoSqu1770640450sum2sq,X_2))
<=> ? [A_2,B_4] :
( is_int(A_2)
& is_int(B_4)
& hAPP_P1175774780nt_int(twoSqu1907779896sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_4)) = X_2 ) ) ) ).
fof(fact_1636_dvd_Oorder__refl,axiom,
! [X_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),X_1)) ).
fof(fact_1637_diffs0__imp__equal,axiom,
! [M,N_1] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1) = zero_zero_nat
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),M) = zero_zero_nat
=> M = N_1 ) ) ).
fof(fact_1638_diff__self__eq__0,axiom,
! [M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),M) = zero_zero_nat ).
fof(fact_1639_minus__nat_Odiff__0,axiom,
! [M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat) = M ).
fof(fact_1640_diff__0__eq__0,axiom,
! [N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N_1) = zero_zero_nat ).
fof(fact_1641_less__imp__diff__less,axiom,
! [N_1,J_2,K_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),K_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),N_1)),K_2)) ) ).
fof(fact_1642_diff__less__mono2,axiom,
! [L_2,M,N_1] :
( 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_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),M))) ) ) ).
fof(fact_1643_dvd__diff__nat,axiom,
! [N_1,K_2,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1))) ) ) ).
fof(fact_1644_diff__cancel2,axiom,
! [M,K_2,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1) ).
fof(fact_1645_Nat_Odiff__cancel,axiom,
! [K_2,M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1) ).
fof(fact_1646_diff__diff__left,axiom,
! [I,J_2,K_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J_2)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),K_2)) ).
fof(fact_1647_diff__add__inverse,axiom,
! [N_1,M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M)),N_1) = M ).
fof(fact_1648_diff__add__inverse2,axiom,
! [M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)),N_1) = M ).
fof(fact_1649_le__diff__iff,axiom,
! [N,K_1,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ) ) ).
fof(fact_1650_Nat_Odiff__diff__eq,axiom,
! [N_1,K_2,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1) ) ) ).
fof(fact_1651_eq__diff__iff,axiom,
! [N,K_1,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),K_1)
<=> Ma = N ) ) ) ).
fof(fact_1652_diff__diff__cancel,axiom,
! [I,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),N_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),I)) = I ) ).
fof(fact_1653_diff__le__mono,axiom,
! [L_2,M,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,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),L_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),L_2))) ) ).
fof(fact_1654_diff__le__mono2,axiom,
! [L_2,M,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,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),M))) ) ).
fof(fact_1655_Nat_Odiff__le__self,axiom,
! [M,N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),M)) ).
fof(fact_1656_diff__mult__distrib2,axiom,
! [K_2,M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)) ).
fof(fact_1657_diff__mult__distrib,axiom,
! [M,N_1,K_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_2)) ).
fof(fact_1658_zero__less__diff,axiom,
! [N,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),Ma)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ).
fof(fact_1659_diff__less,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),M)) ) ) ).
fof(fact_1660_diff__add__0,axiom,
! [N_1,M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M)) = zero_zero_nat ).
fof(fact_1661_diff__is__0__eq,axiom,
! [Ma,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),N) = zero_zero_nat
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ).
fof(fact_1662_diff__is__0__eq_H,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1) = zero_zero_nat ) ).
fof(fact_1663_add__diff__inverse,axiom,
! [M,N_1] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)) = M ) ).
fof(fact_1664_less__diff__conv,axiom,
! [I_1,J,K_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),J)) ) ).
fof(fact_1665_less__diff__iff,axiom,
! [N,K_1,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ) ) ).
fof(fact_1666_diff__less__mono,axiom,
! [C_1,A_3,B_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_3),B_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_1),A_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_3),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,B_2),C_1))) ) ) ).
fof(fact_1667_dvd__diffD1,axiom,
! [K_2,M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N_1)) ) ) ) ).
fof(fact_1668_dvd__diffD,axiom,
! [K_2,M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M)) ) ) ) ).
fof(fact_1669_diff__add__assoc2,axiom,
! [I,K_2,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),I)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)),I) ) ).
fof(fact_1670_add__diff__assoc2,axiom,
! [I,K_2,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)),I) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),I)),K_2) ) ).
fof(fact_1671_diff__add__assoc,axiom,
! [I,K_2,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)) ) ).
fof(fact_1672_le__imp__diff__is__add,axiom,
! [K_1,I_1,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1) = K_1
<=> J = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),I_1) ) ) ).
fof(fact_1673_le__add__diff__inverse2,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1) = M ) ).
fof(fact_1674_le__diff__conv2,axiom,
! [I_1,K_1,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),J)) ) ) ).
fof(fact_1675_add__diff__assoc,axiom,
! [I,K_2,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),K_2) ) ).
fof(fact_1676_le__add__diff__inverse,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)) = M ) ).
fof(fact_1677_le__add__diff,axiom,
! [M,K_2,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M)),K_2))) ) ).
fof(fact_1678_le__diff__conv,axiom,
! [J,K_1,I_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)),I_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1))) ) ).
fof(fact_1679_diff__diff__right,axiom,
! [I,K_2,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),J_2) ) ).
fof(fact_1680_real__le__eq__diff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),Y_2)),zero_zero_real)) ) ).
fof(fact_1681_nat__diff__split__asm,axiom,
! [P_1,A,B] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A),B)))
<=> ~ ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B))
& ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
| ? [D_2] :
( A = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),D_2)
& ~ hBOOL(hAPP_nat_bool(P_1,D_2)) ) ) ) ).
fof(fact_1682_nat__diff__split,axiom,
! [P_1,A,B] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A),B)))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B))
=> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
& ! [D_2] :
( A = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),D_2)
=> hBOOL(hAPP_nat_bool(P_1,D_2)) ) ) ) ).
fof(fact_1683_less__not__refl,axiom,
! [N_1] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1)) ).
fof(fact_1684_nat__neq__iff,axiom,
! [Ma,N] :
( Ma != N
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),Ma)) ) ) ).
fof(fact_1685_linorder__neqE__nat,axiom,
! [X_1,Y_1] :
( 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)) ) ) ).
fof(fact_1686_less__irrefl__nat,axiom,
! [N_1] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1)) ).
fof(fact_1687_less__not__refl2,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M))
=> M != N_1 ) ).
fof(fact_1688_less__not__refl3,axiom,
! [S_1,T_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S_1),T_1))
=> S_1 != T_1 ) ).
fof(fact_1689_nat__less__cases,axiom,
! [P_1,Ma,N] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N),Ma)) )
=> ( ( Ma = N
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N),Ma)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),Ma))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N),Ma)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N),Ma)) ) ) ) ).
fof(fact_1690_dvd_Oless__irrefl,axiom,
! [X_1] :
~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),X_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),X_1)) ) ).
fof(fact_1691_dvd_Oeq__iff,axiom,
! [X_2,Y_2] :
( X_2 = Y_2
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) ) ) ).
fof(fact_1692_dvd_Ole__less,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) )
| X_2 = Y_2 ) ) ).
fof(fact_1693_dvd_Oless__le,axiom,
! [X_2,Y_2] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
& X_2 != Y_2 ) ) ).
fof(fact_1694_dvd_Oless__le__not__le,axiom,
! [X_2,Y_2] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) ) ) ).
fof(fact_1695_dvd_Oneq__le__trans,axiom,
! [A_3,B_2] :
( A_3 != B_2
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3)) ) ) ) ).
fof(fact_1696_dvd_Oeq__refl,axiom,
! [X_1,Y_1] :
( X_1 = Y_1
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) ) ).
fof(fact_1697_dvd_Oantisym__conv,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
<=> X_2 = Y_2 ) ) ).
fof(fact_1698_dvd_Ole__imp__less__or__eq,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
| X_1 = Y_1 ) ) ).
fof(fact_1699_dvd_Ole__neq__trans,axiom,
! [A_3,B_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
=> ( A_3 != B_2
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3)) ) ) ) ).
fof(fact_1700_dvd_Oord__eq__le__trans,axiom,
! [C_1,A_3,B_2] :
( A_3 = B_2
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),C_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1)) ) ) ).
fof(fact_1701_dvd_Oord__le__eq__trans,axiom,
! [C_1,A_3,B_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
=> ( B_2 = C_1
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1)) ) ) ).
fof(fact_1702_dvd__antisym,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),M))
=> M = N_1 ) ) ).
fof(fact_1703_dvd_Oantisym,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1))
=> X_1 = Y_1 ) ) ).
fof(fact_1704_dvd_Oorder__trans,axiom,
! [Z_1,X_1,Y_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),Z_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z_1)) ) ) ).
fof(fact_1705_dvd_Oord__eq__less__trans,axiom,
! [C_1,A_3,B_2] :
( A_3 = B_2
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),C_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_1),B_2)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_1),A_3)) ) ) ) ).
fof(fact_1706_dvd_Ole__less__trans,axiom,
! [Z_1,X_1,Y_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),Z_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),Y_1)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X_1)) ) ) ) ).
fof(fact_1707_dvd_Oless__imp__neq,axiom,
! [X_1,Y_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
=> X_1 != Y_1 ) ).
fof(fact_1708_dvd_Oless__not__sym,axiom,
! [X_1,Y_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) ) ) ).
fof(fact_1709_dvd_Oless__imp__le,axiom,
! [X_1,Y_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) ) ).
fof(fact_1710_dvd_Oless__imp__not__less,axiom,
! [X_1,Y_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) ) ) ).
fof(fact_1711_dvd_Oless__imp__not__eq,axiom,
! [X_1,Y_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
=> X_1 != Y_1 ) ).
fof(fact_1712_dvd_Oless__imp__not__eq2,axiom,
! [X_1,Y_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
=> Y_1 != X_1 ) ).
fof(fact_1713_dvd_Oless__imp__triv,axiom,
! [P_1,X_2,Y_2] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) )
=> hBOOL(P_1) ) ) ).
fof(fact_1714_dvd_Oord__less__eq__trans,axiom,
! [C_1,A_3,B_2] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3)) )
=> ( B_2 = C_1
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_1),A_3)) ) ) ) ).
fof(fact_1715_dvd_Oless__le__trans,axiom,
! [Z_1,X_1,Y_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),Z_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X_1)) ) ) ) ).
fof(fact_1716_dvd_Oless__asym_H,axiom,
! [A_3,B_2] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) ) ) ).
fof(fact_1717_dvd_Oless__trans,axiom,
! [Z_1,X_1,Y_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),Z_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),Y_1)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X_1)) ) ) ) ).
fof(fact_1718_dvd_Oless__asym,axiom,
! [X_1,Y_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) ) ) ).
fof(fact_1719_nat__add__right__cancel,axiom,
! [Ma,K_1,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K_1)
<=> Ma = N ) ).
fof(fact_1720_nat__add__left__cancel,axiom,
! [K_1,Ma,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N)
<=> Ma = N ) ).
fof(fact_1721_nat__add__assoc,axiom,
! [M,N_1,K_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),K_2)) ).
fof(fact_1722_nat__add__left__commute,axiom,
! [X_1,Y_1,Z_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_1),Z_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),Z_1)) ).
fof(fact_1723_nat__add__commute,axiom,
! [M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M) ).
fof(fact_1724_le__antisym,axiom,
! [M,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,N_1),M))
=> M = N_1 ) ) ).
fof(fact_1725_le__trans,axiom,
! [K_2,I,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),K_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),K_2)) ) ) ).
fof(fact_1726_eq__imp__le,axiom,
! [M,N_1] :
( M = N_1
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).
fof(fact_1727_nat__le__linear,axiom,
! [M,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,N_1),M)) ) ).
fof(fact_1728_le__refl,axiom,
! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),N_1)) ).
fof(fact_1729_nat__mult__commute,axiom,
! [M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),M) ).
fof(fact_1730_nat__mult__assoc,axiom,
! [M,N_1,K_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_2)) ).
fof(fact_1731_mult__eq__if,axiom,
! [N_1,M] :
( ( M = zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1) = zero_zero_nat )
& ( M != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N_1)) ) ) ).
fof(fact_1732_power__eq__if,axiom,
! [P_3,M] :
( ( M = zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),M) = one_one_nat )
& ( M != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ).
fof(fact_1733_realpow__minus__mult,axiom,
! [X_21,N_31] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_21),N_31) ) ).
fof(fact_1734_realpow__minus__mult,axiom,
! [X_21,N_31] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_21),N_31) ) ).
fof(fact_1735_realpow__minus__mult,axiom,
! [X_21,N_31] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
=> hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_21),N_31) ) ).
fof(fact_1736_realpow__minus__mult,axiom,
! [X_21,N_31] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_21),N_31) ) ).
fof(fact_1737_realpow__minus__mult,axiom,
! [X_21,N_31] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
=> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_21),N_31) ) ).
fof(fact_1738_realpow__minus__mult,axiom,
! [X_21,N_31] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_21),N_31) ) ).
fof(fact_1739_realpow__minus__mult,axiom,
! [X_21,N_31] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
=> hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_21),N_31) ) ).
fof(fact_1740_not__less0,axiom,
! [N_1] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).
fof(fact_1741_neq0__conv,axiom,
! [N] :
( N != zero_zero_nat
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
fof(fact_1742_less__nat__zero__code,axiom,
! [N_1] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).
fof(fact_1743_gr__implies__not0,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> N_1 != zero_zero_nat ) ).
fof(fact_1744_gr0I,axiom,
! [N_1] :
( N_1 != zero_zero_nat
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).
fof(fact_1745_add__eq__self__zero,axiom,
! [M,N_1] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1) = M
=> N_1 = zero_zero_nat ) ).
fof(fact_1746_add__is__0,axiom,
! [Ma,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N) = zero_zero_nat
<=> ( Ma = zero_zero_nat
& N = zero_zero_nat ) ) ).
fof(fact_1747_Nat_Oadd__0__right,axiom,
! [M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat) = M ).
fof(fact_1748_plus__nat_Oadd__0,axiom,
! [N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N_1) = N_1 ).
fof(fact_1749_conj__le__cong,axiom,
! [P_2,P_1,X_2] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> ( hBOOL(P_1)
<=> hBOOL(P_2) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
& hBOOL(P_1) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
& hBOOL(P_2) ) ) ) ).
fof(fact_1750_imp__le__cong,axiom,
! [P_2,P_1,X_2] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> ( hBOOL(P_1)
<=> hBOOL(P_2) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> hBOOL(P_1) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> hBOOL(P_2) ) ) ) ).
fof(fact_1751_le__0__eq,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),zero_zero_nat))
<=> N = zero_zero_nat ) ).
fof(fact_1752_less__eq__nat_Osimps_I1_J,axiom,
! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1)) ).
fof(fact_1753_mult__0,axiom,
! [N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N_1) = zero_zero_nat ).
fof(fact_1754_mult__0__right,axiom,
! [M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),zero_zero_nat) = zero_zero_nat ).
fof(fact_1755_mult__is__0,axiom,
! [Ma,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N) = zero_zero_nat
<=> ( Ma = zero_zero_nat
| N = zero_zero_nat ) ) ).
fof(fact_1756_mult__cancel1,axiom,
! [K_1,Ma,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)
<=> ( Ma = N
| K_1 = zero_zero_nat ) ) ).
fof(fact_1757_mult__cancel2,axiom,
! [Ma,K_1,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_1)
<=> ( Ma = N
| K_1 = zero_zero_nat ) ) ).
fof(fact_1758_add__lessD1,axiom,
! [I,J_2,K_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),K_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K_2)) ) ).
fof(fact_1759_less__add__eq__less,axiom,
! [M,N_1,K_2,L_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),L_2))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),L_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N_1)
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ) ).
fof(fact_1760_add__less__mono,axiom,
! [K_2,L_2,I,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),L_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),L_2))) ) ) ).
fof(fact_1761_add__less__mono1,axiom,
! [K_2,I,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),K_2))) ) ).
fof(fact_1762_trans__less__add2,axiom,
! [M,I,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),J_2))) ) ).
fof(fact_1763_trans__less__add1,axiom,
! [M,I,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),M))) ) ).
fof(fact_1764_nat__add__left__cancel__less,axiom,
! [K_1,Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ).
fof(fact_1765_not__add__less2,axiom,
! [J_2,I] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),I)),I)) ).
fof(fact_1766_not__add__less1,axiom,
! [I,J_2] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),I)) ).
fof(fact_1767_less__or__eq__imp__le,axiom,
! [M,N_1] :
( ( 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)) ) ).
fof(fact_1768_le__neq__implies__less,axiom,
! [M,N_1] :
( 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)) ) ) ).
fof(fact_1769_less__imp__le__nat,axiom,
! [M,N_1] :
( 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)) ) ).
fof(fact_1770_le__eq__less__or__eq,axiom,
! [Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N))
| Ma = N ) ) ).
fof(fact_1771_nat__less__le,axiom,
! [Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N))
& Ma != N ) ) ).
fof(fact_1772_dvd__reduce,axiom,
! [K_1,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N)) ) ).
fof(fact_1773_add__leE,axiom,
! [M,K_2,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),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_2),N_1)) ) ) ).
fof(fact_1774_add__leD1,axiom,
! [M,K_2,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).
fof(fact_1775_add__leD2,axiom,
! [M,K_2,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_1)) ) ).
fof(fact_1776_add__le__mono,axiom,
! [K_2,L_2,I,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),L_2))) ) ) ).
fof(fact_1777_add__le__mono1,axiom,
! [K_2,I,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),K_2))) ) ).
fof(fact_1778_trans__le__add2,axiom,
! [M,I,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),J_2))) ) ).
fof(fact_1779_trans__le__add1,axiom,
! [M,I,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),M))) ) ).
fof(fact_1780_nat__add__left__cancel__le,axiom,
! [K_1,Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ).
fof(fact_1781_le__iff__add,axiom,
! [Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N))
<=> ? [K] : N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K) ) ).
fof(fact_1782_le__add1,axiom,
! [N_1,M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M))) ).
fof(fact_1783_le__add2,axiom,
! [N_1,M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1))) ).
fof(fact_1784_add__mult__distrib2,axiom,
! [K_2,M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)) ).
fof(fact_1785_add__mult__distrib,axiom,
! [M,N_1,K_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_2)) ).
fof(fact_1786_nat__dvd__1__iff__1,axiom,
! [Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),one_one_nat))
<=> Ma = one_one_nat ) ).
fof(fact_1787_le__square,axiom,
! [M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),M))) ).
fof(fact_1788_le__cube,axiom,
! [M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),M)))) ).
fof(fact_1789_mult__le__mono1,axiom,
! [K_2,I,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),K_2))) ) ).
fof(fact_1790_mult__le__mono2,axiom,
! [K_2,I,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),J_2))) ) ).
fof(fact_1791_mult__le__mono,axiom,
! [K_2,L_2,I,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),L_2))) ) ) ).
fof(fact_1792_diff__square,axiom,
! [X_1,Y_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),Y_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_1),Y_1)) ).
fof(fact_1793_nat__mult__1,axiom,
! [N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),N_1) = N_1 ).
fof(fact_1794_nat__1__eq__mult__iff,axiom,
! [Ma,N] :
( one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N)
<=> ( Ma = one_one_nat
& N = one_one_nat ) ) ).
fof(fact_1795_nat__mult__1__right,axiom,
! [N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),one_one_nat) = N_1 ).
fof(fact_1796_nat__mult__eq__1__iff,axiom,
! [Ma,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N) = one_one_nat
<=> ( Ma = one_one_nat
& N = one_one_nat ) ) ).
fof(fact_1797_nat__dvd__not__less,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),M)) ) ) ).
fof(fact_1798_add__gr__0,axiom,
! [Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
fof(fact_1799_mult__less__mono2,axiom,
! [K_2,I,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),J_2))) ) ) ).
fof(fact_1800_mult__less__mono1,axiom,
! [K_2,I,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),K_2))) ) ) ).
fof(fact_1801_mult__less__cancel2,axiom,
! [Ma,K_1,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),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,Ma),N)) ) ) ).
fof(fact_1802_mult__less__cancel1,axiom,
! [K_1,Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ) ).
fof(fact_1803_nat__0__less__mult__iff,axiom,
! [Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
fof(fact_1804_mult__eq__self__implies__10,axiom,
! [M,N_1] :
( M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)
=> ( N_1 = one_one_nat
| M = zero_zero_nat ) ) ).
fof(fact_1805_zdvd__mono,axiom,
! [Ma,Ta,K_1] :
( is_int(K_1)
=> ( K_1 != zero_zero_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),Ta))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),Ma)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),Ta))) ) ) ) ).
fof(fact_1806_unity__coeff__ex,axiom,
! [P_1,L] :
( ? [X] : hBOOL(hAPP_C24501650l_bool(P_1,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,L),X)))
<=> ? [X] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,L),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X),zero_z126310315umeral)))
& hBOOL(hAPP_C24501650l_bool(P_1,X)) ) ) ).
fof(fact_1807_unity__coeff__ex,axiom,
! [P_1,L] :
( ? [X] : hBOOL(hAPP_real_bool(P_1,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,L),X)))
<=> ? [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,L),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),zero_zero_real)))
& hBOOL(hAPP_real_bool(P_1,X)) ) ) ).
fof(fact_1808_unity__coeff__ex,axiom,
! [P_1,L] :
( ? [X] : hBOOL(hAPP_complex_bool(P_1,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,L),X)))
<=> ? [X] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,L),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),zero_zero_complex)))
& hBOOL(hAPP_complex_bool(P_1,X)) ) ) ).
fof(fact_1809_unity__coeff__ex,axiom,
! [P_1,L] :
( ? [X] : hBOOL(hAPP_Q2096512830t_bool(P_1,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,L),X)))
<=> ? [X] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,L),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X),zero_z891286103de_int)))
& hBOOL(hAPP_Q2096512830t_bool(P_1,X)) ) ) ).
fof(fact_1810_unity__coeff__ex,axiom,
! [P_1,L] :
( ? [X] : hBOOL(hAPP_rat_bool(P_1,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,L),X)))
<=> ? [X] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,L),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),zero_zero_rat)))
& hBOOL(hAPP_rat_bool(P_1,X)) ) ) ).
fof(fact_1811_unity__coeff__ex,axiom,
! [P_1,L] :
( ? [X] :
( is_int(X)
& hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,L),X))) )
<=> ? [X] :
( is_int(X)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,L),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),zero_zero_int)))
& hBOOL(hAPP_int_bool(P_1,X)) ) ) ).
fof(fact_1812_unity__coeff__ex,axiom,
! [P_1,L] :
( ? [X] : hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,L),X)))
<=> ? [X] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,L),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),zero_zero_nat)))
& hBOOL(hAPP_nat_bool(P_1,X)) ) ) ).
fof(fact_1813_inf__period_I3_J,axiom,
! [Ta,D,D_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),D_1))
=> ! [X,K] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Ta)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,K),D_1))),Ta))) ) ) ).
fof(fact_1814_inf__period_I3_J,axiom,
! [Ta,D,D_1] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),D_1))
=> ! [X,K] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Ta)))
<=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,K),D_1))),Ta))) ) ) ).
fof(fact_1815_inf__period_I3_J,axiom,
! [Ta,D,D_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),D_1))
=> ! [X,K] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Ta)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,K),D_1))),Ta))) ) ) ).
fof(fact_1816_inf__period_I3_J,axiom,
! [Ta,D,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1))
=> ! [X,K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Ta)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1))),Ta))) ) ) ).
fof(fact_1817_inf__period_I4_J,axiom,
! [Ta,D,D_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),D_1))
=> ! [X,K] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Ta)))
<=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,K),D_1))),Ta))) ) ) ).
fof(fact_1818_inf__period_I4_J,axiom,
! [Ta,D,D_1] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),D_1))
=> ! [X,K] :
( ~ hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Ta)))
<=> ~ hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,K),D_1))),Ta))) ) ) ).
fof(fact_1819_inf__period_I4_J,axiom,
! [Ta,D,D_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),D_1))
=> ! [X,K] :
( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Ta)))
<=> ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,K),D_1))),Ta))) ) ) ).
fof(fact_1820_inf__period_I4_J,axiom,
! [Ta,D,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1))
=> ! [X,K] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Ta)))
<=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1))),Ta))) ) ) ).
fof(fact_1821_number__of2,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,pls))) ).
fof(fact_1822_dvd__imp__le,axiom,
! [K_2,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_1)) ) ) ).
fof(fact_1823_dvd__mult__cancel,axiom,
! [K_2,M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N_1)) ) ) ).
fof(fact_1824_mult__le__cancel2,axiom,
! [Ma,K_1,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),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,Ma),N)) ) ) ).
fof(fact_1825_mult__le__cancel1,axiom,
! [K_1,Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ) ).
fof(fact_1826_decr__mult__lemma,axiom,
! [K_1,P_1,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X] :
( is_int(X)
=> ( hBOOL(hAPP_int_bool(P_1,X))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D))) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1))
=> ! [X] :
( hBOOL(hAPP_int_bool(P_1,X))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),D)))) ) ) ) ) ).
fof(fact_1827_ex__least__nat__less,axiom,
! [N,P_1] :
( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(P_1,N))
=> ? [K] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),N))
& ! [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),K))
=> ~ hBOOL(hAPP_nat_bool(P_1,I_2)) )
& hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat))) ) ) ) ).
fof(fact_1828_incr__mult__lemma,axiom,
! [K_1,P_1,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X] :
( is_int(X)
=> ( hBOOL(hAPP_int_bool(P_1,X))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D))) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1))
=> ! [X] :
( hBOOL(hAPP_int_bool(P_1,X))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),D)))) ) ) ) ) ).
fof(fact_1829_realpow__num__eq__if,axiom,
! [M_17,N_30] :
( ( N_30 = zero_zero_nat
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_17),N_30) = one_one_int )
& ( N_30 != zero_zero_nat
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_17),N_30) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_17),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat))) ) ) ).
fof(fact_1830_realpow__num__eq__if,axiom,
! [M_17,N_30] :
( ( N_30 = zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_17),N_30) = one_one_nat )
& ( N_30 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_17),N_30) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat))) ) ) ).
fof(fact_1831_realpow__num__eq__if,axiom,
! [M_17,N_30] :
( ( N_30 = zero_zero_nat
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,M_17),N_30) = one_one_real )
& ( N_30 != zero_zero_nat
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,M_17),N_30) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,M_17),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat))) ) ) ).
fof(fact_1832_realpow__num__eq__if,axiom,
! [M_17,N_30] :
( ( N_30 = zero_zero_nat
=> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_17),N_30) = one_on1645066479umeral )
& ( N_30 != zero_zero_nat
=> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_17),N_30) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,M_17),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat))) ) ) ).
fof(fact_1833_realpow__num__eq__if,axiom,
! [M_17,N_30] :
( ( N_30 = zero_zero_nat
=> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_17),N_30) = one_one_complex )
& ( N_30 != zero_zero_nat
=> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_17),N_30) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,M_17),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat))) ) ) ).
fof(fact_1834_realpow__num__eq__if,axiom,
! [M_17,N_30] :
( ( N_30 = zero_zero_nat
=> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_17),N_30) = one_on1684967323de_int )
& ( N_30 != zero_zero_nat
=> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_17),N_30) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,M_17),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat))) ) ) ).
fof(fact_1835_realpow__num__eq__if,axiom,
! [M_17,N_30] :
( ( N_30 = zero_zero_nat
=> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,M_17),N_30) = one_one_rat )
& ( N_30 != zero_zero_nat
=> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,M_17),N_30) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,M_17),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat))) ) ) ).
fof(fact_1836_zprime__factor__exists,axiom,
! [A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3))
=> ? [P_4] :
( is_int(P_4)
& hBOOL(hAPP_int_bool(zprime,P_4))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),A_3)) ) ) ).
fof(fact_1837_zcong__zless__unique,axiom,
! [A_3,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
=> ? [X] :
( is_int(X)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),X),M))
& ! [Y] :
( is_int(Y)
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),M))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),Y),M)) )
=> Y = X ) ) ) ) ).
fof(fact_1838_norR__mem__unique__aux,axiom,
! [A_3,B_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),one_one_int)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)) ) ).
fof(fact_1839_nat__less__add__iff1,axiom,
! [U,Ma,N,J,I_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U)),Ma)),N)) ) ) ).
fof(fact_1840_nat__less__add__iff2,axiom,
! [U,Ma,N,I_1,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)),U)),N))) ) ) ).
fof(fact_1841_diff__commute,axiom,
! [I,J_2,K_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J_2)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),K_2)),J_2) ).
fof(fact_1842_nat__mult__eq__cancel__disj,axiom,
! [K_1,Ma,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)
<=> ( K_1 = zero_zero_nat
| Ma = N ) ) ).
fof(fact_1843_left__add__mult__distrib,axiom,
! [I,U_1,J_2,K_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),U_1)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),U_1)),K_2) ).
fof(fact_1844_nat__mult__eq__cancel1,axiom,
! [Ma,N,K_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)
<=> Ma = N ) ) ).
fof(fact_1845_nat__mult__less__cancel1,axiom,
! [Ma,N,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,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ) ).
fof(fact_1846_nat__mult__dvd__cancel__disj,axiom,
! [K_1,Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)))
<=> ( K_1 = zero_zero_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),N)) ) ) ).
fof(fact_1847_nat__eq__add__iff2,axiom,
! [U,Ma,N,I_1,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N)
<=> Ma = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)),U)),N) ) ) ).
fof(fact_1848_nat__diff__add__eq2,axiom,
! [U_1,M,N_1,I,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),U_1)),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),I)),U_1)),N_1)) ) ).
fof(fact_1849_nat__le__add__iff2,axiom,
! [U,Ma,N,I_1,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)),U)),N))) ) ) ).
fof(fact_1850_nat__eq__add__iff1,axiom,
! [U,Ma,N,J,I_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N)
<=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U)),Ma) = N ) ) ).
fof(fact_1851_nat__diff__add__eq1,axiom,
! [U_1,M,N_1,J_2,I] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),U_1)),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J_2)),U_1)),M)),N_1) ) ).
fof(fact_1852_nat__le__add__iff1,axiom,
! [U,Ma,N,J,I_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U)),Ma)),N)) ) ) ).
fof(fact_1853_nat__mult__dvd__cancel1,axiom,
! [Ma,N,K_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),N)) ) ) ).
fof(fact_1854_nat__mult__le__cancel1,axiom,
! [Ma,N,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,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ) ).
fof(fact_1855_Wilson__Russ,axiom,
! [P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))),hAPP_int_int(number_number_of_int,min)),P_3)) ) ).
fof(fact_1856_add__nonpos__neg,axiom,
! [B_97,A_118] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_118),zero_z126310315umeral))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_97),zero_z126310315umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_118),B_97)),zero_z126310315umeral)) ) ) ).
fof(fact_1857_add__nonpos__neg,axiom,
! [B_97,A_118] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_118),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_97),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_118),B_97)),zero_zero_int)) ) ) ).
fof(fact_1858_add__nonpos__neg,axiom,
! [B_97,A_118] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_118),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_97),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_118),B_97)),zero_zero_nat)) ) ) ).
fof(fact_1859_add__nonpos__neg,axiom,
! [B_97,A_118] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_118),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_97),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_118),B_97)),zero_zero_real)) ) ) ).
fof(fact_1860_add__nonpos__neg,axiom,
! [B_97,A_118] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_118),zero_z891286103de_int))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_97),zero_z891286103de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_118),B_97)),zero_z891286103de_int)) ) ) ).
fof(fact_1861_add__nonpos__neg,axiom,
! [B_97,A_118] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_118),zero_zero_rat))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_97),zero_zero_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_118),B_97)),zero_zero_rat)) ) ) ).
fof(fact_1862_add__neg__nonpos,axiom,
! [B_96,A_117] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_117),zero_z126310315umeral))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_96),zero_z126310315umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_117),B_96)),zero_z126310315umeral)) ) ) ).
fof(fact_1863_add__neg__nonpos,axiom,
! [B_96,A_117] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_117),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_96),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_117),B_96)),zero_zero_int)) ) ) ).
fof(fact_1864_add__neg__nonpos,axiom,
! [B_96,A_117] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_117),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_96),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_117),B_96)),zero_zero_nat)) ) ) ).
fof(fact_1865_add__neg__nonpos,axiom,
! [B_96,A_117] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_117),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_96),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_117),B_96)),zero_zero_real)) ) ) ).
fof(fact_1866_add__neg__nonpos,axiom,
! [B_96,A_117] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_117),zero_z891286103de_int))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_96),zero_z891286103de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_117),B_96)),zero_z891286103de_int)) ) ) ).
fof(fact_1867_add__neg__nonpos,axiom,
! [B_96,A_117] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_117),zero_zero_rat))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_96),zero_zero_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_117),B_96)),zero_zero_rat)) ) ) ).
fof(fact_1868_add__strict__increasing2,axiom,
! [B_95,C_53,A_116] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_116))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_95),C_53))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_95),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_116),C_53))) ) ) ).
fof(fact_1869_add__strict__increasing2,axiom,
! [B_95,C_53,A_116] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_116))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_95),C_53))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_95),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_116),C_53))) ) ) ).
fof(fact_1870_add__strict__increasing2,axiom,
! [B_95,C_53,A_116] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_116))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_95),C_53))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_95),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_116),C_53))) ) ) ).
fof(fact_1871_add__strict__increasing2,axiom,
! [B_95,C_53,A_116] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_116))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_95),C_53))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_95),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_116),C_53))) ) ) ).
fof(fact_1872_add__strict__increasing2,axiom,
! [B_95,C_53,A_116] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_116))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_95),C_53))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_95),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_116),C_53))) ) ) ).
fof(fact_1873_add__strict__increasing2,axiom,
! [B_95,C_53,A_116] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_116))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_95),C_53))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_95),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_116),C_53))) ) ) ).
fof(fact_1874_add__strict__increasing,axiom,
! [B_94,C_52,A_115] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_115))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_94),C_52))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_94),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_115),C_52))) ) ) ).
fof(fact_1875_add__strict__increasing,axiom,
! [B_94,C_52,A_115] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_115))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_94),C_52))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_94),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_115),C_52))) ) ) ).
fof(fact_1876_add__strict__increasing,axiom,
! [B_94,C_52,A_115] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_115))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_94),C_52))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_94),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_115),C_52))) ) ) ).
fof(fact_1877_add__strict__increasing,axiom,
! [B_94,C_52,A_115] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_115))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_94),C_52))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_94),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_115),C_52))) ) ) ).
fof(fact_1878_add__strict__increasing,axiom,
! [B_94,C_52,A_115] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_115))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_94),C_52))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_94),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_115),C_52))) ) ) ).
fof(fact_1879_add__strict__increasing,axiom,
! [B_94,C_52,A_115] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_115))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_94),C_52))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_94),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_115),C_52))) ) ) ).
fof(fact_1880_add__nonneg__pos,axiom,
! [B_93,A_114] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_114))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_93))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_114),B_93))) ) ) ).
fof(fact_1881_add__nonneg__pos,axiom,
! [B_93,A_114] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_114))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_93))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_114),B_93))) ) ) ).
fof(fact_1882_add__nonneg__pos,axiom,
! [B_93,A_114] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_114))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_93))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_114),B_93))) ) ) ).
fof(fact_1883_add__nonneg__pos,axiom,
! [B_93,A_114] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_114))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_93))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_114),B_93))) ) ) ).
fof(fact_1884_add__nonneg__pos,axiom,
! [B_93,A_114] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_114))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_93))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_114),B_93))) ) ) ).
fof(fact_1885_add__nonneg__pos,axiom,
! [B_93,A_114] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_114))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_93))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_114),B_93))) ) ) ).
fof(fact_1886_add__pos__nonneg,axiom,
! [B_92,A_113] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_113))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_92))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_113),B_92))) ) ) ).
fof(fact_1887_add__pos__nonneg,axiom,
! [B_92,A_113] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_113))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_92))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_113),B_92))) ) ) ).
fof(fact_1888_add__pos__nonneg,axiom,
! [B_92,A_113] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_113))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_92))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_113),B_92))) ) ) ).
fof(fact_1889_add__pos__nonneg,axiom,
! [B_92,A_113] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_113))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_92))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_113),B_92))) ) ) ).
fof(fact_1890_add__pos__nonneg,axiom,
! [B_92,A_113] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_113))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_92))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_113),B_92))) ) ) ).
fof(fact_1891_add__pos__nonneg,axiom,
! [B_92,A_113] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_113))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_92))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_113),B_92))) ) ) ).
fof(fact_1892_inv__inv,axiom,
! [A_3,P_3] :
( is_int(A_3)
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),P_3))
=> hAPP_int_int(inv(P_3),hAPP_int_int(inv(P_3),A_3)) = A_3 ) ) ) ) ) ).
fof(fact_1893_MultInvPair__distinct,axiom,
! [J_2,A_3,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_3))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),P_3)) ) ) ) ) ) ).
fof(fact_1894_aux______3,axiom,
! [J_2,K_2,A_3,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),A_3),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),J_2)),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),A_3)),P_3)) ) ).
fof(fact_1895_aux______1,axiom,
! [J_2,A_3,P_3,K_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),K_2)),P_3)) ) ).
fof(fact_1896_zero__reorient,axiom,
! [X_2] :
( is_int(X_2)
=> ( zero_zero_int = X_2
<=> X_2 = zero_zero_int ) ) ).
fof(fact_1897_zero__reorient,axiom,
! [X_2] :
( zero_zero_nat = X_2
<=> X_2 = zero_zero_nat ) ).
fof(fact_1898_zero__reorient,axiom,
! [X_2] :
( zero_zero_real = X_2
<=> X_2 = zero_zero_real ) ).
fof(fact_1899_zero__reorient,axiom,
! [X_2] :
( zero_z126310315umeral = X_2
<=> X_2 = zero_z126310315umeral ) ).
fof(fact_1900_zero__reorient,axiom,
! [X_2] :
( zero_zero_complex = X_2
<=> X_2 = zero_zero_complex ) ).
fof(fact_1901_zero__reorient,axiom,
! [X_2] :
( zero_z891286103de_int = X_2
<=> X_2 = zero_z891286103de_int ) ).
fof(fact_1902_zero__reorient,axiom,
! [X_2] :
( zero_zero_rat = X_2
<=> X_2 = zero_zero_rat ) ).
fof(fact_1903_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_112,B_91,C_51] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_112),B_91)),C_51) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_112),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_91),C_51)) ).
fof(fact_1904_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_112,B_91,C_51] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_112),B_91)),C_51) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_112),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_91),C_51)) ).
fof(fact_1905_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_112,B_91,C_51] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_112),B_91)),C_51) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_112),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_91),C_51)) ).
fof(fact_1906_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_112,B_91,C_51] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_112),B_91)),C_51) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_112),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_91),C_51)) ).
fof(fact_1907_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_112,B_91,C_51] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_112),B_91)),C_51) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_112),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_91),C_51)) ).
fof(fact_1908_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_112,B_91,C_51] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_112),B_91)),C_51) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_112),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_91),C_51)) ).
fof(fact_1909_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_112,B_91,C_51] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_112),B_91)),C_51) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_112),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_91),C_51)) ).
fof(fact_1910_add__right__imp__eq,axiom,
! [B_90,A_111,C_50] :
( hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_90),A_111) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_50),A_111)
=> B_90 = C_50 ) ).
fof(fact_1911_add__right__imp__eq,axiom,
! [B_90,A_111,C_50] :
( ( is_int(B_90)
& is_int(C_50) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_90),A_111) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_50),A_111)
=> B_90 = C_50 ) ) ).
fof(fact_1912_add__right__imp__eq,axiom,
! [B_90,A_111,C_50] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_90),A_111) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_50),A_111)
=> B_90 = C_50 ) ).
fof(fact_1913_add__right__imp__eq,axiom,
! [B_90,A_111,C_50] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_90),A_111) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_50),A_111)
=> B_90 = C_50 ) ).
fof(fact_1914_add__right__imp__eq,axiom,
! [B_90,A_111,C_50] :
( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_90),A_111) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_50),A_111)
=> B_90 = C_50 ) ).
fof(fact_1915_add__right__imp__eq,axiom,
! [B_90,A_111,C_50] :
( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_90),A_111) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_50),A_111)
=> B_90 = C_50 ) ).
fof(fact_1916_add__right__imp__eq,axiom,
! [B_90,A_111,C_50] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_90),A_111) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_50),A_111)
=> B_90 = C_50 ) ).
fof(fact_1917_add__imp__eq,axiom,
! [A_110,B_89,C_49] :
( hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_110),B_89) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_110),C_49)
=> B_89 = C_49 ) ).
fof(fact_1918_add__imp__eq,axiom,
! [A_110,B_89,C_49] :
( ( is_int(B_89)
& is_int(C_49) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_110),B_89) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_110),C_49)
=> B_89 = C_49 ) ) ).
fof(fact_1919_add__imp__eq,axiom,
! [A_110,B_89,C_49] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_110),B_89) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_110),C_49)
=> B_89 = C_49 ) ).
fof(fact_1920_add__imp__eq,axiom,
! [A_110,B_89,C_49] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_110),B_89) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_110),C_49)
=> B_89 = C_49 ) ).
fof(fact_1921_add__imp__eq,axiom,
! [A_110,B_89,C_49] :
( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_110),B_89) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_110),C_49)
=> B_89 = C_49 ) ).
fof(fact_1922_add__imp__eq,axiom,
! [A_110,B_89,C_49] :
( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_110),B_89) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_110),C_49)
=> B_89 = C_49 ) ).
fof(fact_1923_add__imp__eq,axiom,
! [A_110,B_89,C_49] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_110),B_89) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_110),C_49)
=> B_89 = C_49 ) ).
fof(fact_1924_add__left__imp__eq,axiom,
! [A_109,B_88,C_48] :
( hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_109),B_88) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_109),C_48)
=> B_88 = C_48 ) ).
fof(fact_1925_add__left__imp__eq,axiom,
! [A_109,B_88,C_48] :
( ( is_int(B_88)
& is_int(C_48) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_109),B_88) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_109),C_48)
=> B_88 = C_48 ) ) ).
fof(fact_1926_add__left__imp__eq,axiom,
! [A_109,B_88,C_48] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_109),B_88) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_109),C_48)
=> B_88 = C_48 ) ).
fof(fact_1927_add__left__imp__eq,axiom,
! [A_109,B_88,C_48] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_109),B_88) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_109),C_48)
=> B_88 = C_48 ) ).
fof(fact_1928_add__left__imp__eq,axiom,
! [A_109,B_88,C_48] :
( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_109),B_88) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_109),C_48)
=> B_88 = C_48 ) ).
fof(fact_1929_add__left__imp__eq,axiom,
! [A_109,B_88,C_48] :
( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_109),B_88) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_109),C_48)
=> B_88 = C_48 ) ).
fof(fact_1930_add__left__imp__eq,axiom,
! [A_109,B_88,C_48] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_109),B_88) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_109),C_48)
=> B_88 = C_48 ) ).
fof(fact_1931_add__right__cancel,axiom,
! [B,A,C] :
( hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B),A) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),A)
<=> B = C ) ).
fof(fact_1932_add__right__cancel,axiom,
! [B,A,C] :
( ( is_int(B)
& is_int(C) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B),A) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),A)
<=> B = C ) ) ).
fof(fact_1933_add__right__cancel,axiom,
! [B,A,C] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),A)
<=> B = C ) ).
fof(fact_1934_add__right__cancel,axiom,
! [B,A,C] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B),A) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),A)
<=> B = C ) ).
fof(fact_1935_add__right__cancel,axiom,
! [B,A,C] :
( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B),A) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C),A)
<=> B = C ) ).
fof(fact_1936_add__right__cancel,axiom,
! [B,A,C] :
( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B),A) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),A)
<=> B = C ) ).
fof(fact_1937_add__right__cancel,axiom,
! [B,A,C] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B),A) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),A)
<=> B = C ) ).
fof(fact_1938_add__left__cancel,axiom,
! [A,B,C] :
( hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A),B) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A),C)
<=> B = C ) ).
fof(fact_1939_add__left__cancel,axiom,
! [A,B,C] :
( ( is_int(B)
& is_int(C) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),C)
<=> B = C ) ) ).
fof(fact_1940_add__left__cancel,axiom,
! [A,B,C] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),B) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),C)
<=> B = C ) ).
fof(fact_1941_add__left__cancel,axiom,
! [A,B,C] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),B) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),C)
<=> B = C ) ).
fof(fact_1942_add__left__cancel,axiom,
! [A,B,C] :
( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A),B) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A),C)
<=> B = C ) ).
fof(fact_1943_add__left__cancel,axiom,
! [A,B,C] :
( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A),B) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A),C)
<=> B = C ) ).
fof(fact_1944_add__left__cancel,axiom,
! [A,B,C] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),B) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),C)
<=> B = C ) ).
fof(fact_1945_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_108,B_87,C_47] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_108),B_87)),C_47) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_108),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_87),C_47)) ).
fof(fact_1946_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_108,B_87,C_47] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_108),B_87)),C_47) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_108),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_87),C_47)) ).
fof(fact_1947_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_108,B_87,C_47] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_108),B_87)),C_47) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_108),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_87),C_47)) ).
fof(fact_1948_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_108,B_87,C_47] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_108),B_87)),C_47) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_108),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_87),C_47)) ).
fof(fact_1949_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_108,B_87,C_47] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_108),B_87)),C_47) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_108),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_87),C_47)) ).
fof(fact_1950_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_108,B_87,C_47] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_108),B_87)),C_47) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_108),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_87),C_47)) ).
fof(fact_1951_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_108,B_87,C_47] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_108),B_87)),C_47) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_108),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_87),C_47)) ).
fof(fact_1952_one__reorient,axiom,
! [X_2] :
( is_int(X_2)
=> ( one_one_int = X_2
<=> X_2 = one_one_int ) ) ).
fof(fact_1953_one__reorient,axiom,
! [X_2] :
( one_one_nat = X_2
<=> X_2 = one_one_nat ) ).
fof(fact_1954_one__reorient,axiom,
! [X_2] :
( one_one_real = X_2
<=> X_2 = one_one_real ) ).
fof(fact_1955_one__reorient,axiom,
! [X_2] :
( one_on1645066479umeral = X_2
<=> X_2 = one_on1645066479umeral ) ).
fof(fact_1956_one__reorient,axiom,
! [X_2] :
( one_one_complex = X_2
<=> X_2 = one_one_complex ) ).
fof(fact_1957_one__reorient,axiom,
! [X_2] :
( one_on1684967323de_int = X_2
<=> X_2 = one_on1684967323de_int ) ).
fof(fact_1958_one__reorient,axiom,
! [X_2] :
( one_one_rat = X_2
<=> X_2 = one_one_rat ) ).
fof(fact_1959_diff__eq__diff__eq,axiom,
! [A,B,C,D] :
( ( is_int(A)
& is_int(B)
& is_int(C)
& is_int(D) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D)
=> ( A = B
<=> C = D ) ) ) ).
fof(fact_1960_diff__eq__diff__eq,axiom,
! [A,B,C,D] :
( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D)
=> ( A = B
<=> C = D ) ) ).
fof(fact_1961_diff__eq__diff__eq,axiom,
! [A,B,C,D] :
( hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A),B) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,C),D)
=> ( A = B
<=> C = D ) ) ).
fof(fact_1962_diff__eq__diff__eq,axiom,
! [A,B,C,D] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D)
=> ( A = B
<=> C = D ) ) ).
fof(fact_1963_add_Ocomm__neutral,axiom,
! [A_107] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_107),zero_z126310315umeral) = A_107 ).
fof(fact_1964_add_Ocomm__neutral,axiom,
! [A_107] :
( is_int(A_107)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_107),zero_zero_int) = A_107 ) ).
fof(fact_1965_add_Ocomm__neutral,axiom,
! [A_107] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_107),zero_zero_nat) = A_107 ).
fof(fact_1966_add_Ocomm__neutral,axiom,
! [A_107] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_107),zero_zero_real) = A_107 ).
fof(fact_1967_add_Ocomm__neutral,axiom,
! [A_107] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_107),zero_zero_complex) = A_107 ).
fof(fact_1968_add_Ocomm__neutral,axiom,
! [A_107] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_107),zero_z891286103de_int) = A_107 ).
fof(fact_1969_add_Ocomm__neutral,axiom,
! [A_107] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_107),zero_zero_rat) = A_107 ).
fof(fact_1970_add__0__right,axiom,
! [A_106] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_106),zero_z126310315umeral) = A_106 ).
fof(fact_1971_add__0__right,axiom,
! [A_106] :
( is_int(A_106)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_106),zero_zero_int) = A_106 ) ).
fof(fact_1972_add__0__right,axiom,
! [A_106] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_106),zero_zero_nat) = A_106 ).
fof(fact_1973_add__0__right,axiom,
! [A_106] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_106),zero_zero_real) = A_106 ).
fof(fact_1974_add__0__right,axiom,
! [A_106] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_106),zero_zero_complex) = A_106 ).
fof(fact_1975_add__0__right,axiom,
! [A_106] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_106),zero_z891286103de_int) = A_106 ).
fof(fact_1976_add__0__right,axiom,
! [A_106] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_106),zero_zero_rat) = A_106 ).
fof(fact_1977_double__zero__sym,axiom,
! [A] :
( is_int(A)
=> ( zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A)
<=> A = zero_zero_int ) ) ).
fof(fact_1978_double__zero__sym,axiom,
! [A] :
( zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A)
<=> A = zero_zero_real ) ).
fof(fact_1979_double__zero__sym,axiom,
! [A] :
( zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A)
<=> A = zero_zero_rat ) ).
fof(fact_1980_add__0,axiom,
! [A_105] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_105) = A_105 ).
fof(fact_1981_add__0,axiom,
! [A_105] :
( is_int(A_105)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_105) = A_105 ) ).
fof(fact_1982_add__0,axiom,
! [A_105] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_105) = A_105 ).
fof(fact_1983_add__0,axiom,
! [A_105] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_105) = A_105 ).
fof(fact_1984_add__0,axiom,
! [A_105] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_105) = A_105 ).
fof(fact_1985_add__0,axiom,
! [A_105] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_105) = A_105 ).
fof(fact_1986_add__0,axiom,
! [A_105] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_105) = A_105 ).
fof(fact_1987_add__0__left,axiom,
! [A_104] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_104) = A_104 ).
fof(fact_1988_add__0__left,axiom,
! [A_104] :
( is_int(A_104)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_104) = A_104 ) ).
fof(fact_1989_add__0__left,axiom,
! [A_104] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_104) = A_104 ).
fof(fact_1990_add__0__left,axiom,
! [A_104] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_104) = A_104 ).
fof(fact_1991_add__0__left,axiom,
! [A_104] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_104) = A_104 ).
fof(fact_1992_add__0__left,axiom,
! [A_104] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_104) = A_104 ).
fof(fact_1993_add__0__left,axiom,
! [A_104] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_104) = A_104 ).
fof(fact_1994_add__le__imp__le__left,axiom,
! [C_46,A_103,B_86] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_46),A_103)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_46),B_86)))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_103),B_86)) ) ).
fof(fact_1995_add__le__imp__le__left,axiom,
! [C_46,A_103,B_86] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_46),A_103)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_46),B_86)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_103),B_86)) ) ).
fof(fact_1996_add__le__imp__le__left,axiom,
! [C_46,A_103,B_86] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_46),A_103)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_46),B_86)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_103),B_86)) ) ).
fof(fact_1997_add__le__imp__le__left,axiom,
! [C_46,A_103,B_86] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_46),A_103)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_46),B_86)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_103),B_86)) ) ).
fof(fact_1998_add__le__imp__le__left,axiom,
! [C_46,A_103,B_86] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_46),A_103)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_46),B_86)))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_103),B_86)) ) ).
fof(fact_1999_add__le__imp__le__left,axiom,
! [C_46,A_103,B_86] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_46),A_103)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_46),B_86)))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_103),B_86)) ) ).
fof(fact_2000_add__le__imp__le__right,axiom,
! [A_102,C_45,B_85] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_102),C_45)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_85),C_45)))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_102),B_85)) ) ).
fof(fact_2001_add__le__imp__le__right,axiom,
! [A_102,C_45,B_85] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_102),C_45)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_85),C_45)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_102),B_85)) ) ).
fof(fact_2002_add__le__imp__le__right,axiom,
! [A_102,C_45,B_85] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_102),C_45)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_85),C_45)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_102),B_85)) ) ).
fof(fact_2003_add__le__imp__le__right,axiom,
! [A_102,C_45,B_85] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_102),C_45)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_85),C_45)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_102),B_85)) ) ).
fof(fact_2004_add__le__imp__le__right,axiom,
! [A_102,C_45,B_85] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_102),C_45)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_85),C_45)))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_102),B_85)) ) ).
fof(fact_2005_add__le__imp__le__right,axiom,
! [A_102,C_45,B_85] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_102),C_45)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_85),C_45)))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_102),B_85)) ) ).
fof(fact_2006_add__mono,axiom,
! [C_44,D_8,A_101,B_84] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_101),B_84))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_44),D_8))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_101),C_44)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_84),D_8))) ) ) ).
fof(fact_2007_add__mono,axiom,
! [C_44,D_8,A_101,B_84] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_101),B_84))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_44),D_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_101),C_44)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_84),D_8))) ) ) ).
fof(fact_2008_add__mono,axiom,
! [C_44,D_8,A_101,B_84] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_101),B_84))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_44),D_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_101),C_44)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_84),D_8))) ) ) ).
fof(fact_2009_add__mono,axiom,
! [C_44,D_8,A_101,B_84] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_101),B_84))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_44),D_8))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_101),C_44)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_84),D_8))) ) ) ).
fof(fact_2010_add__mono,axiom,
! [C_44,D_8,A_101,B_84] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_101),B_84))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_44),D_8))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_101),C_44)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_84),D_8))) ) ) ).
fof(fact_2011_add__mono,axiom,
! [C_44,D_8,A_101,B_84] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_101),B_84))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_44),D_8))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_101),C_44)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_84),D_8))) ) ) ).
fof(fact_2012_add__left__mono,axiom,
! [C_43,A_100,B_83] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_100),B_83))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_43),A_100)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_43),B_83))) ) ).
fof(fact_2013_add__left__mono,axiom,
! [C_43,A_100,B_83] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_100),B_83))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_43),A_100)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_43),B_83))) ) ).
fof(fact_2014_add__left__mono,axiom,
! [C_43,A_100,B_83] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_100),B_83))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_43),A_100)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_43),B_83))) ) ).
fof(fact_2015_add__left__mono,axiom,
! [C_43,A_100,B_83] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_100),B_83))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_43),A_100)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_43),B_83))) ) ).
fof(fact_2016_add__left__mono,axiom,
! [C_43,A_100,B_83] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_100),B_83))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_43),A_100)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_43),B_83))) ) ).
fof(fact_2017_add__left__mono,axiom,
! [C_43,A_100,B_83] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_100),B_83))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_43),A_100)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_43),B_83))) ) ).
fof(fact_2018_add__right__mono,axiom,
! [C_42,A_99,B_82] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_99),B_82))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_99),C_42)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_82),C_42))) ) ).
fof(fact_2019_add__right__mono,axiom,
! [C_42,A_99,B_82] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_99),B_82))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_99),C_42)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_82),C_42))) ) ).
fof(fact_2020_add__right__mono,axiom,
! [C_42,A_99,B_82] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_99),B_82))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_99),C_42)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_82),C_42))) ) ).
fof(fact_2021_add__right__mono,axiom,
! [C_42,A_99,B_82] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_99),B_82))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_99),C_42)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_82),C_42))) ) ).
fof(fact_2022_add__right__mono,axiom,
! [C_42,A_99,B_82] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_99),B_82))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_99),C_42)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_82),C_42))) ) ).
fof(fact_2023_add__right__mono,axiom,
! [C_42,A_99,B_82] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_99),B_82))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_99),C_42)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_82),C_42))) ) ).
fof(fact_2024_add__le__cancel__left,axiom,
! [C,A,B] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),A)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),B)))
<=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A),B)) ) ).
fof(fact_2025_add__le__cancel__left,axiom,
! [C,A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B)) ) ).
fof(fact_2026_add__le__cancel__left,axiom,
! [C,A,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),B)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B)) ) ).
fof(fact_2027_add__le__cancel__left,axiom,
! [C,A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),B)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) ) ).
fof(fact_2028_add__le__cancel__left,axiom,
! [C,A,B] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),A)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),B)))
<=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A),B)) ) ).
fof(fact_2029_add__le__cancel__left,axiom,
! [C,A,B] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),B)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B)) ) ).
fof(fact_2030_add__le__cancel__right,axiom,
! [A,C,B] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A),C)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B),C)))
<=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A),B)) ) ).
fof(fact_2031_add__le__cancel__right,axiom,
! [A,C,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B),C)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B)) ) ).
fof(fact_2032_add__le__cancel__right,axiom,
! [A,C,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),C)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B)) ) ).
fof(fact_2033_add__le__cancel__right,axiom,
! [A,C,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B),C)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) ) ).
fof(fact_2034_add__le__cancel__right,axiom,
! [A,C,B] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A),C)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B),C)))
<=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A),B)) ) ).
fof(fact_2035_add__le__cancel__right,axiom,
! [A,C,B] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B),C)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B)) ) ).
fof(fact_2036_add__less__imp__less__left,axiom,
! [C_41,A_98,B_81] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_41),A_98)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_41),B_81)))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_98),B_81)) ) ).
fof(fact_2037_add__less__imp__less__left,axiom,
! [C_41,A_98,B_81] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_41),A_98)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_41),B_81)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_98),B_81)) ) ).
fof(fact_2038_add__less__imp__less__left,axiom,
! [C_41,A_98,B_81] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_41),A_98)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_41),B_81)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_98),B_81)) ) ).
fof(fact_2039_add__less__imp__less__left,axiom,
! [C_41,A_98,B_81] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_41),A_98)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_41),B_81)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_98),B_81)) ) ).
fof(fact_2040_add__less__imp__less__left,axiom,
! [C_41,A_98,B_81] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_41),A_98)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_41),B_81)))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_98),B_81)) ) ).
fof(fact_2041_add__less__imp__less__left,axiom,
! [C_41,A_98,B_81] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_41),A_98)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_41),B_81)))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_98),B_81)) ) ).
fof(fact_2042_add__less__imp__less__right,axiom,
! [A_97,C_40,B_80] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_97),C_40)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_80),C_40)))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_97),B_80)) ) ).
fof(fact_2043_add__less__imp__less__right,axiom,
! [A_97,C_40,B_80] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_97),C_40)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_80),C_40)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_97),B_80)) ) ).
fof(fact_2044_add__less__imp__less__right,axiom,
! [A_97,C_40,B_80] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_97),C_40)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_80),C_40)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_97),B_80)) ) ).
fof(fact_2045_add__less__imp__less__right,axiom,
! [A_97,C_40,B_80] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_97),C_40)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_80),C_40)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_97),B_80)) ) ).
fof(fact_2046_add__less__imp__less__right,axiom,
! [A_97,C_40,B_80] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_97),C_40)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_80),C_40)))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_97),B_80)) ) ).
fof(fact_2047_add__less__imp__less__right,axiom,
! [A_97,C_40,B_80] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_97),C_40)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_80),C_40)))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_97),B_80)) ) ).
fof(fact_2048_add__strict__mono,axiom,
! [C_39,D_7,A_96,B_79] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_96),B_79))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_39),D_7))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_96),C_39)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_79),D_7))) ) ) ).
fof(fact_2049_add__strict__mono,axiom,
! [C_39,D_7,A_96,B_79] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_96),B_79))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_39),D_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_96),C_39)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_79),D_7))) ) ) ).
fof(fact_2050_add__strict__mono,axiom,
! [C_39,D_7,A_96,B_79] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_96),B_79))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_39),D_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_96),C_39)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_79),D_7))) ) ) ).
fof(fact_2051_add__strict__mono,axiom,
! [C_39,D_7,A_96,B_79] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_96),B_79))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_39),D_7))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_96),C_39)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_79),D_7))) ) ) ).
fof(fact_2052_add__strict__mono,axiom,
! [C_39,D_7,A_96,B_79] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_96),B_79))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_39),D_7))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_96),C_39)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_79),D_7))) ) ) ).
fof(fact_2053_add__strict__mono,axiom,
! [C_39,D_7,A_96,B_79] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_96),B_79))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_39),D_7))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_96),C_39)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_79),D_7))) ) ) ).
fof(fact_2054_add__strict__left__mono,axiom,
! [C_38,A_95,B_78] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_95),B_78))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_38),A_95)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_38),B_78))) ) ).
fof(fact_2055_add__strict__left__mono,axiom,
! [C_38,A_95,B_78] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_95),B_78))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_38),A_95)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_38),B_78))) ) ).
fof(fact_2056_add__strict__left__mono,axiom,
! [C_38,A_95,B_78] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_95),B_78))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_38),A_95)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_38),B_78))) ) ).
fof(fact_2057_add__strict__left__mono,axiom,
! [C_38,A_95,B_78] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_95),B_78))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_38),A_95)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_38),B_78))) ) ).
fof(fact_2058_add__strict__left__mono,axiom,
! [C_38,A_95,B_78] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_95),B_78))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_38),A_95)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_38),B_78))) ) ).
fof(fact_2059_add__strict__left__mono,axiom,
! [C_38,A_95,B_78] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_95),B_78))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_38),A_95)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_38),B_78))) ) ).
fof(fact_2060_add__strict__right__mono,axiom,
! [C_37,A_94,B_77] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_94),B_77))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_94),C_37)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_77),C_37))) ) ).
fof(fact_2061_add__strict__right__mono,axiom,
! [C_37,A_94,B_77] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_94),B_77))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_94),C_37)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_77),C_37))) ) ).
fof(fact_2062_add__strict__right__mono,axiom,
! [C_37,A_94,B_77] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_94),B_77))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_94),C_37)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_77),C_37))) ) ).
fof(fact_2063_add__strict__right__mono,axiom,
! [C_37,A_94,B_77] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_94),B_77))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_94),C_37)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_77),C_37))) ) ).
fof(fact_2064_add__strict__right__mono,axiom,
! [C_37,A_94,B_77] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_94),B_77))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_94),C_37)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_77),C_37))) ) ).
fof(fact_2065_add__strict__right__mono,axiom,
! [C_37,A_94,B_77] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_94),B_77))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_94),C_37)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_77),C_37))) ) ).
fof(fact_2066_add__less__cancel__left,axiom,
! [C,A,B] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),A)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),B)))
<=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A),B)) ) ).
fof(fact_2067_add__less__cancel__left,axiom,
! [C,A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) ) ).
fof(fact_2068_add__less__cancel__left,axiom,
! [C,A,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),B)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B)) ) ).
fof(fact_2069_add__less__cancel__left,axiom,
! [C,A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),B)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) ) ).
fof(fact_2070_add__less__cancel__left,axiom,
! [C,A,B] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),A)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),B)))
<=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A),B)) ) ).
fof(fact_2071_add__less__cancel__left,axiom,
! [C,A,B] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),B)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)) ) ).
fof(fact_2072_add__less__cancel__right,axiom,
! [A,C,B] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A),C)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B),C)))
<=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A),B)) ) ).
fof(fact_2073_add__less__cancel__right,axiom,
! [A,C,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B),C)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) ) ).
fof(fact_2074_add__less__cancel__right,axiom,
! [A,C,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),C)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B)) ) ).
fof(fact_2075_add__less__cancel__right,axiom,
! [A,C,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B),C)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) ) ).
fof(fact_2076_add__less__cancel__right,axiom,
! [A,C,B] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A),C)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B),C)))
<=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A),B)) ) ).
fof(fact_2077_add__less__cancel__right,axiom,
! [A,C,B] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B),C)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)) ) ).
fof(fact_2078_diff__0__right,axiom,
! [A_93] :
( is_int(A_93)
=> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_93),zero_zero_int) = A_93 ) ).
fof(fact_2079_diff__0__right,axiom,
! [A_93] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_93),zero_zero_real) = A_93 ).
fof(fact_2080_diff__0__right,axiom,
! [A_93] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_93),zero_zero_complex) = A_93 ).
fof(fact_2081_diff__0__right,axiom,
! [A_93] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_93),zero_zero_rat) = A_93 ).
fof(fact_2082_diff__self,axiom,
! [A_92] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_92),A_92) = zero_zero_int ).
fof(fact_2083_diff__self,axiom,
! [A_92] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_92),A_92) = zero_zero_real ).
fof(fact_2084_diff__self,axiom,
! [A_92] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_92),A_92) = zero_zero_complex ).
fof(fact_2085_diff__self,axiom,
! [A_92] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_92),A_92) = zero_zero_rat ).
fof(fact_2086_eq__iff__diff__eq__0,axiom,
! [A,B] :
( ( is_int(A)
& is_int(B) )
=> ( A = B
<=> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int ) ) ).
fof(fact_2087_eq__iff__diff__eq__0,axiom,
! [A,B] :
( A = B
<=> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) = zero_zero_real ) ).
fof(fact_2088_eq__iff__diff__eq__0,axiom,
! [A,B] :
( A = B
<=> hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A),B) = zero_zero_complex ) ).
fof(fact_2089_eq__iff__diff__eq__0,axiom,
! [A,B] :
( A = B
<=> hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) = zero_zero_rat ) ).
fof(fact_2090_right__minus__eq,axiom,
! [A,B] :
( ( is_int(A)
& is_int(B) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int
<=> A = B ) ) ).
fof(fact_2091_right__minus__eq,axiom,
! [A,B] :
( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) = zero_zero_real
<=> A = B ) ).
fof(fact_2092_right__minus__eq,axiom,
! [A,B] :
( hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A),B) = zero_zero_complex
<=> A = B ) ).
fof(fact_2093_right__minus__eq,axiom,
! [A,B] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) = zero_zero_rat
<=> A = B ) ).
fof(fact_2094_diff__eq__diff__less__eq,axiom,
! [A,B,C,D] :
( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C),D)) ) ) ).
fof(fact_2095_diff__eq__diff__less__eq,axiom,
! [A,B,C,D] :
( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D)
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C),D)) ) ) ).
fof(fact_2096_diff__eq__diff__less__eq,axiom,
! [A,B,C,D] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D)
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C),D)) ) ) ).
fof(fact_2097_diff__eq__diff__less,axiom,
! [A,B,C,D] :
( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),D)) ) ) ).
fof(fact_2098_diff__eq__diff__less,axiom,
! [A,B,C,D] :
( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D)
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),D)) ) ) ).
fof(fact_2099_diff__eq__diff__less,axiom,
! [A,B,C,D] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D)
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),D)) ) ) ).
fof(fact_2100_mult_Ocomm__neutral,axiom,
! [A_91] :
( is_int(A_91)
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_91),one_one_int) = A_91 ) ).
fof(fact_2101_mult_Ocomm__neutral,axiom,
! [A_91] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_91),one_one_nat) = A_91 ).
fof(fact_2102_mult_Ocomm__neutral,axiom,
! [A_91] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_91),one_one_real) = A_91 ).
fof(fact_2103_mult_Ocomm__neutral,axiom,
! [A_91] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_91),one_on1645066479umeral) = A_91 ).
fof(fact_2104_mult_Ocomm__neutral,axiom,
! [A_91] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_91),one_one_complex) = A_91 ).
fof(fact_2105_mult_Ocomm__neutral,axiom,
! [A_91] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_91),one_on1684967323de_int) = A_91 ).
fof(fact_2106_mult_Ocomm__neutral,axiom,
! [A_91] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_91),one_one_rat) = A_91 ).
fof(fact_2107_mult__1__right,axiom,
! [A_90] :
( is_int(A_90)
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_90),one_one_int) = A_90 ) ).
fof(fact_2108_mult__1__right,axiom,
! [A_90] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_90),one_one_nat) = A_90 ).
fof(fact_2109_mult__1__right,axiom,
! [A_90] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_90),one_one_real) = A_90 ).
fof(fact_2110_mult__1__right,axiom,
! [A_90] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_90),one_on1645066479umeral) = A_90 ).
fof(fact_2111_mult__1__right,axiom,
! [A_90] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_90),one_one_complex) = A_90 ).
fof(fact_2112_mult__1__right,axiom,
! [A_90] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_90),one_on1684967323de_int) = A_90 ).
fof(fact_2113_mult__1__right,axiom,
! [A_90] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_90),one_one_rat) = A_90 ).
fof(fact_2114_mult__1,axiom,
! [A_89] :
( is_int(A_89)
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_89) = A_89 ) ).
fof(fact_2115_mult__1,axiom,
! [A_89] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_89) = A_89 ).
fof(fact_2116_mult__1,axiom,
! [A_89] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_89) = A_89 ).
fof(fact_2117_mult__1,axiom,
! [A_89] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_89) = A_89 ).
fof(fact_2118_mult__1,axiom,
! [A_89] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_89) = A_89 ).
fof(fact_2119_mult__1,axiom,
! [A_89] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_89) = A_89 ).
fof(fact_2120_mult__1,axiom,
! [A_89] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_89) = A_89 ).
fof(fact_2121_mult__1__left,axiom,
! [A_88] :
( is_int(A_88)
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_88) = A_88 ) ).
fof(fact_2122_mult__1__left,axiom,
! [A_88] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_88) = A_88 ).
fof(fact_2123_mult__1__left,axiom,
! [A_88] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_88) = A_88 ).
fof(fact_2124_mult__1__left,axiom,
! [A_88] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_88) = A_88 ).
fof(fact_2125_mult__1__left,axiom,
! [A_88] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_88) = A_88 ).
fof(fact_2126_mult__1__left,axiom,
! [A_88] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_88) = A_88 ).
fof(fact_2127_mult__1__left,axiom,
! [A_88] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_88) = A_88 ).
fof(fact_2128_inv__distinct,axiom,
! [A_3,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))
=> A_3 != hAPP_int_int(inv(P_3),A_3) ) ) ) ).
fof(fact_2129_inv__not__1,axiom,
! [A_3,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))
=> hAPP_int_int(inv(P_3),A_3) != one_one_int ) ) ) ).
fof(fact_2130_inv__not__p__minus__1,axiom,
! [A_3,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))
=> hAPP_int_int(inv(P_3),A_3) != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int) ) ) ) ).
fof(fact_2131_inv__g__1,axiom,
! [A_3,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(inv(P_3),A_3))) ) ) ) ).
fof(fact_2132_inv__less__p__minus__1,axiom,
! [A_3,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(inv(P_3),A_3)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))) ) ) ) ).
fof(fact_2133_add__diff__cancel,axiom,
! [A_87,B_76] :
( is_int(A_87)
=> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_87),B_76)),B_76) = A_87 ) ).
fof(fact_2134_add__diff__cancel,axiom,
! [A_87,B_76] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_87),B_76)),B_76) = A_87 ).
fof(fact_2135_add__diff__cancel,axiom,
! [A_87,B_76] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_87),B_76)),B_76) = A_87 ).
fof(fact_2136_add__diff__cancel,axiom,
! [A_87,B_76] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_87),B_76)),B_76) = A_87 ).
fof(fact_2137_diff__add__cancel,axiom,
! [A_86,B_75] :
( is_int(A_86)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_86),B_75)),B_75) = A_86 ) ).
fof(fact_2138_diff__add__cancel,axiom,
! [A_86,B_75] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_86),B_75)),B_75) = A_86 ).
fof(fact_2139_diff__add__cancel,axiom,
! [A_86,B_75] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_86),B_75)),B_75) = A_86 ).
fof(fact_2140_diff__add__cancel,axiom,
! [A_86,B_75] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_86),B_75)),B_75) = A_86 ).
fof(fact_2141_MultInv__prop1,axiom,
! [X_1,Y_1,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),Y_1),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),X_1)),hAPP_int_int(multInv(P_3),Y_1)),P_3)) ) ) ).
fof(fact_2142_inv__not__0,axiom,
! [A_3,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))
=> hAPP_int_int(inv(P_3),A_3) != zero_zero_int ) ) ) ).
fof(fact_2143_MultInv__zcong__prop1,axiom,
! [A_3,J_2,K_2,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),K_2),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),P_3)) ) ) ).
fof(fact_2144_MultInv__prop5,axiom,
! [Y_1,X_1,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Y_1),zero_zero_int),P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),X_1)),hAPP_int_int(multInv(P_3),Y_1)),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),Y_1),P_3)) ) ) ) ) ) ).
fof(fact_2145_MultInv__prop4,axiom,
! [X_1,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X_1))),X_1),P_3)) ) ) ) ).
fof(fact_2146_MultInv__prop3,axiom,
! [X_1,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),X_1)),zero_zero_int),P_3)) ) ) ) ).
fof(fact_2147_inv__is__inv,axiom,
! [A_3,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(inv(P_3),A_3))),one_one_int),P_3)) ) ) ) ).
fof(fact_2148_add__nonpos__nonpos,axiom,
! [B_74,A_85] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_85),zero_z126310315umeral))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_74),zero_z126310315umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_85),B_74)),zero_z126310315umeral)) ) ) ).
fof(fact_2149_add__nonpos__nonpos,axiom,
! [B_74,A_85] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_85),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_74),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_85),B_74)),zero_zero_int)) ) ) ).
fof(fact_2150_add__nonpos__nonpos,axiom,
! [B_74,A_85] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_85),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_74),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_85),B_74)),zero_zero_nat)) ) ) ).
fof(fact_2151_add__nonpos__nonpos,axiom,
! [B_74,A_85] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_85),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_74),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_85),B_74)),zero_zero_real)) ) ) ).
fof(fact_2152_add__nonpos__nonpos,axiom,
! [B_74,A_85] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_85),zero_z891286103de_int))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_74),zero_z891286103de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_85),B_74)),zero_z891286103de_int)) ) ) ).
fof(fact_2153_add__nonpos__nonpos,axiom,
! [B_74,A_85] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_85),zero_zero_rat))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_74),zero_zero_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_85),B_74)),zero_zero_rat)) ) ) ).
fof(fact_2154_add__increasing2,axiom,
! [B_73,A_84,C_36] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_36))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_73),A_84))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_73),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_84),C_36))) ) ) ).
fof(fact_2155_add__increasing2,axiom,
! [B_73,A_84,C_36] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_36))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_73),A_84))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_73),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_84),C_36))) ) ) ).
fof(fact_2156_add__increasing2,axiom,
! [B_73,A_84,C_36] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_36))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_73),A_84))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_73),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_84),C_36))) ) ) ).
fof(fact_2157_add__increasing2,axiom,
! [B_73,A_84,C_36] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_36))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_73),A_84))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_73),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_84),C_36))) ) ) ).
fof(fact_2158_add__increasing2,axiom,
! [B_73,A_84,C_36] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_36))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_73),A_84))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_73),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_84),C_36))) ) ) ).
fof(fact_2159_add__increasing2,axiom,
! [B_73,A_84,C_36] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_36))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_73),A_84))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_73),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_84),C_36))) ) ) ).
fof(fact_2160_add__increasing,axiom,
! [B_72,C_35,A_83] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_83))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_72),C_35))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_72),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_83),C_35))) ) ) ).
fof(fact_2161_add__increasing,axiom,
! [B_72,C_35,A_83] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_83))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_72),C_35))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_72),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_83),C_35))) ) ) ).
fof(fact_2162_add__increasing,axiom,
! [B_72,C_35,A_83] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_83))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_72),C_35))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_72),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_83),C_35))) ) ) ).
fof(fact_2163_add__increasing,axiom,
! [B_72,C_35,A_83] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_83))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_72),C_35))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_72),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_83),C_35))) ) ) ).
fof(fact_2164_add__increasing,axiom,
! [B_72,C_35,A_83] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_83))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_72),C_35))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_72),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_83),C_35))) ) ) ).
fof(fact_2165_add__increasing,axiom,
! [B_72,C_35,A_83] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_83))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_72),C_35))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_72),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_83),C_35))) ) ) ).
fof(fact_2166_add__nonneg__eq__0__iff,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),X_2))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Y_2))
=> ( hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),Y_2) = zero_z126310315umeral
<=> ( X_2 = zero_z126310315umeral
& Y_2 = zero_z126310315umeral ) ) ) ) ).
fof(fact_2167_add__nonneg__eq__0__iff,axiom,
! [Y_2,X_2] :
( ( is_int(Y_2)
& is_int(X_2) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Y_2) = zero_zero_int
<=> ( X_2 = zero_zero_int
& Y_2 = zero_zero_int ) ) ) ) ) ).
fof(fact_2168_add__nonneg__eq__0__iff,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),X_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_2))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Y_2) = zero_zero_nat
<=> ( X_2 = zero_zero_nat
& Y_2 = zero_zero_nat ) ) ) ) ).
fof(fact_2169_add__nonneg__eq__0__iff,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2))
=> ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Y_2) = zero_zero_real
<=> ( X_2 = zero_zero_real
& Y_2 = zero_zero_real ) ) ) ) ).
fof(fact_2170_add__nonneg__eq__0__iff,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),X_2))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Y_2))
=> ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),Y_2) = zero_z891286103de_int
<=> ( X_2 = zero_z891286103de_int
& Y_2 = zero_z891286103de_int ) ) ) ) ).
fof(fact_2171_add__nonneg__eq__0__iff,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_2))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_2))
=> ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Y_2) = zero_zero_rat
<=> ( X_2 = zero_zero_rat
& Y_2 = zero_zero_rat ) ) ) ) ).
fof(fact_2172_add__nonneg__nonneg,axiom,
! [B_71,A_82] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_82))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_71))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_82),B_71))) ) ) ).
fof(fact_2173_add__nonneg__nonneg,axiom,
! [B_71,A_82] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_82))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_71))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_82),B_71))) ) ) ).
fof(fact_2174_add__nonneg__nonneg,axiom,
! [B_71,A_82] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_82))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_71))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_82),B_71))) ) ) ).
fof(fact_2175_add__nonneg__nonneg,axiom,
! [B_71,A_82] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_82))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_71))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_82),B_71))) ) ) ).
fof(fact_2176_add__nonneg__nonneg,axiom,
! [B_71,A_82] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_82))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_71))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_82),B_71))) ) ) ).
fof(fact_2177_add__nonneg__nonneg,axiom,
! [B_71,A_82] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_82))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_71))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_82),B_71))) ) ) ).
fof(fact_2178_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int)) ) ).
fof(fact_2179_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).
fof(fact_2180_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).
fof(fact_2181_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) ) ).
fof(fact_2182_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).
fof(fact_2183_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).
fof(fact_2184_add__neg__neg,axiom,
! [B_70,A_81] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_81),zero_z126310315umeral))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_70),zero_z126310315umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_81),B_70)),zero_z126310315umeral)) ) ) ).
fof(fact_2185_add__neg__neg,axiom,
! [B_70,A_81] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_81),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_70),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_81),B_70)),zero_zero_int)) ) ) ).
fof(fact_2186_add__neg__neg,axiom,
! [B_70,A_81] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_81),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_70),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_81),B_70)),zero_zero_nat)) ) ) ).
fof(fact_2187_add__neg__neg,axiom,
! [B_70,A_81] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_81),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_70),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_81),B_70)),zero_zero_real)) ) ) ).
fof(fact_2188_add__neg__neg,axiom,
! [B_70,A_81] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_81),zero_z891286103de_int))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_70),zero_z891286103de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_81),B_70)),zero_z891286103de_int)) ) ) ).
fof(fact_2189_add__neg__neg,axiom,
! [B_70,A_81] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_81),zero_zero_rat))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_70),zero_zero_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_81),B_70)),zero_zero_rat)) ) ) ).
fof(fact_2190_add__pos__pos,axiom,
! [B_69,A_80] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_80))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_69))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_80),B_69))) ) ) ).
fof(fact_2191_add__pos__pos,axiom,
! [B_69,A_80] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_80))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_69))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_80),B_69))) ) ) ).
fof(fact_2192_add__pos__pos,axiom,
! [B_69,A_80] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_80))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_69))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_80),B_69))) ) ) ).
fof(fact_2193_add__pos__pos,axiom,
! [B_69,A_80] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_80))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_69))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_80),B_69))) ) ) ).
fof(fact_2194_add__pos__pos,axiom,
! [B_69,A_80] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_80))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_69))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_80),B_69))) ) ) ).
fof(fact_2195_add__pos__pos,axiom,
! [B_69,A_80] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_80))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_69))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_80),B_69))) ) ) ).
fof(fact_2196_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),zero_zero_int)) ) ).
fof(fact_2197_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),zero_zero_real)) ) ).
fof(fact_2198_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),zero_zero_rat)) ) ).
fof(fact_2199_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) ) ).
fof(fact_2200_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A)) ) ).
fof(fact_2201_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A)) ) ).
fof(fact_2202_add__le__less__mono,axiom,
! [C_34,D_6,A_79,B_68] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_79),B_68))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_34),D_6))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_79),C_34)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_68),D_6))) ) ) ).
fof(fact_2203_add__le__less__mono,axiom,
! [C_34,D_6,A_79,B_68] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_79),B_68))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_34),D_6))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_79),C_34)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_68),D_6))) ) ) ).
fof(fact_2204_add__le__less__mono,axiom,
! [C_34,D_6,A_79,B_68] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_79),B_68))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_34),D_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_79),C_34)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_68),D_6))) ) ) ).
fof(fact_2205_add__le__less__mono,axiom,
! [C_34,D_6,A_79,B_68] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_79),B_68))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_34),D_6))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_79),C_34)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_68),D_6))) ) ) ).
fof(fact_2206_add__le__less__mono,axiom,
! [C_34,D_6,A_79,B_68] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_79),B_68))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_34),D_6))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_79),C_34)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_68),D_6))) ) ) ).
fof(fact_2207_add__le__less__mono,axiom,
! [C_34,D_6,A_79,B_68] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_79),B_68))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_34),D_6))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_79),C_34)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_68),D_6))) ) ) ).
fof(fact_2208_add__less__le__mono,axiom,
! [C_33,D_5,A_78,B_67] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_78),B_67))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_33),D_5))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_78),C_33)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_67),D_5))) ) ) ).
fof(fact_2209_add__less__le__mono,axiom,
! [C_33,D_5,A_78,B_67] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_78),B_67))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_33),D_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_78),C_33)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_67),D_5))) ) ) ).
fof(fact_2210_add__less__le__mono,axiom,
! [C_33,D_5,A_78,B_67] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_78),B_67))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_33),D_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_78),C_33)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_67),D_5))) ) ) ).
fof(fact_2211_add__less__le__mono,axiom,
! [C_33,D_5,A_78,B_67] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_78),B_67))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_33),D_5))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_78),C_33)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_67),D_5))) ) ) ).
fof(fact_2212_add__less__le__mono,axiom,
! [C_33,D_5,A_78,B_67] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_78),B_67))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_33),D_5))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_78),C_33)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_67),D_5))) ) ) ).
fof(fact_2213_add__less__le__mono,axiom,
! [C_33,D_5,A_78,B_67] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_78),B_67))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_33),D_5))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_78),C_33)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_67),D_5))) ) ) ).
fof(fact_2214_le__iff__diff__le__0,axiom,
! [A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int)) ) ).
fof(fact_2215_le__iff__diff__le__0,axiom,
! [A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),zero_zero_real)) ) ).
fof(fact_2216_le__iff__diff__le__0,axiom,
! [A,B] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat)) ) ).
fof(fact_2217_less__iff__diff__less__0,axiom,
! [A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int)) ) ).
fof(fact_2218_less__iff__diff__less__0,axiom,
! [A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),zero_zero_real)) ) ).
fof(fact_2219_less__iff__diff__less__0,axiom,
! [A,B] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat)) ) ).
fof(fact_2220_aux______4,axiom,
! [K_2,A_3,J_2,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),J_2)),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),A_3)),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),P_3)) ) ) ) ) ).
fof(fact_2221_aux______2,axiom,
! [J_2,A_3,K_2,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),zero_zero_int),P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),K_2)),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),A_3),P_3)) ) ) ) ) ).
fof(fact_2222_MultInv__zcong__prop2,axiom,
! [A_3,J_2,K_2,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),zero_zero_int),P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),P_3)) ) ) ) ) ) ).
fof(fact_2223_MultInv__zcong__prop3,axiom,
! [J_2,K_2,A_3,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),zero_zero_int),P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),K_2),P_3)) ) ) ) ) ) ) ).
fof(fact_2224_Int2_Oaux____2,axiom,
! [X_1,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),hAPP_int_int(multInv(P_3),X_1))),hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X_1)))),X_1),P_3)) ) ) ) ).
fof(fact_2225_Int2_Oaux____1,axiom,
! [X_1,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),hAPP_int_int(multInv(P_3),X_1))),hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X_1)))),P_3)) ) ) ) ).
fof(fact_2226_MultInv__prop2a,axiom,
! [X_1,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),X_1)),X_1)),one_one_int),P_3)) ) ) ) ).
fof(fact_2227_MultInv__prop2,axiom,
! [X_1,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),hAPP_int_int(multInv(P_3),X_1))),one_one_int),P_3)) ) ) ) ).
fof(fact_2228_zfact_Osimps,axiom,
! [N_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),zero_zero_int))
=> hAPP_int_int(zfact,N_1) = one_one_int )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),zero_zero_int))
=> hAPP_int_int(zfact,N_1) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_1),hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N_1),one_one_int))) ) ) ).
fof(fact_2229_wset__mem__inv__mem,axiom,
! [B,A,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5)))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(inv(P_5),B)),hAPP_i1948725293t_bool(wset(A),P_5))) ) ) ) ) ).
fof(fact_2230_wset__inv__mem__mem,axiom,
! [B,A,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(inv(P_5),B)),hAPP_i1948725293t_bool(wset(A),P_5)))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5))) ) ) ) ) ) ) ).
fof(fact_2231_int__le__induct,axiom,
! [P_1,I_1,K_1] :
( 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] :
( is_int(I_2)
=> ( 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(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))) ) ) )
=> hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).
fof(fact_2232_d22set__induct__old,axiom,
! [X_2,P_1] :
( ! [A_2] :
( is_int(A_2)
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_2),one_one_int))) )
=> hBOOL(hAPP_int_bool(P_1,A_2)) ) )
=> hBOOL(hAPP_int_bool(P_1,X_2)) ) ).
fof(fact_2233_int__less__induct,axiom,
! [P_1,I_1,K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),K_1))
=> ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_1),one_one_int)))
=> ( ! [I_2] :
( is_int(I_2)
=> ( 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(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))) ) ) )
=> hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).
fof(fact_2234_int__gr__induct,axiom,
! [P_1,K_1,I_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_1))
=> ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_1),one_one_int)))
=> ( ! [I_2] :
( is_int(I_2)
=> ( 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(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))) ) ) )
=> hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).
fof(fact_2235_mono__nat__linear__lb,axiom,
! [Ma,K_1,F] :
( ! [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,hAPP_nat_nat(F,M_1)),hAPP_nat_nat(F,N_2))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(F,Ma)),K_1)),hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K_1)))) ) ).
fof(fact_2236_wset__mem__mem,axiom,
! [P_5,A] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),hAPP_i1948725293t_bool(wset(A),P_5))) ) ).
fof(fact_2237_wset__subset,axiom,
! [B,P_5,A] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),one_one_int)),P_5)))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5))) ) ) ).
fof(fact_2238_wset__less,axiom,
! [B,A,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))) ) ) ) ).
fof(fact_2239_wset__g__1,axiom,
! [B,A,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)) ) ) ) ).
fof(fact_2240_wset__mem__imp__or,axiom,
! [B,P_5,A] :
( ( is_int(B)
& is_int(A) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),one_one_int)),P_5)))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5)))
=> ( B = A
| B = hAPP_int_int(inv(P_5),A) ) ) ) ) ) ).
fof(fact_2241_wset__mem,axiom,
! [B,A,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5))) ) ) ) ) ).
fof(fact_2242_d22set__eq__wset,axiom,
! [P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> hAPP_i1948725293t_bool(d22set,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_5) ) ).
fof(fact_2243_ex__least__nat__le,axiom,
! [N,P_1] :
( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(P_1,N))
=> ? [K] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N))
& ! [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),K))
=> ~ hBOOL(hAPP_nat_bool(P_1,I_2)) )
& hBOOL(hAPP_nat_bool(P_1,K)) ) ) ) ).
fof(fact_2244_less__imp__add__positive,axiom,
! [I,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
=> ? [K] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
& hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K) = J_2 ) ) ).
fof(fact_2245_pinf_I9_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,Z),X))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X),Sa)))
<=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X),Sa))) ) ) ).
fof(fact_2246_pinf_I9_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Sa)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Sa))) ) ) ).
fof(fact_2247_pinf_I9_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z),X))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X),Sa)))
<=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X),Sa))) ) ) ).
fof(fact_2248_pinf_I9_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z),X))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Sa)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Sa))) ) ) ).
fof(fact_2249_pinf_I9_J,axiom,
! [D,Sa] :
? [Z] :
( is_int(Z)
& ! [X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Sa)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Sa))) ) ) ) ).
fof(fact_2250_pinf_I9_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z),X))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Sa)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Sa))) ) ) ).
fof(fact_2251_minf_I9_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,X),Z))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X),Sa)))
<=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X),Sa))) ) ) ).
fof(fact_2252_minf_I9_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Z))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Sa)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Sa))) ) ) ).
fof(fact_2253_minf_I9_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X),Z))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X),Sa)))
<=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X),Sa))) ) ) ).
fof(fact_2254_minf_I9_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X),Z))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Sa)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Sa))) ) ) ).
fof(fact_2255_minf_I9_J,axiom,
! [D,Sa] :
? [Z] :
( is_int(Z)
& ! [X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Z))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Sa)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Sa))) ) ) ) ).
fof(fact_2256_minf_I9_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Z))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Sa)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Sa))) ) ) ).
fof(fact_2257_pinf_I10_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,Z),X))
=> ( ~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X),Sa)))
<=> ~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X),Sa))) ) ) ).
fof(fact_2258_pinf_I10_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),X))
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Sa)))
<=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Sa))) ) ) ).
fof(fact_2259_pinf_I10_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z),X))
=> ( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X),Sa)))
<=> ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X),Sa))) ) ) ).
fof(fact_2260_pinf_I10_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z),X))
=> ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Sa)))
<=> ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Sa))) ) ) ).
fof(fact_2261_pinf_I10_J,axiom,
! [D,Sa] :
? [Z] :
( is_int(Z)
& ! [X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Sa)))
<=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Sa))) ) ) ) ).
fof(fact_2262_pinf_I10_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z),X))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Sa)))
<=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Sa))) ) ) ).
fof(fact_2263_minf_I10_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,X),Z))
=> ( ~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X),Sa)))
<=> ~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X),Sa))) ) ) ).
fof(fact_2264_minf_I10_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Z))
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Sa)))
<=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Sa))) ) ) ).
fof(fact_2265_minf_I10_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X),Z))
=> ( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X),Sa)))
<=> ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X),Sa))) ) ) ).
fof(fact_2266_minf_I10_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X),Z))
=> ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Sa)))
<=> ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Sa))) ) ) ).
fof(fact_2267_minf_I10_J,axiom,
! [D,Sa] :
? [Z] :
( is_int(Z)
& ! [X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Z))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Sa)))
<=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Sa))) ) ) ) ).
fof(fact_2268_minf_I10_J,axiom,
! [D,Sa] :
? [Z] :
! [X] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Z))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Sa)))
<=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Sa))) ) ) ).
fof(fact_2269_pow__divides__eq__int,axiom,
! [A,B,N] :
( N != zero_zero_nat
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),N)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),N)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B)) ) ) ).
fof(fact_2270_d22set__le__swap,axiom,
! [A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B))
=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(d22set,A))) ) ).
fof(fact_2271_d22set__le,axiom,
! [B,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(d22set,A)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) ) ).
fof(fact_2272_d22set__g__1,axiom,
! [B,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(d22set,A)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)) ) ).
fof(fact_2273_d22set__mem,axiom,
! [A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(d22set,A))) ) ) ).
fof(fact_2274_gcd__lcm__complete__lattice__nat_Otop__le,axiom,
! [A_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_3))
=> A_3 = zero_zero_nat ) ).
fof(fact_2275_gcd__lcm__complete__lattice__nat_Otop__unique,axiom,
! [A] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A))
<=> A = zero_zero_nat ) ).
fof(fact_2276_gcd__lcm__complete__lattice__nat_Oless__top,axiom,
! [A] :
( A != zero_zero_nat
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),zero_zero_nat))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A)) ) ) ).
fof(fact_2277_gcd__lcm__complete__lattice__nat_Otop__greatest,axiom,
! [A_3] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),zero_zero_nat)) ).
fof(fact_2278_gcd__lcm__complete__lattice__nat_Onot__top__less,axiom,
! [A_3] :
~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_3))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),zero_zero_nat)) ) ).
fof(fact_2279_gcd__lcm__complete__lattice__nat_Ole__bot,axiom,
! [A_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),one_one_nat))
=> A_3 = one_one_nat ) ).
fof(fact_2280_gcd__lcm__complete__lattice__nat_Obot__unique,axiom,
! [A] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),one_one_nat))
<=> A = one_one_nat ) ).
fof(fact_2281_gcd__lcm__complete__lattice__nat_Obot__less,axiom,
! [A] :
( A != one_one_nat
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),one_one_nat)) ) ) ).
fof(fact_2282_gcd__lcm__complete__lattice__nat_Obot__least,axiom,
! [A_3] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_3)) ).
fof(fact_2283_gcd__lcm__complete__lattice__nat_Onot__less__bot,axiom,
! [A_3] :
~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),one_one_nat))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_3)) ) ).
fof(fact_2284_dvd__pos__nat,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) ) ) ).
fof(fact_2285_pow__divides__eq__nat,axiom,
! [A,B,N] :
( N != zero_zero_nat
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) ) ) ).
fof(fact_2286_pow__divides__pow__int,axiom,
! [A_3,N_1,B_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N_1)))
=> ( N_1 != zero_zero_nat
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_3),B_2)) ) ) ).
fof(fact_2287_divides__le,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
| N_1 = zero_zero_nat ) ) ).
fof(fact_2288_mult__left__cancel,axiom,
! [N_1,M,K_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)
=> N_1 = M ) ) ).
fof(fact_2289_mult_Oprod__diff__prod,axiom,
! [X_20,Y_16,A_77,B_66] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_20),Y_16)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_77),B_66)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_20),A_77)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_16),B_66))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_20),A_77)),B_66))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_77),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_16),B_66))) ).
fof(fact_2290_mult_Oprod__diff__prod,axiom,
! [X_20,Y_16,A_77,B_66] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_20),Y_16)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_77),B_66)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_20),A_77)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_16),B_66))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_20),A_77)),B_66))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_77),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_16),B_66))) ).
fof(fact_2291_SR__def,axiom,
! [P_5] : hAPP_i1948725293t_bool(sr,P_5) = collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),P_5))) ).
fof(fact_2292_neg__zmod__mult__2,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int))
=> hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),one_one_int)),A_3))),one_one_int) ) ).
fof(fact_2293_zmod__number__of__Bit1,axiom,
! [V_2,W] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
=> hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)))),one_one_int) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
=> hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_2)),one_one_int)),hAPP_int_int(number_number_of_int,W)))),one_one_int) ) ) ).
fof(fact_2294_StandardRes__prop4,axiom,
! [X_1,Y_1,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(standardRes(M),X_1)),hAPP_int_int(standardRes(M),Y_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1)),M)) ) ).
fof(fact_2295_negDivAlg__eqn__1__number__of,axiom,
! [W] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W))))
=> negDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W))) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W))))
=> negDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,W)),negDivAlg(one_one_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,W)))) ) ) ) ).
fof(fact_2296_StandardRes__def,axiom,
! [M,X_1] : hAPP_int_int(standardRes(M),X_1) = hAPP_int_int(div_mod_int(X_1),M) ).
fof(fact_2297_StandardRes__SR__prop,axiom,
! [X_2,P_5] :
( is_int(X_2)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(sr,P_5)))
=> hAPP_int_int(standardRes(P_5),X_2) = X_2 ) ) ).
fof(fact_2298_mod__mod__trivial,axiom,
! [A_76,B_65] : hAPP_int_int(div_mod_int(hAPP_int_int(div_mod_int(A_76),B_65)),B_65) = hAPP_int_int(div_mod_int(A_76),B_65) ).
fof(fact_2299_mod__mod__trivial,axiom,
! [A_76,B_65] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(div_mod_nat(A_76),B_65)),B_65) = hAPP_nat_nat(div_mod_nat(A_76),B_65) ).
fof(fact_2300_mod__mod__trivial,axiom,
! [A_76,B_65] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(div_mo1740067990umeral(A_76),B_65)),B_65) = hAPP_C498520661umeral(div_mo1740067990umeral(A_76),B_65) ).
fof(fact_2301_mod__mod__trivial,axiom,
! [A_76,B_65] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(div_mo231679042de_int(A_76),B_65)),B_65) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_76),B_65) ).
fof(fact_2302_mod__self,axiom,
! [A_75] : hAPP_int_int(div_mod_int(A_75),A_75) = zero_zero_int ).
fof(fact_2303_mod__self,axiom,
! [A_75] : hAPP_nat_nat(div_mod_nat(A_75),A_75) = zero_zero_nat ).
fof(fact_2304_mod__self,axiom,
! [A_75] : hAPP_C498520661umeral(div_mo1740067990umeral(A_75),A_75) = zero_z126310315umeral ).
fof(fact_2305_mod__self,axiom,
! [A_75] : hAPP_Q1762011733de_int(div_mo231679042de_int(A_75),A_75) = zero_z891286103de_int ).
fof(fact_2306_mod__by__0,axiom,
! [A_74] :
( is_int(A_74)
=> hAPP_int_int(div_mod_int(A_74),zero_zero_int) = A_74 ) ).
fof(fact_2307_mod__by__0,axiom,
! [A_74] : hAPP_nat_nat(div_mod_nat(A_74),zero_zero_nat) = A_74 ).
fof(fact_2308_mod__by__0,axiom,
! [A_74] : hAPP_C498520661umeral(div_mo1740067990umeral(A_74),zero_z126310315umeral) = A_74 ).
fof(fact_2309_mod__by__0,axiom,
! [A_74] : hAPP_Q1762011733de_int(div_mo231679042de_int(A_74),zero_z891286103de_int) = A_74 ).
fof(fact_2310_mod__0,axiom,
! [A_73] : hAPP_int_int(div_mod_int(zero_zero_int),A_73) = zero_zero_int ).
fof(fact_2311_mod__0,axiom,
! [A_73] : hAPP_nat_nat(div_mod_nat(zero_zero_nat),A_73) = zero_zero_nat ).
fof(fact_2312_mod__0,axiom,
! [A_73] : hAPP_C498520661umeral(div_mo1740067990umeral(zero_z126310315umeral),A_73) = zero_z126310315umeral ).
fof(fact_2313_mod__0,axiom,
! [A_73] : hAPP_Q1762011733de_int(div_mo231679042de_int(zero_z891286103de_int),A_73) = zero_z891286103de_int ).
fof(fact_2314_mod__mult__right__eq,axiom,
! [A_72,B_64,C_32] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_72),B_64)),C_32) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_72),hAPP_int_int(div_mod_int(B_64),C_32))),C_32) ).
fof(fact_2315_mod__mult__right__eq,axiom,
! [A_72,B_64,C_32] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_72),B_64)),C_32) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_72),hAPP_nat_nat(div_mod_nat(B_64),C_32))),C_32) ).
fof(fact_2316_mod__mult__right__eq,axiom,
! [A_72,B_64,C_32] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_72),B_64)),C_32) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_72),hAPP_C498520661umeral(div_mo1740067990umeral(B_64),C_32))),C_32) ).
fof(fact_2317_mod__mult__right__eq,axiom,
! [A_72,B_64,C_32] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_72),B_64)),C_32) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_72),hAPP_Q1762011733de_int(div_mo231679042de_int(B_64),C_32))),C_32) ).
fof(fact_2318_mod__mult__left__eq,axiom,
! [A_71,B_63,C_31] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_71),B_63)),C_31) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_71),C_31)),B_63)),C_31) ).
fof(fact_2319_mod__mult__left__eq,axiom,
! [A_71,B_63,C_31] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_71),B_63)),C_31) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_71),C_31)),B_63)),C_31) ).
fof(fact_2320_mod__mult__left__eq,axiom,
! [A_71,B_63,C_31] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_71),B_63)),C_31) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_71),C_31)),B_63)),C_31) ).
fof(fact_2321_mod__mult__left__eq,axiom,
! [A_71,B_63,C_31] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_71),B_63)),C_31) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_71),C_31)),B_63)),C_31) ).
fof(fact_2322_mod__mult__eq,axiom,
! [A_70,B_62,C_30] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_70),B_62)),C_30) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_70),C_30)),hAPP_int_int(div_mod_int(B_62),C_30))),C_30) ).
fof(fact_2323_mod__mult__eq,axiom,
! [A_70,B_62,C_30] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_70),B_62)),C_30) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_70),C_30)),hAPP_nat_nat(div_mod_nat(B_62),C_30))),C_30) ).
fof(fact_2324_mod__mult__eq,axiom,
! [A_70,B_62,C_30] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_70),B_62)),C_30) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_70),C_30)),hAPP_C498520661umeral(div_mo1740067990umeral(B_62),C_30))),C_30) ).
fof(fact_2325_mod__mult__eq,axiom,
! [A_70,B_62,C_30] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_70),B_62)),C_30) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_70),C_30)),hAPP_Q1762011733de_int(div_mo231679042de_int(B_62),C_30))),C_30) ).
fof(fact_2326_mod__mult__mult1,axiom,
! [C_29,A_69,B_61] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_29),A_69)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_29),B_61)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_29),hAPP_int_int(div_mod_int(A_69),B_61)) ).
fof(fact_2327_mod__mult__mult1,axiom,
! [C_29,A_69,B_61] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_29),A_69)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_29),B_61)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_29),hAPP_nat_nat(div_mod_nat(A_69),B_61)) ).
fof(fact_2328_mod__mult__mult1,axiom,
! [C_29,A_69,B_61] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_29),A_69)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_29),B_61)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_29),hAPP_C498520661umeral(div_mo1740067990umeral(A_69),B_61)) ).
fof(fact_2329_mod__mult__mult1,axiom,
! [C_29,A_69,B_61] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_29),A_69)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_29),B_61)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_29),hAPP_Q1762011733de_int(div_mo231679042de_int(A_69),B_61)) ).
fof(fact_2330_mod__mult__mult2,axiom,
! [A_68,C_28,B_60] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_68),C_28)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_60),C_28)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_68),B_60)),C_28) ).
fof(fact_2331_mod__mult__mult2,axiom,
! [A_68,C_28,B_60] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_68),C_28)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_60),C_28)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_68),B_60)),C_28) ).
fof(fact_2332_mod__mult__mult2,axiom,
! [A_68,C_28,B_60] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_68),C_28)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_60),C_28)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_68),B_60)),C_28) ).
fof(fact_2333_mod__mult__mult2,axiom,
! [A_68,C_28,B_60] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_68),C_28)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_60),C_28)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_68),B_60)),C_28) ).
fof(fact_2334_zmod__simps_I4_J,axiom,
! [A_67,C_27,B_59] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_67),C_27)),B_59)),C_27) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_67),B_59)),C_27) ).
fof(fact_2335_zmod__simps_I4_J,axiom,
! [A_67,C_27,B_59] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_67),C_27)),B_59)),C_27) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_67),B_59)),C_27) ).
fof(fact_2336_zmod__simps_I4_J,axiom,
! [A_67,C_27,B_59] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_67),C_27)),B_59)),C_27) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_67),B_59)),C_27) ).
fof(fact_2337_zmod__simps_I4_J,axiom,
! [A_67,C_27,B_59] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_67),C_27)),B_59)),C_27) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_67),B_59)),C_27) ).
fof(fact_2338_mod__mult__cong,axiom,
! [B_58,B_57,A_66,C_26,A_65] :
( hAPP_int_int(div_mod_int(A_66),C_26) = hAPP_int_int(div_mod_int(A_65),C_26)
=> ( hAPP_int_int(div_mod_int(B_58),C_26) = hAPP_int_int(div_mod_int(B_57),C_26)
=> hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_66),B_58)),C_26) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_65),B_57)),C_26) ) ) ).
fof(fact_2339_mod__mult__cong,axiom,
! [B_58,B_57,A_66,C_26,A_65] :
( hAPP_nat_nat(div_mod_nat(A_66),C_26) = hAPP_nat_nat(div_mod_nat(A_65),C_26)
=> ( hAPP_nat_nat(div_mod_nat(B_58),C_26) = hAPP_nat_nat(div_mod_nat(B_57),C_26)
=> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_66),B_58)),C_26) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_65),B_57)),C_26) ) ) ).
fof(fact_2340_mod__mult__cong,axiom,
! [B_58,B_57,A_66,C_26,A_65] :
( hAPP_C498520661umeral(div_mo1740067990umeral(A_66),C_26) = hAPP_C498520661umeral(div_mo1740067990umeral(A_65),C_26)
=> ( hAPP_C498520661umeral(div_mo1740067990umeral(B_58),C_26) = hAPP_C498520661umeral(div_mo1740067990umeral(B_57),C_26)
=> hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_66),B_58)),C_26) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_65),B_57)),C_26) ) ) ).
fof(fact_2341_mod__mult__cong,axiom,
! [B_58,B_57,A_66,C_26,A_65] :
( hAPP_Q1762011733de_int(div_mo231679042de_int(A_66),C_26) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_65),C_26)
=> ( hAPP_Q1762011733de_int(div_mo231679042de_int(B_58),C_26) = hAPP_Q1762011733de_int(div_mo231679042de_int(B_57),C_26)
=> hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_66),B_58)),C_26) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_65),B_57)),C_26) ) ) ).
fof(fact_2342_mod__add__self2,axiom,
! [A_64,B_56] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_64),B_56)),B_56) = hAPP_int_int(div_mod_int(A_64),B_56) ).
fof(fact_2343_mod__add__self2,axiom,
! [A_64,B_56] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_64),B_56)),B_56) = hAPP_nat_nat(div_mod_nat(A_64),B_56) ).
fof(fact_2344_mod__add__self2,axiom,
! [A_64,B_56] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_64),B_56)),B_56) = hAPP_C498520661umeral(div_mo1740067990umeral(A_64),B_56) ).
fof(fact_2345_mod__add__self2,axiom,
! [A_64,B_56] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_64),B_56)),B_56) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_64),B_56) ).
fof(fact_2346_mod__add__self1,axiom,
! [B_55,A_63] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_55),A_63)),B_55) = hAPP_int_int(div_mod_int(A_63),B_55) ).
fof(fact_2347_mod__add__self1,axiom,
! [B_55,A_63] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_55),A_63)),B_55) = hAPP_nat_nat(div_mod_nat(A_63),B_55) ).
fof(fact_2348_mod__add__self1,axiom,
! [B_55,A_63] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_55),A_63)),B_55) = hAPP_C498520661umeral(div_mo1740067990umeral(A_63),B_55) ).
fof(fact_2349_mod__add__self1,axiom,
! [B_55,A_63] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_55),A_63)),B_55) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_63),B_55) ).
fof(fact_2350_mod__add__right__eq,axiom,
! [A_62,B_54,C_25] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_62),B_54)),C_25) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_62),hAPP_int_int(div_mod_int(B_54),C_25))),C_25) ).
fof(fact_2351_mod__add__right__eq,axiom,
! [A_62,B_54,C_25] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_62),B_54)),C_25) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_62),hAPP_nat_nat(div_mod_nat(B_54),C_25))),C_25) ).
fof(fact_2352_mod__add__right__eq,axiom,
! [A_62,B_54,C_25] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_62),B_54)),C_25) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_62),hAPP_C498520661umeral(div_mo1740067990umeral(B_54),C_25))),C_25) ).
fof(fact_2353_mod__add__right__eq,axiom,
! [A_62,B_54,C_25] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_62),B_54)),C_25) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_62),hAPP_Q1762011733de_int(div_mo231679042de_int(B_54),C_25))),C_25) ).
fof(fact_2354_mod__add__left__eq,axiom,
! [A_61,B_53,C_24] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_61),B_53)),C_24) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_61),C_24)),B_53)),C_24) ).
fof(fact_2355_mod__add__left__eq,axiom,
! [A_61,B_53,C_24] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_61),B_53)),C_24) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_61),C_24)),B_53)),C_24) ).
fof(fact_2356_mod__add__left__eq,axiom,
! [A_61,B_53,C_24] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_61),B_53)),C_24) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_61),C_24)),B_53)),C_24) ).
fof(fact_2357_mod__add__left__eq,axiom,
! [A_61,B_53,C_24] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_61),B_53)),C_24) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_61),C_24)),B_53)),C_24) ).
fof(fact_2358_mod__add__eq,axiom,
! [A_60,B_52,C_23] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_60),B_52)),C_23) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_60),C_23)),hAPP_int_int(div_mod_int(B_52),C_23))),C_23) ).
fof(fact_2359_mod__add__eq,axiom,
! [A_60,B_52,C_23] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_60),B_52)),C_23) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_60),C_23)),hAPP_nat_nat(div_mod_nat(B_52),C_23))),C_23) ).
fof(fact_2360_mod__add__eq,axiom,
! [A_60,B_52,C_23] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_60),B_52)),C_23) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_60),C_23)),hAPP_C498520661umeral(div_mo1740067990umeral(B_52),C_23))),C_23) ).
fof(fact_2361_mod__add__eq,axiom,
! [A_60,B_52,C_23] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_60),B_52)),C_23) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_60),C_23)),hAPP_Q1762011733de_int(div_mo231679042de_int(B_52),C_23))),C_23) ).
fof(fact_2362_zmod__simps_I2_J,axiom,
! [A_59,B_51,C_22] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_59),hAPP_int_int(div_mod_int(B_51),C_22))),C_22) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_59),B_51)),C_22) ).
fof(fact_2363_zmod__simps_I2_J,axiom,
! [A_59,B_51,C_22] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_59),hAPP_nat_nat(div_mod_nat(B_51),C_22))),C_22) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_59),B_51)),C_22) ).
fof(fact_2364_zmod__simps_I2_J,axiom,
! [A_59,B_51,C_22] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_59),hAPP_C498520661umeral(div_mo1740067990umeral(B_51),C_22))),C_22) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_59),B_51)),C_22) ).
fof(fact_2365_zmod__simps_I2_J,axiom,
! [A_59,B_51,C_22] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_59),hAPP_Q1762011733de_int(div_mo231679042de_int(B_51),C_22))),C_22) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_59),B_51)),C_22) ).
fof(fact_2366_zmod__simps_I1_J,axiom,
! [A_58,C_21,B_50] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_58),C_21)),B_50)),C_21) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_58),B_50)),C_21) ).
fof(fact_2367_zmod__simps_I1_J,axiom,
! [A_58,C_21,B_50] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_58),C_21)),B_50)),C_21) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_58),B_50)),C_21) ).
fof(fact_2368_zmod__simps_I1_J,axiom,
! [A_58,C_21,B_50] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_58),C_21)),B_50)),C_21) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_58),B_50)),C_21) ).
fof(fact_2369_zmod__simps_I1_J,axiom,
! [A_58,C_21,B_50] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_58),C_21)),B_50)),C_21) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_58),B_50)),C_21) ).
fof(fact_2370_mod__add__cong,axiom,
! [B_49,B_48,A_57,C_20,A_56] :
( hAPP_int_int(div_mod_int(A_57),C_20) = hAPP_int_int(div_mod_int(A_56),C_20)
=> ( hAPP_int_int(div_mod_int(B_49),C_20) = hAPP_int_int(div_mod_int(B_48),C_20)
=> hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_57),B_49)),C_20) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_56),B_48)),C_20) ) ) ).
fof(fact_2371_mod__add__cong,axiom,
! [B_49,B_48,A_57,C_20,A_56] :
( hAPP_nat_nat(div_mod_nat(A_57),C_20) = hAPP_nat_nat(div_mod_nat(A_56),C_20)
=> ( hAPP_nat_nat(div_mod_nat(B_49),C_20) = hAPP_nat_nat(div_mod_nat(B_48),C_20)
=> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_57),B_49)),C_20) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_56),B_48)),C_20) ) ) ).
fof(fact_2372_mod__add__cong,axiom,
! [B_49,B_48,A_57,C_20,A_56] :
( hAPP_C498520661umeral(div_mo1740067990umeral(A_57),C_20) = hAPP_C498520661umeral(div_mo1740067990umeral(A_56),C_20)
=> ( hAPP_C498520661umeral(div_mo1740067990umeral(B_49),C_20) = hAPP_C498520661umeral(div_mo1740067990umeral(B_48),C_20)
=> hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_57),B_49)),C_20) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_56),B_48)),C_20) ) ) ).
fof(fact_2373_mod__add__cong,axiom,
! [B_49,B_48,A_57,C_20,A_56] :
( hAPP_Q1762011733de_int(div_mo231679042de_int(A_57),C_20) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_56),C_20)
=> ( hAPP_Q1762011733de_int(div_mo231679042de_int(B_49),C_20) = hAPP_Q1762011733de_int(div_mo231679042de_int(B_48),C_20)
=> hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_57),B_49)),C_20) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_56),B_48)),C_20) ) ) ).
fof(fact_2374_mod__diff__right__eq,axiom,
! [A_55,B_47,C_19] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_55),B_47)),C_19) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_55),hAPP_int_int(div_mod_int(B_47),C_19))),C_19) ).
fof(fact_2375_mod__diff__left__eq,axiom,
! [A_54,B_46,C_18] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_54),B_46)),C_18) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_54),C_18)),B_46)),C_18) ).
fof(fact_2376_mod__diff__eq,axiom,
! [A_53,B_45,C_17] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_53),B_45)),C_17) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_53),C_17)),hAPP_int_int(div_mod_int(B_45),C_17))),C_17) ).
fof(fact_2377_mod__diff__cong,axiom,
! [B_44,B_43,A_52,C_16,A_51] :
( hAPP_int_int(div_mod_int(A_52),C_16) = hAPP_int_int(div_mod_int(A_51),C_16)
=> ( hAPP_int_int(div_mod_int(B_44),C_16) = hAPP_int_int(div_mod_int(B_43),C_16)
=> hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_52),B_44)),C_16) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_51),B_43)),C_16) ) ) ).
fof(fact_2378_dvd__mod__imp__dvd,axiom,
! [K_6,M_16,N_29] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_6),hAPP_C498520661umeral(div_mo1740067990umeral(M_16),N_29)))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_6),N_29))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_6),M_16)) ) ) ).
fof(fact_2379_dvd__mod__imp__dvd,axiom,
! [K_6,M_16,N_29] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_6),hAPP_Q1762011733de_int(div_mo231679042de_int(M_16),N_29)))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_6),N_29))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_6),M_16)) ) ) ).
fof(fact_2380_dvd__mod__imp__dvd,axiom,
! [K_6,M_16,N_29] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_6),hAPP_int_int(div_mod_int(M_16),N_29)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_6),N_29))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_6),M_16)) ) ) ).
fof(fact_2381_dvd__mod__imp__dvd,axiom,
! [K_6,M_16,N_29] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_6),hAPP_nat_nat(div_mod_nat(M_16),N_29)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_6),N_29))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_6),M_16)) ) ) ).
fof(fact_2382_dvd__mod,axiom,
! [N_28,K_5,M_15] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_5),M_15))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_5),N_28))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_5),hAPP_C498520661umeral(div_mo1740067990umeral(M_15),N_28))) ) ) ).
fof(fact_2383_dvd__mod,axiom,
! [N_28,K_5,M_15] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_5),M_15))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_5),N_28))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_5),hAPP_Q1762011733de_int(div_mo231679042de_int(M_15),N_28))) ) ) ).
fof(fact_2384_dvd__mod,axiom,
! [N_28,K_5,M_15] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_5),M_15))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_5),N_28))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_5),hAPP_int_int(div_mod_int(M_15),N_28))) ) ) ).
fof(fact_2385_dvd__mod,axiom,
! [N_28,K_5,M_15] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_5),M_15))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_5),N_28))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_5),hAPP_nat_nat(div_mod_nat(M_15),N_28))) ) ) ).
fof(fact_2386_mod__mod__cancel,axiom,
! [A_50,C_15,B_42] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,C_15),B_42))
=> hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(div_mo1740067990umeral(A_50),B_42)),C_15) = hAPP_C498520661umeral(div_mo1740067990umeral(A_50),C_15) ) ).
fof(fact_2387_mod__mod__cancel,axiom,
! [A_50,C_15,B_42] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,C_15),B_42))
=> hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(div_mo231679042de_int(A_50),B_42)),C_15) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_50),C_15) ) ).
fof(fact_2388_mod__mod__cancel,axiom,
! [A_50,C_15,B_42] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C_15),B_42))
=> hAPP_int_int(div_mod_int(hAPP_int_int(div_mod_int(A_50),B_42)),C_15) = hAPP_int_int(div_mod_int(A_50),C_15) ) ).
fof(fact_2389_mod__mod__cancel,axiom,
! [A_50,C_15,B_42] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_15),B_42))
=> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(div_mod_nat(A_50),B_42)),C_15) = hAPP_nat_nat(div_mod_nat(A_50),C_15) ) ).
fof(fact_2390_dvd__mod__iff,axiom,
! [Ma,K_1,N] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_1),N))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_1),hAPP_C498520661umeral(div_mo1740067990umeral(Ma),N)))
<=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_1),Ma)) ) ) ).
fof(fact_2391_dvd__mod__iff,axiom,
! [Ma,K_1,N] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_1),N))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_1),hAPP_Q1762011733de_int(div_mo231679042de_int(Ma),N)))
<=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_1),Ma)) ) ) ).
fof(fact_2392_dvd__mod__iff,axiom,
! [Ma,K_1,N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(div_mod_int(Ma),N)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),Ma)) ) ) ).
fof(fact_2393_dvd__mod__iff,axiom,
! [Ma,K_1,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(div_mod_nat(Ma),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),Ma)) ) ) ).
fof(fact_2394_zmod__self,axiom,
! [A_3] : hAPP_int_int(div_mod_int(A_3),A_3) = zero_zero_int ).
fof(fact_2395_zmod__zero,axiom,
! [B_2] : hAPP_int_int(div_mod_int(zero_zero_int),B_2) = zero_zero_int ).
fof(fact_2396_zmod__zmult1__eq,axiom,
! [A_3,B_2,C_1] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),C_1) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(div_mod_int(B_2),C_1))),C_1) ).
fof(fact_2397_zmod__simps_I3_J,axiom,
! [A_3,B_2,C_1] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(div_mod_int(B_2),C_1))),C_1) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),C_1) ).
fof(fact_2398_zdiff__zmod__right,axiom,
! [X_1,Y_1,M] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),hAPP_int_int(div_mod_int(Y_1),M))),M) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_1)),M) ).
fof(fact_2399_zdiff__zmod__left,axiom,
! [X_1,M,Y_1] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(X_1),M)),Y_1)),M) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_1)),M) ).
fof(fact_2400_zcong__zmod,axiom,
! [A,B,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(div_mod_int(A),Ma)),hAPP_int_int(div_mod_int(B),Ma)),Ma)) ) ).
fof(fact_2401_Residues_Oaux,axiom,
! [X_1,M,Y_1] :
( hAPP_int_int(div_mod_int(X_1),M) = hAPP_int_int(div_mod_int(Y_1),M)
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),Y_1),M)) ) ).
fof(fact_2402_mod__mod__is__mod,axiom,
! [X_1,M] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),hAPP_int_int(div_mod_int(X_1),M)),M)) ).
fof(fact_2403_zdvd__zmod__imp__zdvd,axiom,
! [K_2,M,N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(div_mod_int(M),N_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) ) ) ).
fof(fact_2404_zdvd__zmod,axiom,
! [N_1,F_3,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F_3),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F_3),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F_3),hAPP_int_int(div_mod_int(M),N_1))) ) ) ).
fof(fact_2405_zpower__zmod,axiom,
! [X_1,M,Y_1] : hAPP_int_int(div_mod_int(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(div_mod_int(X_1),M)),Y_1)),M) = hAPP_int_int(div_mod_int(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),Y_1)),M) ).
fof(fact_2406_StandardRes__prop1,axiom,
! [X_1,M] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),hAPP_int_int(standardRes(M),X_1)),M)) ).
fof(fact_2407_mod__mult__self2__is__0,axiom,
! [A_49,B_41] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_49),B_41)),B_41) = zero_zero_int ).
fof(fact_2408_mod__mult__self2__is__0,axiom,
! [A_49,B_41] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_49),B_41)),B_41) = zero_zero_nat ).
fof(fact_2409_mod__mult__self2__is__0,axiom,
! [A_49,B_41] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_49),B_41)),B_41) = zero_z126310315umeral ).
fof(fact_2410_mod__mult__self2__is__0,axiom,
! [A_49,B_41] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_49),B_41)),B_41) = zero_z891286103de_int ).
fof(fact_2411_mod__mult__self1__is__0,axiom,
! [B_40,A_48] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_40),A_48)),B_40) = zero_zero_int ).
fof(fact_2412_mod__mult__self1__is__0,axiom,
! [B_40,A_48] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_40),A_48)),B_40) = zero_zero_nat ).
fof(fact_2413_mod__mult__self1__is__0,axiom,
! [B_40,A_48] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_40),A_48)),B_40) = zero_z126310315umeral ).
fof(fact_2414_mod__mult__self1__is__0,axiom,
! [B_40,A_48] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_40),A_48)),B_40) = zero_z891286103de_int ).
fof(fact_2415_mod__by__1,axiom,
! [A_47] : hAPP_int_int(div_mod_int(A_47),one_one_int) = zero_zero_int ).
fof(fact_2416_mod__by__1,axiom,
! [A_47] : hAPP_nat_nat(div_mod_nat(A_47),one_one_nat) = zero_zero_nat ).
fof(fact_2417_mod__by__1,axiom,
! [A_47] : hAPP_C498520661umeral(div_mo1740067990umeral(A_47),one_on1645066479umeral) = zero_z126310315umeral ).
fof(fact_2418_mod__by__1,axiom,
! [A_47] : hAPP_Q1762011733de_int(div_mo231679042de_int(A_47),one_on1684967323de_int) = zero_z891286103de_int ).
fof(fact_2419_mod__mult__self2,axiom,
! [A_46,B_39,C_14] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_46),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_39),C_14))),B_39) = hAPP_int_int(div_mod_int(A_46),B_39) ).
fof(fact_2420_mod__mult__self2,axiom,
! [A_46,B_39,C_14] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_46),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_39),C_14))),B_39) = hAPP_nat_nat(div_mod_nat(A_46),B_39) ).
fof(fact_2421_mod__mult__self2,axiom,
! [A_46,B_39,C_14] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_46),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_39),C_14))),B_39) = hAPP_C498520661umeral(div_mo1740067990umeral(A_46),B_39) ).
fof(fact_2422_mod__mult__self2,axiom,
! [A_46,B_39,C_14] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_46),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_39),C_14))),B_39) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_46),B_39) ).
fof(fact_2423_mod__mult__self1,axiom,
! [A_45,C_13,B_38] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_45),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_13),B_38))),B_38) = hAPP_int_int(div_mod_int(A_45),B_38) ).
fof(fact_2424_mod__mult__self1,axiom,
! [A_45,C_13,B_38] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_45),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_13),B_38))),B_38) = hAPP_nat_nat(div_mod_nat(A_45),B_38) ).
fof(fact_2425_mod__mult__self1,axiom,
! [A_45,C_13,B_38] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_45),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_13),B_38))),B_38) = hAPP_C498520661umeral(div_mo1740067990umeral(A_45),B_38) ).
fof(fact_2426_mod__mult__self1,axiom,
! [A_45,C_13,B_38] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_45),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_13),B_38))),B_38) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_45),B_38) ).
fof(fact_2427_dvd__imp__mod__0,axiom,
! [A_44,B_37] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_44),B_37))
=> hAPP_C498520661umeral(div_mo1740067990umeral(B_37),A_44) = zero_z126310315umeral ) ).
fof(fact_2428_dvd__imp__mod__0,axiom,
! [A_44,B_37] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_44),B_37))
=> hAPP_Q1762011733de_int(div_mo231679042de_int(B_37),A_44) = zero_z891286103de_int ) ).
fof(fact_2429_dvd__imp__mod__0,axiom,
! [A_44,B_37] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_44),B_37))
=> hAPP_int_int(div_mod_int(B_37),A_44) = zero_zero_int ) ).
fof(fact_2430_dvd__imp__mod__0,axiom,
! [A_44,B_37] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_44),B_37))
=> hAPP_nat_nat(div_mod_nat(B_37),A_44) = zero_zero_nat ) ).
fof(fact_2431_dvd__eq__mod__eq__0,axiom,
! [A,B] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),B))
<=> hAPP_C498520661umeral(div_mo1740067990umeral(B),A) = zero_z126310315umeral ) ).
fof(fact_2432_dvd__eq__mod__eq__0,axiom,
! [A,B] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),B))
<=> hAPP_Q1762011733de_int(div_mo231679042de_int(B),A) = zero_z891286103de_int ) ).
fof(fact_2433_dvd__eq__mod__eq__0,axiom,
! [A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B))
<=> hAPP_int_int(div_mod_int(B),A) = zero_zero_int ) ).
fof(fact_2434_dvd__eq__mod__eq__0,axiom,
! [A,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))
<=> hAPP_nat_nat(div_mod_nat(B),A) = zero_zero_nat ) ).
fof(fact_2435_pos__mod__bound,axiom,
! [A_3,B_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(div_mod_int(A_3),B_2)),B_2)) ) ).
fof(fact_2436_neg__mod__bound,axiom,
! [A_3,B_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),hAPP_int_int(div_mod_int(A_3),B_2))) ) ).
fof(fact_2437_zmod__le__nonneg__dividend,axiom,
! [K_2,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(M),K_2)),M)) ) ).
fof(fact_2438_Divides_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(X_1),Y_1))) ) ) ).
fof(fact_2439_zmod__eq__0__iff,axiom,
! [Ma,D] :
( is_int(Ma)
=> ( hAPP_int_int(div_mod_int(Ma),D) = zero_zero_int
<=> ? [Q_4] :
( is_int(Q_4)
& Ma = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D),Q_4) ) ) ) ).
fof(fact_2440_zmod__eq__dvd__iff,axiom,
! [X_2,N,Y_2] :
( hAPP_int_int(div_mod_int(X_2),N) = hAPP_int_int(div_mod_int(Y_2),N)
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),Y_2))) ) ).
fof(fact_2441_dvd__eq__mod__eq__0__number__of,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_i769753017umeral(number1443263063umeral,X_2)),hAPP_i769753017umeral(number1443263063umeral,Y_2)))
<=> hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_i769753017umeral(number1443263063umeral,Y_2)),hAPP_i769753017umeral(number1443263063umeral,X_2)) = zero_z126310315umeral ) ).
fof(fact_2442_dvd__eq__mod__eq__0__number__of,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_i1732201573de_int(number1226105091de_int,X_2)),hAPP_i1732201573de_int(number1226105091de_int,Y_2)))
<=> hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_i1732201573de_int(number1226105091de_int,Y_2)),hAPP_i1732201573de_int(number1226105091de_int,X_2)) = zero_z891286103de_int ) ).
fof(fact_2443_dvd__eq__mod__eq__0__number__of,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(number_number_of_int,X_2)),hAPP_int_int(number_number_of_int,Y_2)))
<=> hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,Y_2)),hAPP_int_int(number_number_of_int,X_2)) = zero_zero_int ) ).
fof(fact_2444_dvd__eq__mod__eq__0__number__of,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(number_number_of_nat,X_2)),hAPP_int_nat(number_number_of_nat,Y_2)))
<=> hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,Y_2)),hAPP_int_nat(number_number_of_nat,X_2)) = zero_zero_nat ) ).
fof(fact_2445_StandardRes__ubound,axiom,
! [X_1,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(standardRes(P_3),X_1)),P_3)) ) ).
fof(fact_2446_StandardRes__eq__zcong,axiom,
! [Ma,X_2] :
( hAPP_int_int(standardRes(Ma),X_2) = zero_zero_int
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_2),zero_zero_int),Ma)) ) ).
fof(fact_2447_StandardRes__prop3,axiom,
! [X_2,P_5] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_2),zero_zero_int),P_5))
<=> hAPP_int_int(standardRes(P_5),X_2) != zero_zero_int ) ).
fof(fact_2448_mod__neg__neg__trivial,axiom,
! [B_2,A_3] :
( is_int(A_3)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_3))
=> hAPP_int_int(div_mod_int(A_3),B_2) = A_3 ) ) ) ).
fof(fact_2449_neg__mod__conj,axiom,
! [A_3,B_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(A_3),B_2)),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),hAPP_int_int(div_mod_int(A_3),B_2))) ) ) ).
fof(fact_2450_neg__mod__sign,axiom,
! [A_3,B_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(A_3),B_2)),zero_zero_int)) ) ).
fof(fact_2451_mod__pos__pos__trivial,axiom,
! [B_2,A_3] :
( is_int(A_3)
=> ( 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_int,A_3),B_2))
=> hAPP_int_int(div_mod_int(A_3),B_2) = A_3 ) ) ) ).
fof(fact_2452_pos__mod__conj,axiom,
! [A_3,B_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(A_3),B_2)))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(div_mod_int(A_3),B_2)),B_2)) ) ) ).
fof(fact_2453_pos__mod__sign,axiom,
! [A_3,B_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(A_3),B_2))) ) ).
fof(fact_2454_zcong__zmod__eq,axiom,
! [A,B,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma))
<=> hAPP_int_int(div_mod_int(A),Ma) = hAPP_int_int(div_mod_int(B),Ma) ) ) ).
fof(fact_2455_zmod__zdvd__zmod,axiom,
! [A_3,B_2,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),B_2))
=> hAPP_int_int(div_mod_int(hAPP_int_int(div_mod_int(A_3),B_2)),M) = hAPP_int_int(div_mod_int(A_3),M) ) ) ).
fof(fact_2456_zmod__minus1__right,axiom,
! [A_3] : hAPP_int_int(div_mod_int(A_3),hAPP_int_int(number_number_of_int,min)) = zero_zero_int ).
fof(fact_2457_zdvd__iff__zmod__eq__0__number__of,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(number_number_of_int,X_2)),hAPP_int_int(number_number_of_int,Y_2)))
<=> hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,Y_2)),hAPP_int_int(number_number_of_int,X_2)) = zero_zero_int ) ).
fof(fact_2458_mod__pos__neg__trivial,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),zero_zero_int))
=> hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2) ) ) ).
fof(fact_2459_StandardRes__lbound,axiom,
! [X_1,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(standardRes(P_3),X_1))) ) ).
fof(fact_2460_StandardRes__prop2,axiom,
! [X1_2,X2,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( hAPP_int_int(standardRes(Ma),X1_2) = hAPP_int_int(standardRes(Ma),X2)
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X1_2),X2),Ma)) ) ) ).
fof(fact_2461_zmod__number__of__Bit0,axiom,
! [V_2,W] : hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W))) ).
fof(fact_2462_split__zmod,axiom,
! [P_1,N,K_1] :
( ( is_int(N)
& is_int(K_1) )
=> ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(div_mod_int(N),K_1)))
<=> ( ( K_1 = zero_zero_int
=> hBOOL(hAPP_int_bool(P_1,N)) )
& ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
=> ! [I_2,J_1] :
( ( is_int(I_2)
& is_int(J_1) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),J_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),K_1))
& N = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) )
=> hBOOL(hAPP_int_bool(P_1,J_1)) ) ) )
& ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int))
=> ! [I_2,J_1] :
( ( is_int(I_2)
& is_int(J_1) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),J_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int))
& N = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) )
=> hBOOL(hAPP_int_bool(P_1,J_1)) ) ) ) ) ) ) ).
fof(fact_2463_zmult2__lemma__aux3,axiom,
! [Q_2,B_2,R_3,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_eq_int,zero_zero_int),R_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3))) ) ) ) ).
fof(fact_2464_zmult2__lemma__aux4,axiom,
! [Q_2,B_2,R_3,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_eq_int,zero_zero_int),R_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1))) ) ) ) ).
fof(fact_2465_zmult2__lemma__aux1,axiom,
! [Q_2,B_2,R_3,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,B_2),R_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3))) ) ) ) ).
fof(fact_2466_zmult2__lemma__aux2,axiom,
! [Q_2,B_2,R_3,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,B_2),R_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3)),zero_zero_int)) ) ) ) ).
fof(fact_2467_divmod__int__rel__mod__eq,axiom,
! [A_13,B_10,Q_9,Y_1] :
( ( is_int(B_10)
& is_int(Y_1) )
=> ( A_13 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_10),Q_9)),Y_1)
=> ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),B_10)) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_10),Y_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),zero_zero_int)) ) ) )
=> ( B_10 != zero_zero_int
=> hAPP_int_int(div_mod_int(A_13),B_10) = Y_1 ) ) ) ) ).
fof(fact_2468_neq__one__mod__two,axiom,
! [X_2] :
( hAPP_int_int(div_mod_int(X_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_zero_int
<=> hAPP_int_int(div_mod_int(X_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_int ) ).
fof(fact_2469_zmod__minus1,axiom,
! [B_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,min)),B_2) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),one_one_int) ) ).
fof(fact_2470_mult_Ozero__left,axiom,
! [B_36] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),B_36) = zero_zero_real ).
fof(fact_2471_mult_Ozero__left,axiom,
! [B_36] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),B_36) = zero_zero_complex ).
fof(fact_2472_mult__left_Ozero,axiom,
! [Y_15] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),Y_15) = zero_zero_real ).
fof(fact_2473_mult__left_Ozero,axiom,
! [Y_15] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),Y_15) = zero_zero_complex ).
fof(fact_2474_mult_Ozero__right,axiom,
! [A_43] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_43),zero_zero_real) = zero_zero_real ).
fof(fact_2475_mult_Ozero__right,axiom,
! [A_43] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_43),zero_zero_complex) = zero_zero_complex ).
fof(fact_2476_mult__right_Ozero,axiom,
! [X_19] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_19),zero_zero_real) = zero_zero_real ).
fof(fact_2477_mult__right_Ozero,axiom,
! [X_19] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_19),zero_zero_complex) = zero_zero_complex ).
fof(fact_2478_mult_Oadd__right,axiom,
! [A_42,B_35,B_34] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_42),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_35),B_34)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_42),B_35)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_42),B_34)) ).
fof(fact_2479_mult_Oadd__right,axiom,
! [A_42,B_35,B_34] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_42),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_35),B_34)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_42),B_35)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_42),B_34)) ).
fof(fact_2480_mult__right_Oadd,axiom,
! [Xa_4,X_18,Y_14] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_4),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_18),Y_14)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_4),X_18)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_4),Y_14)) ).
fof(fact_2481_mult__right_Oadd,axiom,
! [Xa_4,X_18,Y_14] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_4),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_18),Y_14)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_4),X_18)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_4),Y_14)) ).
fof(fact_2482_mult_Oadd__left,axiom,
! [A_41,A_40,B_33] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_41),A_40)),B_33) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_41),B_33)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_40),B_33)) ).
fof(fact_2483_mult_Oadd__left,axiom,
! [A_41,A_40,B_33] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_41),A_40)),B_33) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_41),B_33)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_40),B_33)) ).
fof(fact_2484_mult__left_Oadd,axiom,
! [X_17,Y_13,Ya_2] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_17),Y_13)),Ya_2) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_17),Ya_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_13),Ya_2)) ).
fof(fact_2485_mult__left_Oadd,axiom,
! [X_17,Y_13,Ya_2] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_17),Y_13)),Ya_2) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_17),Ya_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Y_13),Ya_2)) ).
fof(fact_2486_mult_Odiff__right,axiom,
! [A_39,B_32,B_31] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_39),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_32),B_31)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_39),B_32)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_39),B_31)) ).
fof(fact_2487_mult_Odiff__right,axiom,
! [A_39,B_32,B_31] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_39),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,B_32),B_31)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_39),B_32)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_39),B_31)) ).
fof(fact_2488_mult__right_Odiff,axiom,
! [Xa_3,X_16,Y_12] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_3),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_16),Y_12)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_3),X_16)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_3),Y_12)) ).
fof(fact_2489_mult__right_Odiff,axiom,
! [Xa_3,X_16,Y_12] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_3),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_16),Y_12)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_3),X_16)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_3),Y_12)) ).
fof(fact_2490_mult_Odiff__left,axiom,
! [A_38,A_37,B_30] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_38),A_37)),B_30) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_38),B_30)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_37),B_30)) ).
fof(fact_2491_mult_Odiff__left,axiom,
! [A_38,A_37,B_30] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_38),A_37)),B_30) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_38),B_30)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_37),B_30)) ).
fof(fact_2492_mult__left_Odiff,axiom,
! [X_15,Y_11,Ya_1] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_15),Y_11)),Ya_1) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_15),Ya_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_11),Ya_1)) ).
fof(fact_2493_mult__left_Odiff,axiom,
! [X_15,Y_11,Ya_1] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_15),Y_11)),Ya_1) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_15),Ya_1)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Y_11),Ya_1)) ).
fof(fact_2494_negDivAlg__minus1,axiom,
! [B_2] : negDivAlg(hAPP_int_int(number_number_of_int,min),B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),one_one_int)) ).
fof(fact_2495_negDivAlg_Osimps,axiom,
! [A_3,B_2] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) )
=> negDivAlg(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)) )
& ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) )
=> negDivAlg(A_3,B_2) = hAPP_P1975530577nt_int(adjust(B_2),negDivAlg(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))) ) ) ).
fof(fact_2496_negDivAlg__eqn,axiom,
! [A_3,B_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)))
=> negDivAlg(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)))
=> negDivAlg(A_3,B_2) = hAPP_P1975530577nt_int(adjust(B_2),negDivAlg(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))) ) ) ) ).
fof(fact_2497_negDivAlg__eqn__number__of,axiom,
! [W,V_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_2)))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2))))
=> negDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2))) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2))))
=> negDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,V_2)),negDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,V_2)))) ) ) ) ).
fof(fact_2498_pos__zmod__mult__2,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3))
=> hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(B_2),A_3))) ) ).
fof(fact_2499_SRStar__mult__prop2,axiom,
! [X_2,A,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(sRStar,P_5)))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),hAPP_int_int(multInv(P_5),X_2)))),hAPP_i1948725293t_bool(sRStar,P_5))) ) ) ) ) ).
fof(fact_2500_posDivAlg_Osimps,axiom,
! [A_3,B_2] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) )
=> posDivAlg(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),A_3) )
& ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) )
=> posDivAlg(A_3,B_2) = hAPP_P1975530577nt_int(adjust(B_2),posDivAlg(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))) ) ) ).
fof(fact_2501_posDivAlg__eqn__1__number__of,axiom,
! [W] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(number_number_of_int,W)))
=> posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(number_number_of_int,W)))
=> posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,W)),posDivAlg(one_one_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,W)))) ) ) ) ).
fof(fact_2502_posDivAlg__eqn,axiom,
! [A_3,B_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2))
=> posDivAlg(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),A_3) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2))
=> posDivAlg(A_3,B_2) = hAPP_P1975530577nt_int(adjust(B_2),posDivAlg(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))) ) ) ) ).
fof(fact_2503_posDivAlg__eqn__number__of,axiom,
! [W,V_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_2)))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2)))
=> posDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2)))
=> posDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,V_2)),posDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,V_2)))) ) ) ) ).
fof(fact_2504_zmod__eq__0D,axiom,
! [M_14,D_4] :
( is_int(M_14)
=> ( hAPP_int_int(div_mod_int(M_14),D_4) = zero_zero_int
=> ? [Q_4] :
( is_int(Q_4)
& M_14 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D_4),Q_4) ) ) ) ).
fof(fact_2505_StandardRes__SRStar__prop2,axiom,
! [X_2,P_5] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
=> ( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(sRStar,P_5)))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),hAPP_int_int(multInv(P_5),X_2))),hAPP_i1948725293t_bool(sRStar,P_5))) ) ) ) ).
fof(fact_2506_SRStar__mult__prop1,axiom,
! [Y_2,X_2,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(sRStar,P_5)))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),hAPP_i1948725293t_bool(sRStar,P_5)))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Y_2))),hAPP_i1948725293t_bool(sRStar,P_5))) ) ) ) ) ).
fof(fact_2507_mod__less,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hAPP_nat_nat(div_mod_nat(M),N_1) = M ) ).
fof(fact_2508_mod__less__eq__dividend,axiom,
! [M,N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(div_mod_nat(M),N_1)),M)) ).
fof(fact_2509_mod__mult__distrib2,axiom,
! [K_2,M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(div_mod_nat(M),N_1)) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)) ).
fof(fact_2510_mod__mult__distrib,axiom,
! [M,N_1,K_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(M),N_1)),K_2) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_2)) ).
fof(fact_2511_StandardRes__SRStar__prop3,axiom,
! [X_2,P_5] :
( is_int(X_2)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(sRStar,P_5)))
=> hAPP_int_int(standardRes(P_5),X_2) = X_2 ) ) ).
fof(fact_2512_mod__less__divisor,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(div_mod_nat(M),N_1)),N_1)) ) ).
fof(fact_2513_mod__eq__0__iff,axiom,
! [Ma,D] :
( hAPP_nat_nat(div_mod_nat(Ma),D) = zero_zero_nat
<=> ? [Q_4] : Ma = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,D),Q_4) ) ).
fof(fact_2514_mod__geq,axiom,
! [M,N_1] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hAPP_nat_nat(div_mod_nat(M),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1) ) ).
fof(fact_2515_mod__if,axiom,
! [M,N_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hAPP_nat_nat(div_mod_nat(M),N_1) = M )
& ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hAPP_nat_nat(div_mod_nat(M),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1) ) ) ).
fof(fact_2516_mod__mult__self3,axiom,
! [K_2,N_1,M] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)),M)),N_1) = hAPP_nat_nat(div_mod_nat(M),N_1) ).
fof(fact_2517_le__mod__geq,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> hAPP_nat_nat(div_mod_nat(M),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1) ) ).
fof(fact_2518_SRStar__SR__prop,axiom,
! [X_2,P_5] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(sRStar,P_5)))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(sr,P_5))) ) ).
fof(fact_2519_mod__le__divisor,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(div_mod_nat(M),N_1)),N_1)) ) ).
fof(fact_2520_posDivAlg__0,axiom,
! [B_2] : posDivAlg(zero_zero_int,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int) ).
fof(fact_2521_SRStar__def,axiom,
! [P_5] : hAPP_i1948725293t_bool(sRStar,P_5) = collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),P_5))) ).
fof(fact_2522_StandardRes__SRStar__prop1a,axiom,
! [X_2,P_5] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(sRStar,P_5)))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_2),zero_zero_int),P_5)) ) ).
fof(fact_2523_mod__lemma,axiom,
! [Q_2,R_3,B_2,C_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_3),B_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),hAPP_nat_nat(div_mod_nat(Q_2),C_1))),R_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1))) ) ) ).
fof(fact_2524_split__mod,axiom,
! [P_1,N,K_1] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(div_mod_nat(N),K_1)))
<=> ( ( K_1 = zero_zero_nat
=> hBOOL(hAPP_nat_bool(P_1,N)) )
& ( K_1 != zero_zero_nat
=> ! [I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K_1))
=> ( N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),I_2)),J_1)
=> hBOOL(hAPP_nat_bool(P_1,J_1)) ) ) ) ) ) ).
fof(fact_2525_mod2__gr__0,axiom,
! [Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(div_mod_nat(Ma),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
<=> hAPP_nat_nat(div_mod_nat(Ma),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_nat ) ).
fof(fact_2526_StandardRes__SRStar__prop4,axiom,
! [X_2,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(sRStar,P_5)))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),X_2)),hAPP_i1948725293t_bool(sRStar,P_5))) ) ) ) ).
fof(fact_2527_StandardRes__SRStar__prop1,axiom,
! [X_2,P_5] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),X_2)),hAPP_i1948725293t_bool(sRStar,P_5)))
<=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_2),zero_zero_int),P_5)) ) ) ).
fof(fact_2528_mod__exhaust__less__4,axiom,
! [M] :
( hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_nat
| hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_nat
| hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))
| hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) ) ).
fof(fact_2529_nat__mod__eq__lemma,axiom,
! [X_1,N_1,Y_1] :
( hAPP_nat_nat(div_mod_nat(X_1),N_1) = hAPP_nat_nat(div_mod_nat(Y_1),N_1)
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1))
=> ? [Q_4] : X_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),Q_4)) ) ) ).
fof(fact_2530_negDivAlg_Opsimps,axiom,
! [A,B] :
( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(negDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A),B)))
=> ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) )
=> negDivAlg(A,B) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)) )
& ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) )
=> negDivAlg(A,B) = hAPP_P1975530577nt_int(adjust(B),negDivAlg(A,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B))) ) ) ) ).
fof(fact_2531_posDivAlg_Opsimps,axiom,
! [A,B] :
( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(posDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A),B)))
=> ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) )
=> posDivAlg(A,B) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),A) )
& ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) )
=> posDivAlg(A,B) = hAPP_P1975530577nt_int(adjust(B),posDivAlg(A,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B))) ) ) ) ).
fof(fact_2532_zOddI,axiom,
! [X_2,K_1] :
( X_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_1)),one_one_int)
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd)) ) ).
fof(fact_2533_neg__zdiv__mult__2,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),one_one_int)),A_3) ) ).
fof(fact_2534_pos__zdiv__mult__2,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),A_3) ) ).
fof(fact_2535_nat__mod__eq__iff,axiom,
! [X_2,N,Y_2] :
( hAPP_nat_nat(div_mod_nat(X_2),N) = hAPP_nat_nat(div_mod_nat(Y_2),N)
<=> ? [Q1,Q2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Q1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Q2)) ) ).
fof(fact_2536_zdiv__zero,axiom,
! [B_2] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,zero_zero_int),B_2) = zero_zero_int ).
fof(fact_2537_div__by__0,axiom,
! [A_36] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_36),zero_zero_int) = zero_zero_int ).
fof(fact_2538_div__by__0,axiom,
! [A_36] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_36),zero_zero_nat) = zero_zero_nat ).
fof(fact_2539_div__by__0,axiom,
! [A_36] : hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_36),zero_z126310315umeral) = zero_z126310315umeral ).
fof(fact_2540_div__by__0,axiom,
! [A_36] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_36),zero_z891286103de_int) = zero_z891286103de_int ).
fof(fact_2541_div__0,axiom,
! [A_35] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,zero_zero_int),A_35) = zero_zero_int ).
fof(fact_2542_div__0,axiom,
! [A_35] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,zero_zero_nat),A_35) = zero_zero_nat ).
fof(fact_2543_div__0,axiom,
! [A_35] : hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,zero_z126310315umeral),A_35) = zero_z126310315umeral ).
fof(fact_2544_div__0,axiom,
! [A_35] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,zero_z891286103de_int),A_35) = zero_z891286103de_int ).
fof(fact_2545_div__by__1,axiom,
! [A_34] :
( is_int(A_34)
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_34),one_one_int) = A_34 ) ).
fof(fact_2546_div__by__1,axiom,
! [A_34] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_34),one_one_nat) = A_34 ).
fof(fact_2547_div__by__1,axiom,
! [A_34] : hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_34),one_on1645066479umeral) = A_34 ).
fof(fact_2548_div__by__1,axiom,
! [A_34] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_34),one_on1684967323de_int) = A_34 ).
fof(fact_2549_div__dvd__div,axiom,
! [C,A,B] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),B))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),C))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B),A)),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,C),A)))
<=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,B),C)) ) ) ) ).
fof(fact_2550_div__dvd__div,axiom,
! [C,A,B] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),B))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),C))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B),A)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,C),A)))
<=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B),C)) ) ) ) ).
fof(fact_2551_div__dvd__div,axiom,
! [C,A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),C))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B),A)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,C),A)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B),C)) ) ) ) ).
fof(fact_2552_div__dvd__div,axiom,
! [C,A,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,C),A)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B),C)) ) ) ) ).
fof(fact_2553_EvenOdd_Oodd__times__odd,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zOdd))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Y_2)),zOdd)) ) ) ).
fof(fact_2554_odd__mult__odd__prop,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Y_2)),zOdd))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd)) ) ).
fof(fact_2555_pos__imp__zdiv__neg__iff,axiom,
! [A,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,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),zero_zero_int)) ) ) ).
fof(fact_2556_neg__imp__zdiv__neg__iff,axiom,
! [A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) ) ) ).
fof(fact_2557_div__neg__pos__less0,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),zero_zero_int)) ) ) ).
fof(fact_2558_zdiv__self,axiom,
! [A_3] :
( is_int(A_3)
=> ( A_3 != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),A_3) = one_one_int ) ) ).
fof(fact_2559_Divides_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),Y_1))) ) ) ).
fof(fact_2560_zdiv__number__of__Bit0,axiom,
! [V_2,W] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) ).
fof(fact_2561_DIVISION__BY__ZERO,axiom,
! [A_3] :
( is_int(A_3)
=> ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),zero_zero_int) = zero_zero_int
& hAPP_int_int(div_mod_int(A_3),zero_zero_int) = A_3 ) ) ).
fof(fact_2562_zmod__zdiv__trivial,axiom,
! [A_3,B_2] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(div_mod_int(A_3),B_2)),B_2) = zero_zero_int ).
fof(fact_2563_zdvd__mult__div__cancel,axiom,
! [N_1,M] :
( is_int(M)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),M))
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_1),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,M),N_1)) = M ) ) ).
fof(fact_2564_zdiv__zadd1__eq,axiom,
! [A_3,B_2,C_1] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),C_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),C_1))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_3),C_1)),hAPP_int_int(div_mod_int(B_2),C_1))),C_1)) ).
fof(fact_2565_div__mult__mult1__if,axiom,
! [A_33,B_29,C_12] :
( is_int(C_12)
=> ( ( C_12 = zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_12),A_33)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_12),B_29)) = zero_zero_int )
& ( C_12 != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_12),A_33)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_12),B_29)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_33),B_29) ) ) ) ).
fof(fact_2566_div__mult__mult1__if,axiom,
! [A_33,B_29,C_12] :
( ( C_12 = zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_12),A_33)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_12),B_29)) = zero_zero_nat )
& ( C_12 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_12),A_33)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_12),B_29)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_33),B_29) ) ) ).
fof(fact_2567_div__mult__mult1__if,axiom,
! [A_33,B_29,C_12] :
( ( C_12 = zero_z126310315umeral
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_12),A_33)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_12),B_29)) = zero_z126310315umeral )
& ( C_12 != zero_z126310315umeral
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_12),A_33)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_12),B_29)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_33),B_29) ) ) ).
fof(fact_2568_div__mult__mult1__if,axiom,
! [A_33,B_29,C_12] :
( ( C_12 = zero_z891286103de_int
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_12),A_33)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_12),B_29)) = zero_z891286103de_int )
& ( C_12 != zero_z891286103de_int
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_12),A_33)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_12),B_29)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_33),B_29) ) ) ).
fof(fact_2569_div__mult__self2__is__id,axiom,
! [A_32,B_28] :
( ( is_int(A_32)
& is_int(B_28) )
=> ( B_28 != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_32),B_28)),B_28) = A_32 ) ) ).
fof(fact_2570_div__mult__self2__is__id,axiom,
! [A_32,B_28] :
( B_28 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_32),B_28)),B_28) = A_32 ) ).
fof(fact_2571_div__mult__self2__is__id,axiom,
! [A_32,B_28] :
( B_28 != zero_z126310315umeral
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_32),B_28)),B_28) = A_32 ) ).
fof(fact_2572_div__mult__self2__is__id,axiom,
! [A_32,B_28] :
( B_28 != zero_z891286103de_int
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_32),B_28)),B_28) = A_32 ) ).
fof(fact_2573_div__mult__self1__is__id,axiom,
! [A_31,B_27] :
( ( is_int(A_31)
& is_int(B_27) )
=> ( B_27 != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_27),A_31)),B_27) = A_31 ) ) ).
fof(fact_2574_div__mult__self1__is__id,axiom,
! [A_31,B_27] :
( B_27 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_27),A_31)),B_27) = A_31 ) ).
fof(fact_2575_div__mult__self1__is__id,axiom,
! [A_31,B_27] :
( B_27 != zero_z126310315umeral
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_27),A_31)),B_27) = A_31 ) ).
fof(fact_2576_div__mult__self1__is__id,axiom,
! [A_31,B_27] :
( B_27 != zero_z891286103de_int
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_27),A_31)),B_27) = A_31 ) ).
fof(fact_2577_div__mult__mult2,axiom,
! [A_30,B_26,C_11] :
( is_int(C_11)
=> ( C_11 != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_30),C_11)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_26),C_11)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_30),B_26) ) ) ).
fof(fact_2578_div__mult__mult2,axiom,
! [A_30,B_26,C_11] :
( C_11 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_30),C_11)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_26),C_11)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_30),B_26) ) ).
fof(fact_2579_div__mult__mult2,axiom,
! [A_30,B_26,C_11] :
( C_11 != zero_z126310315umeral
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_30),C_11)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_26),C_11)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_30),B_26) ) ).
fof(fact_2580_div__mult__mult2,axiom,
! [A_30,B_26,C_11] :
( C_11 != zero_z891286103de_int
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_30),C_11)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_26),C_11)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_30),B_26) ) ).
fof(fact_2581_div__mult__mult1,axiom,
! [A_29,B_25,C_10] :
( is_int(C_10)
=> ( C_10 != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_10),A_29)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_10),B_25)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_29),B_25) ) ) ).
fof(fact_2582_div__mult__mult1,axiom,
! [A_29,B_25,C_10] :
( C_10 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_10),A_29)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_10),B_25)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_29),B_25) ) ).
fof(fact_2583_div__mult__mult1,axiom,
! [A_29,B_25,C_10] :
( C_10 != zero_z126310315umeral
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_10),A_29)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_10),B_25)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_29),B_25) ) ).
fof(fact_2584_div__mult__mult1,axiom,
! [A_29,B_25,C_10] :
( C_10 != zero_z891286103de_int
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_10),A_29)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_10),B_25)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_29),B_25) ) ).
fof(fact_2585_div__self,axiom,
! [A_28] :
( is_int(A_28)
=> ( A_28 != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_28),A_28) = one_one_int ) ) ).
fof(fact_2586_div__self,axiom,
! [A_28] :
( A_28 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_28),A_28) = one_one_nat ) ).
fof(fact_2587_div__self,axiom,
! [A_28] :
( A_28 != zero_z126310315umeral
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_28),A_28) = one_on1645066479umeral ) ).
fof(fact_2588_div__self,axiom,
! [A_28] :
( A_28 != zero_z891286103de_int
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_28),A_28) = one_on1684967323de_int ) ).
fof(fact_2589_dvd__mult__div__cancel,axiom,
! [A_27,B_24] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_27),B_24))
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_27),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B_24),A_27)) = B_24 ) ).
fof(fact_2590_dvd__mult__div__cancel,axiom,
! [A_27,B_24] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_27),B_24))
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_27),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B_24),A_27)) = B_24 ) ).
fof(fact_2591_dvd__mult__div__cancel,axiom,
! [A_27,B_24] :
( is_int(B_24)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_27),B_24))
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_27),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_24),A_27)) = B_24 ) ) ).
fof(fact_2592_dvd__mult__div__cancel,axiom,
! [A_27,B_24] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_27),B_24))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_27),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_24),A_27)) = B_24 ) ).
fof(fact_2593_div__mult__swap,axiom,
! [A_26,C_9,B_23] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,C_9),B_23))
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_26),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B_23),C_9)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_26),B_23)),C_9) ) ).
fof(fact_2594_div__mult__swap,axiom,
! [A_26,C_9,B_23] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,C_9),B_23))
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_26),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B_23),C_9)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_26),B_23)),C_9) ) ).
fof(fact_2595_div__mult__swap,axiom,
! [A_26,C_9,B_23] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C_9),B_23))
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_26),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_23),C_9)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_26),B_23)),C_9) ) ).
fof(fact_2596_div__mult__swap,axiom,
! [A_26,C_9,B_23] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_9),B_23))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_26),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_23),C_9)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_26),B_23)),C_9) ) ).
fof(fact_2597_dvd__div__mult__self,axiom,
! [A_25,B_22] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_25),B_22))
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B_22),A_25)),A_25) = B_22 ) ).
fof(fact_2598_dvd__div__mult__self,axiom,
! [A_25,B_22] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_25),B_22))
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B_22),A_25)),A_25) = B_22 ) ).
fof(fact_2599_dvd__div__mult__self,axiom,
! [A_25,B_22] :
( is_int(B_22)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_25),B_22))
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_22),A_25)),A_25) = B_22 ) ) ).
fof(fact_2600_dvd__div__mult__self,axiom,
! [A_25,B_22] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_25),B_22))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_22),A_25)),A_25) = B_22 ) ).
fof(fact_2601_dvd__div__mult,axiom,
! [C_8,A_24,B_21] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_24),B_21))
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B_21),A_24)),C_8) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_21),C_8)),A_24) ) ).
fof(fact_2602_dvd__div__mult,axiom,
! [C_8,A_24,B_21] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_24),B_21))
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B_21),A_24)),C_8) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_21),C_8)),A_24) ) ).
fof(fact_2603_dvd__div__mult,axiom,
! [C_8,A_24,B_21] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_24),B_21))
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_21),A_24)),C_8) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_21),C_8)),A_24) ) ).
fof(fact_2604_dvd__div__mult,axiom,
! [C_8,A_24,B_21] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_24),B_21))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_21),A_24)),C_8) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_21),C_8)),A_24) ) ).
fof(fact_2605_div__mult__div__if__dvd,axiom,
! [Z_7,W_2,Y_10,X_14] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Y_10),X_14))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Z_7),W_2))
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,X_14),Y_10)),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,W_2),Z_7)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_14),W_2)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Y_10),Z_7)) ) ) ).
fof(fact_2606_div__mult__div__if__dvd,axiom,
! [Z_7,W_2,Y_10,X_14] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Y_10),X_14))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Z_7),W_2))
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,X_14),Y_10)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,W_2),Z_7)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_14),W_2)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Y_10),Z_7)) ) ) ).
fof(fact_2607_div__mult__div__if__dvd,axiom,
! [Z_7,W_2,Y_10,X_14] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y_10),X_14))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_7),W_2))
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_14),Y_10)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,W_2),Z_7)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_14),W_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_10),Z_7)) ) ) ).
fof(fact_2608_div__mult__div__if__dvd,axiom,
! [Z_7,W_2,Y_10,X_14] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_10),X_14))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_7),W_2))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_14),Y_10)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,W_2),Z_7)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_14),W_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Y_10),Z_7)) ) ) ).
fof(fact_2609_div__add,axiom,
! [Y_9,Z_6,X_13] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Z_6),X_13))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Z_6),Y_9))
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_13),Y_9)),Z_6) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,X_13),Z_6)),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,Y_9),Z_6)) ) ) ).
fof(fact_2610_div__add,axiom,
! [Y_9,Z_6,X_13] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Z_6),X_13))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Z_6),Y_9))
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_13),Y_9)),Z_6) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,X_13),Z_6)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,Y_9),Z_6)) ) ) ).
fof(fact_2611_div__add,axiom,
! [Y_9,Z_6,X_13] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_6),X_13))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_6),Y_9))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_13),Y_9)),Z_6) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_13),Z_6)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Y_9),Z_6)) ) ) ).
fof(fact_2612_div__add,axiom,
! [Y_9,Z_6,X_13] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_6),X_13))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_6),Y_9))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_13),Y_9)),Z_6) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_13),Z_6)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Y_9),Z_6)) ) ) ).
fof(fact_2613_mod__div__trivial,axiom,
! [A_23,B_20] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(div_mod_int(A_23),B_20)),B_20) = zero_zero_int ).
fof(fact_2614_mod__div__trivial,axiom,
! [A_23,B_20] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(div_mod_nat(A_23),B_20)),B_20) = zero_zero_nat ).
fof(fact_2615_mod__div__trivial,axiom,
! [A_23,B_20] : hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_23),B_20)),B_20) = zero_z126310315umeral ).
fof(fact_2616_mod__div__trivial,axiom,
! [A_23,B_20] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_23),B_20)),B_20) = zero_z891286103de_int ).
fof(fact_2617_div__power,axiom,
! [N_27,Y_8,X_12] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Y_8),X_12))
=> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,X_12),Y_8)),N_27) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_12),N_27)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_8),N_27)) ) ).
fof(fact_2618_div__power,axiom,
! [N_27,Y_8,X_12] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Y_8),X_12))
=> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,X_12),Y_8)),N_27) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_12),N_27)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_8),N_27)) ) ).
fof(fact_2619_div__power,axiom,
! [N_27,Y_8,X_12] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y_8),X_12))
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_12),Y_8)),N_27) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_12),N_27)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_8),N_27)) ) ).
fof(fact_2620_div__power,axiom,
! [N_27,Y_8,X_12] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_8),X_12))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_12),Y_8)),N_27) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_12),N_27)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_8),N_27)) ) ).
fof(fact_2621_int__div__less__self,axiom,
! [K_2,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),K_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),K_2)),X_1)) ) ) ).
fof(fact_2622_zdiv__mono1__neg,axiom,
! [B_2,A_3,A_5] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),A_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_5),B_2)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))) ) ) ).
fof(fact_2623_zdiv__mono1,axiom,
! [B_2,A_3,A_5] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),A_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_5),B_2))) ) ) ).
fof(fact_2624_div__neg__neg__trivial,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_3))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = zero_zero_int ) ) ).
fof(fact_2625_zdiv__mono2__neg,axiom,
! [B_2,B_5,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_5)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))) ) ) ) ).
fof(fact_2626_div__nonpos__pos__le0,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),zero_zero_int)) ) ) ).
fof(fact_2627_neg__imp__zdiv__nonneg__iff,axiom,
! [A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int)) ) ) ).
fof(fact_2628_div__pos__pos__trivial,axiom,
! [B_2,A_3] :
( 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_int,A_3),B_2))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = zero_zero_int ) ) ).
fof(fact_2629_div__nonneg__neg__le0,axiom,
! [B_2,A_3] :
( 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_int,B_2),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),zero_zero_int)) ) ) ).
fof(fact_2630_zdiv__mono2,axiom,
! [B_2,B_5,A_3] :
( 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_int,zero_zero_int),B_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_5))) ) ) ) ).
fof(fact_2631_nonneg1__imp__zdiv__pos__iff,axiom,
! [B,A] :
( 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),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) ) ) ) ).
fof(fact_2632_pos__imp__zdiv__pos__iff,axiom,
! [I_1,K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,I_1),K_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_1)) ) ) ).
fof(fact_2633_pos__imp__zdiv__nonneg__iff,axiom,
! [A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).
fof(fact_2634_zdiv__eq__0__iff,axiom,
! [I_1,K_1] :
( is_int(K_1)
=> ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,I_1),K_1) = zero_zero_int
<=> ( K_1 = zero_zero_int
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),I_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),K_1)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_1)) ) ) ) ) ).
fof(fact_2635_div__prop1,axiom,
! [X_1,Y_1,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_1),Z_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),Z_1)),Y_1)) ) ) ).
fof(fact_2636_zdiv__zmult2__eq,axiom,
! [A_3,B_2,C_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),C_1) ) ).
fof(fact_2637_zmod__zdiv__equality,axiom,
! [A_3,B_2] :
( is_int(A_3)
=> A_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),hAPP_int_int(div_mod_int(A_3),B_2)) ) ).
fof(fact_2638_zdiv__zmult1__eq,axiom,
! [A_3,B_2,C_1] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),C_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),C_1))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(div_mod_int(B_2),C_1))),C_1)) ).
fof(fact_2639_zdiv__zmod__equality,axiom,
! [B_2,A_3,K_2] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),hAPP_int_int(div_mod_int(A_3),B_2))),K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),K_2) ).
fof(fact_2640_zdiv__zmod__equality2,axiom,
! [A_3,B_2,K_2] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),B_2)),hAPP_int_int(div_mod_int(A_3),B_2))),K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),K_2) ).
fof(fact_2641_zmod__zdiv__equality_H,axiom,
! [M,N_1] : hAPP_int_int(div_mod_int(M),N_1) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,M),N_1)),N_1)) ).
fof(fact_2642_zmult__div__cancel,axiom,
! [N_1,M] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_1),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,M),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),hAPP_int_int(div_mod_int(M),N_1)) ).
fof(fact_2643_div__mult__self2,axiom,
! [A_22,C_7,B_19] :
( is_int(B_19)
=> ( B_19 != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_22),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_19),C_7))),B_19) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_7),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_22),B_19)) ) ) ).
fof(fact_2644_div__mult__self2,axiom,
! [A_22,C_7,B_19] :
( B_19 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_22),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_19),C_7))),B_19) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_7),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_22),B_19)) ) ).
fof(fact_2645_div__mult__self2,axiom,
! [A_22,C_7,B_19] :
( B_19 != zero_z126310315umeral
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_22),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_19),C_7))),B_19) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_7),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_22),B_19)) ) ).
fof(fact_2646_div__mult__self2,axiom,
! [A_22,C_7,B_19] :
( B_19 != zero_z891286103de_int
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_22),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_19),C_7))),B_19) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_7),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_22),B_19)) ) ).
fof(fact_2647_div__mult__self1,axiom,
! [A_21,C_6,B_18] :
( is_int(B_18)
=> ( B_18 != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_21),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_6),B_18))),B_18) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_6),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_21),B_18)) ) ) ).
fof(fact_2648_div__mult__self1,axiom,
! [A_21,C_6,B_18] :
( B_18 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_6),B_18))),B_18) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_21),B_18)) ) ).
fof(fact_2649_div__mult__self1,axiom,
! [A_21,C_6,B_18] :
( B_18 != zero_z126310315umeral
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_21),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_6),B_18))),B_18) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_6),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_21),B_18)) ) ).
fof(fact_2650_div__mult__self1,axiom,
! [A_21,C_6,B_18] :
( B_18 != zero_z891286103de_int
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_21),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_6),B_18))),B_18) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_6),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_21),B_18)) ) ).
fof(fact_2651_div__add__self2,axiom,
! [A_20,B_17] :
( is_int(B_17)
=> ( B_17 != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_20),B_17)),B_17) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_20),B_17)),one_one_int) ) ) ).
fof(fact_2652_div__add__self2,axiom,
! [A_20,B_17] :
( B_17 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_20),B_17)),B_17) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_20),B_17)),one_one_nat) ) ).
fof(fact_2653_div__add__self2,axiom,
! [A_20,B_17] :
( B_17 != zero_z126310315umeral
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_20),B_17)),B_17) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_20),B_17)),one_on1645066479umeral) ) ).
fof(fact_2654_div__add__self2,axiom,
! [A_20,B_17] :
( B_17 != zero_z891286103de_int
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_20),B_17)),B_17) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_20),B_17)),one_on1684967323de_int) ) ).
fof(fact_2655_div__add__self1,axiom,
! [A_19,B_16] :
( is_int(B_16)
=> ( B_16 != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_16),A_19)),B_16) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_19),B_16)),one_one_int) ) ) ).
fof(fact_2656_div__add__self1,axiom,
! [A_19,B_16] :
( B_16 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_16),A_19)),B_16) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_19),B_16)),one_one_nat) ) ).
fof(fact_2657_div__add__self1,axiom,
! [A_19,B_16] :
( B_16 != zero_z126310315umeral
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_16),A_19)),B_16) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_19),B_16)),one_on1645066479umeral) ) ).
fof(fact_2658_div__add__self1,axiom,
! [A_19,B_16] :
( B_16 != zero_z891286103de_int
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_16),A_19)),B_16) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_19),B_16)),one_on1684967323de_int) ) ).
fof(fact_2659_dvd__div__div__eq__mult,axiom,
! [D,B,C,A] :
( A != zero_z126310315umeral
=> ( C != zero_z126310315umeral
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),B))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,C),D))
=> ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B),A) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,D),C)
<=> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B),C) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A),D) ) ) ) ) ) ).
fof(fact_2660_dvd__div__div__eq__mult,axiom,
! [D,B,C,A] :
( A != zero_z891286103de_int
=> ( C != zero_z891286103de_int
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),B))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,C),D))
=> ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B),A) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,D),C)
<=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B),C) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A),D) ) ) ) ) ) ).
fof(fact_2661_dvd__div__div__eq__mult,axiom,
! [D,B,C,A] :
( ( is_int(C)
& is_int(A) )
=> ( A != zero_zero_int
=> ( C != zero_zero_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C),D))
=> ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B),A) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,D),C)
<=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),C) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D) ) ) ) ) ) ) ).
fof(fact_2662_dvd__div__div__eq__mult,axiom,
! [D,B,C,A] :
( A != zero_zero_nat
=> ( C != zero_zero_nat
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C),D))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,D),C)
<=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),C) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ).
fof(fact_2663_dvd__div__eq__mult,axiom,
! [C,B,A] :
( A != zero_z126310315umeral
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),B))
=> ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B),A) = C
<=> B = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C),A) ) ) ) ).
fof(fact_2664_dvd__div__eq__mult,axiom,
! [C,B,A] :
( A != zero_z891286103de_int
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),B))
=> ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B),A) = C
<=> B = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C),A) ) ) ) ).
fof(fact_2665_dvd__div__eq__mult,axiom,
! [C,B,A] :
( ( is_int(C)
& is_int(B)
& is_int(A) )
=> ( A != zero_zero_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B))
=> ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B),A) = C
<=> B = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A) ) ) ) ) ).
fof(fact_2666_dvd__div__eq__mult,axiom,
! [C,B,A] :
( A != zero_zero_nat
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B),A) = C
<=> B = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ).
fof(fact_2667_semiring__div__class_Omod__div__equality_H,axiom,
! [A_18,B_15] :
( is_int(A_18)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_18),B_15)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_18),B_15)),B_15)) = A_18 ) ).
fof(fact_2668_semiring__div__class_Omod__div__equality_H,axiom,
! [A_18,B_15] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_18),B_15)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_18),B_15)),B_15)) = A_18 ).
fof(fact_2669_semiring__div__class_Omod__div__equality_H,axiom,
! [A_18,B_15] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_18),B_15)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_18),B_15)),B_15)) = A_18 ).
fof(fact_2670_semiring__div__class_Omod__div__equality_H,axiom,
! [A_18,B_15] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_18),B_15)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_18),B_15)),B_15)) = A_18 ).
fof(fact_2671_mod__div__equality2,axiom,
! [B_14,A_17] :
( is_int(A_17)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_14),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_17),B_14))),hAPP_int_int(div_mod_int(A_17),B_14)) = A_17 ) ).
fof(fact_2672_mod__div__equality2,axiom,
! [B_14,A_17] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_14),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_17),B_14))),hAPP_nat_nat(div_mod_nat(A_17),B_14)) = A_17 ).
fof(fact_2673_mod__div__equality2,axiom,
! [B_14,A_17] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_14),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_17),B_14))),hAPP_C498520661umeral(div_mo1740067990umeral(A_17),B_14)) = A_17 ).
fof(fact_2674_mod__div__equality2,axiom,
! [B_14,A_17] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_14),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_17),B_14))),hAPP_Q1762011733de_int(div_mo231679042de_int(A_17),B_14)) = A_17 ).
fof(fact_2675_mod__div__equality,axiom,
! [A_16,B_13] :
( is_int(A_16)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_16),B_13)),B_13)),hAPP_int_int(div_mod_int(A_16),B_13)) = A_16 ) ).
fof(fact_2676_mod__div__equality,axiom,
! [A_16,B_13] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_16),B_13)),B_13)),hAPP_nat_nat(div_mod_nat(A_16),B_13)) = A_16 ).
fof(fact_2677_mod__div__equality,axiom,
! [A_16,B_13] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_16),B_13)),B_13)),hAPP_C498520661umeral(div_mo1740067990umeral(A_16),B_13)) = A_16 ).
fof(fact_2678_mod__div__equality,axiom,
! [A_16,B_13] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_16),B_13)),B_13)),hAPP_Q1762011733de_int(div_mo231679042de_int(A_16),B_13)) = A_16 ).
fof(fact_2679_div__mod__equality2,axiom,
! [B_12,A_15,C_5] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_12),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_15),B_12))),hAPP_int_int(div_mod_int(A_15),B_12))),C_5) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_15),C_5) ).
fof(fact_2680_div__mod__equality2,axiom,
! [B_12,A_15,C_5] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_12),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_15),B_12))),hAPP_nat_nat(div_mod_nat(A_15),B_12))),C_5) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_15),C_5) ).
fof(fact_2681_div__mod__equality2,axiom,
! [B_12,A_15,C_5] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_12),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_15),B_12))),hAPP_C498520661umeral(div_mo1740067990umeral(A_15),B_12))),C_5) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_15),C_5) ).
fof(fact_2682_div__mod__equality2,axiom,
! [B_12,A_15,C_5] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_12),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_15),B_12))),hAPP_Q1762011733de_int(div_mo231679042de_int(A_15),B_12))),C_5) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_15),C_5) ).
fof(fact_2683_div__mod__equality,axiom,
! [A_14,B_11,C_4] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_14),B_11)),B_11)),hAPP_int_int(div_mod_int(A_14),B_11))),C_4) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_14),C_4) ).
fof(fact_2684_div__mod__equality,axiom,
! [A_14,B_11,C_4] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_14),B_11)),B_11)),hAPP_nat_nat(div_mod_nat(A_14),B_11))),C_4) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_14),C_4) ).
fof(fact_2685_div__mod__equality,axiom,
! [A_14,B_11,C_4] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_14),B_11)),B_11)),hAPP_C498520661umeral(div_mo1740067990umeral(A_14),B_11))),C_4) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_14),C_4) ).
fof(fact_2686_div__mod__equality,axiom,
! [A_14,B_11,C_4] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_14),B_11)),B_11)),hAPP_Q1762011733de_int(div_mo231679042de_int(A_14),B_11))),C_4) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_14),C_4) ).
fof(fact_2687_zdiv__leq__prop,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_1),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),Y_1))),X_1)) ) ).
fof(fact_2688_power__preserves__odd,axiom,
! [X_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_2),N)),zOdd))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd)) ) ) ).
fof(fact_2689_div__eq__minus1,axiom,
! [B_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,min)),B_2) = hAPP_int_int(number_number_of_int,min) ) ).
fof(fact_2690_xzgcda__linear__aux2,axiom,
! [R_3,S_1,T_1,R_5,S_3,M,T_2,N_1] :
( R_5 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_3),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_2),N_1))
=> ( R_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_1),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_1),N_1))
=> hAPP_int_int(div_mod_int(R_5),R_3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,S_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),S_1))),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,T_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),T_1))),N_1)) ) ) ).
fof(fact_2691_zcong__zmod__aux,axiom,
! [A_3,B_2,M] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),M)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),M)))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_3),M)),hAPP_int_int(div_mod_int(B_2),M))) ).
fof(fact_2692_Euler_Oaux__2,axiom,
! [P_5] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,P_5),zOdd))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).
fof(fact_2693_split__zdiv,axiom,
! [P_1,N,K_1] :
( ( is_int(N)
& is_int(K_1) )
=> ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),K_1)))
<=> ( ( K_1 = zero_zero_int
=> hBOOL(hAPP_int_bool(P_1,zero_zero_int)) )
& ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
=> ! [I_2,J_1] :
( ( is_int(I_2)
& is_int(J_1) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),J_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),K_1))
& N = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) )
=> hBOOL(hAPP_int_bool(P_1,I_2)) ) ) )
& ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int))
=> ! [I_2,J_1] :
( ( is_int(I_2)
& is_int(J_1) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),J_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int))
& N = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) )
=> hBOOL(hAPP_int_bool(P_1,I_2)) ) ) ) ) ) ) ).
fof(fact_2694_div__prop2,axiom,
! [X_1,Y_1,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_1),Z_1)),Z_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),Z_1)),Y_1)) ) ) ).
fof(fact_2695_divmod__int__rel__div__eq,axiom,
! [A_13,B_10,Y_1,R_7] :
( ( is_int(B_10)
& is_int(Y_1) )
=> ( A_13 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_10),Y_1)),R_7)
=> ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_7))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_7),B_10)) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_10),R_7))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_7),zero_zero_int)) ) ) )
=> ( B_10 != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_13),B_10) = Y_1 ) ) ) ) ).
fof(fact_2696_zmod__zmult2__eq,axiom,
! [A_3,B_2,C_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
=> hAPP_int_int(div_mod_int(A_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),C_1))),hAPP_int_int(div_mod_int(A_3),B_2)) ) ).
fof(fact_2697_posDivAlg__div__mod,axiom,
! [L_2,K_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2))
=> posDivAlg(K_2,L_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),L_2)),hAPP_int_int(div_mod_int(K_2),L_2)) ) ) ).
fof(fact_2698_negDivAlg__div__mod,axiom,
! [L_2,K_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),L_2))
=> negDivAlg(K_2,L_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),L_2)),hAPP_int_int(div_mod_int(K_2),L_2)) ) ) ).
fof(fact_2699_div__pos__neg__trivial,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),zero_zero_int))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_int_int(number_number_of_int,min) ) ) ).
fof(fact_2700_split__neg__lemma,axiom,
! [P_1,N,K_1] :
( is_int(N)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),K_1)),hAPP_int_int(div_mod_int(N),K_1)))
<=> ! [I_2,J_1] :
( ( is_int(I_2)
& is_int(J_1) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),J_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int))
& N = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1)) ) ) ) ) ) ).
fof(fact_2701_split__pos__lemma,axiom,
! [P_1,N,K_1] :
( is_int(N)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),K_1)),hAPP_int_int(div_mod_int(N),K_1)))
<=> ! [I_2,J_1] :
( ( is_int(I_2)
& is_int(J_1) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),J_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),K_1))
& N = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1)) ) ) ) ) ) ).
fof(fact_2702_zdiv__number__of__Bit1,axiom,
! [V_2,W] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_2)),one_one_int)),hAPP_int_int(number_number_of_int,W)) ) ) ).
fof(fact_2703_eq__diff__eq_H,axiom,
! [X_2,Y_2,Z_2] :
( X_2 = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_2),Z_2)
<=> Y_2 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Z_2) ) ).
fof(fact_2704_zprime__zOdd__eq__grt__2,axiom,
! [P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,P_5),zOdd))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) ) ) ).
fof(fact_2705_negDivAlg_Opinduct,axiom,
! [P_1,A0,A1] :
( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(negDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1)))
=> ( ! [A_2,B_4] :
( ( is_int(A_2)
& is_int(B_4) )
=> ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(negDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_4)))
=> ( ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_4)))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_4),zero_zero_int)) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_4))) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),B_4)) ) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1)) ) ) ).
fof(fact_2706_posDivAlg_Opinduct,axiom,
! [P_1,A0,A1] :
( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(posDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1)))
=> ( ! [A_2,B_4] :
( ( is_int(A_2)
& is_int(B_4) )
=> ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(posDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_4)))
=> ( ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_4))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_4),zero_zero_int)) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_4))) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),B_4)) ) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1)) ) ) ).
fof(fact_2707_zOddE,axiom,
! [X_2] :
( is_int(X_2)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd))
=> ~ ! [K] :
( is_int(K)
=> X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K)),one_one_int) ) ) ) ).
fof(fact_2708_mod__div__decomp,axiom,
! [A_12,B_9] :
( is_int(A_12)
=> A_12 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_12),B_9)),B_9)),hAPP_int_int(div_mod_int(A_12),B_9)) ) ).
fof(fact_2709_mod__div__decomp,axiom,
! [A_12,B_9] : A_12 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_12),B_9)),B_9)),hAPP_nat_nat(div_mod_nat(A_12),B_9)) ).
fof(fact_2710_mod__div__decomp,axiom,
! [A_12,B_9] : A_12 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_12),B_9)),B_9)),hAPP_C498520661umeral(div_mo1740067990umeral(A_12),B_9)) ).
fof(fact_2711_mod__div__decomp,axiom,
! [A_12,B_9] : A_12 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_12),B_9)),B_9)),hAPP_Q1762011733de_int(div_mo231679042de_int(A_12),B_9)) ).
fof(fact_2712_zfact__prop,axiom,
! [A_3,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3)) ) ) ) ) ).
fof(fact_2713_Euler__part1,axiom,
! [X_1,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),X_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_int_int(number_number_of_int,min)),P_3)) ) ) ) ) ).
fof(fact_2714_Euler__part3,axiom,
! [X_1,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),X_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),one_one_int),P_3)) ) ) ) ) ).
fof(fact_2715_nat__if__cong,axiom,
! [X_2,Y_2,P_1] :
( ( hBOOL(P_1)
=> hAPP_int_nat(nat,X_2) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,P_1),X_2),Y_2)) )
& ( ~ hBOOL(P_1)
=> hAPP_int_nat(nat,Y_2) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,P_1),X_2),Y_2)) ) ) ).
fof(fact_2716_div__le__mono,axiom,
! [K_2,M,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,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),K_2))) ) ).
fof(fact_2717_div__le__dividend,axiom,
! [M,N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1)),M)) ).
fof(fact_2718_div__mult2__eq,axiom,
! [A_3,B_2,C_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),B_2)),C_1) ).
fof(fact_2719_Divides_Otransfer__nat__int__functions_I1_J,axiom,
! [Y_1,X_1] :
( 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_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(nat,X_1)),hAPP_int_nat(nat,Y_1)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),Y_1)) ) ) ).
fof(fact_2720_nat__div__distrib,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),Y_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(nat,X_1)),hAPP_int_nat(nat,Y_1)) ) ).
fof(fact_2721_div__less,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1) = zero_zero_nat ) ).
fof(fact_2722_nat__mult__div__cancel__disj,axiom,
! [M,N_1,K_2] :
( ( K_2 = zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)) = zero_zero_nat )
& ( K_2 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1) ) ) ).
fof(fact_2723_transfer__nat__int__numerals_I1_J,axiom,
zero_zero_nat = hAPP_int_nat(nat,zero_zero_int) ).
fof(fact_2724_nat__0,axiom,
hAPP_int_nat(nat,zero_zero_int) = zero_zero_nat ).
fof(fact_2725_transfer__nat__int__relations_I1_J,axiom,
! [Y_2,X_2] :
( ( is_int(Y_2)
& is_int(X_2) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
=> ( hAPP_int_nat(nat,X_2) = hAPP_int_nat(nat,Y_2)
<=> X_2 = Y_2 ) ) ) ) ).
fof(fact_2726_all__nat,axiom,
! [P_1] :
( ! [X1] : hBOOL(hAPP_nat_bool(P_1,X1))
<=> ! [X] :
( is_int(X)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat,X))) ) ) ) ).
fof(fact_2727_ex__nat,axiom,
! [P_1] :
( ? [X1] : hBOOL(hAPP_nat_bool(P_1,X1))
<=> ? [X] :
( is_int(X)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
& hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat,X))) ) ) ).
fof(fact_2728_eq__nat__nat__iff,axiom,
! [Z_3,Z_2] :
( ( is_int(Z_3)
& is_int(Z_2) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_3))
=> ( hAPP_int_nat(nat,Z_2) = hAPP_int_nat(nat,Z_3)
<=> Z_2 = Z_3 ) ) ) ) ).
fof(fact_2729_nat__number__of,axiom,
! [W] : hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,W)) = hAPP_int_nat(number_number_of_nat,W) ).
fof(fact_2730_nat__number__of__def,axiom,
! [V_2] : hAPP_int_nat(number_number_of_nat,V_2) = hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,V_2)) ).
fof(fact_2731_transfer__nat__int__numerals_I2_J,axiom,
one_one_nat = hAPP_int_nat(nat,one_one_int) ).
fof(fact_2732_div__add1__eq,axiom,
! [A_3,B_2,C_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_3),B_2)),C_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_2),C_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_3),C_1)),hAPP_nat_nat(div_mod_nat(B_2),C_1))),C_1)) ).
fof(fact_2733_div__le__mono2,axiom,
! [K_2,N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> ( 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(hAPP_nat_fun_nat_nat(div_div_nat,K_2),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,K_2),M))) ) ) ).
fof(fact_2734_div__mult__self__is__m,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)),N_1) = M ) ).
fof(fact_2735_div__mult__self1__is__m,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),M)),N_1) = M ) ).
fof(fact_2736_nat__mult__div__cancel1,axiom,
! [M,N_1,K_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1) ) ).
fof(fact_2737_div__less__dividend,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_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(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1)),M)) ) ) ).
fof(fact_2738_nat__le__0,axiom,
! [Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),zero_zero_int))
=> hAPP_int_nat(nat,Z_1) = zero_zero_nat ) ).
fof(fact_2739_nat__0__iff,axiom,
! [I_1] :
( hAPP_int_nat(nat,I_1) = zero_zero_nat
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),zero_zero_int)) ) ).
fof(fact_2740_nat__mono__iff,axiom,
! [W_1,Z_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,W_1)),hAPP_int_nat(nat,Z_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) ) ) ).
fof(fact_2741_zless__nat__conj,axiom,
! [W_1,Z_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,W_1)),hAPP_int_nat(nat,Z_2)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) ) ) ).
fof(fact_2742_transfer__nat__int__relations_I3_J,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(nat,X_2)),hAPP_int_nat(nat,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_2)) ) ) ) ).
fof(fact_2743_div__mult1__eq,axiom,
! [A_3,B_2,C_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)),C_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_2),C_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),hAPP_nat_nat(div_mod_nat(B_2),C_1))),C_1)) ).
fof(fact_2744_mod__mult2__eq,axiom,
! [A_3,B_2,C_1] : hAPP_nat_nat(div_mod_nat(A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),B_2)),C_1))),hAPP_nat_nat(div_mod_nat(A_3),B_2)) ).
fof(fact_2745_div__mod__equality_H,axiom,
! [M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(div_mod_nat(M),N_1)) ).
fof(fact_2746_mult__div__cancel,axiom,
! [N_1,M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(div_mod_nat(M),N_1)) ).
fof(fact_2747_Divides_Omod__div__equality_H,axiom,
! [M,N_1] : hAPP_nat_nat(div_mod_nat(M),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1)),N_1)) ).
fof(fact_2748_split__div,axiom,
! [P_1,N,K_1] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),K_1)))
<=> ( ( K_1 = zero_zero_nat
=> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
& ( K_1 != zero_zero_nat
=> ! [I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K_1))
=> ( N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),I_2)),J_1)
=> hBOOL(hAPP_nat_bool(P_1,I_2)) ) ) ) ) ) ).
fof(fact_2749_zero__less__nat__eq,axiom,
! [Z_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(nat,Z_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2)) ) ).
fof(fact_2750_transfer__nat__int__numerals_I4_J,axiom,
hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) = hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) ).
fof(fact_2751_transfer__nat__int__relations_I2_J,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,X_2)),hAPP_int_nat(nat,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Y_2)) ) ) ) ).
fof(fact_2752_nat__less__eq__zless,axiom,
! [Z_2,W_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,W_1)),hAPP_int_nat(nat,Z_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) ) ) ).
fof(fact_2753_nat__le__eq__zle,axiom,
! [Z_2,W_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),W_1))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(nat,W_1)),hAPP_int_nat(nat,Z_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_2)) ) ) ).
fof(fact_2754_Nat__Transfer_Otransfer__nat__int__functions_I2_J,axiom,
! [Y_1,X_1] :
( 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_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,X_1)),hAPP_int_nat(nat,Y_1)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1)) ) ) ).
fof(fact_2755_nat__mult__distrib,axiom,
! [Z_4,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
=> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,Z_1)),hAPP_int_nat(nat,Z_4)) ) ).
fof(fact_2756_Nat__Transfer_Otransfer__nat__int__functions_I1_J,axiom,
! [Y_1,X_1] :
( 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_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,X_1)),hAPP_int_nat(nat,Y_1)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_1)) ) ) ).
fof(fact_2757_nat__add__distrib,axiom,
! [Z_4,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_4))
=> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,Z_1)),hAPP_int_nat(nat,Z_4)) ) ) ).
fof(fact_2758_transfer__nat__int__relations_I4_J,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat,X_2)),hAPP_int_nat(nat,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_2),Y_2)) ) ) ) ).
fof(fact_2759_nat__diff__distrib,axiom,
! [Z_1,Z_4] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_4),Z_1))
=> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,Z_1)),hAPP_int_nat(nat,Z_4)) ) ) ).
fof(fact_2760_Nat__Transfer_Otransfer__nat__int__functions_I4_J,axiom,
! [N_1,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(nat,X_1)),N_1) = hAPP_int_nat(nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N_1)) ) ).
fof(fact_2761_nat__power__eq,axiom,
! [N_1,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
=> hAPP_int_nat(nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(nat,Z_1)),N_1) ) ).
fof(fact_2762_Divides_Otransfer__nat__int__functions_I2_J,axiom,
! [Y_1,X_1] :
( 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_nat_nat(div_mod_nat(hAPP_int_nat(nat,X_1)),hAPP_int_nat(nat,Y_1)) = hAPP_int_nat(nat,hAPP_int_int(div_mod_int(X_1),Y_1)) ) ) ).
fof(fact_2763_nat__mod__distrib,axiom,
! [Y_1,X_1] :
( 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_nat(nat,hAPP_int_int(div_mod_int(X_1),Y_1)) = hAPP_nat_nat(div_mod_nat(hAPP_int_nat(nat,X_1)),hAPP_int_nat(nat,Y_1)) ) ) ).
fof(fact_2764_add__self__div__2,axiom,
! [M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),M)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = M ).
fof(fact_2765_transfer__nat__int__numerals_I3_J,axiom,
hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) = hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_2766_Euler_Oaux__1,axiom,
! [A_3,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_3))
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat,P_3)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,P_3)),one_one_nat))) ) ).
fof(fact_2767_EvenOdd_Oneg__one__odd__power,axiom,
! [X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,X_2)) = hAPP_int_int(number_number_of_int,min) ) ) ).
fof(fact_2768_Little__Fermat,axiom,
! [X_1,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),X_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))),one_one_int),P_3)) ) ) ).
fof(fact_2769_div__2__gt__zero,axiom,
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).
fof(fact_2770_MultInv__def,axiom,
! [P_3,X_1] : hAPP_int_int(multInv(P_3),X_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_2771_Suc__n__div__2__gt__zero,axiom,
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).
fof(fact_2772_Int2_Oaux__2,axiom,
! [P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).
fof(fact_2773_Int2_Oaux__1,axiom,
! [P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,P_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) ).
fof(fact_2774_WilsonRuss_Oinv__def,axiom,
! [P_3,A_3] : hAPP_int_int(inv(P_3),A_3) = hAPP_int_int(div_mod_int(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3) ).
fof(fact_2775_Euler_Oaux____2,axiom,
! [P_3] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_2776_Euler__part2,axiom,
! [A_3,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3)) ) ) ) ).
fof(fact_2777_Euler__Criterion,axiom,
! [A_3,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(legendre(A_3),P_3)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3)) ) ) ).
fof(fact_2778_nat__aux__def,axiom,
! [I,N_1] : hAPP_nat_nat(nat_aux(I),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,I)),N_1) ).
fof(fact_2779_mod__eqD,axiom,
! [M,D_3,R_3] :
( hAPP_nat_nat(div_mod_nat(M),D_3) = R_3
=> ? [Q_4] : M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,R_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Q_4),D_3)) ) ).
fof(fact_2780_mod__eq__0D,axiom,
! [M_14,D_4] :
( hAPP_nat_nat(div_mod_nat(M_14),D_4) = zero_zero_nat
=> ? [Q_4] : M_14 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,D_4),Q_4) ) ).
fof(fact_2781_phi__prime,axiom,
! [P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> hAPP_int_nat(phi,P_3) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)) ) ).
fof(fact_2782_transfer__morphism__nat__int,axiom,
hBOOL(hAPP_f448129468l_bool(nat_tr876908586nt_nat(nat),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))) ).
fof(fact_2783_zmult2__lemma,axiom,
! [C_1,A_3,B_2,Q_2,R_3] :
( is_int(B_2)
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)))
=> ( B_2 != zero_zero_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Q_2),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3)))) ) ) ) ) ).
fof(fact_2784_Nat__Transfer_Otransfer__nat__int__functions_I3_J,axiom,
! [Y_1,X_1] :
( 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_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,X_1)),hAPP_int_nat(nat,Y_1)) = hAPP_int_nat(nat,hAPP_int_int(nat_tsub(X_1),Y_1)) ) ) ).
fof(fact_2785_transfer__morphismI,axiom,
! [F,A_1] : hBOOL(hAPP_f448129468l_bool(nat_tr876908586nt_nat(F),A_1)) ).
fof(fact_2786_transfer__morphismI,axiom,
! [F,A_1] : hBOOL(hAPP_f54304608l_bool(nat_tr160667106at_int(F),A_1)) ).
fof(fact_2787_transfer__morphism__def,axiom,
! [F,A_1] : hBOOL(hAPP_f448129468l_bool(nat_tr876908586nt_nat(F),A_1)) ).
fof(fact_2788_transfer__morphism__def,axiom,
! [F,A_1] : hBOOL(hAPP_f54304608l_bool(nat_tr160667106at_int(F),A_1)) ).
fof(fact_2789_unique__remainder,axiom,
! [Q_8,R_5,A_3,B_2,Q_2,R_3] :
( ( is_int(R_5)
& is_int(B_2)
& is_int(R_3) )
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)))
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_8),R_5)))
=> ( B_2 != zero_zero_int
=> R_3 = R_5 ) ) ) ) ).
fof(fact_2790_unique__quotient,axiom,
! [Q_8,R_5,A_3,B_2,Q_2,R_3] :
( ( is_int(Q_8)
& is_int(B_2)
& is_int(Q_2) )
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)))
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_8),R_5)))
=> ( B_2 != zero_zero_int
=> Q_2 = Q_8 ) ) ) ) ).
fof(fact_2791_self__remainder,axiom,
! [A_3,Q_2,R_3] :
( ( is_int(A_3)
& is_int(R_3) )
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,A_3),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)))
=> ( A_3 != zero_zero_int
=> R_3 = zero_zero_int ) ) ) ).
fof(fact_2792_divmod__int__rel__0,axiom,
! [B_2] :
( is_int(B_2)
=> ( B_2 != zero_zero_int
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(zero_zero_int,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int))) ) ) ).
fof(fact_2793_self__quotient,axiom,
! [A_3,Q_2,R_3] :
( ( is_int(A_3)
& is_int(Q_2) )
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,A_3),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)))
=> ( A_3 != zero_zero_int
=> Q_2 = one_one_int ) ) ) ).
fof(fact_2794_divmod__int__rel__div,axiom,
! [A_3,B_2,Q_2,R_3] :
( ( is_int(B_2)
& is_int(Q_2) )
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)))
=> ( B_2 != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = Q_2 ) ) ) ).
fof(fact_2795_divmod__int__rel__mod,axiom,
! [A_3,B_2,Q_2,R_3] :
( ( is_int(B_2)
& is_int(R_3) )
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)))
=> ( B_2 != zero_zero_int
=> hAPP_int_int(div_mod_int(A_3),B_2) = R_3 ) ) ) ).
fof(fact_2796_negDivAlg__correct,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),negDivAlg(A_3,B_2))) ) ) ).
fof(fact_2797_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(nat_tsub(X_1),Y_1))) ) ) ).
fof(fact_2798_tsub__eq,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1))
=> hAPP_int_int(nat_tsub(X_1),Y_1) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_1) ) ).
fof(fact_2799_divmod__int__rel__div__mod,axiom,
! [A_3,B_2] :
( is_int(B_2)
=> ( B_2 != zero_zero_int
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),hAPP_int_int(div_mod_int(A_3),B_2)))) ) ) ).
fof(fact_2800_posDivAlg__correct,axiom,
! [B_2,A_3] :
( 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_int,zero_zero_int),B_2))
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),posDivAlg(A_3,B_2))) ) ) ).
fof(fact_2801_zadd1__lemma,axiom,
! [B_2,Bq,Br,A_3,C_1,Aq,Ar] :
( is_int(C_1)
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Aq),Ar)))
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(B_2,C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Bq),Br)))
=> ( C_1 != zero_zero_int
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2),C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aq),Bq)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ar),Br)),C_1))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ar),Br)),C_1)))) ) ) ) ) ).
fof(fact_2802_tsub__def,axiom,
! [Y_1,X_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1))
=> hAPP_int_int(nat_tsub(X_1),Y_1) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_1) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1))
=> hAPP_int_int(nat_tsub(X_1),Y_1) = zero_zero_int ) ) ).
fof(fact_2803_divmod__int__relI,axiom,
! [A_3,B_2,Q_2,R_3] :
( A_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3)
=> ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2)) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_3))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int)) ) ) )
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3))) ) ) ).
fof(fact_2804_zmult1__lemma,axiom,
! [A_3,B_2,C_1,Q_2,R_3] :
( is_int(C_1)
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(B_2,C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)))
=> ( C_1 != zero_zero_int
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2),C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_2)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),R_3)),C_1))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),R_3)),C_1)))) ) ) ) ).
fof(fact_2805_inv__inv__aux,axiom,
! [P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_3))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))))) ) ).
fof(fact_2806_two__times__odd__div__two__plus__one,axiom,
! [X_1] :
( is_int(X_1)
=> ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_int) = X_1 ) ) ).
fof(fact_2807_Code__Numeral_Oint__of__code,axiom,
! [K_2] :
( ( K_2 = zero_z126310315umeral
=> code_int_of(K_2) = zero_zero_int )
& ( K_2 != zero_z126310315umeral
=> ( ( hAPP_C498520661umeral(div_mo1740067990umeral(K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_z126310315umeral
=> code_int_of(K_2) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),code_int_of(hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) )
& ( hAPP_C498520661umeral(div_mo1740067990umeral(K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_z126310315umeral
=> code_int_of(K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),code_int_of(hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),one_one_int) ) ) ) ) ).
fof(fact_2808_EvenOdd_Oneg__one__even__power,axiom,
! [X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,X_2)) = one_one_int ) ) ).
fof(fact_2809_of__nat__double,axiom,
! [X_11] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_11))
=> hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_11))) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_int_nat(nat,X_11))),hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_int_nat(nat,X_11))) ) ).
fof(fact_2810_of__nat__double,axiom,
! [X_11] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_11))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_11))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,X_11))),hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,X_11))) ) ).
fof(fact_2811_of__nat__double,axiom,
! [X_11] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_11))
=> hAPP_nat_nat(semiri984289939at_nat,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_11))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(semiri984289939at_nat,hAPP_int_nat(nat,X_11))),hAPP_nat_nat(semiri984289939at_nat,hAPP_int_nat(nat,X_11))) ) ).
fof(fact_2812_of__nat__double,axiom,
! [X_11] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_11))
=> hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_11))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(nat,X_11))),hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(nat,X_11))) ) ).
fof(fact_2813_of__nat__double,axiom,
! [X_11] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_11))
=> hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_11))) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(nat,X_11))),hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(nat,X_11))) ) ).
fof(fact_2814_of__nat__double,axiom,
! [X_11] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_11))
=> hAPP_n522471361de_int(semiri1424489471de_int,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_11))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_n522471361de_int(semiri1424489471de_int,hAPP_int_nat(nat,X_11))),hAPP_n522471361de_int(semiri1424489471de_int,hAPP_int_nat(nat,X_11))) ) ).
fof(fact_2815_of__nat__double,axiom,
! [X_11] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_11))
=> hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_11))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(nat,X_11))),hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(nat,X_11))) ) ).
fof(fact_2816_less__mono__imp__le__mono,axiom,
! [I_1,J,F] :
( ! [I_2,J_1] :
( 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))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,I_1)),hAPP_nat_nat(F,J))) ) ) ).
fof(fact_2817_field__le__mult__one__interval,axiom,
! [X_10,Y_7] :
( ! [Z] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z),X_10)),Y_7)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_10),Y_7)) ) ).
fof(fact_2818_field__le__mult__one__interval,axiom,
! [X_10,Y_7] :
( ! [Z] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Z))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z),one_one_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Z),X_10)),Y_7)) ) )
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_10),Y_7)) ) ).
fof(fact_2819_lessI,axiom,
! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,N_1))) ).
fof(fact_2820_Suc__mono,axiom,
! [M,N_1] :
( 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))) ) ).
fof(fact_2821_zero__less__Suc,axiom,
! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(suc,N_1))) ).
fof(fact_2822_dvd__1__left,axiom,
! [K_2] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(suc,zero_zero_nat)),K_2)) ).
fof(fact_2823_code__numeral__zero__minus__one,axiom,
hAPP_C498520661umeral(hAPP_C1594335432umeral(minus_1690775515umeral,zero_z126310315umeral),one_on1645066479umeral) = zero_z126310315umeral ).
fof(fact_2824_number__of__int,axiom,
! [N_26] : hAPP_int_nat(number_number_of_nat,hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_nat(semiri984289939at_nat,N_26) ).
fof(fact_2825_number__of__int,axiom,
! [N_26] : hAPP_int_int(number_number_of_int,hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_int(semiri1621563631at_int,N_26) ).
fof(fact_2826_number__of__int,axiom,
! [N_26] : hAPP_int_real(number267125858f_real,hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_real(semiri132038758t_real,N_26) ).
fof(fact_2827_number__of__int,axiom,
! [N_26] : hAPP_int_complex(number528085621omplex,hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_complex(semiri2020571505omplex,N_26) ).
fof(fact_2828_number__of__int,axiom,
! [N_26] : hAPP_int_rat(number_number_of_rat,hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_rat(semiri151668891at_rat,N_26) ).
fof(fact_2829_Suc__inject,axiom,
! [X_1,Y_1] :
( hAPP_nat_nat(suc,X_1) = hAPP_nat_nat(suc,Y_1)
=> X_1 = Y_1 ) ).
fof(fact_2830_of__nat__eq__iff,axiom,
! [Ma,N] :
( hAPP_n522471361de_int(semiri1424489471de_int,Ma) = hAPP_n522471361de_int(semiri1424489471de_int,N)
<=> Ma = N ) ).
fof(fact_2831_of__nat__eq__iff,axiom,
! [Ma,N] :
( hAPP_n1108039445umeral(semiri1619134803umeral,Ma) = hAPP_n1108039445umeral(semiri1619134803umeral,N)
<=> Ma = N ) ).
fof(fact_2832_of__nat__eq__iff,axiom,
! [Ma,N] :
( hAPP_nat_nat(semiri984289939at_nat,Ma) = hAPP_nat_nat(semiri984289939at_nat,N)
<=> Ma = N ) ).
fof(fact_2833_of__nat__eq__iff,axiom,
! [Ma,N] :
( hAPP_nat_int(semiri1621563631at_int,Ma) = hAPP_nat_int(semiri1621563631at_int,N)
<=> Ma = N ) ).
fof(fact_2834_of__nat__eq__iff,axiom,
! [Ma,N] :
( hAPP_nat_real(semiri132038758t_real,Ma) = hAPP_nat_real(semiri132038758t_real,N)
<=> Ma = N ) ).
fof(fact_2835_of__nat__eq__iff,axiom,
! [Ma,N] :
( hAPP_nat_complex(semiri2020571505omplex,Ma) = hAPP_nat_complex(semiri2020571505omplex,N)
<=> Ma = N ) ).
fof(fact_2836_of__nat__eq__iff,axiom,
! [Ma,N] :
( hAPP_nat_rat(semiri151668891at_rat,Ma) = hAPP_nat_rat(semiri151668891at_rat,N)
<=> Ma = N ) ).
fof(fact_2837_nat_Oinject,axiom,
! [Nat_5,Nat_4] :
( hAPP_nat_nat(suc,Nat_5) = hAPP_nat_nat(suc,Nat_4)
<=> Nat_5 = Nat_4 ) ).
fof(fact_2838_Suc__n__not__n,axiom,
! [N_1] : hAPP_nat_nat(suc,N_1) != N_1 ).
fof(fact_2839_n__not__Suc__n,axiom,
! [N_1] : N_1 != hAPP_nat_nat(suc,N_1) ).
fof(fact_2840_of__nat__Suc,axiom,
! [M_13] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,M_13)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,M_13)) ).
fof(fact_2841_of__nat__Suc,axiom,
! [M_13] : hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(suc,M_13)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_nat_nat(semiri984289939at_nat,M_13)) ).
fof(fact_2842_of__nat__Suc,axiom,
! [M_13] : hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(suc,M_13)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(semiri132038758t_real,M_13)) ).
fof(fact_2843_of__nat__Suc,axiom,
! [M_13] : hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_nat_nat(suc,M_13)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,one_on1645066479umeral),hAPP_n1108039445umeral(semiri1619134803umeral,M_13)) ).
fof(fact_2844_of__nat__Suc,axiom,
! [M_13] : hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(suc,M_13)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),hAPP_nat_complex(semiri2020571505omplex,M_13)) ).
fof(fact_2845_of__nat__Suc,axiom,
! [M_13] : hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(suc,M_13)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,one_on1684967323de_int),hAPP_n522471361de_int(semiri1424489471de_int,M_13)) ).
fof(fact_2846_of__nat__Suc,axiom,
! [M_13] : hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(suc,M_13)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),hAPP_nat_rat(semiri151668891at_rat,M_13)) ).
fof(fact_2847_int__Suc0__eq__1,axiom,
hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,zero_zero_nat)) = one_one_int ).
fof(fact_2848_zless__iff__Suc__zadd,axiom,
! [W_1,Z_2] :
( is_int(Z_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2))
<=> ? [N_2] : Z_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2))) ) ) ).
fof(fact_2849_int__Suc,axiom,
! [M] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,M)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,M)) ).
fof(fact_2850_Suc__neq__Zero,axiom,
! [M] : hAPP_nat_nat(suc,M) != zero_zero_nat ).
fof(fact_2851_Zero__neq__Suc,axiom,
! [M] : zero_zero_nat != hAPP_nat_nat(suc,M) ).
fof(fact_2852_nat_Osimps_I3_J,axiom,
! [Nat_3] : hAPP_nat_nat(suc,Nat_3) != zero_zero_nat ).
fof(fact_2853_Suc__not__Zero,axiom,
! [M] : hAPP_nat_nat(suc,M) != zero_zero_nat ).
fof(fact_2854_nat_Osimps_I2_J,axiom,
! [Nat_2] : zero_zero_nat != hAPP_nat_nat(suc,Nat_2) ).
fof(fact_2855_Zero__not__Suc,axiom,
! [M] : zero_zero_nat != hAPP_nat_nat(suc,M) ).
fof(fact_2856_nat__int,axiom,
! [N_1] : hAPP_int_nat(nat,hAPP_nat_int(semiri1621563631at_int,N_1)) = N_1 ).
fof(fact_2857_not__less__eq,axiom,
! [Ma,N] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,Ma))) ) ).
fof(fact_2858_less__Suc__eq,axiom,
! [Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(suc,N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N))
| Ma = N ) ) ).
fof(fact_2859_Suc__less__eq,axiom,
! [Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,Ma)),hAPP_nat_nat(suc,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ).
fof(fact_2860_not__less__less__Suc__eq,axiom,
! [N,Ma] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,Ma)))
<=> N = Ma ) ) ).
fof(fact_2861_less__antisym,axiom,
! [N_1,M] :
( ~ 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 ) ) ).
fof(fact_2862_less__SucI,axiom,
! [M,N_1] :
( 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))) ) ).
fof(fact_2863_Suc__lessI,axiom,
! [M,N_1] :
( 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)) ) ) ).
fof(fact_2864_less__trans__Suc,axiom,
! [K_2,I,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),K_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I)),K_2)) ) ) ).
fof(fact_2865_less__SucE,axiom,
! [M,N_1] :
( 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 ) ) ).
fof(fact_2866_Suc__lessD,axiom,
! [M,N_1] :
( 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)) ) ).
fof(fact_2867_Suc__less__SucD,axiom,
! [M,N_1] :
( 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)) ) ).
fof(fact_2868_add__Suc__shift,axiom,
! [M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(suc,N_1)) ).
fof(fact_2869_add__Suc,axiom,
! [M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M)),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)) ).
fof(fact_2870_add__Suc__right,axiom,
! [M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)) ).
fof(fact_2871_Suc__leD,axiom,
! [M,N_1] :
( 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)) ) ).
fof(fact_2872_le__SucE,axiom,
! [M,N_1] :
( 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) ) ) ).
fof(fact_2873_le__SucI,axiom,
! [M,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,M),hAPP_nat_nat(suc,N_1))) ) ).
fof(fact_2874_Suc__le__mono,axiom,
! [N,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),hAPP_nat_nat(suc,Ma)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),Ma)) ) ).
fof(fact_2875_le__Suc__eq,axiom,
! [Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),hAPP_nat_nat(suc,N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N))
| Ma = hAPP_nat_nat(suc,N) ) ) ).
fof(fact_2876_not__less__eq__eq,axiom,
! [Ma,N] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),Ma)) ) ).
fof(fact_2877_Suc__n__not__le__n,axiom,
! [N_1] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),N_1)) ).
fof(fact_2878_Suc__mult__cancel1,axiom,
! [K_1,Ma,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),N)
<=> Ma = N ) ).
fof(fact_2879_diff__Suc__Suc,axiom,
! [M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1) ).
fof(fact_2880_Suc__diff__diff,axiom,
! [M,N_1,K_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N_1)),hAPP_nat_nat(suc,K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),K_2) ).
fof(fact_2881_mod__Suc__eq__Suc__mod,axiom,
! [M,N_1] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N_1))),N_1) ).
fof(fact_2882_exp__mono__eq,axiom,
! [X_2,N,Y_2] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_2),hAPP_nat_nat(suc,N))
<=> X_2 = Y_2 ) ).
fof(fact_2883_even__zero__int,axiom,
hBOOL(hAPP_int_bool(even_odd_even_int,zero_zero_int)) ).
fof(fact_2884_odd__one__int,axiom,
~ hBOOL(hAPP_int_bool(even_odd_even_int,one_one_int)) ).
fof(fact_2885_Parity_Oodd__times__odd,axiom,
! [Y_1,X_1] :
( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
=> ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,Y_1))
=> ~ hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1))) ) ) ).
fof(fact_2886_anything__times__even,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_int_bool(even_odd_even_int,Y_1))
=> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1))) ) ).
fof(fact_2887_even__times__anything,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
=> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1))) ) ).
fof(fact_2888_Parity_Oeven__product,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Y_2)))
<=> ( hBOOL(hAPP_int_bool(even_odd_even_int,X_2))
| hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) ) ) ).
fof(fact_2889_Parity_Oodd__plus__odd,axiom,
! [Y_1,X_1] :
( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
=> ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,Y_1))
=> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_1))) ) ) ).
fof(fact_2890_odd__plus__even,axiom,
! [Y_1,X_1] :
( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
=> ( hBOOL(hAPP_int_bool(even_odd_even_int,Y_1))
=> ~ hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_1))) ) ) ).
fof(fact_2891_Parity_Oeven__plus__odd,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
=> ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,Y_1))
=> ~ hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_1))) ) ) ).
fof(fact_2892_Parity_Oeven__plus__even,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
=> ( hBOOL(hAPP_int_bool(even_odd_even_int,Y_1))
=> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_1))) ) ) ).
fof(fact_2893_even__sum,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Y_2)))
<=> ( ( hBOOL(hAPP_int_bool(even_odd_even_int,X_2))
& hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) )
| ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X_2))
& ~ hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) ) ) ) ).
fof(fact_2894_even__difference,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),Y_2)))
<=> ( ( hBOOL(hAPP_int_bool(even_odd_even_int,X_2))
& hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) )
| ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X_2))
& ~ hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) ) ) ) ).
fof(fact_2895_odd__pow,axiom,
! [N_1,X_1] :
( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
=> ~ hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N_1))) ) ).
fof(fact_2896_one__not__even,axiom,
~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,one_one_int),zEven)) ).
fof(fact_2897_EvenOdd_Oeven__product,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Y_2)),zEven))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven))
| hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) ) ) ).
fof(fact_2898_even__times__either,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Y_2)),zEven)) ) ).
fof(fact_2899_EvenOdd_Oeven__plus__even,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Y_2)),zEven)) ) ) ).
fof(fact_2900_even__diff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),Y_2)),zEven))
<=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) ) ) ).
fof(fact_2901_even__minus__even,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),Y_2)),zEven)) ) ) ).
fof(fact_2902_not__odd__impl__even,axiom,
! [X_2] :
( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) ) ).
fof(fact_2903_even__odd__disj,axiom,
! [X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd))
| hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) ) ).
fof(fact_2904_odd__iff__not__even,axiom,
! [X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd))
<=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) ) ).
fof(fact_2905_even__odd__conj,axiom,
! [X_2] :
~ ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd))
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) ) ).
fof(fact_2906_of__nat__0__le__iff,axiom,
! [N_25] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(semiri151668891at_rat,N_25))) ).
fof(fact_2907_of__nat__0__le__iff,axiom,
! [N_25] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_n522471361de_int(semiri1424489471de_int,N_25))) ).
fof(fact_2908_of__nat__0__le__iff,axiom,
! [N_25] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_n1108039445umeral(semiri1619134803umeral,N_25))) ).
fof(fact_2909_of__nat__0__le__iff,axiom,
! [N_25] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,N_25))) ).
fof(fact_2910_of__nat__0__le__iff,axiom,
! [N_25] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,N_25))) ).
fof(fact_2911_of__nat__0__le__iff,axiom,
! [N_25] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_25))) ).
fof(fact_2912_zero__le__imp__of__nat,axiom,
! [M_12] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(semiri151668891at_rat,M_12))) ).
fof(fact_2913_zero__le__imp__of__nat,axiom,
! [M_12] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_n522471361de_int(semiri1424489471de_int,M_12))) ).
fof(fact_2914_zero__le__imp__of__nat,axiom,
! [M_12] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_n1108039445umeral(semiri1619134803umeral,M_12))) ).
fof(fact_2915_zero__le__imp__of__nat,axiom,
! [M_12] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,M_12))) ).
fof(fact_2916_zero__le__imp__of__nat,axiom,
! [M_12] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,M_12))) ).
fof(fact_2917_zero__le__imp__of__nat,axiom,
! [M_12] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,M_12))) ).
fof(fact_2918_of__nat__less__0__iff,axiom,
! [M_11] : ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,M_11)),zero_zero_rat)) ).
fof(fact_2919_of__nat__less__0__iff,axiom,
! [M_11] : ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_11)),zero_z891286103de_int)) ).
fof(fact_2920_of__nat__less__0__iff,axiom,
! [M_11] : ~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_11)),zero_z126310315umeral)) ).
fof(fact_2921_of__nat__less__0__iff,axiom,
! [M_11] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_11)),zero_zero_real)) ).
fof(fact_2922_of__nat__less__0__iff,axiom,
! [M_11] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_11)),zero_zero_nat)) ).
fof(fact_2923_of__nat__less__0__iff,axiom,
! [M_11] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_11)),zero_zero_int)) ).
fof(fact_2924_of__nat__0,axiom,
hAPP_nat_rat(semiri151668891at_rat,zero_zero_nat) = zero_zero_rat ).
fof(fact_2925_of__nat__0,axiom,
hAPP_n522471361de_int(semiri1424489471de_int,zero_zero_nat) = zero_z891286103de_int ).
fof(fact_2926_of__nat__0,axiom,
hAPP_nat_complex(semiri2020571505omplex,zero_zero_nat) = zero_zero_complex ).
fof(fact_2927_of__nat__0,axiom,
hAPP_n1108039445umeral(semiri1619134803umeral,zero_zero_nat) = zero_z126310315umeral ).
fof(fact_2928_of__nat__0,axiom,
hAPP_nat_real(semiri132038758t_real,zero_zero_nat) = zero_zero_real ).
fof(fact_2929_of__nat__0,axiom,
hAPP_nat_nat(semiri984289939at_nat,zero_zero_nat) = zero_zero_nat ).
fof(fact_2930_of__nat__0,axiom,
hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) = zero_zero_int ).
fof(fact_2931_of__nat__less__iff,axiom,
! [Ma,N] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(semiri1619134803umeral,Ma)),hAPP_n1108039445umeral(semiri1619134803umeral,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ).
fof(fact_2932_of__nat__less__iff,axiom,
! [Ma,N] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,Ma)),hAPP_nat_rat(semiri151668891at_rat,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ).
fof(fact_2933_of__nat__less__iff,axiom,
! [Ma,N] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,Ma)),hAPP_n522471361de_int(semiri1424489471de_int,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ).
fof(fact_2934_of__nat__less__iff,axiom,
! [Ma,N] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ).
fof(fact_2935_of__nat__less__iff,axiom,
! [Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ).
fof(fact_2936_of__nat__less__iff,axiom,
! [Ma,N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ).
fof(fact_2937_less__imp__of__nat__less,axiom,
! [M_10,N_24] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_10)),hAPP_n1108039445umeral(semiri1619134803umeral,N_24))) ) ).
fof(fact_2938_less__imp__of__nat__less,axiom,
! [M_10,N_24] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,M_10)),hAPP_nat_rat(semiri151668891at_rat,N_24))) ) ).
fof(fact_2939_less__imp__of__nat__less,axiom,
! [M_10,N_24] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_10)),hAPP_n522471361de_int(semiri1424489471de_int,N_24))) ) ).
fof(fact_2940_less__imp__of__nat__less,axiom,
! [M_10,N_24] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_10)),hAPP_nat_real(semiri132038758t_real,N_24))) ) ).
fof(fact_2941_less__imp__of__nat__less,axiom,
! [M_10,N_24] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_10)),hAPP_nat_nat(semiri984289939at_nat,N_24))) ) ).
fof(fact_2942_less__imp__of__nat__less,axiom,
! [M_10,N_24] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_10)),hAPP_nat_int(semiri1621563631at_int,N_24))) ) ).
fof(fact_2943_of__nat__less__imp__less,axiom,
! [M_9,N_23] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_9)),hAPP_n1108039445umeral(semiri1619134803umeral,N_23)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23)) ) ).
fof(fact_2944_of__nat__less__imp__less,axiom,
! [M_9,N_23] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,M_9)),hAPP_nat_rat(semiri151668891at_rat,N_23)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23)) ) ).
fof(fact_2945_of__nat__less__imp__less,axiom,
! [M_9,N_23] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_9)),hAPP_n522471361de_int(semiri1424489471de_int,N_23)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23)) ) ).
fof(fact_2946_of__nat__less__imp__less,axiom,
! [M_9,N_23] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_9)),hAPP_nat_real(semiri132038758t_real,N_23)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23)) ) ).
fof(fact_2947_of__nat__less__imp__less,axiom,
! [M_9,N_23] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_9)),hAPP_nat_nat(semiri984289939at_nat,N_23)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23)) ) ).
fof(fact_2948_of__nat__less__imp__less,axiom,
! [M_9,N_23] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_9)),hAPP_nat_int(semiri1621563631at_int,N_23)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23)) ) ).
fof(fact_2949_of__nat__le__iff,axiom,
! [Ma,N] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(semiri1619134803umeral,Ma)),hAPP_n1108039445umeral(semiri1619134803umeral,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ).
fof(fact_2950_of__nat__le__iff,axiom,
! [Ma,N] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(semiri1424489471de_int,Ma)),hAPP_n522471361de_int(semiri1424489471de_int,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ).
fof(fact_2951_of__nat__le__iff,axiom,
! [Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ).
fof(fact_2952_of__nat__le__iff,axiom,
! [Ma,N] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(semiri151668891at_rat,Ma)),hAPP_nat_rat(semiri151668891at_rat,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ).
fof(fact_2953_of__nat__le__iff,axiom,
! [Ma,N] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ).
fof(fact_2954_of__nat__le__iff,axiom,
! [Ma,N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ).
fof(fact_2955_of__nat__add,axiom,
! [M_8,N_22] : hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_8)),hAPP_n1108039445umeral(semiri1619134803umeral,N_22)) ).
fof(fact_2956_of__nat__add,axiom,
! [M_8,N_22] : hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(semiri151668891at_rat,M_8)),hAPP_nat_rat(semiri151668891at_rat,N_22)) ).
fof(fact_2957_of__nat__add,axiom,
! [M_8,N_22] : hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_8)),hAPP_n522471361de_int(semiri1424489471de_int,N_22)) ).
fof(fact_2958_of__nat__add,axiom,
! [M_8,N_22] : hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_nat_complex(semiri2020571505omplex,M_8)),hAPP_nat_complex(semiri2020571505omplex,N_22)) ).
fof(fact_2959_of__nat__add,axiom,
! [M_8,N_22] : hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(semiri132038758t_real,M_8)),hAPP_nat_real(semiri132038758t_real,N_22)) ).
fof(fact_2960_of__nat__add,axiom,
! [M_8,N_22] : hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(semiri984289939at_nat,M_8)),hAPP_nat_nat(semiri984289939at_nat,N_22)) ).
fof(fact_2961_of__nat__add,axiom,
! [M_8,N_22] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M_8)),hAPP_nat_int(semiri1621563631at_int,N_22)) ).
fof(fact_2962_of__nat__mult,axiom,
! [M_7,N_21] : hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_7)),hAPP_n1108039445umeral(semiri1619134803umeral,N_21)) ).
fof(fact_2963_of__nat__mult,axiom,
! [M_7,N_21] : hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(semiri151668891at_rat,M_7)),hAPP_nat_rat(semiri151668891at_rat,N_21)) ).
fof(fact_2964_of__nat__mult,axiom,
! [M_7,N_21] : hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_7)),hAPP_n522471361de_int(semiri1424489471de_int,N_21)) ).
fof(fact_2965_of__nat__mult,axiom,
! [M_7,N_21] : hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(semiri2020571505omplex,M_7)),hAPP_nat_complex(semiri2020571505omplex,N_21)) ).
fof(fact_2966_of__nat__mult,axiom,
! [M_7,N_21] : hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(semiri132038758t_real,M_7)),hAPP_nat_real(semiri132038758t_real,N_21)) ).
fof(fact_2967_of__nat__mult,axiom,
! [M_7,N_21] : hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(semiri984289939at_nat,M_7)),hAPP_nat_nat(semiri984289939at_nat,N_21)) ).
fof(fact_2968_of__nat__mult,axiom,
! [M_7,N_21] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M_7)),hAPP_nat_int(semiri1621563631at_int,N_21)) ).
fof(fact_2969_of__nat__1,axiom,
hAPP_nat_rat(semiri151668891at_rat,one_one_nat) = one_one_rat ).
fof(fact_2970_of__nat__1,axiom,
hAPP_n522471361de_int(semiri1424489471de_int,one_one_nat) = one_on1684967323de_int ).
fof(fact_2971_of__nat__1,axiom,
hAPP_nat_complex(semiri2020571505omplex,one_one_nat) = one_one_complex ).
fof(fact_2972_of__nat__1,axiom,
hAPP_n1108039445umeral(semiri1619134803umeral,one_one_nat) = one_on1645066479umeral ).
fof(fact_2973_of__nat__1,axiom,
hAPP_nat_real(semiri132038758t_real,one_one_nat) = one_one_real ).
fof(fact_2974_of__nat__1,axiom,
hAPP_nat_nat(semiri984289939at_nat,one_one_nat) = one_one_nat ).
fof(fact_2975_of__nat__1,axiom,
hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int ).
fof(fact_2976_transfer__int__nat__numerals_I1_J,axiom,
zero_zero_int = hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) ).
fof(fact_2977_int__eq__0__conv,axiom,
! [N] :
( hAPP_nat_int(semiri1621563631at_int,N) = zero_zero_int
<=> N = zero_zero_nat ) ).
fof(fact_2978_int__0,axiom,
hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) = zero_zero_int ).
fof(fact_2979_of__nat__power,axiom,
! [M_6,N_20] : hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_6)),N_20) ).
fof(fact_2980_of__nat__power,axiom,
! [M_6,N_20] : hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_6)),N_20) ).
fof(fact_2981_of__nat__power,axiom,
! [M_6,N_20] : hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_nat_rat(semiri151668891at_rat,M_6)),N_20) ).
fof(fact_2982_of__nat__power,axiom,
! [M_6,N_20] : hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(semiri2020571505omplex,M_6)),N_20) ).
fof(fact_2983_of__nat__power,axiom,
! [M_6,N_20] : hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(semiri984289939at_nat,M_6)),N_20) ).
fof(fact_2984_of__nat__power,axiom,
! [M_6,N_20] : hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(semiri132038758t_real,M_6)),N_20) ).
fof(fact_2985_of__nat__power,axiom,
! [M_6,N_20] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M_6)),N_20) ).
fof(fact_2986_int__less__0__conv,axiom,
! [K_2] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,K_2)),zero_zero_int)) ).
fof(fact_2987_transfer__int__nat__quantifiers_I1_J,axiom,
! [P_1] :
( ! [X] :
( is_int(X)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> hBOOL(hAPP_int_bool(P_1,X)) ) )
<=> ! [X] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X))) ) ).
fof(fact_2988_transfer__int__nat__quantifiers_I2_J,axiom,
! [P_1] :
( ? [X] :
( is_int(X)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
& hBOOL(hAPP_int_bool(P_1,X)) )
<=> ? [X] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X))) ) ).
fof(fact_2989_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z_1] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Z_1))) ).
fof(fact_2990_zero__zle__int,axiom,
! [N_1] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_1))) ).
fof(fact_2991_Nat__Transfer_Otransfer__int__nat__relations_I2_J,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,X_2)),hAPP_nat_int(semiri1621563631at_int,Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) ) ).
fof(fact_2992_zless__int,axiom,
! [Ma,N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ).
fof(fact_2993_power__0__Suc,axiom,
! [N_19] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),hAPP_nat_nat(suc,N_19)) = zero_zero_rat ).
fof(fact_2994_power__0__Suc,axiom,
! [N_19] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),hAPP_nat_nat(suc,N_19)) = zero_z891286103de_int ).
fof(fact_2995_power__0__Suc,axiom,
! [N_19] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_nat_nat(suc,N_19)) = zero_zero_complex ).
fof(fact_2996_power__0__Suc,axiom,
! [N_19] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_nat_nat(suc,N_19)) = zero_z126310315umeral ).
fof(fact_2997_power__0__Suc,axiom,
! [N_19] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),hAPP_nat_nat(suc,N_19)) = zero_zero_real ).
fof(fact_2998_power__0__Suc,axiom,
! [N_19] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_nat_nat(suc,N_19)) = zero_zero_nat ).
fof(fact_2999_power__0__Suc,axiom,
! [N_19] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),hAPP_nat_nat(suc,N_19)) = zero_zero_int ).
fof(fact_3000_Nat__Transfer_Otransfer__int__nat__relations_I3_J,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,X_2)),hAPP_nat_int(semiri1621563631at_int,Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2)) ) ).
fof(fact_3001_zle__int,axiom,
! [Ma,N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ).
fof(fact_3002_zle__iff__zadd,axiom,
! [W_1,Z_2] :
( is_int(Z_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_2))
<=> ? [N_2] : Z_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),hAPP_nat_int(semiri1621563631at_int,N_2)) ) ) ).
fof(fact_3003_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
! [X_1,Y_1] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Y_1)) ).
fof(fact_3004_int__mult,axiom,
! [M,N_1] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N_1)) ).
fof(fact_3005_zmult__int,axiom,
! [M,N_1] : hAPP_int_int(hAPP_int_fun_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(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)) ).
fof(fact_3006_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
! [X_9,Q_7] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_9),Q_7)),X_9) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_9),hAPP_nat_nat(suc,Q_7)) ).
fof(fact_3007_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
! [X_9,Q_7] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_9),Q_7)),X_9) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_9),hAPP_nat_nat(suc,Q_7)) ).
fof(fact_3008_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
! [X_9,Q_7] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_9),Q_7)),X_9) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_9),hAPP_nat_nat(suc,Q_7)) ).
fof(fact_3009_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
! [X_9,Q_7] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_9),Q_7)),X_9) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_9),hAPP_nat_nat(suc,Q_7)) ).
fof(fact_3010_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
! [X_9,Q_7] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_9),Q_7)),X_9) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_9),hAPP_nat_nat(suc,Q_7)) ).
fof(fact_3011_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
! [X_9,Q_7] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_9),Q_7)),X_9) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_9),hAPP_nat_nat(suc,Q_7)) ).
fof(fact_3012_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
! [X_9,Q_7] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_9),Q_7)),X_9) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_9),hAPP_nat_nat(suc,Q_7)) ).
fof(fact_3013_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
! [X_8,Q_6] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_8),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_8),Q_6)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_8),hAPP_nat_nat(suc,Q_6)) ).
fof(fact_3014_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
! [X_8,Q_6] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_8),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_8),Q_6)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_8),hAPP_nat_nat(suc,Q_6)) ).
fof(fact_3015_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
! [X_8,Q_6] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_8),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_8),Q_6)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_8),hAPP_nat_nat(suc,Q_6)) ).
fof(fact_3016_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
! [X_8,Q_6] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_8),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_8),Q_6)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_8),hAPP_nat_nat(suc,Q_6)) ).
fof(fact_3017_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
! [X_8,Q_6] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_8),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_8),Q_6)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_8),hAPP_nat_nat(suc,Q_6)) ).
fof(fact_3018_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
! [X_8,Q_6] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_8),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_8),Q_6)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_8),hAPP_nat_nat(suc,Q_6)) ).
fof(fact_3019_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
! [X_8,Q_6] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_8),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_8),Q_6)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_8),hAPP_nat_nat(suc,Q_6)) ).
fof(fact_3020_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
! [X_7,Q_5] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_7),hAPP_nat_nat(suc,Q_5)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_7),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_7),Q_5)) ).
fof(fact_3021_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
! [X_7,Q_5] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_7),hAPP_nat_nat(suc,Q_5)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_7),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_7),Q_5)) ).
fof(fact_3022_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
! [X_7,Q_5] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_7),hAPP_nat_nat(suc,Q_5)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_7),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_7),Q_5)) ).
fof(fact_3023_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
! [X_7,Q_5] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_7),hAPP_nat_nat(suc,Q_5)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_7),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_7),Q_5)) ).
fof(fact_3024_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
! [X_7,Q_5] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_7),hAPP_nat_nat(suc,Q_5)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_7),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_7),Q_5)) ).
fof(fact_3025_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
! [X_7,Q_5] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_7),hAPP_nat_nat(suc,Q_5)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_7),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_7),Q_5)) ).
fof(fact_3026_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
! [X_7,Q_5] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_7),hAPP_nat_nat(suc,Q_5)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_7),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_7),Q_5)) ).
fof(fact_3027_power__Suc2,axiom,
! [A_11,N_18] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_11),hAPP_nat_nat(suc,N_18)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_11),N_18)),A_11) ).
fof(fact_3028_power__Suc2,axiom,
! [A_11,N_18] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_11),hAPP_nat_nat(suc,N_18)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_11),N_18)),A_11) ).
fof(fact_3029_power__Suc2,axiom,
! [A_11,N_18] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_11),hAPP_nat_nat(suc,N_18)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_11),N_18)),A_11) ).
fof(fact_3030_power__Suc2,axiom,
! [A_11,N_18] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_11),hAPP_nat_nat(suc,N_18)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_11),N_18)),A_11) ).
fof(fact_3031_power__Suc2,axiom,
! [A_11,N_18] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_11),hAPP_nat_nat(suc,N_18)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_11),N_18)),A_11) ).
fof(fact_3032_power__Suc2,axiom,
! [A_11,N_18] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_11),hAPP_nat_nat(suc,N_18)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_11),N_18)),A_11) ).
fof(fact_3033_power__Suc2,axiom,
! [A_11,N_18] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_11),hAPP_nat_nat(suc,N_18)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_11),N_18)),A_11) ).
fof(fact_3034_power__Suc,axiom,
! [A_10,N_17] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_10),hAPP_nat_nat(suc,N_17)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_10),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_10),N_17)) ).
fof(fact_3035_power__Suc,axiom,
! [A_10,N_17] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_10),hAPP_nat_nat(suc,N_17)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_10),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_10),N_17)) ).
fof(fact_3036_power__Suc,axiom,
! [A_10,N_17] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_10),hAPP_nat_nat(suc,N_17)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_10),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_10),N_17)) ).
fof(fact_3037_power__Suc,axiom,
! [A_10,N_17] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_10),hAPP_nat_nat(suc,N_17)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_10),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_10),N_17)) ).
fof(fact_3038_power__Suc,axiom,
! [A_10,N_17] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_10),hAPP_nat_nat(suc,N_17)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_10),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_10),N_17)) ).
fof(fact_3039_power__Suc,axiom,
! [A_10,N_17] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_10),hAPP_nat_nat(suc,N_17)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_10),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_10),N_17)) ).
fof(fact_3040_power__Suc,axiom,
! [A_10,N_17] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_10),hAPP_nat_nat(suc,N_17)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_10),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_10),N_17)) ).
fof(fact_3041_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
! [X_1,Y_1] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),Y_1)) ).
fof(fact_3042_zadd__int,axiom,
! [M,N_1] : hAPP_int_int(hAPP_int_fun_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(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)) ).
fof(fact_3043_zadd__int__left,axiom,
! [M,N_1,Z_1] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,N_1)),Z_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1))),Z_1) ).
fof(fact_3044_transfer__int__nat__numerals_I2_J,axiom,
one_one_int = hAPP_nat_int(semiri1621563631at_int,one_one_nat) ).
fof(fact_3045_int__1,axiom,
hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int ).
fof(fact_3046_Nat__Transfer_Otransfer__int__nat__relations_I4_J,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,X_2)),hAPP_nat_int(semiri1621563631at_int,Y_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) ) ).
fof(fact_3047_zdvd__int,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,X_2)),hAPP_nat_int(semiri1621563631at_int,Y_2))) ) ).
fof(fact_3048_gr0__conv__Suc,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
<=> ? [M_1] : N = hAPP_nat_nat(suc,M_1) ) ).
fof(fact_3049_less__Suc0,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,zero_zero_nat)))
<=> N = zero_zero_nat ) ).
fof(fact_3050_less__Suc__eq__0__disj,axiom,
! [Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(suc,N)))
<=> ( Ma = zero_zero_nat
| ? [J_1] :
( Ma = hAPP_nat_nat(suc,J_1)
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),N)) ) ) ) ).
fof(fact_3051_nat__lt__two__imp__zero__or__one,axiom,
! [X_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))
=> ( X_1 = zero_zero_nat
| X_1 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).
fof(fact_3052_dvd__1__iff__1,axiom,
! [Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),hAPP_nat_nat(suc,zero_zero_nat)))
<=> Ma = hAPP_nat_nat(suc,zero_zero_nat) ) ).
fof(fact_3053_one__is__add,axiom,
! [Ma,N] :
( hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N)
<=> ( ( Ma = hAPP_nat_nat(suc,zero_zero_nat)
& N = zero_zero_nat )
| ( Ma = zero_zero_nat
& N = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).
fof(fact_3054_add__is__1,axiom,
! [Ma,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N) = hAPP_nat_nat(suc,zero_zero_nat)
<=> ( ( Ma = hAPP_nat_nat(suc,zero_zero_nat)
& N = zero_zero_nat )
| ( Ma = zero_zero_nat
& N = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).
fof(fact_3055_mult__eq__1__iff,axiom,
! [Ma,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N) = hAPP_nat_nat(suc,zero_zero_nat)
<=> ( Ma = hAPP_nat_nat(suc,zero_zero_nat)
& N = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).
fof(fact_3056_less__iff__Suc__add,axiom,
! [Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N))
<=> ? [K] : N = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K)) ) ).
fof(fact_3057_less__add__Suc2,axiom,
! [I,M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),I)))) ).
fof(fact_3058_less__add__Suc1,axiom,
! [I,M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),M)))) ).
fof(fact_3059_One__nat__def,axiom,
one_one_nat = hAPP_nat_nat(suc,zero_zero_nat) ).
fof(fact_3060_Suc__le__lessD,axiom,
! [M,N_1] :
( 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)) ) ).
fof(fact_3061_le__less__Suc__eq,axiom,
! [Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,Ma)))
<=> N = Ma ) ) ).
fof(fact_3062_Suc__leI,axiom,
! [M,N_1] :
( 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)) ) ).
fof(fact_3063_le__imp__less__Suc,axiom,
! [M,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_nat,M),hAPP_nat_nat(suc,N_1))) ) ).
fof(fact_3064_Suc__le__eq,axiom,
! [Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Ma)),N))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ).
fof(fact_3065_less__Suc__eq__le,axiom,
! [Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(suc,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ).
fof(fact_3066_less__eq__Suc__le,axiom,
! [N,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),Ma))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),Ma)) ) ).
fof(fact_3067_Suc__mult__less__cancel1,axiom,
! [K_1,Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) ) ).
fof(fact_3068_diff__less__Suc,axiom,
! [M,N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),hAPP_nat_nat(suc,M))) ).
fof(fact_3069_mult__Suc__right,axiom,
! [M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)) ).
fof(fact_3070_mult__Suc,axiom,
! [M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,M)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)) ).
fof(fact_3071_Suc__mult__le__cancel1,axiom,
! [K_1,Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) ) ).
fof(fact_3072_Suc__eq__plus1,axiom,
! [N_1] : hAPP_nat_nat(suc,N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat) ).
fof(fact_3073_Suc__eq__plus1__left,axiom,
! [N_1] : hAPP_nat_nat(suc,N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N_1) ).
fof(fact_3074_Suc__diff__le,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)) ) ).
fof(fact_3075_diff__Suc__1,axiom,
! [N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_1)),one_one_nat) = N_1 ).
fof(fact_3076_diff__Suc__eq__diff__pred,axiom,
! [M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N_1) ).
fof(fact_3077_zdiv__int,axiom,
! [A_3,B_2] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),B_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(semiri1621563631at_int,A_3)),hAPP_nat_int(semiri1621563631at_int,B_2)) ).
fof(fact_3078_Divides_Otransfer__int__nat__functions_I1_J,axiom,
! [X_1,Y_1] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_1),Y_1)) ).
fof(fact_3079_div__1,axiom,
! [M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),hAPP_nat_nat(suc,zero_zero_nat)) = M ).
fof(fact_3080_mod__1,axiom,
! [M] : hAPP_nat_nat(div_mod_nat(M),hAPP_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).
fof(fact_3081_mod__Suc,axiom,
! [M,N_1] :
( ( hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N_1)) = N_1
=> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N_1) = zero_zero_nat )
& ( hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N_1)) != N_1
=> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N_1)) ) ) ).
fof(fact_3082_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
! [X_1,N_1] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,X_1)),N_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1)) ).
fof(fact_3083_int__power,axiom,
! [M,N_1] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N_1)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M)),N_1) ).
fof(fact_3084_zpower__int,axiom,
! [M,N_1] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M)),N_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N_1)) ).
fof(fact_3085_power__Suc__0,axiom,
! [N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1) = hAPP_nat_nat(suc,zero_zero_nat) ).
fof(fact_3086_nat__power__eq__Suc__0__iff,axiom,
! [X_2,Ma] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),Ma) = hAPP_nat_nat(suc,zero_zero_nat)
<=> ( Ma = zero_zero_nat
| X_2 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).
fof(fact_3087_zmod__int,axiom,
! [A_3,B_2] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(div_mod_nat(A_3),B_2)) = hAPP_int_int(div_mod_int(hAPP_nat_int(semiri1621563631at_int,A_3)),hAPP_nat_int(semiri1621563631at_int,B_2)) ).
fof(fact_3088_Divides_Otransfer__int__nat__functions_I2_J,axiom,
! [X_1,Y_1] : hAPP_int_int(div_mod_int(hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(div_mod_nat(X_1),Y_1)) ).
fof(fact_3089_exp__mono__lt,axiom,
! [X_2,N,Y_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),hAPP_nat_nat(suc,N))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_2),hAPP_nat_nat(suc,N))))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) ) ).
fof(fact_3090_divides__rexp,axiom,
! [N_1,X_1,Y_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_1),hAPP_nat_nat(suc,N_1)))) ) ).
fof(fact_3091_exp__mono__le,axiom,
! [X_2,N,Y_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),hAPP_nat_nat(suc,N))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_2),hAPP_nat_nat(suc,N))))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2)) ) ).
fof(fact_3092_even__power,axiom,
! [X_2,N] :
( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_2),N)))
<=> ( hBOOL(hAPP_int_bool(even_odd_even_int,X_2))
& N != zero_zero_nat ) ) ).
fof(fact_3093_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
! [X_1,Y_1] : hAPP_int_int(nat_tsub(hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_1),Y_1)) ).
fof(fact_3094_even__plus__odd__prop2,axiom,
! [A,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)),zOdd))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),zEven))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),zOdd)) ) ) ).
fof(fact_3095_even__plus__odd__prop1,axiom,
! [A,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)),zOdd))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),zOdd))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),zEven)) ) ) ).
fof(fact_3096_IntNatAux_Oeven__plus__odd,axiom,
! [B,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),zEven))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),zOdd))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)),zOdd)) ) ) ).
fof(fact_3097_IntNatAux_Oodd__plus__odd,axiom,
! [B,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),zOdd))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),zOdd))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)),zEven)) ) ) ).
fof(fact_3098_even__minus__odd,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zOdd))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),Y_2)),zOdd)) ) ) ).
fof(fact_3099_odd__minus__even,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),Y_2)),zOdd)) ) ) ).
fof(fact_3100_odd__minus__odd,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zOdd))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),Y_2)),zEven)) ) ) ).
fof(fact_3101_of__nat__diff,axiom,
! [N_16,M_5] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_16),M_5))
=> hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_5),N_16)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_nat_rat(semiri151668891at_rat,M_5)),hAPP_nat_rat(semiri151668891at_rat,N_16)) ) ).
fof(fact_3102_of__nat__diff,axiom,
! [N_16,M_5] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_16),M_5))
=> hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_5),N_16)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_nat_complex(semiri2020571505omplex,M_5)),hAPP_nat_complex(semiri2020571505omplex,N_16)) ) ).
fof(fact_3103_of__nat__diff,axiom,
! [N_16,M_5] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_16),M_5))
=> hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_5),N_16)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(semiri132038758t_real,M_5)),hAPP_nat_real(semiri132038758t_real,N_16)) ) ).
fof(fact_3104_of__nat__diff,axiom,
! [N_16,M_5] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_16),M_5))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_5),N_16)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_5)),hAPP_nat_int(semiri1621563631at_int,N_16)) ) ).
fof(fact_3105_int__le__0__conv,axiom,
! [N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,N)),zero_zero_int))
<=> N = zero_zero_nat ) ).
fof(fact_3106_power__le__imp__le__base,axiom,
! [A_9,N_15,B_8] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_9),hAPP_nat_nat(suc,N_15))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_8),hAPP_nat_nat(suc,N_15))))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_8))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_9),B_8)) ) ) ).
fof(fact_3107_power__le__imp__le__base,axiom,
! [A_9,N_15,B_8] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_9),hAPP_nat_nat(suc,N_15))),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_8),hAPP_nat_nat(suc,N_15))))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_8))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_9),B_8)) ) ) ).
fof(fact_3108_power__le__imp__le__base,axiom,
! [A_9,N_15,B_8] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_9),hAPP_nat_nat(suc,N_15))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_8),hAPP_nat_nat(suc,N_15))))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_8))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_9),B_8)) ) ) ).
fof(fact_3109_power__le__imp__le__base,axiom,
! [A_9,N_15,B_8] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_9),hAPP_nat_nat(suc,N_15))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_8),hAPP_nat_nat(suc,N_15))))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_8))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_9),B_8)) ) ) ).
fof(fact_3110_power__le__imp__le__base,axiom,
! [A_9,N_15,B_8] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_9),hAPP_nat_nat(suc,N_15))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_8),hAPP_nat_nat(suc,N_15))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_9),B_8)) ) ) ).
fof(fact_3111_power__le__imp__le__base,axiom,
! [A_9,N_15,B_8] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_9),hAPP_nat_nat(suc,N_15))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_8),hAPP_nat_nat(suc,N_15))))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_9),B_8)) ) ) ).
fof(fact_3112_power__inject__base,axiom,
! [A_8,N_14,B_7] :
( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_8),hAPP_nat_nat(suc,N_14)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_7),hAPP_nat_nat(suc,N_14))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_8))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_7))
=> A_8 = B_7 ) ) ) ).
fof(fact_3113_power__inject__base,axiom,
! [A_8,N_14,B_7] :
( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_8),hAPP_nat_nat(suc,N_14)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_7),hAPP_nat_nat(suc,N_14))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_8))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_7))
=> A_8 = B_7 ) ) ) ).
fof(fact_3114_power__inject__base,axiom,
! [A_8,N_14,B_7] :
( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_8),hAPP_nat_nat(suc,N_14)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_7),hAPP_nat_nat(suc,N_14))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_8))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_7))
=> A_8 = B_7 ) ) ) ).
fof(fact_3115_power__inject__base,axiom,
! [A_8,N_14,B_7] :
( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_8),hAPP_nat_nat(suc,N_14)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_7),hAPP_nat_nat(suc,N_14))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_8))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_7))
=> A_8 = B_7 ) ) ) ).
fof(fact_3116_power__inject__base,axiom,
! [A_8,N_14,B_7] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_8),hAPP_nat_nat(suc,N_14)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_7),hAPP_nat_nat(suc,N_14))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_8))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_7))
=> A_8 = B_7 ) ) ) ).
fof(fact_3117_power__inject__base,axiom,
! [A_8,N_14,B_7] :
( ( is_int(A_8)
& is_int(B_7) )
=> ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_8),hAPP_nat_nat(suc,N_14)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_7),hAPP_nat_nat(suc,N_14))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_8))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_7))
=> A_8 = B_7 ) ) ) ) ).
fof(fact_3118_power__gt1,axiom,
! [N_13,A_7] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_7))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_7),hAPP_nat_nat(suc,N_13)))) ) ).
fof(fact_3119_power__gt1,axiom,
! [N_13,A_7] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_7))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_7),hAPP_nat_nat(suc,N_13)))) ) ).
fof(fact_3120_power__gt1,axiom,
! [N_13,A_7] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_7))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_7),hAPP_nat_nat(suc,N_13)))) ) ).
fof(fact_3121_power__gt1,axiom,
! [N_13,A_7] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_7))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_7),hAPP_nat_nat(suc,N_13)))) ) ).
fof(fact_3122_power__gt1,axiom,
! [N_13,A_7] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_7),hAPP_nat_nat(suc,N_13)))) ) ).
fof(fact_3123_power__gt1,axiom,
! [N_13,A_7] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_7),hAPP_nat_nat(suc,N_13)))) ) ).
fof(fact_3124_nat__0__le,axiom,
! [Z_1] :
( is_int(Z_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,Z_1)) = Z_1 ) ) ).
fof(fact_3125_int__eq__iff,axiom,
! [Ma,Z_2] :
( is_int(Z_2)
=> ( hAPP_nat_int(semiri1621563631at_int,Ma) = Z_2
<=> ( Ma = hAPP_int_nat(nat,Z_2)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) ) ) ) ).
fof(fact_3126_int__nat__eq,axiom,
! [Z_1] :
( is_int(Z_1)
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,Z_1)) = Z_1 )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,Z_1)) = zero_zero_int ) ) ) ).
fof(fact_3127_zless__nat__eq__int__zless,axiom,
! [Ma,Z_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_int_nat(nat,Z_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),Z_2)) ) ).
fof(fact_3128_zdiff__int,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> hAPP_int_int(hAPP_int_fun_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(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)) ) ).
fof(fact_3129_n__less__m__mult__n,axiom,
! [M,N_1] :
( 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(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1))) ) ) ).
fof(fact_3130_n__less__n__mult__m,axiom,
! [M,N_1] :
( 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(hAPP_nat_fun_nat_nat(times_times_nat,N_1),M))) ) ) ).
fof(fact_3131_one__less__mult,axiom,
! [M,N_1] :
( 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(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1))) ) ) ).
fof(fact_3132_one__less__m,axiom,
! [M,K_2] :
( M != hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2)
=> ( 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)) ) ) ).
fof(fact_3133_one__less__k,axiom,
! [M,K_2] :
( M != hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2)
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),K_2)) ) ) ).
fof(fact_3134_prod__mn__less__k,axiom,
! [M,K_2,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1) = K_2
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),K_2)) ) ) ) ) ).
fof(fact_3135_mn__eq__m__one,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1) = M
=> N_1 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).
fof(fact_3136_Suc__pred,axiom,
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat))) = N_1 ) ).
fof(fact_3137_diff__Suc__less,axiom,
! [I,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,I))),N_1)) ) ).
fof(fact_3138_one__le__mult__iff,axiom,
! [Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),Ma))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),N)) ) ) ).
fof(fact_3139_nat__1,axiom,
hAPP_int_nat(nat,one_one_int) = hAPP_nat_nat(suc,zero_zero_nat) ).
fof(fact_3140_diff__Suc__diff__eq2,axiom,
! [M,K_2,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2))),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,J_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),M)) ) ).
fof(fact_3141_diff__Suc__diff__eq1,axiom,
! [M,K_2,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),hAPP_nat_nat(suc,J_2)) ) ).
fof(fact_3142_nat__one__le__power,axiom,
! [N_1,I] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),I))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N_1))) ) ).
fof(fact_3143_mod__mult__self4,axiom,
! [K_2,N_1,M] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)),M))),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N_1) ).
fof(fact_3144_zero__code__numeral__code,axiom,
zero_z126310315umeral = hAPP_i769753017umeral(number1443263063umeral,pls) ).
fof(fact_3145_of__nat__0__less__iff,axiom,
! [N] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_nat_rat(semiri151668891at_rat,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
fof(fact_3146_of__nat__0__less__iff,axiom,
! [N] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_n522471361de_int(semiri1424489471de_int,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
fof(fact_3147_of__nat__0__less__iff,axiom,
! [N] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_n1108039445umeral(semiri1619134803umeral,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
fof(fact_3148_of__nat__0__less__iff,axiom,
! [N] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
fof(fact_3149_of__nat__0__less__iff,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
fof(fact_3150_of__nat__0__less__iff,axiom,
! [N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
fof(fact_3151_power__preserves__even,axiom,
! [X_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_2),N)),zEven))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) ) ) ).
fof(fact_3152_zero__less__int__conv,axiom,
! [N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
fof(fact_3153_odd__minus__one__even,axiom,
! [X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),one_one_int)),zEven)) ) ).
fof(fact_3154_transfer__int__nat__numerals_I4_J,axiom,
hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) ).
fof(fact_3155_zmult__zless__mono2__lemma,axiom,
! [K_2,I,J_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,K_2)),I)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,K_2)),J_2))) ) ) ).
fof(fact_3156_realpow__Suc__le__self,axiom,
! [N_12,R_6] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),R_6))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,R_6),one_one_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,R_6),hAPP_nat_nat(suc,N_12))),R_6)) ) ) ).
fof(fact_3157_realpow__Suc__le__self,axiom,
! [N_12,R_6] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),R_6))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,R_6),one_on1684967323de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,R_6),hAPP_nat_nat(suc,N_12))),R_6)) ) ) ).
fof(fact_3158_realpow__Suc__le__self,axiom,
! [N_12,R_6] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),R_6))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,R_6),one_on1645066479umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,R_6),hAPP_nat_nat(suc,N_12))),R_6)) ) ) ).
fof(fact_3159_realpow__Suc__le__self,axiom,
! [N_12,R_6] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_6))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R_6),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_6),hAPP_nat_nat(suc,N_12))),R_6)) ) ) ).
fof(fact_3160_realpow__Suc__le__self,axiom,
! [N_12,R_6] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),R_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,R_6),one_one_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,R_6),hAPP_nat_nat(suc,N_12))),R_6)) ) ) ).
fof(fact_3161_realpow__Suc__le__self,axiom,
! [N_12,R_6] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_6),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,R_6),hAPP_nat_nat(suc,N_12))),R_6)) ) ) ).
fof(fact_3162_power__Suc__less__one,axiom,
! [N_11,A_6] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_6))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_6),one_one_rat))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_6),hAPP_nat_nat(suc,N_11))),one_one_rat)) ) ) ).
fof(fact_3163_power__Suc__less__one,axiom,
! [N_11,A_6] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_6))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_6),one_on1684967323de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_6),hAPP_nat_nat(suc,N_11))),one_on1684967323de_int)) ) ) ).
fof(fact_3164_power__Suc__less__one,axiom,
! [N_11,A_6] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_6))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_6),one_on1645066479umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_6),hAPP_nat_nat(suc,N_11))),one_on1645066479umeral)) ) ) ).
fof(fact_3165_power__Suc__less__one,axiom,
! [N_11,A_6] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_6))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_6),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_6),hAPP_nat_nat(suc,N_11))),one_one_real)) ) ) ).
fof(fact_3166_power__Suc__less__one,axiom,
! [N_11,A_6] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_6),one_one_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_6),hAPP_nat_nat(suc,N_11))),one_one_nat)) ) ) ).
fof(fact_3167_power__Suc__less__one,axiom,
! [N_11,A_6] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_6),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_6),hAPP_nat_nat(suc,N_11))),one_one_int)) ) ) ).
fof(fact_3168_split__nat,axiom,
! [P_1,I_1] :
( is_int(I_1)
=> ( hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat,I_1)))
<=> ( ! [N_2] :
( I_1 = hAPP_nat_int(semiri1621563631at_int,N_2)
=> hBOOL(hAPP_nat_bool(P_1,N_2)) )
& ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),zero_zero_int))
=> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) ) ) ) ) ).
fof(fact_3169_nat__eq__iff,axiom,
! [W_1,Ma] :
( is_int(W_1)
=> ( hAPP_int_nat(nat,W_1) = Ma
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1))
=> W_1 = hAPP_nat_int(semiri1621563631at_int,Ma) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1))
=> Ma = zero_zero_nat ) ) ) ) ).
fof(fact_3170_nat__eq__iff2,axiom,
! [Ma,W_1] :
( is_int(W_1)
=> ( Ma = hAPP_int_nat(nat,W_1)
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1))
=> W_1 = hAPP_nat_int(semiri1621563631at_int,Ma) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1))
=> Ma = zero_zero_nat ) ) ) ) ).
fof(fact_3171_int__eq__iff__number__of,axiom,
! [Ma,V] :
( hAPP_nat_int(semiri1621563631at_int,Ma) = hAPP_int_int(number_number_of_int,V)
<=> ( Ma = hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,V))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V))) ) ) ).
fof(fact_3172_numeral__3__eq__3,axiom,
hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) ).
fof(fact_3173_numeral__1__eq__Suc__0,axiom,
hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)) = hAPP_nat_nat(suc,zero_zero_nat) ).
fof(fact_3174_Suc3__eq__add__3,axiom,
! [N_1] : hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),N_1) ).
fof(fact_3175_lemma__realpow__diff,axiom,
! [Y_6,P_6,N_10] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10))
=> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6) ) ).
fof(fact_3176_lemma__realpow__diff,axiom,
! [Y_6,P_6,N_10] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10))
=> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6) ) ).
fof(fact_3177_lemma__realpow__diff,axiom,
! [Y_6,P_6,N_10] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10))
=> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6) ) ).
fof(fact_3178_lemma__realpow__diff,axiom,
! [Y_6,P_6,N_10] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10))
=> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6) ) ).
fof(fact_3179_lemma__realpow__diff,axiom,
! [Y_6,P_6,N_10] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6) ) ).
fof(fact_3180_lemma__realpow__diff,axiom,
! [Y_6,P_6,N_10] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10))
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6) ) ).
fof(fact_3181_lemma__realpow__diff,axiom,
! [Y_6,P_6,N_10] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10))
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6) ) ).
fof(fact_3182_Suc__diff__1,axiom,
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)) = N_1 ) ).
fof(fact_3183_Suc__pred_H,axiom,
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> N_1 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)) ) ).
fof(fact_3184_add__eq__if,axiom,
! [N_1,M] :
( ( M = zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1) = N_1 )
& ( M != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N_1)) ) ) ).
fof(fact_3185_div__if,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1) = zero_zero_nat )
& ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1)) ) ) ) ).
fof(fact_3186_div__geq,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1)) ) ) ).
fof(fact_3187_Suc__times__mod__eq,axiom,
! [M,K_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),K_2))
=> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M))),K_2) = one_one_nat ) ).
fof(fact_3188_div__Suc,axiom,
! [A_3,C_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,A_3)),C_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,zero_zero_nat)),C_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_3),C_1)),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,zero_zero_nat)),C_1))),C_1)) ).
fof(fact_3189_transfer__int__nat__numerals_I3_J,axiom,
hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_3190_nat__less__iff,axiom,
! [Ma,W_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,W_1)),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_nat_int(semiri1621563631at_int,Ma))) ) ) ).
fof(fact_3191_realpow__two__diff,axiom,
! [X_6,Y_5] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_6),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_6),Y_5)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_6),Y_5)) ).
fof(fact_3192_realpow__two__diff,axiom,
! [X_6,Y_5] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_6),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_6),Y_5)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_6),Y_5)) ).
fof(fact_3193_realpow__two__diff,axiom,
! [X_6,Y_5] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_6),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_6),Y_5)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_6),Y_5)) ).
fof(fact_3194_realpow__two__diff,axiom,
! [X_6,Y_5] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_6),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_6),Y_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_6),Y_5)) ).
fof(fact_3195_zdiff__int__split,axiom,
! [P_1,X_2,Y_2] :
( hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_2),Y_2))))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),X_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,X_2)),hAPP_nat_int(semiri1621563631at_int,Y_2)))) )
& ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2))
=> hBOOL(hAPP_int_bool(P_1,zero_zero_int)) ) ) ) ).
fof(fact_3196_numeral__2__eq__2,axiom,
hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)) ).
fof(fact_3197_semiring__norm_I115_J,axiom,
hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).
fof(fact_3198_add__2__eq__Suc,axiom,
! [N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,N_1)) ).
fof(fact_3199_add__2__eq__Suc_H,axiom,
! [N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,N_1)) ).
fof(fact_3200_Suc__diff__eq__diff__pred,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,pls)),N_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)))) ) ).
fof(fact_3201_one__less__nat__eq,axiom,
! [Z_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(nat,Z_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Z_2)) ) ).
fof(fact_3202_expand__Suc,axiom,
! [V_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,V_2)))
=> hAPP_int_nat(number_number_of_nat,V_2) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat)) ) ).
fof(fact_3203_div2__Suc__Suc,axiom,
! [M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_3204_mod2__Suc__Suc,axiom,
! [M] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(suc,M))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_3205_Suc__nat__eq__nat__zadd1,axiom,
! [Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
=> hAPP_nat_nat(suc,hAPP_int_nat(nat,Z_1)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_1)) ) ).
fof(fact_3206_Suc__div__eq__add3__div__number__of,axiom,
! [M,V_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M)))),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M)),hAPP_int_nat(number_number_of_nat,V_2)) ).
fof(fact_3207_Suc__div__eq__add3__div,axiom,
! [M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M)))),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M)),N_1) ).
fof(fact_3208_div__Suc__eq__div__add3,axiom,
! [M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N_1)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),N_1)) ).
fof(fact_3209_Suc__mod__eq__add3__mod__number__of,axiom,
! [M,V_2] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M)))),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M)),hAPP_int_nat(number_number_of_nat,V_2)) ).
fof(fact_3210_Suc__mod__eq__add3__mod,axiom,
! [M,N_1] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M)))),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M)),N_1) ).
fof(fact_3211_mod__Suc__eq__mod__add3,axiom,
! [M,N_1] : hAPP_nat_nat(div_mod_nat(M),hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N_1)))) = hAPP_nat_nat(div_mod_nat(M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),N_1)) ).
fof(fact_3212_split__div_H,axiom,
! [P_1,Ma,N] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Ma),N)))
<=> ( ( N = zero_zero_nat
& hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
| ? [Q_4] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Q_4)),Ma))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),hAPP_nat_nat(suc,Q_4))))
& hBOOL(hAPP_nat_bool(P_1,Q_4)) ) ) ) ).
fof(fact_3213_split__div__lemma,axiom,
! [Q_1,Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Q_1)),Ma))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),hAPP_nat_nat(suc,Q_1)))) )
<=> Q_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Ma),N) ) ) ).
fof(fact_3214_le__div__geq,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1)) ) ) ).
fof(fact_3215_of__nat__less__two__power,axiom,
! [N_9] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,N_9)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_9))) ).
fof(fact_3216_of__nat__less__two__power,axiom,
! [N_9] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,N_9)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_9))) ).
fof(fact_3217_of__nat__less__two__power,axiom,
! [N_9] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N_9)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_9))) ).
fof(fact_3218_of__nat__number__of__lemma,axiom,
! [V_4] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4)))
=> hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(number_number_of_nat,V_4)) = hAPP_int_rat(number_number_of_rat,V_4) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4)))
=> hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(number_number_of_nat,V_4)) = zero_zero_rat ) ) ).
fof(fact_3219_of__nat__number__of__lemma,axiom,
! [V_4] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4)))
=> hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(number_number_of_nat,V_4)) = hAPP_int_complex(number528085621omplex,V_4) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4)))
=> hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(number_number_of_nat,V_4)) = zero_zero_complex ) ) ).
fof(fact_3220_of__nat__number__of__lemma,axiom,
! [V_4] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4)))
=> hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(number_number_of_nat,V_4)) = hAPP_int_real(number267125858f_real,V_4) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4)))
=> hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(number_number_of_nat,V_4)) = zero_zero_real ) ) ).
fof(fact_3221_of__nat__number__of__lemma,axiom,
! [V_4] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4)))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_4)) = hAPP_int_int(number_number_of_int,V_4) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4)))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_4)) = zero_zero_int ) ) ).
fof(fact_3222_even__equiv__def,axiom,
! [X_2] :
( is_int(X_2)
=> ( hBOOL(hAPP_int_bool(even_odd_even_int,X_2))
<=> ? [Y] :
( is_int(Y)
& X_2 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Y) ) ) ) ).
fof(fact_3223_int__even__iff__2__dvd,axiom,
! [X_2] :
( hBOOL(hAPP_int_bool(even_odd_even_int,X_2))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2)) ) ).
fof(fact_3224_nat__dvd__iff,axiom,
! [Z_2,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat,Z_2)),Ma))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_2),hAPP_nat_int(semiri1621563631at_int,Ma))) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2))
=> Ma = zero_zero_nat ) ) ) ).
fof(fact_3225_zEvenI,axiom,
! [X_2,K_1] :
( X_2 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_1)
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven)) ) ).
fof(fact_3226_less__2__cases,axiom,
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
=> ( N_1 = zero_zero_nat
| N_1 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).
fof(fact_3227_nat__2,axiom,
hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)) ).
fof(fact_3228_even__def,axiom,
! [X_2] :
( hBOOL(hAPP_int_bool(even_odd_even_int,X_2))
<=> hAPP_int_int(div_mod_int(X_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_int ) ).
fof(fact_3229_two__times__even__div__two,axiom,
! [X_1] :
( is_int(X_1)
=> ( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = X_1 ) ) ).
fof(fact_3230_one__code__numeral__code,axiom,
one_on1645066479umeral = hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit1,pls)) ).
fof(fact_3231_comm__semiring__1__class_Onormalizing__semiring__rules_I37_J,axiom,
! [X_5,N_8] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_5),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_8))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_5),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_5),N_8)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_5),N_8))) ).
fof(fact_3232_lemma__Suc__Suc__4n__diff__2,axiom,
! [N_1] :
( N_1 != zero_zero_nat
=> hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N_1) ) ).
fof(fact_3233_Suc__Suc__mult__two__diff__two,axiom,
! [N_1] :
( N_1 != zero_zero_nat
=> hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1) ) ).
fof(fact_3234_even__div__2__prop1,axiom,
! [X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven))
=> hAPP_int_int(div_mod_int(X_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_int ) ).
fof(fact_3235_even__div__2__l,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Y_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Y_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).
fof(fact_3236_int__power__div__base,axiom,
! [K_2,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_2))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_2),M)),K_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,zero_zero_nat))) ) ) ).
fof(fact_3237_even__prod__div__2,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Y_2)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Y_2) ) ).
fof(fact_3238_even__div__2__prop2,axiom,
! [X_2] :
( is_int(X_2)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven))
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = X_2 ) ) ).
fof(fact_3239_even__sum__div__2,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Y_2)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Y_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) ) ).
fof(fact_3240_one__div__nat__number__of,axiom,
! [V_3] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,one_one_int),hAPP_int_int(number_number_of_int,V_3))) ).
fof(fact_3241_neg__one__power__parity,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
=> ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) )
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,X_2)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,Y_2)) ) ) ) ).
fof(fact_3242_odd__equiv__def,axiom,
! [X_2] :
( is_int(X_2)
=> ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X_2))
<=> ? [Y] :
( is_int(Y)
& X_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Y)),one_one_int) ) ) ) ).
fof(fact_3243_odd__plus__one__div__two,axiom,
! [X_1] :
( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int) ) ).
fof(fact_3244_even__plus__one__div__two,axiom,
! [X_1] :
( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) ).
fof(fact_3245_Suc__mult__two__diff__one,axiom,
! [N_1] :
( N_1 != zero_zero_nat
=> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)),one_one_nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1) ) ).
fof(fact_3246_inv__is__inv__aux,axiom,
! [M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M))
=> hAPP_nat_nat(suc,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),one_one_int)) ) ).
fof(fact_3247_zEvenE,axiom,
! [X_2] :
( is_int(X_2)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven))
=> ~ ! [K] :
( is_int(K)
=> X_2 != hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K) ) ) ) ).
fof(fact_3248_mod__induct__0,axiom,
! [I_1,P_1,P_5] :
( ! [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),P_5))
=> ( hBOOL(hAPP_nat_bool(P_1,I_2))
=> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,I_2)),P_5))) ) )
=> ( hBOOL(hAPP_nat_bool(P_1,I_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),P_5))
=> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) ) ) ) ).
fof(fact_3249_realpow__pos__nth2,axiom,
! [N_1,A_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
=> ? [R_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))
& hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_1),hAPP_nat_nat(suc,N_1)) = A_3 ) ) ).
fof(fact_3250_zero__less__imp__eq__int,axiom,
! [K_2] :
( is_int(K_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_2))
=> ? [N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2))
& K_2 = hAPP_nat_int(semiri1621563631at_int,N_2) ) ) ) ).
fof(fact_3251_mod__induct,axiom,
! [J,I_1,P_1,P_5] :
( ! [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),P_5))
=> ( hBOOL(hAPP_nat_bool(P_1,I_2))
=> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,I_2)),P_5))) ) )
=> ( hBOOL(hAPP_nat_bool(P_1,I_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),P_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),P_5))
=> hBOOL(hAPP_nat_bool(P_1,J)) ) ) ) ) ).
fof(fact_3252_dvd_Olift__Suc__mono__le,axiom,
! [N,N_7,F] :
( ! [N_2] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_2)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),N_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N)),hAPP_nat_nat(F,N_7))) ) ) ).
fof(fact_3253_int__int__eq,axiom,
! [Ma,N] :
( hAPP_nat_int(semiri1621563631at_int,Ma) = hAPP_nat_int(semiri1621563631at_int,N)
<=> Ma = N ) ).
fof(fact_3254_int__if__cong,axiom,
! [X_2,Y_2,P_1] :
( ( hBOOL(P_1)
=> hAPP_nat_int(semiri1621563631at_int,X_2) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,P_1),X_2),Y_2)) )
& ( ~ hBOOL(P_1)
=> hAPP_nat_int(semiri1621563631at_int,Y_2) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,P_1),X_2),Y_2)) ) ) ).
fof(fact_3255_Nat__Transfer_Otransfer__int__nat__relations_I1_J,axiom,
! [X_2,Y_2] :
( hAPP_nat_int(semiri1621563631at_int,X_2) = hAPP_nat_int(semiri1621563631at_int,Y_2)
<=> X_2 = Y_2 ) ).
fof(fact_3256_nat__of__aux__code,axiom,
! [N_1,I] :
( ( I = zero_z126310315umeral
=> hAPP_nat_nat(code_nat_of_aux(I),N_1) = N_1 )
& ( I != zero_z126310315umeral
=> hAPP_nat_nat(code_nat_of_aux(I),N_1) = hAPP_nat_nat(code_nat_of_aux(hAPP_C498520661umeral(hAPP_C1594335432umeral(minus_1690775515umeral,I),one_on1645066479umeral)),hAPP_nat_nat(suc,N_1)) ) ) ).
fof(fact_3257_nonneg__eq__int,axiom,
! [Z_1] :
( is_int(Z_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
=> ~ ! [M_1] : Z_1 != hAPP_nat_int(semiri1621563631at_int,M_1) ) ) ).
fof(fact_3258_nonneg__int__cases,axiom,
! [K_2] :
( is_int(K_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
=> ~ ! [N_2] : K_2 != hAPP_nat_int(semiri1621563631at_int,N_2) ) ) ).
fof(fact_3259_zero__le__imp__eq__int,axiom,
! [K_2] :
( is_int(K_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
=> ? [N_2] : K_2 = hAPP_nat_int(semiri1621563631at_int,N_2) ) ) ).
fof(fact_3260_inc__induct,axiom,
! [P_1,I_1,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> ( hBOOL(hAPP_nat_bool(P_1,J))
=> ( ! [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J))
=> ( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,I_2)))
=> hBOOL(hAPP_nat_bool(P_1,I_2)) ) )
=> hBOOL(hAPP_nat_bool(P_1,I_1)) ) ) ) ).
fof(fact_3261_less__imp__Suc__add,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> ? [K] : N_1 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)) ) ).
fof(fact_3262_dvd_Olift__Suc__mono__less,axiom,
! [N,N_7,F] :
( ! [N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_2)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))),hAPP_nat_nat(F,N_2))) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N_7))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N)),hAPP_nat_nat(F,N_7)))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_7)),hAPP_nat_nat(F,N))) ) ) ) ).
fof(fact_3263_dvd_Olift__Suc__mono__less__iff,axiom,
! [N,Ma,F] :
( ! [N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_2)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))),hAPP_nat_nat(F,N_2))) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N)),hAPP_nat_nat(F,Ma)))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,Ma)),hAPP_nat_nat(F,N))) )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),Ma)) ) ) ).
fof(fact_3264_gr0__implies__Suc,axiom,
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ? [M_1] : N_1 = hAPP_nat_nat(suc,M_1) ) ).
fof(fact_3265_one__mod__nat__number__of,axiom,
! [V_3] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3)))
=> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_nat_nat(suc,zero_zero_nat) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3)))
=> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(nat,hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,V_3))) ) ) ).
fof(fact_3266_SRStar__card,axiom,
! [P_5] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(sRStar,P_5))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int) ) ).
fof(fact_3267_int__diff__cases,axiom,
! [Z_1] :
( is_int(Z_1)
=> ~ ! [M_1,N_2] : Z_1 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_2)) ) ).
fof(fact_3268_zero__induct__lemma,axiom,
! [I_1,P_1,K_1] :
( hBOOL(hAPP_nat_bool(P_1,K_1))
=> ( ! [N_2] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2)))
=> hBOOL(hAPP_nat_bool(P_1,N_2)) )
=> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,K_1),I_1))) ) ) ).
fof(fact_3269_Suc__le__D,axiom,
! [N_1,M_4] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),M_4))
=> ? [M_1] : M_4 = hAPP_nat_nat(suc,M_1) ) ).
fof(fact_3270_not__neg__0,axiom,
~ hBOOL(hAPP_int_bool(nat_neg,zero_zero_int)) ).
fof(fact_3271_not__neg__1,axiom,
~ hBOOL(hAPP_int_bool(nat_neg,one_one_int)) ).
fof(fact_3272_not__neg__int,axiom,
! [N_1] : ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_nat_int(semiri1621563631at_int,N_1))) ).
fof(fact_3273_neg__def,axiom,
! [Z_5] :
( hBOOL(hAPP_int_bool(nat_neg,Z_5))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_5),zero_zero_int)) ) ).
fof(fact_3274_not__neg__eq__ge__0,axiom,
! [X_2] :
( ~ hBOOL(hAPP_int_bool(nat_neg,X_2))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) ) ).
fof(fact_3275_neg__number__of__Bit1,axiom,
! [W_1] :
( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,W_1))))
<=> hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W_1))) ) ).
fof(fact_3276_not__neg__number__of__Pls,axiom,
~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,pls))) ).
fof(fact_3277_neg__number__of__Bit0,axiom,
! [W_1] :
( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W_1))))
<=> hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W_1))) ) ).
fof(fact_3278_neg__nat,axiom,
! [Z_1] :
( hBOOL(hAPP_int_bool(nat_neg,Z_1))
=> hAPP_int_nat(nat,Z_1) = zero_zero_nat ) ).
fof(fact_3279_neg__number__of__Min,axiom,
hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,min))) ).
fof(fact_3280_not__neg__nat,axiom,
! [Z_1] :
( is_int(Z_1)
=> ( ~ hBOOL(hAPP_int_bool(nat_neg,Z_1))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,Z_1)) = Z_1 ) ) ).
fof(fact_3281_neg__imp__number__of__eq__0,axiom,
! [V_2] :
( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> hAPP_int_nat(number_number_of_nat,V_2) = zero_zero_nat ) ).
fof(fact_3282_eq__nat__number__of,axiom,
! [V,V_1] :
( ( is_int(V)
& is_int(V_1) )
=> ( hAPP_int_nat(number_number_of_nat,V) = hAPP_int_nat(number_number_of_nat,V_1)
<=> ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,V_1)),zero_zero_int)) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V)))
=> ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
=> hAPP_int_int(number_number_of_int,V) = zero_zero_int )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
=> V = V_1 ) ) ) ) ) ) ).
fof(fact_3283_nat__number__of__add__left,axiom,
! [V_3,K_2,V_2] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),K_2) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),V_3))),K_2) ) ) ) ) ).
fof(fact_3284_int__nat__number__of,axiom,
! [V_2] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_2)) = zero_zero_int )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_int(number_number_of_int,V_2) ) ) ).
fof(fact_3285_div__nat__number__of,axiom,
! [V_3,V_2] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = zero_zero_nat )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,V_3))) ) ) ).
fof(fact_3286_power__nat__number__of__number__of,axiom,
! [W,V_2] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,W)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,W)) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,W)) = hAPP_int_nat(nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_nat(number_number_of_nat,W))) ) ) ).
fof(fact_3287_power__nat__number__of,axiom,
! [N_1,V_2] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N_1) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),N_1) = hAPP_int_nat(nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,V_2)),N_1)) ) ) ).
fof(fact_3288_mod__nat__number__of,axiom,
! [V_3,V_2] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = zero_zero_nat )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3)))
=> hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,V_2) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3)))
=> hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(nat,hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,V_3))) ) ) ) ) ).
fof(fact_3289_MultInvPair__card__two,axiom,
! [J,A,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J),zero_zero_int),P_5))
=> hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(multInvPair(A,P_5),J)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ) ) ) ) ) ).
fof(fact_3290_SetS__elems__card,axiom,
! [A,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A))
=> ! [X] :
( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X),setS(A,P_5)))
=> hAPP_f957591787ol_nat(finite_card_int,X) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ) ) ) ) ) ).
fof(fact_3291_Suc__nat__number__of__add,axiom,
! [N_1,V_2] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N_1) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2))),N_1) ) ) ).
fof(fact_3292_lessE,axiom,
! [I,K_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K_2))
=> ( K_2 != hAPP_nat_nat(suc,I)
=> ~ ! [J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1))
=> K_2 != hAPP_nat_nat(suc,J_1) ) ) ) ).
fof(fact_3293_Suc__lessE,axiom,
! [I,K_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I)),K_2))
=> ~ ! [J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1))
=> K_2 != hAPP_nat_nat(suc,J_1) ) ) ).
fof(fact_3294_real__lbound__gt__zero,axiom,
! [D2,D1] :
( 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] :
( 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)) ) ) ) ).
fof(fact_3295_succ__Pls,axiom,
hAPP_int_int(succ,pls) = hAPP_int_int(bit1,pls) ).
fof(fact_3296_succ__Bit0,axiom,
! [K_2] : hAPP_int_int(succ,hAPP_int_int(bit0,K_2)) = hAPP_int_int(bit1,K_2) ).
fof(fact_3297_succ__Bit1,axiom,
! [K_2] : hAPP_int_int(succ,hAPP_int_int(bit1,K_2)) = hAPP_int_int(bit0,hAPP_int_int(succ,K_2)) ).
fof(fact_3298_succ__def,axiom,
! [K_2] : hAPP_int_int(succ,K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),one_one_int) ).
fof(fact_3299_succ__Min,axiom,
hAPP_int_int(succ,min) = pls ).
fof(fact_3300_diff__bin__simps_I2_J,axiom,
! [K_2] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),min) = hAPP_int_int(succ,K_2) ).
fof(fact_3301_add__Bit1__Bit1,axiom,
! [K_2,L_2] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),hAPP_int_int(succ,L_2))) ).
fof(fact_3302_nat__1__add__number__of,axiom,
! [V_2] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_int_nat(number_number_of_nat,V_2)) = one_one_nat )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2)) ) ) ).
fof(fact_3303_nat__number__of__add__1,axiom,
! [V_2] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat) = one_one_nat )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2)) ) ) ).
fof(fact_3304_Suc__nat__number__of,axiom,
! [V_2] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> hAPP_nat_nat(suc,hAPP_int_nat(number_number_of_nat,V_2)) = one_one_nat )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> hAPP_nat_nat(suc,hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2)) ) ) ).
fof(fact_3305_SetS__card,axiom,
! [A,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_f521865025ol_nat(finite1876863882t_bool,setS(A,P_5))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) ) ) ) ).
fof(fact_3306_MultInvPair__prop2,axiom,
! [A,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5))
=> comple219730294t_bool(setS(A,P_5)) = hAPP_i1948725293t_bool(sRStar,P_5) ) ) ) ).
fof(fact_3307_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X_1,Y_1,Xa_2,Ya] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) ).
fof(fact_3308_not0__implies__Suc,axiom,
! [N_1] :
( N_1 != zero_zero_nat
=> ? [M_1] : N_1 = hAPP_nat_nat(suc,M_1) ) ).
fof(fact_3309_nat__induct,axiom,
! [N,P_1] :
( hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
=> ( ! [N_2] :
( hBOOL(hAPP_nat_bool(P_1,N_2))
=> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2))) )
=> hBOOL(hAPP_nat_bool(P_1,N)) ) ) ).
fof(fact_3310_zero__induct,axiom,
! [P_1,K_1] :
( hBOOL(hAPP_nat_bool(P_1,K_1))
=> ( ! [N_2] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2)))
=> hBOOL(hAPP_nat_bool(P_1,N_2)) )
=> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) ) ) ).
fof(fact_3311_real__sqrt__eq__iff,axiom,
! [X_2,Y_2] :
( hAPP_real_real(sqrt,X_2) = hAPP_real_real(sqrt,Y_2)
<=> X_2 = Y_2 ) ).
fof(fact_3312_real__sqrt__le__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X_2)),hAPP_real_real(sqrt,Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)) ) ).
fof(fact_3313_real__sqrt__le__mono,axiom,
! [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,hAPP_real_real(sqrt,X_1)),hAPP_real_real(sqrt,Y_1))) ) ).
fof(fact_3314_real__sqrt__zero,axiom,
hAPP_real_real(sqrt,zero_zero_real) = zero_zero_real ).
fof(fact_3315_real__sqrt__eq__0__iff,axiom,
! [X_2] :
( hAPP_real_real(sqrt,X_2) = zero_zero_real
<=> X_2 = zero_zero_real ) ).
fof(fact_3316_real__sqrt__less__mono,axiom,
! [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,hAPP_real_real(sqrt,X_1)),hAPP_real_real(sqrt,Y_1))) ) ).
fof(fact_3317_real__sqrt__less__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X_2)),hAPP_real_real(sqrt,Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) ) ).
fof(fact_3318_real__sqrt__mult,axiom,
! [X_1,Y_1] : hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sqrt,X_1)),hAPP_real_real(sqrt,Y_1)) ).
fof(fact_3319_real__sqrt__power,axiom,
! [X_1,K_2] : hAPP_real_real(sqrt,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),K_2)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X_1)),K_2) ).
fof(fact_3320_real__sqrt__one,axiom,
hAPP_real_real(sqrt,one_one_real) = one_one_real ).
fof(fact_3321_real__sqrt__eq__1__iff,axiom,
! [X_2] :
( hAPP_real_real(sqrt,X_2) = one_one_real
<=> X_2 = one_one_real ) ).
fof(fact_3322_real__sqrt__eq__zero__cancel,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> ( hAPP_real_real(sqrt,X_1) = zero_zero_real
=> X_1 = zero_zero_real ) ) ).
fof(fact_3323_real__sqrt__ge__zero,axiom,
! [X_1] :
( 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),hAPP_real_real(sqrt,X_1))) ) ).
fof(fact_3324_real__sqrt__le__0__iff,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X_2)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),zero_zero_real)) ) ).
fof(fact_3325_real__sqrt__ge__0__iff,axiom,
! [Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2)) ) ).
fof(fact_3326_real__sqrt__gt__zero,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sqrt,X_1))) ) ).
fof(fact_3327_real__sqrt__not__eq__zero,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hAPP_real_real(sqrt,X_1) != zero_zero_real ) ).
fof(fact_3328_real__sqrt__lt__0__iff,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X_2)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),zero_zero_real)) ) ).
fof(fact_3329_real__sqrt__gt__0__iff,axiom,
! [Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sqrt,Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2)) ) ).
fof(fact_3330_real__sqrt__ge__1__iff,axiom,
! [Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(sqrt,Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Y_2)) ) ).
fof(fact_3331_real__sqrt__le__1__iff,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X_2)),one_one_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),one_one_real)) ) ).
fof(fact_3332_real__sqrt__ge__one,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(sqrt,X_1))) ) ).
fof(fact_3333_real__sqrt__lt__1__iff,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X_2)),one_one_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),one_one_real)) ) ).
fof(fact_3334_real__sqrt__gt__1__iff,axiom,
! [Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(sqrt,Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Y_2)) ) ).
fof(fact_3335_le__real__sqrt__sumsq,axiom,
! [X_1,Y_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_1),Y_1))))) ).
fof(fact_3336_real__sqrt__mult__self__sum__ge__zero,axiom,
! [X_1,Y_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_1),Y_1))))) ).
fof(fact_3337_real__sqrt__two__ge__zero,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_3338_real__sqrt__two__gt__zero,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_3339_real__sqrt__two__gt__one,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_3340_real__sqrt__unique,axiom,
! [Y_1,X_1] :
( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = X_1
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1))
=> hAPP_real_real(sqrt,X_1) = Y_1 ) ) ).
fof(fact_3341_real__sqrt__pow2,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = X_1 ) ).
fof(fact_3342_real__sqrt__pow2__iff,axiom,
! [X_2] :
( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = X_2
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2)) ) ).
fof(fact_3343_real__sqrt__pow2__gt__zero,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).
fof(fact_3344_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A_3,C_1,B_2,D_3] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),C_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_2),D_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,C_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,D_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) ).
fof(fact_3345_real__sqrt__sum__squares__ge2,axiom,
! [Y_1,X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).
fof(fact_3346_real__sqrt__sum__squares__ge1,axiom,
! [X_1,Y_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).
fof(fact_3347_real__sqrt__sum__squares__eq__cancel2,axiom,
! [X_1,Y_1] :
( hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = Y_1
=> X_1 = zero_zero_real ) ).
fof(fact_3348_real__sqrt__sum__squares__eq__cancel,axiom,
! [X_1,Y_1] :
( hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = X_1
=> Y_1 = zero_zero_real ) ).
fof(fact_3349_real__sqrt__sum__squares__mult__squared__eq,axiom,
! [X_1,Y_1,Xa_2,Ya] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_3350_real__sqrt__sum__squares__ge__zero,axiom,
! [X_1,Y_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).
fof(fact_3351_lemma__sqrt__hcomplex__capprox,axiom,
! [Y_1,X_1,U_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),U_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> ( 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))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),U_1)) ) ) ) ) ) ).
fof(fact_3352_Union__SetS__finite,axiom,
! [A,P_5] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,comple219730294t_bool(setS(A,P_5)))) ) ).
fof(fact_3353_nat_Oexhaust,axiom,
! [Y_1] :
( Y_1 != zero_zero_nat
=> ~ ! [Nat_1] : Y_1 != hAPP_nat_nat(suc,Nat_1) ) ).
fof(fact_3354_real__divide__square__eq,axiom,
! [R_3,A_3] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),A_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),R_3)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,A_3),R_3) ).
fof(fact_3355_real__sqrt__divide,axiom,
! [X_1,Y_1] : hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,X_1)),hAPP_real_real(sqrt,Y_1)) ).
fof(fact_3356_wset__fin,axiom,
! [A,P_5] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(wset(A),P_5))) ).
fof(fact_3357_SetS__elems__finite,axiom,
! [A,P_5,X] :
( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X),setS(A,P_5)))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,X)) ) ).
fof(fact_3358_d22set__fin,axiom,
! [A] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(d22set,A))) ).
fof(fact_3359_real__0__le__divide__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_2),Y_2)))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),zero_zero_real))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2)) )
& ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),zero_zero_real)) ) ) ) ).
fof(fact_3360_real__less__half__sum,axiom,
! [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,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).
fof(fact_3361_real__gt__half__sum,axiom,
! [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,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Y_1)) ) ).
fof(fact_3362_SRStar__finite,axiom,
! [P_5] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(sRStar,P_5))) ) ).
fof(fact_3363_lemma__real__divide__sqrt__less,axiom,
! [U_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),U_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),U_1)) ) ).
fof(fact_3364_real__average__minus__first,axiom,
! [A_3,B_2] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),A_3) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_2),A_3)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_3365_real__average__minus__second,axiom,
! [B_2,A_3] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_2),A_3)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),A_3) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_2),A_3)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_3366_real__sum__of__halves,axiom,
! [X_1] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = X_1 ).
fof(fact_3367_eq__divide__2__times__iff,axiom,
! [X_2,Y_2,Z_2] :
( X_2 = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Y_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_2))
<=> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Y_2),Z_2) ) ).
fof(fact_3368_complex__mod__triangle__ineq2,axiom,
! [B_2,A_3] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_2),A_3))),hAPP_complex_real(norm_norm_complex,B_2))),hAPP_complex_real(norm_norm_complex,A_3))) ).
fof(fact_3369_SetS__finite,axiom,
! [A,P_5] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
=> hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,setS(A,P_5))) ) ).
fof(fact_3370_le__Suc__ex__iff,axiom,
! [K_1,L] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),L))
<=> ? [N_2] : L = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_2) ) ).
fof(fact_3371_lemma__MVT,axiom,
! [F,A,B] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A))),A)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A))),B)) ).
fof(fact_3372_arctan__half,axiom,
! [X_1] : hAPP_real_real(arctan,X_1) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) ).
fof(fact_3373_lemma__f__mono__add,axiom,
! [Ma,No,F] :
( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),hAPP_nat_real(F,hAPP_nat_nat(suc,N_2))))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,Ma)),hAPP_nat_real(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),No)))) ) ).
fof(fact_3374_arctan__monotone_H,axiom,
! [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,hAPP_real_real(arctan,X_1)),hAPP_real_real(arctan,Y_1))) ) ).
fof(fact_3375_arctan__zero__zero,axiom,
hAPP_real_real(arctan,zero_zero_real) = zero_zero_real ).
fof(fact_3376_arctan__monotone,axiom,
! [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,hAPP_real_real(arctan,X_1)),hAPP_real_real(arctan,Y_1))) ) ).
fof(fact_3377_natceiling__add__number__of,axiom,
! [X_1,N_1] :
( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N_1)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_int_real(number267125858f_real,N_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X_1)),hAPP_int_nat(number_number_of_nat,N_1)) ) ) ).
fof(fact_3378_real__sqrt__sum__squares__less,axiom,
! [Y_1,X_1,U_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,Y_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),U_1)) ) ) ).
fof(fact_3379_cos__x__y__le__one,axiom,
! [X_1,Y_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),one_one_real)) ).
fof(fact_3380_natfloor__add__number__of,axiom,
! [X_1,N_1] :
( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N_1)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_int_real(number267125858f_real,N_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X_1)),hAPP_int_nat(number_number_of_nat,N_1)) ) ) ).
fof(fact_3381_real__norm__def,axiom,
! [R_3] : hAPP_real_real(norm_norm_real,R_3) = hAPP_real_real(abs_abs_real,R_3) ).
fof(fact_3382_real__sqrt__abs2,axiom,
! [X_1] : hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)) = hAPP_real_real(abs_abs_real,X_1) ).
fof(fact_3383_natfloor__zero,axiom,
natfloor(zero_zero_real) = zero_zero_nat ).
fof(fact_3384_zero__le__natfloor,axiom,
! [X_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),natfloor(X_1))) ).
fof(fact_3385_natfloor__mono,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natfloor(X_1)),natfloor(Y_1))) ) ).
fof(fact_3386_natfloor__number__of__eq,axiom,
! [N_1] : natfloor(hAPP_int_real(number267125858f_real,N_1)) = hAPP_int_nat(number_number_of_nat,N_1) ).
fof(fact_3387_natfloor__one,axiom,
natfloor(one_one_real) = one_one_nat ).
fof(fact_3388_natceiling__zero,axiom,
natceiling(zero_zero_real) = zero_zero_nat ).
fof(fact_3389_zero__le__natceiling,axiom,
! [X_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),natceiling(X_1))) ).
fof(fact_3390_natceiling__mono,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_1)),natceiling(Y_1))) ) ).
fof(fact_3391_natceiling__number__of__eq,axiom,
! [N_1] : natceiling(hAPP_int_real(number267125858f_real,N_1)) = hAPP_int_nat(number_number_of_nat,N_1) ).
fof(fact_3392_natceiling__one,axiom,
natceiling(one_one_real) = one_one_nat ).
fof(fact_3393_rabs__ratiotest__lemma,axiom,
! [X_1,Y_1,C_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_1),hAPP_real_real(abs_abs_real,Y_1))))
=> X_1 = zero_zero_real ) ) ).
fof(fact_3394_abs__add__one__not__less__self,axiom,
! [X_1] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real)),X_1)) ).
fof(fact_3395_sin__bound__lemma,axiom,
! [U_1,V_2,X_1,Y_1] :
( X_1 = Y_1
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,U_1)),V_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),U_1)),Y_1))),V_2)) ) ) ).
fof(fact_3396_natfloor__neg,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real))
=> natfloor(X_1) = zero_zero_nat ) ).
fof(fact_3397_abs__add__one__gt__zero,axiom,
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_real_real(abs_abs_real,X_1)))) ).
fof(fact_3398_natceiling__neg,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real))
=> natceiling(X_1) = zero_zero_nat ) ).
fof(fact_3399_le__natfloor__eq__one,axiom,
! [X_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),natfloor(X_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_2)) ) ).
fof(fact_3400_natceiling__le__eq__one,axiom,
! [X_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_2)),one_one_nat))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),one_one_real)) ) ).
fof(fact_3401_real__sqrt__abs,axiom,
! [X_1] : hAPP_real_real(sqrt,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_real_real(abs_abs_real,X_1) ).
fof(fact_3402_le__mult__natfloor,axiom,
! [B_2,A_3] :
( 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,zero_zero_real),B_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,natfloor(A_3)),natfloor(B_2))),natfloor(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),B_2)))) ) ) ).
fof(fact_3403_less__one__imp__sqr__less__one,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_real)) ) ).
fof(fact_3404_natfloor__add__one,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),one_one_real)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X_1)),one_one_nat) ) ).
fof(fact_3405_real__sqrt__ge__abs1,axiom,
! [X_1,Y_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).
fof(fact_3406_real__sqrt__ge__abs2,axiom,
! [Y_1,X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y_1)),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).
fof(fact_3407_arctan__add,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,Y_1)),one_one_real))
=> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(arctan,X_1)),hAPP_real_real(arctan,Y_1)) = hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_1)))) ) ) ).
fof(fact_3408_natceiling__add__one,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),one_one_real)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X_1)),one_one_nat) ) ).
fof(fact_3409_le__natfloor__eq__number__of,axiom,
! [X_2,N] :
( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,N)),natfloor(X_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,N)),X_2)) ) ) ) ).
fof(fact_3410_natceiling__le__eq__number__of,axiom,
! [X_2,N] :
( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_2)),hAPP_int_nat(number_number_of_nat,N)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),hAPP_int_real(number267125858f_real,N))) ) ) ) ).
fof(fact_3411_lemma__interval,axiom,
! [B_2,A_3,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),B_2))
=> ? [D_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
& ! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y))),D_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_3),Y))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),B_2)) ) ) ) ) ) ).
fof(fact_3412_lemma__interval__lt,axiom,
! [B_2,A_3,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),B_2))
=> ? [D_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
& ! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y))),D_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),Y))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),B_2)) ) ) ) ) ) ).
fof(fact_3413_abs__ln__one__plus__x__minus__x__bound,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1))),X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).
fof(fact_3414_machin,axiom,
hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,hAPP_int_int(bit1,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))))))))))) ).
fof(fact_3415_pi__neq__zero,axiom,
pi != zero_zero_real ).
fof(fact_3416_abs__int__eq,axiom,
! [M] : hAPP_int_int(abs_abs_int,hAPP_nat_int(semiri1621563631at_int,M)) = hAPP_nat_int(semiri1621563631at_int,M) ).
fof(fact_3417_zdvd__antisym__abs,axiom,
! [A_3,B_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_3),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_2),A_3))
=> hAPP_int_int(abs_abs_int,A_3) = hAPP_int_int(abs_abs_int,B_2) ) ) ).
fof(fact_3418_zdvd__self__abs2,axiom,
! [D_3] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(abs_abs_int,D_3)),D_3)) ).
fof(fact_3419_zdvd__self__abs1,axiom,
! [D_3] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),hAPP_int_int(abs_abs_int,D_3))) ).
fof(fact_3420_abs__zmult__eq__1,axiom,
! [M,N_1] :
( hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N_1)) = one_one_int
=> hAPP_int_int(abs_abs_int,M) = one_one_int ) ).
fof(fact_3421_pi__ge__zero,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),pi)) ).
fof(fact_3422_pi__gt__zero,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),pi)) ).
fof(fact_3423_pi__not__less__zero,axiom,
~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,pi),zero_zero_real)) ).
fof(fact_3424_zdvd1__eq,axiom,
! [X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_2),one_one_int))
<=> hAPP_int_int(abs_abs_int,X_2) = one_one_int ) ).
fof(fact_3425_int__nat__abs,axiom,
! [X_1] : hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,X_1))) = hAPP_int_int(abs_abs_int,X_1) ).
fof(fact_3426_ln__less__cancel__iff,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X_2)),hAPP_real_real(ln,Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) ) ) ) ).
fof(fact_3427_ln__inj__iff,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2))
=> ( hAPP_real_real(ln,X_2) = hAPP_real_real(ln,Y_2)
<=> X_2 = Y_2 ) ) ) ).
fof(fact_3428_ln__less__self,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X_1)),X_1)) ) ).
fof(fact_3429_abs__div,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y_1),X_1))
=> hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),Y_1)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(abs_abs_int,X_1)),hAPP_int_int(abs_abs_int,Y_1)) ) ).
fof(fact_3430_ln__one,axiom,
hAPP_real_real(ln,one_one_real) = zero_zero_real ).
fof(fact_3431_ln__eq__minus__one,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hAPP_real_real(ln,X_1) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),one_one_real)
=> X_1 = one_one_real ) ) ).
fof(fact_3432_zabs__less__one__iff,axiom,
! [Z_2] :
( is_int(Z_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(abs_abs_int,Z_2)),one_one_int))
<=> Z_2 = zero_zero_int ) ) ).
fof(fact_3433_dvd__imp__le__int,axiom,
! [D_3,I] :
( is_int(I)
=> ( I != zero_zero_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),I))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,D_3)),hAPP_int_int(abs_abs_int,I))) ) ) ) ).
fof(fact_3434_zero__le__zpower__abs,axiom,
! [X_1,N_1] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,X_1)),N_1))) ).
fof(fact_3435_nat__abs__mult__distrib,axiom,
! [W,Z_1] : hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,W))),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Z_1))) ).
fof(fact_3436_abs__eq__1__iff,axiom,
! [Z_2] :
( is_int(Z_2)
=> ( hAPP_int_int(abs_abs_int,Z_2) = one_one_int
<=> ( Z_2 = one_one_int
| Z_2 = hAPP_int_int(number_number_of_int,min) ) ) ) ).
fof(fact_3437_ln__le__cancel__iff,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X_2)),hAPP_real_real(ln,Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)) ) ) ) ).
fof(fact_3438_ln__ge__zero,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(ln,X_1))) ) ).
fof(fact_3439_ln__gt__zero__imp__gt__one,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(ln,X_1)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1)) ) ) ).
fof(fact_3440_ln__less__zero,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X_1)),zero_zero_real)) ) ) ).
fof(fact_3441_ln__less__zero__iff,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X_2)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),one_one_real)) ) ) ).
fof(fact_3442_ln__eq__zero__iff,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> ( hAPP_real_real(ln,X_2) = zero_zero_real
<=> X_2 = one_one_real ) ) ).
fof(fact_3443_ln__gt__zero__iff,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(ln,X_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_2)) ) ) ).
fof(fact_3444_ln__gt__zero,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(ln,X_1))) ) ).
fof(fact_3445_ln__le__minus__one,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X_1)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),one_one_real))) ) ).
fof(fact_3446_zero__less__zpower__abs__iff,axiom,
! [X_2,N] :
( is_int(X_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,X_2)),N)))
<=> ( X_2 != zero_zero_int
| N = zero_zero_nat ) ) ) ).
fof(fact_3447_abs__power3__distrib,axiom,
! [X_1] : hAPP_int_int(abs_abs_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) ).
fof(fact_3448_zdvd__mult__cancel1,axiom,
! [N,Ma] :
( is_int(Ma)
=> ( Ma != zero_zero_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),N)),Ma))
<=> hAPP_int_int(abs_abs_int,N) = one_one_int ) ) ) ).
fof(fact_3449_dvd__int__iff,axiom,
! [Z_2,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_2),hAPP_nat_int(semiri1621563631at_int,Ma)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Z_2))),Ma)) ) ).
fof(fact_3450_int__dvd__iff,axiom,
! [Ma,Z_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,Ma)),Z_2))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Z_2)))) ) ).
fof(fact_3451_ln__ge__zero__imp__ge__one,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(ln,X_1)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1)) ) ) ).
fof(fact_3452_ln__ge__zero__iff,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(ln,X_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_2)) ) ) ).
fof(fact_3453_ln__mult,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1))
=> hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(ln,X_1)),hAPP_real_real(ln,Y_1)) ) ) ).
fof(fact_3454_ln__add__one__self__le__self,axiom,
! [X_1] :
( 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,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1))),X_1)) ) ).
fof(fact_3455_ln__div,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1))
=> hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,X_1)),hAPP_real_real(ln,Y_1)) ) ) ).
fof(fact_3456_ln__add__one__self__le__self2,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1))),X_1)) ) ).
fof(fact_3457_power2__eq__iff__abs__eq,axiom,
! [A,B] :
( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))
<=> hAPP_int_int(abs_abs_int,A) = hAPP_int_int(abs_abs_int,B) ) ).
fof(fact_3458_abs__power2__distrib,axiom,
! [A_3] : hAPP_int_int(abs_abs_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,A_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_3459_pi__ge__two,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi)) ).
fof(fact_3460_pi__less__4,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_3461_pi__half__neq__two,axiom,
hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).
fof(fact_3462_power2__eq1__iff,axiom,
! [A_3] :
( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_int
=> hAPP_int_int(abs_abs_int,A_3) = one_one_int ) ).
fof(fact_3463_decr__lemma,axiom,
! [X_1,Z_1,D_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Z_1))),one_one_int)),D_3))),Z_1)) ) ).
fof(fact_3464_incr__lemma,axiom,
! [Z_1,X_1,D_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Z_1))),one_one_int)),D_3)))) ) ).
fof(fact_3465_pi__half__le__two,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_3466_pi__half__neq__zero,axiom,
hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_zero_real ).
fof(fact_3467_pi__half__less__two,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_3468_pi__half__ge__zero,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_3469_pi__half__gt__zero,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_3470_arctan__ubound,axiom,
! [Y_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,Y_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_3471_arctan1__eq__pi4,axiom,
hAPP_real_real(arctan,one_one_real) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_3472_ln__one__plus__pos__lower__bound,axiom,
! [X_1] :
( 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,X_1),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1)))) ) ) ).
fof(fact_3473_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
! [X_1] :
( 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,X_1),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1))),X_1))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).
fof(fact_3474_best__odd__division__abs,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_2))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd))
=> ? [N_2] :
( is_int(N_2)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_2),X_2))))),X_2)) ) ) ) ).
fof(fact_3475_best__division__abs,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_1))
=> ? [N_2] :
( is_int(N_2)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_2),X_1))))),X_1)) ) ) ).
fof(fact_3476_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1))),X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ) ).
fof(fact_3477_nat0__intermed__int__val,axiom,
! [K_1,F,N] :
( is_int(K_1)
=> ( ! [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_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)))
=> ? [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N))
& hAPP_nat_int(F,I_2) = K_1 ) ) ) ) ) ).
fof(fact_3478_int__val__lemma,axiom,
! [K_1,F,N] :
( is_int(K_1)
=> ( ! [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_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)))
=> ? [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N))
& hAPP_nat_int(F,I_2) = K_1 ) ) ) ) ) ).
fof(fact_3479_real__sqrt__minus,axiom,
! [X_1] : hAPP_real_real(sqrt,hAPP_real_real(uminus_uminus_real,X_1)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sqrt,X_1)) ).
fof(fact_3480_arctan__minus,axiom,
! [X_1] : hAPP_real_real(arctan,hAPP_real_real(uminus_uminus_real,X_1)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(arctan,X_1)) ).
fof(fact_3481_real__minus__mult__self__le,axiom,
! [U_1,X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_1),U_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1))) ).
fof(fact_3482_real__add__minus__iff,axiom,
! [X_2,A] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),hAPP_real_real(uminus_uminus_real,A)) = zero_zero_real
<=> X_2 = A ) ).
fof(fact_3483_real__add__eq__0__iff,axiom,
! [X_2,Y_2] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Y_2) = zero_zero_real
<=> Y_2 = hAPP_real_real(uminus_uminus_real,X_2) ) ).
fof(fact_3484_abs__le__interval__iff,axiom,
! [X_2,R_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_2)),R_2))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,R_2)),X_2))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),R_2)) ) ) ).
fof(fact_3485_real__diff__def,axiom,
! [R_3,S_1] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_3),S_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_3),hAPP_real_real(uminus_uminus_real,S_1)) ).
fof(fact_3486_minus__real__def,axiom,
! [X_1,Y_1] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(uminus_uminus_real,Y_1)) ).
fof(fact_3487_abs__minus__add__cancel,axiom,
! [X_1,Y_1] : hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(uminus_uminus_real,Y_1))) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y_1),hAPP_real_real(uminus_uminus_real,X_1))) ).
fof(fact_3488_complex__mod__minus__le__complex__mod,axiom,
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(norm_norm_complex,X_1))),hAPP_complex_real(norm_norm_complex,X_1))) ).
fof(fact_3489_real__add__le__0__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Y_2)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),hAPP_real_real(uminus_uminus_real,X_2))) ) ).
fof(fact_3490_real__0__le__add__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,X_2)),Y_2)) ) ).
fof(fact_3491_real__add__less__0__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Y_2)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),hAPP_real_real(uminus_uminus_real,X_2))) ) ).
fof(fact_3492_real__0__less__add__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,X_2)),Y_2)) ) ).
fof(fact_3493_real__abs__def,axiom,
! [R_3] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),zero_zero_real))
=> hAPP_real_real(abs_abs_real,R_3) = hAPP_real_real(uminus_uminus_real,R_3) )
& ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),zero_zero_real))
=> hAPP_real_real(abs_abs_real,R_3) = R_3 ) ) ).
fof(fact_3494_abs__real__def,axiom,
! [A_3] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),zero_zero_real))
=> hAPP_real_real(abs_abs_real,A_3) = hAPP_real_real(uminus_uminus_real,A_3) )
& ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),zero_zero_real))
=> hAPP_real_real(abs_abs_real,A_3) = A_3 ) ) ).
fof(fact_3495_abs__sum__triangle__ineq,axiom,
! [X_1,Y_1,L_2,M] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(uminus_uminus_real,L_2)),hAPP_real_real(uminus_uminus_real,M))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(uminus_uminus_real,L_2)))),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y_1),hAPP_real_real(uminus_uminus_real,M)))))) ).
fof(fact_3496_realpow__square__minus__le,axiom,
! [U_1,X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,U_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_3497_ln__one__minus__pos__upper__bound,axiom,
! [X_1] :
( 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_real,X_1),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X_1))),hAPP_real_real(uminus_uminus_real,X_1))) ) ) ).
fof(fact_3498_aux5,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real))
=> hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X_1)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X_1))))) ) ).
fof(fact_3499_m2pi__less__pi,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi))),pi)) ).
fof(fact_3500_minus__pi__half__less__zero,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_real)) ).
fof(fact_3501_arctan__bounded,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arctan,Y_1)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,Y_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).
fof(fact_3502_arctan__lbound,axiom,
! [Y_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arctan,Y_1))) ).
fof(fact_3503_ln__one__minus__pos__lower__bound,axiom,
! [X_1] :
( 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,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(uminus_uminus_real,X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X_1)))) ) ) ).
fof(fact_3504_nat__intermed__int__val,axiom,
! [K_1,F,N,Ma] :
( is_int(K_1)
=> ( ! [I_2] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),I_2))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N)) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_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,Ma),N))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,Ma)),K_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N)))
=> ? [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),I_2))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N))
& hAPP_nat_int(F,I_2) = K_1 ) ) ) ) ) ) ).
fof(fact_3505_arcsin__lt__bounded,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),one_one_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y_1)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arcsin,Y_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ) ).
fof(fact_3506_arcsin__lbound,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y_1))) ) ) ).
fof(fact_3507_arcsin__bounded,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y_1)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ) ).
fof(fact_3508_arcsin__ubound,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).
fof(fact_3509_negative__zle,axiom,
! [N_1,M] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N_1))),hAPP_nat_int(semiri1621563631at_int,M))) ).
fof(fact_3510_negative__zless,axiom,
! [N_1,M] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1)))),hAPP_nat_int(semiri1621563631at_int,M))) ).
fof(fact_3511_minus__Pls,axiom,
hAPP_int_int(uminus_uminus_int,pls) = pls ).
fof(fact_3512_minus__Bit0,axiom,
! [K_2] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(bit0,K_2)) = hAPP_int_int(bit0,hAPP_int_int(uminus_uminus_int,K_2)) ).
fof(fact_3513_zminus__0,axiom,
hAPP_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).
fof(fact_3514_zmult__zminus,axiom,
! [Z_1,W] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(uminus_uminus_int,Z_1)),W) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),W)) ).
fof(fact_3515_minus__numeral__code_I5_J,axiom,
! [W] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(number_number_of_int,hAPP_int_int(uminus_uminus_int,W)) ).
fof(fact_3516_zminus__zadd__distrib,axiom,
! [Z_1,W] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),W)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,Z_1)),hAPP_int_int(uminus_uminus_int,W)) ).
fof(fact_3517_zminus__zminus,axiom,
! [Z_1] :
( is_int(Z_1)
=> hAPP_int_int(uminus_uminus_int,hAPP_int_int(uminus_uminus_int,Z_1)) = Z_1 ) ).
fof(fact_3518_zcong__zminus,axiom,
! [A,B,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),hAPP_int_int(uminus_uminus_int,Ma)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)) ) ).
fof(fact_3519_uminus__dvd__conv_I1_J,axiom,
! [D,Ta] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),Ta))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(uminus_uminus_int,D)),Ta)) ) ).
fof(fact_3520_uminus__dvd__conv_I2_J,axiom,
! [D,Ta] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),Ta))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(uminus_uminus_int,Ta))) ) ).
fof(fact_3521_zdiv__zminus2,axiom,
! [A_3,B_2] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_3)),B_2) ).
fof(fact_3522_zdiv__zminus__zminus,axiom,
! [A_3,B_2] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_3)),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) ).
fof(fact_3523_zmod__zminus2,axiom,
! [A_3,B_2] : hAPP_int_int(div_mod_int(A_3),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),B_2)) ).
fof(fact_3524_zmod__zminus__zminus,axiom,
! [A_3,B_2] : hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(A_3),B_2)) ).
fof(fact_3525_zminus__zmod,axiom,
! [X_1,M] : hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(X_1),M))),M) = hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,X_1)),M) ).
fof(fact_3526_even__neg,axiom,
! [X_2] :
( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(uminus_uminus_int,X_2)))
<=> hBOOL(hAPP_int_bool(even_odd_even_int,X_2)) ) ).
fof(fact_3527_complex__diff__def,axiom,
! [X_1,Y_1] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_1),Y_1) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_1),hAPP_complex_complex(uminus473333897omplex,Y_1)) ).
fof(fact_3528_zadd__zminus__inverse2,axiom,
! [Z_1] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,Z_1)),Z_1) = zero_zero_int ).
fof(fact_3529_negative__eq__positive,axiom,
! [N,Ma] :
( hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,Ma)
<=> ( N = zero_zero_nat
& Ma = zero_zero_nat ) ) ).
fof(fact_3530_not__int__zless__negative,axiom,
! [N_1,M] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N_1)),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,M)))) ).
fof(fact_3531_Int_OMin__def,axiom,
min = hAPP_int_int(uminus_uminus_int,one_one_int) ).
fof(fact_3532_zmod__zminus1__not__zero,axiom,
! [K_2,L_2] :
( hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,K_2)),L_2) != zero_zero_int
=> hAPP_int_int(div_mod_int(K_2),L_2) != zero_zero_int ) ).
fof(fact_3533_zmod__zminus2__not__zero,axiom,
! [K_2,L_2] :
( hAPP_int_int(div_mod_int(K_2),hAPP_int_int(uminus_uminus_int,L_2)) != zero_zero_int
=> hAPP_int_int(div_mod_int(K_2),L_2) != zero_zero_int ) ).
fof(fact_3534_mult__Min,axiom,
! [K_2] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,min),K_2) = hAPP_int_int(uminus_uminus_int,K_2) ).
fof(fact_3535_diff__int__def,axiom,
! [Z_1,W] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_1),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),hAPP_int_int(uminus_uminus_int,W)) ).
fof(fact_3536_diff__int__def__symmetric,axiom,
! [Z_1,W] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),hAPP_int_int(uminus_uminus_int,W)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_1),W) ).
fof(fact_3537_minus__Min,axiom,
hAPP_int_int(uminus_uminus_int,min) = hAPP_int_int(bit1,pls) ).
fof(fact_3538_int__zle__neg,axiom,
! [N,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,N)),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,Ma))))
<=> ( N = zero_zero_nat
& Ma = zero_zero_nat ) ) ).
fof(fact_3539_negative__zle__0,axiom,
! [N_1] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N_1))),zero_zero_int)) ).
fof(fact_3540_nat__zminus__int,axiom,
! [N_1] : hAPP_int_nat(nat,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N_1))) = zero_zero_nat ).
fof(fact_3541_zabs__def,axiom,
! [I] :
( is_int(I)
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),zero_zero_int))
=> hAPP_int_int(abs_abs_int,I) = hAPP_int_int(uminus_uminus_int,I) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),zero_zero_int))
=> hAPP_int_int(abs_abs_int,I) = I ) ) ) ).
fof(fact_3542_minus__numeral__code_I6_J,axiom,
! [V_2,W] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),hAPP_int_int(uminus_uminus_int,W))) ).
fof(fact_3543_zmod__zminus2__eq__if,axiom,
! [A_3,B_2] :
( ( hAPP_int_int(div_mod_int(A_3),B_2) = zero_zero_int
=> hAPP_int_int(div_mod_int(A_3),hAPP_int_int(uminus_uminus_int,B_2)) = zero_zero_int )
& ( hAPP_int_int(div_mod_int(A_3),B_2) != zero_zero_int
=> hAPP_int_int(div_mod_int(A_3),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_3),B_2)),B_2) ) ) ).
fof(fact_3544_zmod__zminus1__eq__if,axiom,
! [A_3,B_2] :
( ( hAPP_int_int(div_mod_int(A_3),B_2) = zero_zero_int
=> hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),B_2) = zero_zero_int )
& ( hAPP_int_int(div_mod_int(A_3),B_2) != zero_zero_int
=> hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),B_2) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),hAPP_int_int(div_mod_int(A_3),B_2)) ) ) ).
fof(fact_3545_zdiv__minus1__right,axiom,
! [A_3] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(number_number_of_int,min)) = hAPP_int_int(uminus_uminus_int,A_3) ).
fof(fact_3546_neg__zminus__int,axiom,
! [N_1] : hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1))))) ).
fof(fact_3547_negative__zless__0,axiom,
! [N_1] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1)))),zero_zero_int)) ).
fof(fact_3548_not__zle__0__negative,axiom,
! [N_1] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1))))) ).
fof(fact_3549_power3__minus,axiom,
! [A_3] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(uminus_uminus_int,A_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_int_int(uminus_uminus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) ).
fof(fact_3550_nat__mult__distrib__neg,axiom,
! [Z_4,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),zero_zero_int))
=> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,hAPP_int_int(uminus_uminus_int,Z_1))),hAPP_int_nat(nat,hAPP_int_int(uminus_uminus_int,Z_4))) ) ).
fof(fact_3551_zdiv__zminus1__eq__if,axiom,
! [A_3,B_2] :
( is_int(B_2)
=> ( B_2 != zero_zero_int
=> ( ( hAPP_int_int(div_mod_int(A_3),B_2) = zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_3)),B_2) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)) )
& ( hAPP_int_int(div_mod_int(A_3),B_2) != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_3)),B_2) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),one_one_int) ) ) ) ) ).
fof(fact_3552_zdiv__zminus2__eq__if,axiom,
! [A_3,B_2] :
( is_int(B_2)
=> ( B_2 != zero_zero_int
=> ( ( hAPP_int_int(div_mod_int(A_3),B_2) = zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)) )
& ( hAPP_int_int(div_mod_int(A_3),B_2) != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),one_one_int) ) ) ) ) ).
fof(fact_3553_neg__even__power,axiom,
! [A,X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zEven))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(uminus_uminus_int,A)),hAPP_int_nat(nat,X_2)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(nat,X_2)) ) ) ).
fof(fact_3554_neg__odd__power,axiom,
! [A,X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(uminus_uminus_int,A)),hAPP_int_nat(nat,X_2)) = hAPP_int_int(uminus_uminus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(nat,X_2))) ) ) ).
fof(fact_3555_zminus1__lemma,axiom,
! [A,B,Q_1,R_2] :
( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,B),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_2)))
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(uminus_uminus_int,A),B),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,R_2),zero_zero_int)),hAPP_int_int(uminus_uminus_int,Q_1)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(uminus_uminus_int,Q_1)),one_one_int))),hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,R_2),zero_zero_int)),zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B),R_2))))) ) ).
fof(fact_3556_negD,axiom,
! [X_1] :
( is_int(X_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),zero_zero_int))
=> ? [N_2] : X_1 = hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2))) ) ) ).
fof(fact_3557_arctan__inverse,axiom,
! [X_1] :
( X_1 != zero_zero_real
=> hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),X_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sgn_sgn_real,X_1)),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_real_real(arctan,X_1)) ) ).
fof(fact_3558_cos__arcsin,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real))
=> hAPP_real_real(cos,hAPP_real_real(arcsin,X_1)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).
fof(fact_3559_ln__bound,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X_1)),X_1)) ) ).
fof(fact_3560_cos__minus,axiom,
! [X_1] : hAPP_real_real(cos,hAPP_real_real(uminus_uminus_real,X_1)) = hAPP_real_real(cos,X_1) ).
fof(fact_3561_cos__le__one,axiom,
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,X_1)),one_one_real)) ).
fof(fact_3562_cos__zero,axiom,
hAPP_real_real(cos,zero_zero_real) = one_one_real ).
fof(fact_3563_cos__arctan__not__zero,axiom,
! [X_1] : hAPP_real_real(cos,hAPP_real_real(arctan,X_1)) != zero_zero_real ).
fof(fact_3564_real__sgn__eq,axiom,
! [X_1] : hAPP_real_real(sgn_sgn_real,X_1) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),hAPP_real_real(abs_abs_real,X_1)) ).
fof(fact_3565_cos__monotone__0__pi_H,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1))
=> ( 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),pi))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,X_1)),hAPP_real_real(cos,Y_1))) ) ) ) ).
fof(fact_3566_cos__ge__minus__one,axiom,
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),hAPP_real_real(cos,X_1))) ).
fof(fact_3567_abs__cos__le__one,axiom,
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(cos,X_1))),one_one_real)) ).
fof(fact_3568_cos__pi,axiom,
hAPP_real_real(cos,pi) = hAPP_int_real(number267125858f_real,min) ).
fof(fact_3569_cos__periodic__pi,axiom,
! [X_1] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),pi)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(cos,X_1)) ).
fof(fact_3570_real__sgn__pos,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hAPP_real_real(sgn_sgn_real,X_1) = one_one_real ) ).
fof(fact_3571_cos__monotone__0__pi,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),pi))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,X_1)),hAPP_real_real(cos,Y_1))) ) ) ) ).
fof(fact_3572_cos__monotone__minus__pi__0_H,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,pi)),Y_1))
=> ( 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),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,Y_1)),hAPP_real_real(cos,X_1))) ) ) ) ).
fof(fact_3573_real__sgn__neg,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real))
=> hAPP_real_real(sgn_sgn_real,X_1) = hAPP_int_real(number267125858f_real,min) ) ).
fof(fact_3574_sgn__real__def,axiom,
! [A_3] :
( ( A_3 = zero_zero_real
=> hAPP_real_real(sgn_sgn_real,A_3) = zero_zero_real )
& ( A_3 != zero_zero_real
=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
=> hAPP_real_real(sgn_sgn_real,A_3) = one_one_real )
& ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
=> hAPP_real_real(sgn_sgn_real,A_3) = hAPP_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ).
fof(fact_3575_real__sgn__def,axiom,
! [X_1] :
( ( X_1 = zero_zero_real
=> hAPP_real_real(sgn_sgn_real,X_1) = zero_zero_real )
& ( X_1 != zero_zero_real
=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hAPP_real_real(sgn_sgn_real,X_1) = one_one_real )
& ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hAPP_real_real(sgn_sgn_real,X_1) = hAPP_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ).
fof(fact_3576_cos__two__neq__zero,axiom,
hAPP_real_real(cos,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_zero_real ).
fof(fact_3577_cos__monotone__minus__pi__0,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,pi)),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,Y_1)),hAPP_real_real(cos,X_1))) ) ) ) ).
fof(fact_3578_cos__two__le__zero,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_real)) ).
fof(fact_3579_cos__two__less__zero,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_real)) ).
fof(fact_3580_cos__pi__half,axiom,
hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real ).
fof(fact_3581_cos__two__pi,axiom,
hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi)) = one_one_real ).
fof(fact_3582_cos__periodic,axiom,
! [X_1] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi))) = hAPP_real_real(cos,X_1) ).
fof(fact_3583_cos__60,axiom,
hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_3584_cos__45,axiom,
hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_3585_cos__30,axiom,
hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_3586_cos__double__less__one,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1))),one_one_real)) ) ) ).
fof(fact_3587_cos__gt__zero,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(cos,X_1))) ) ) ).
fof(fact_3588_cos__3over2__pi,axiom,
hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),pi)) = zero_zero_real ).
fof(fact_3589_cos__ge__zero,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(cos,X_1))) ) ) ).
fof(fact_3590_cos__gt__zero__pi,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(cos,X_1))) ) ) ).
fof(fact_3591_cos__total,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real))
=> ? [X] :
( 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,X),pi))
& hAPP_real_real(cos,X) = Y_1
& ! [Y] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),pi))
& hAPP_real_real(cos,Y) = Y_1 )
=> Y = X ) ) ) ) ).
fof(fact_3592_cos__is__zero,axiom,
? [X] :
( 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,X),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
& hAPP_real_real(cos,X) = zero_zero_real
& ! [Y] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
& hAPP_real_real(cos,Y) = zero_zero_real )
=> Y = X ) ) ).
fof(fact_3593_tan__double,axiom,
! [X_1] :
( hAPP_real_real(cos,X_1) != zero_zero_real
=> ( hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1)) != zero_zero_real
=> hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(tan,X_1))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(tan,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).
fof(fact_3594_cos__arccos__lemma1,axiom,
! [X_4,Y_4] : hAPP_real_real(cos,hAPP_real_real(arccos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).
fof(fact_3595_tan__minus,axiom,
! [X_1] : hAPP_real_real(tan,hAPP_real_real(uminus_uminus_real,X_1)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(tan,X_1)) ).
fof(fact_3596_tan__zero,axiom,
hAPP_real_real(tan,zero_zero_real) = zero_zero_real ).
fof(fact_3597_tan__arctan,axiom,
! [Y_1] : hAPP_real_real(tan,hAPP_real_real(arctan,Y_1)) = Y_1 ).
fof(fact_3598_tan__pi,axiom,
hAPP_real_real(tan,pi) = zero_zero_real ).
fof(fact_3599_tan__periodic__pi,axiom,
! [X_1] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),pi)) = hAPP_real_real(tan,X_1) ).
fof(fact_3600_tan__periodic__n,axiom,
! [X_1,N_1] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,N_1)),pi))) = hAPP_real_real(tan,X_1) ).
fof(fact_3601_zsgn__def,axiom,
! [I] :
( is_int(I)
=> ( ( I = zero_zero_int
=> hAPP_int_int(sgn_sgn_int,I) = zero_zero_int )
& ( I != zero_zero_int
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),I))
=> hAPP_int_int(sgn_sgn_int,I) = one_one_int )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),I))
=> hAPP_int_int(sgn_sgn_int,I) = hAPP_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).
fof(fact_3602_arccos__cos,axiom,
! [X_1] :
( 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,X_1),pi))
=> hAPP_real_real(arccos,hAPP_real_real(cos,X_1)) = X_1 ) ) ).
fof(fact_3603_cos__arccos__abs,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y_1)),one_one_real))
=> hAPP_real_real(cos,hAPP_real_real(arccos,Y_1)) = Y_1 ) ).
fof(fact_3604_tan__60,axiom,
hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) = hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) ).
fof(fact_3605_arccos__lbound,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(arccos,Y_1))) ) ) ).
fof(fact_3606_cos__arccos,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real))
=> hAPP_real_real(cos,hAPP_real_real(arccos,Y_1)) = Y_1 ) ) ).
fof(fact_3607_arccos__ubound,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arccos,Y_1)),pi)) ) ) ).
fof(fact_3608_arccos__cos2,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,pi)),X_1))
=> hAPP_real_real(arccos,hAPP_real_real(cos,X_1)) = hAPP_real_real(uminus_uminus_real,X_1) ) ) ).
fof(fact_3609_tan__periodic,axiom,
! [X_1] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi))) = hAPP_real_real(tan,X_1) ).
fof(fact_3610_tan__45,axiom,
hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = one_one_real ).
fof(fact_3611_tan__add,axiom,
! [Y_1,X_1] :
( hAPP_real_real(cos,X_1) != zero_zero_real
=> ( hAPP_real_real(cos,Y_1) != zero_zero_real
=> ( hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1)) != zero_zero_real
=> hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(tan,X_1)),hAPP_real_real(tan,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(tan,X_1)),hAPP_real_real(tan,Y_1)))) ) ) ) ).
fof(fact_3612_lemma__tan__add1,axiom,
! [Y_1,X_1] :
( hAPP_real_real(cos,X_1) != zero_zero_real
=> ( hAPP_real_real(cos,Y_1) != zero_zero_real
=> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(tan,X_1)),hAPP_real_real(tan,Y_1))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(cos,Y_1))) ) ) ).
fof(fact_3613_tan__gt__zero,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(tan,X_1))) ) ) ).
fof(fact_3614_arccos__bounded,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(arccos,Y_1)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arccos,Y_1)),pi)) ) ) ) ).
fof(fact_3615_tan__monotone_H,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_2))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(tan,Y_2)),hAPP_real_real(tan,X_2))) ) ) ) ) ) ).
fof(fact_3616_tan__monotone,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y_1))
=> ( 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),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(tan,Y_1)),hAPP_real_real(tan,X_1))) ) ) ) ).
fof(fact_3617_arccos__lt__bounded,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),one_one_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(arccos,Y_1)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arccos,Y_1)),pi)) ) ) ) ).
fof(fact_3618_tan__inverse,axiom,
! [Y_1] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(tan,Y_1)) = hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Y_1)) ).
fof(fact_3619_tan__30,axiom,
hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) ).
fof(fact_3620_tan__less__zero,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(tan,X_1)),zero_zero_real)) ) ) ).
fof(fact_3621_arctan__tan,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> hAPP_real_real(arctan,hAPP_real_real(tan,X_1)) = X_1 ) ) ).
fof(fact_3622_arctan,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arctan,Y_1)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,Y_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
& hAPP_real_real(tan,hAPP_real_real(arctan,Y_1)) = Y_1 ) ).
fof(fact_3623_arccos,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(arccos,Y_1)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arccos,Y_1)),pi))
& hAPP_real_real(cos,hAPP_real_real(arccos,Y_1)) = Y_1 ) ) ) ).
fof(fact_3624_tan__total__pi4,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real))
=> ? [Z] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),Z))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))
& hAPP_real_real(tan,Z) = X_1 ) ) ).
fof(fact_3625_tan__total__pos,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1))
=> ? [X] :
( 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_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
& hAPP_real_real(tan,X) = Y_1 ) ) ).
fof(fact_3626_tan__total,axiom,
! [Y_1] :
? [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
& hAPP_real_real(tan,X) = Y_1
& ! [Y] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
& hAPP_real_real(tan,Y) = Y_1 )
=> Y = X ) ) ).
fof(fact_3627_lemma__tan__total1,axiom,
! [Y_1] :
? [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
& hAPP_real_real(tan,X) = Y_1 ) ).
fof(fact_3628_lemma__tan__total,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1))
=> ? [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),hAPP_real_real(tan,X))) ) ) ).
fof(fact_3629_tan__half,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> hAPP_real_real(tan,X_1) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1))),one_one_real)) ) ) ).
fof(fact_3630_cos__pi__eq__zero,axiom,
! [M] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real ).
fof(fact_3631_sin__arccos,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real))
=> hAPP_real_real(sin,hAPP_real_real(arccos,X_1)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).
fof(fact_3632_real__of__nat__ge__zero,axiom,
! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(real_nat,N_1))) ).
fof(fact_3633_sin__minus,axiom,
! [X_1] : hAPP_real_real(sin,hAPP_real_real(uminus_uminus_real,X_1)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1)) ).
fof(fact_3634_abs__real__of__nat__cancel,axiom,
! [X_1] : hAPP_real_real(abs_abs_real,hAPP_nat_real(real_nat,X_1)) = hAPP_nat_real(real_nat,X_1) ).
fof(fact_3635_real__of__nat__inject,axiom,
! [N,Ma] :
( hAPP_nat_real(real_nat,N) = hAPP_nat_real(real_nat,Ma)
<=> N = Ma ) ).
fof(fact_3636_sin__zero,axiom,
hAPP_real_real(sin,zero_zero_real) = zero_zero_real ).
fof(fact_3637_natfloor__real__of__nat,axiom,
! [N_1] : natfloor(hAPP_nat_real(real_nat,N_1)) = N_1 ).
fof(fact_3638_natceiling__real__of__nat,axiom,
! [N_1] : natceiling(hAPP_nat_real(real_nat,N_1)) = N_1 ).
fof(fact_3639_sin__npi2,axiom,
! [N_1] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,N_1))) = zero_zero_real ).
fof(fact_3640_sin__npi,axiom,
! [N_1] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),pi)) = zero_zero_real ).
fof(fact_3641_real__of__nat__def,axiom,
real_nat = semiri132038758t_real ).
fof(fact_3642_real__eq__of__nat,axiom,
real_nat = semiri132038758t_real ).
fof(fact_3643_sin__le__one,axiom,
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sin,X_1)),one_one_real)) ).
fof(fact_3644_real__of__nat__zero__iff,axiom,
! [N] :
( hAPP_nat_real(real_nat,N) = zero_zero_real
<=> N = zero_zero_nat ) ).
fof(fact_3645_real__of__nat__zero,axiom,
hAPP_nat_real(real_nat,zero_zero_nat) = zero_zero_real ).
fof(fact_3646_not__real__of__nat__less__zero,axiom,
! [N_1] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),zero_zero_real)) ).
fof(fact_3647_real__of__nat__less__iff,axiom,
! [N,Ma] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(real_nat,Ma)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),Ma)) ) ).
fof(fact_3648_sin__pi,axiom,
hAPP_real_real(sin,pi) = zero_zero_real ).
fof(fact_3649_real__of__nat__le__iff,axiom,
! [N,Ma] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(real_nat,Ma)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),Ma)) ) ).
fof(fact_3650_real__of__nat__mult,axiom,
! [M,N_1] : hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N_1)) ).
fof(fact_3651_real__of__nat__add,axiom,
! [M,N_1] : hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N_1)) ).
fof(fact_3652_real__of__nat__1,axiom,
hAPP_nat_real(real_nat,one_one_nat) = one_one_real ).
fof(fact_3653_power__real__of__nat,axiom,
! [M,N_1] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(real_nat,M)),N_1) = hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N_1)) ).
fof(fact_3654_real__of__nat__power,axiom,
! [M,N_1] : hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N_1)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(real_nat,M)),N_1) ).
fof(fact_3655_real__natceiling__ge,axiom,
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_nat_real(real_nat,natceiling(X_1)))) ).
fof(fact_3656_sin__ge__zero,axiom,
! [X_1] :
( 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,X_1),pi))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sin,X_1))) ) ) ).
fof(fact_3657_sin__gt__zero__pi,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),pi))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sin,X_1))) ) ) ).
fof(fact_3658_real__of__nat__le__zero__cancel__iff,axiom,
! [N] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N)),zero_zero_real))
<=> N = zero_zero_nat ) ).
fof(fact_3659_sin__ge__minus__one,axiom,
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),hAPP_real_real(sin,X_1))) ).
fof(fact_3660_real__of__nat__Suc__gt__zero,axiom,
! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1)))) ).
fof(fact_3661_cos__one__sin__zero,axiom,
! [X_1] :
( hAPP_real_real(cos,X_1) = one_one_real
=> hAPP_real_real(sin,X_1) = zero_zero_real ) ).
fof(fact_3662_abs__sin__le__one,axiom,
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(sin,X_1))),one_one_real)) ).
fof(fact_3663_real__of__nat__one,axiom,
hAPP_nat_real(real_nat,hAPP_nat_nat(suc,zero_zero_nat)) = one_one_real ).
fof(fact_3664_sin__add,axiom,
! [X_1,Y_1] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X_1)),hAPP_real_real(cos,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(sin,Y_1))) ).
fof(fact_3665_sin__diff2,axiom,
! [X_1,Y_1] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,Y_1)),hAPP_real_real(sin,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,Y_1)),hAPP_real_real(cos,X_1))) ).
fof(fact_3666_sin__diff,axiom,
! [X_1,Y_1] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X_1)),hAPP_real_real(cos,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(sin,Y_1))) ).
fof(fact_3667_real__of__nat__Suc,axiom,
! [N_1] : hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N_1)),one_one_real) ).
fof(fact_3668_sin__periodic__pi,axiom,
! [X_1] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),pi)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1)) ).
fof(fact_3669_sin__periodic__pi2,axiom,
! [X_1] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,pi),X_1)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1)) ).
fof(fact_3670_real__of__nat__diff,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N_1)) ) ).
fof(fact_3671_real__of__nat__div4,axiom,
! [N_1,X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),X_1))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(real_nat,X_1)))) ).
fof(fact_3672_tan__def,axiom,
! [X_1] : hAPP_real_real(tan,X_1) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,X_1)),hAPP_real_real(cos,X_1)) ).
fof(fact_3673_real__natfloor__le,axiom,
! [X_1] :
( 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,hAPP_nat_real(real_nat,natfloor(X_1))),X_1)) ) ).
fof(fact_3674_le__natfloor,axiom,
! [X_1,A_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,X_1)),A_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),natfloor(A_3))) ) ).
fof(fact_3675_natceiling__le,axiom,
! [X_1,A_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_nat_real(real_nat,A_3)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_1)),A_3)) ) ).
fof(fact_3676_sin__2npi,axiom,
! [N_1] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(real_nat,N_1))),pi)) = zero_zero_real ).
fof(fact_3677_natfloor__power,axiom,
! [N_1,X_1] :
( X_1 = hAPP_nat_real(real_nat,natfloor(X_1))
=> natfloor(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,natfloor(X_1)),N_1) ) ).
fof(fact_3678_real__of__nat__gt__zero__cancel__iff,axiom,
! [N] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
fof(fact_3679_sin__cos__squared__add3,axiom,
! [X_1] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(cos,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X_1)),hAPP_real_real(sin,X_1))) = one_one_real ).
fof(fact_3680_sin__zero__abs__cos__one,axiom,
! [X_1] :
( hAPP_real_real(sin,X_1) = zero_zero_real
=> hAPP_real_real(abs_abs_real,hAPP_real_real(cos,X_1)) = one_one_real ) ).
fof(fact_3681_cos__diff2,axiom,
! [X_1,Y_1] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,Y_1)),hAPP_real_real(cos,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,Y_1)),hAPP_real_real(sin,X_1))) ).
fof(fact_3682_cos__diff,axiom,
! [X_1,Y_1] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(cos,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X_1)),hAPP_real_real(sin,Y_1))) ).
fof(fact_3683_cos__add,axiom,
! [X_1,Y_1] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(cos,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X_1)),hAPP_real_real(sin,Y_1))) ).
fof(fact_3684_nat__less__real__le,axiom,
! [N,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),Ma))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N)),one_one_real)),hAPP_nat_real(real_nat,Ma))) ) ).
fof(fact_3685_nat__le__real__less,axiom,
! [N,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),Ma))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,Ma)),one_one_real))) ) ).
fof(fact_3686_le__natfloor__eq,axiom,
! [A,X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),natfloor(X_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A)),X_2)) ) ) ).
fof(fact_3687_tan__npi,axiom,
! [N_1] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),pi)) = zero_zero_real ).
fof(fact_3688_natceiling__le__eq,axiom,
! [A,X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_2)),A))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),hAPP_nat_real(real_nat,A))) ) ) ).
fof(fact_3689_real__natfloor__add__one__gt,axiom,
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,natfloor(X_1))),one_one_real))) ).
fof(fact_3690_tan__periodic__nat,axiom,
! [X_1,N_1] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),pi))) = hAPP_real_real(tan,X_1) ).
fof(fact_3691_natfloor__subtract,axiom,
! [A_3,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A_3)),X_1))
=> natfloor(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),hAPP_nat_real(real_nat,A_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,natfloor(X_1)),A_3) ) ).
fof(fact_3692_real__natfloor__gt__diff__one,axiom,
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),one_one_real)),hAPP_nat_real(real_nat,natfloor(X_1)))) ).
fof(fact_3693_natceiling__subtract,axiom,
! [A_3,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A_3)),X_1))
=> natceiling(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),hAPP_nat_real(real_nat,A_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,natceiling(X_1)),A_3) ) ).
fof(fact_3694_sin__expansion__lemma,axiom,
! [X_1,M] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,M))),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,M)),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_3695_real__of__nat__div,axiom,
! [N_1,D_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),D_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),N_1))
=> hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),D_3)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(real_nat,D_3)) ) ) ).
fof(fact_3696_ln__realpow,axiom,
! [N_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hAPP_real_real(ln,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_real_real(ln,X_1)) ) ).
fof(fact_3697_less__natfloor,axiom,
! [N_1,X_1] :
( 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_real,X_1),hAPP_nat_real(real_nat,N_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,natfloor(X_1)),N_1)) ) ) ).
fof(fact_3698_real__of__nat__div2,axiom,
! [N_1,X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(real_nat,X_1))),hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),X_1))))) ).
fof(fact_3699_real__of__nat__number__of,axiom,
! [V_2] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> hAPP_nat_real(real_nat,hAPP_int_nat(number_number_of_nat,V_2)) = zero_zero_real )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> hAPP_nat_real(real_nat,hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_real(number267125858f_real,V_2) ) ) ).
fof(fact_3700_real__of__nat__div3,axiom,
! [N_1,X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(real_nat,X_1))),hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),X_1)))),one_one_real)) ).
fof(fact_3701_natfloor__add,axiom,
! [A_3,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_nat_real(real_nat,A_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X_1)),A_3) ) ).
fof(fact_3702_sin__arcsin,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real))
=> hAPP_real_real(sin,hAPP_real_real(arcsin,Y_1)) = Y_1 ) ) ).
fof(fact_3703_ge__natfloor__plus__one__imp__gt,axiom,
! [Z_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_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))) ) ).
fof(fact_3704_natfloor__eq,axiom,
! [N_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_1)),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N_1)),one_one_real)))
=> natfloor(X_1) = N_1 ) ) ).
fof(fact_3705_natceiling__add,axiom,
! [A_3,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_nat_real(real_nat,A_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X_1)),A_3) ) ).
fof(fact_3706_cos__expansion__lemma,axiom,
! [X_1,M] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,M))),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,M)),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).
fof(fact_3707_sin__gt__zero,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sin,X_1))) ) ) ).
fof(fact_3708_sin__cos__npi,axiom,
! [N_1] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)))),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),N_1) ).
fof(fact_3709_sin__double,axiom,
! [X_1] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(sin,X_1))),hAPP_real_real(cos,X_1)) ).
fof(fact_3710_two__realpow__gt,axiom,
! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1))) ).
fof(fact_3711_LIMSEQ__inverse__realpow__zero__lemma,axiom,
! [N_1,X_1] :
( 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,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),X_1)),one_one_real)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),one_one_real)),N_1))) ) ).
fof(fact_3712_add__tan__eq,axiom,
! [Y_1,X_1] :
( hAPP_real_real(cos,X_1) != zero_zero_real
=> ( hAPP_real_real(cos,Y_1) != zero_zero_real
=> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(tan,X_1)),hAPP_real_real(tan,Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(cos,Y_1))) ) ) ).
fof(fact_3713_real__of__nat__div__aux,axiom,
! [X_1,D_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),D_3))
=> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,X_1)),hAPP_nat_real(real_nat,D_3)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_1),D_3))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,hAPP_nat_nat(div_mod_nat(X_1),D_3))),hAPP_nat_real(real_nat,D_3))) ) ).
fof(fact_3714_cos__npi,axiom,
! [N_1] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),pi)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),N_1) ).
fof(fact_3715_cos__npi2,axiom,
! [N_1] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,N_1))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),N_1) ).
fof(fact_3716_sin__two__pi,axiom,
hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi)) = zero_zero_real ).
fof(fact_3717_sin__periodic,axiom,
! [X_1] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi))) = hAPP_real_real(sin,X_1) ).
fof(fact_3718_sin__pi__half,axiom,
hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_real ).
fof(fact_3719_sin__30,axiom,
hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_3720_sin__45,axiom,
hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_3721_sin__60,axiom,
hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_3722_fact__lemma,axiom,
! [N_1] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N_1)) ).
fof(fact_3723_natfloor__div__nat,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Y_1))
=> natfloor(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),hAPP_nat_real(real_nat,Y_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,natfloor(X_1)),Y_1) ) ) ).
fof(fact_3724_natceiling__eq,axiom,
! [N_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N_1)),one_one_real)))
=> natceiling(X_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat) ) ) ).
fof(fact_3725_sin__gt__zero2,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sin,X_1))) ) ) ).
fof(fact_3726_sin__cos__squared__add2__mult,axiom,
! [R_3,A_3] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(cos,A_3))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(sin,A_3))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_3727_sin__cos__squared__add,axiom,
! [X_1] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_real ).
fof(fact_3728_sin__cos__squared__add2,axiom,
! [X_1] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_real ).
fof(fact_3729_sin__squared__eq,axiom,
! [X_1] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_3730_cos__squared__eq,axiom,
! [X_1] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_3731_sin__monotone__2pi_H,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y_1))
=> ( 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),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sin,Y_1)),hAPP_real_real(sin,X_1))) ) ) ) ).
fof(fact_3732_cos__sin__eq,axiom,
! [X_1] : hAPP_real_real(cos,X_1) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_1)) ).
fof(fact_3733_sin__cos__eq,axiom,
! [X_1] : hAPP_real_real(sin,X_1) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_1)) ).
fof(fact_3734_cos__2npi,axiom,
! [N_1] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(real_nat,N_1))),pi)) = one_one_real ).
fof(fact_3735_sin__less__zero,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sin,X_1)),zero_zero_real)) ) ) ).
fof(fact_3736_cos__double,axiom,
! [X_1] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_3737_sin__3over2__pi,axiom,
hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),pi)) = hAPP_real_real(uminus_uminus_real,one_one_real) ).
fof(fact_3738_minus__sin__cos__eq,axiom,
! [X_1] : hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1)) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_3739_arcsin__sin,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> hAPP_real_real(arcsin,hAPP_real_real(sin,X_1)) = X_1 ) ) ).
fof(fact_3740_sin__cos__add,axiom,
! [X_1,Y_1] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X_1)),hAPP_real_real(cos,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(sin,Y_1))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_1))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(cos,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X_1)),hAPP_real_real(sin,Y_1))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real ).
fof(fact_3741_sin__cos__minus,axiom,
! [X_1] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(sin,hAPP_real_real(uminus_uminus_real,X_1))),hAPP_real_real(sin,X_1))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(cos,hAPP_real_real(uminus_uminus_real,X_1))),hAPP_real_real(cos,X_1))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real ).
fof(fact_3742_sin__arccos__abs,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y_1)),one_one_real))
=> hAPP_real_real(sin,hAPP_real_real(arccos,Y_1)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).
fof(fact_3743_arcsin,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y_1)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
& hAPP_real_real(sin,hAPP_real_real(arcsin,Y_1)) = Y_1 ) ) ) ).
fof(fact_3744_arcsin__pi,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y_1)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y_1)),pi))
& hAPP_real_real(sin,hAPP_real_real(arcsin,Y_1)) = Y_1 ) ) ) ).
fof(fact_3745_sin__arccos__lemma1,axiom,
! [X_4,Y_4] : hAPP_real_real(sin,hAPP_real_real(arccos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_3746_sin__total,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),one_one_real))
=> ? [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
& hAPP_real_real(sin,X) = Y_1
& ! [Y] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
& hAPP_real_real(sin,Y) = Y_1 )
=> Y = X ) ) ) ) ).
fof(fact_3747_reals__Archimedean6,axiom,
! [R_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_3))
=> ? [N_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_2),one_one_nat))),R_3))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_nat_real(real_nat,N_2))) ) ) ).
fof(fact_3748_reals__Archimedean4,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> ? [N_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),Y_1)),X_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_2))),Y_1))) ) ) ) ).
fof(fact_3749_reals__Archimedean6a,axiom,
! [R_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_3))
=> ? [N_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_2)),R_3))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_2)))) ) ) ).
fof(fact_3750_reals__Archimedean3,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ! [Y] :
? [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),X_1))) ) ).
fof(fact_3751_lemma__STAR__cos,axiom,
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N_1)) = zero_zero_real ) ).
fof(fact_3752_sin__zero__iff,axiom,
! [X_2] :
( hAPP_real_real(sin,X_2) = zero_zero_real
<=> ( ? [N_2] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))
& X_2 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) )
| ? [N_2] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))
& X_2 = hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ) ).
fof(fact_3753_cos__zero__iff,axiom,
! [X_2] :
( hAPP_real_real(cos,X_2) = zero_zero_real
<=> ( ? [N_2] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))
& X_2 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) )
| ? [N_2] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))
& X_2 = hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ) ).
fof(fact_3754_odd__add,axiom,
! [Ma,N] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N)))
<=> ~ ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Ma))
<=> ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) ) ) ).
fof(fact_3755_even__add,axiom,
! [Ma,N] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N)))
<=> ( hBOOL(hAPP_nat_bool(even_odd_even_nat,Ma))
<=> hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) ) ) ).
fof(fact_3756_even__sum__nat,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Y_2)))
<=> ( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
& hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) )
| ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
& ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) ) ) ) ).
fof(fact_3757_even__zero__nat,axiom,
hBOOL(hAPP_nat_bool(even_odd_even_nat,zero_zero_nat)) ).
fof(fact_3758_even__Suc,axiom,
! [X_2] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(suc,X_2)))
<=> ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)) ) ).
fof(fact_3759_odd__1__nat,axiom,
~ hBOOL(hAPP_nat_bool(even_odd_even_nat,one_one_nat)) ).
fof(fact_3760_even__product__nat,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_2),Y_2)))
<=> ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
| hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) ) ) ).
fof(fact_3761_real__of__nat__fact__not__zero,axiom,
! [N_1] : hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)) != zero_zero_real ).
fof(fact_3762_odd__pos,axiom,
! [N_1] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).
fof(fact_3763_even__difference__nat,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_2),Y_2)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2))
| ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
& hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) )
| ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
& ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) ) ) ) ).
fof(fact_3764_Parity_Otransfer__int__nat__relations,axiom,
! [X_2] :
( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(semiri1621563631at_int,X_2)))
<=> hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)) ) ).
fof(fact_3765_even__nat__def,axiom,
! [X_2] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
<=> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(semiri1621563631at_int,X_2))) ) ).
fof(fact_3766_real__of__nat__fact__ge__zero,axiom,
! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))) ).
fof(fact_3767_real__of__nat__fact__gt__zero,axiom,
! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))) ).
fof(fact_3768_even__nat__equiv__def2,axiom,
! [X_2] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
<=> ? [Y] : X_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),Y) ) ).
fof(fact_3769_odd__nat__equiv__def2,axiom,
! [X_2] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
<=> ? [Y] : X_2 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),Y)) ) ).
fof(fact_3770_odd__nat__plus__one__div__two,axiom,
! [X_1] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,X_1)),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) ) ).
fof(fact_3771_even__nat__plus__one__div__two,axiom,
! [X_1] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,X_1)),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) ) ).
fof(fact_3772_odd__nat__mod__two__eq__one,axiom,
! [X_1] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
=> hAPP_nat_nat(div_mod_nat(X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(suc,zero_zero_nat) ) ).
fof(fact_3773_even__nat__mod__two__eq__zero,axiom,
! [X_1] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
=> hAPP_nat_nat(div_mod_nat(X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = zero_zero_nat ) ).
fof(fact_3774_odd__nat__equiv__def,axiom,
! [X_2] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
<=> hAPP_nat_nat(div_mod_nat(X_2),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(suc,zero_zero_nat) ) ).
fof(fact_3775_even__nat__equiv__def,axiom,
! [X_2] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
<=> hAPP_nat_nat(div_mod_nat(X_2),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = zero_zero_nat ) ).
fof(fact_3776_even__power__nat,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_2),Y_2)))
<=> ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Y_2)) ) ) ).
fof(fact_3777_fact__diff__Suc,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,M)))
=> hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N_1)),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1))) ) ).
fof(fact_3778_even__num__iff,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
<=> ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).
fof(fact_3779_even__nat__div__two__times__two,axiom,
! [X_1] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) = X_1 ) ).
fof(fact_3780_odd__nat__div__two__times__two__plus__one,axiom,
! [X_1] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
=> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) = X_1 ) ).
fof(fact_3781_pos__int__even__equiv__nat__even,axiom,
! [X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> ( hBOOL(hAPP_int_bool(even_odd_even_int,X_2))
<=> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_int_nat(nat,X_2))) ) ) ).
fof(fact_3782_nat__even__iff__2__dvd,axiom,
! [X_2] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2)) ) ).
fof(fact_3783_even__dvd,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)) ) ).
fof(fact_3784_even__mult__two__ex,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
<=> ? [M_1] : N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M_1) ) ).
fof(fact_3785_even__even__mod__4__iff,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
<=> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(div_mod_nat(N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).
fof(fact_3786_odd__Suc__mult__two__ex,axiom,
! [N] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
<=> ? [M_1] : N = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M_1)) ) ).
fof(fact_3787_lemma__even__div2,axiom,
! [N_1] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) ).
fof(fact_3788_lemma__STAR__sin,axiom,
! [N] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,hAPP_nat_bool(even_odd_even_nat,N)),zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N)) = zero_zero_real ).
fof(fact_3789_lemma__not__even__div2,axiom,
! [N_1] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) ).
fof(fact_3790_lemma__odd__mod__4__div__2,axiom,
! [N_1] :
( hAPP_nat_nat(div_mod_nat(N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))
=> ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).
fof(fact_3791_lemma__even__mod__4__div__2,axiom,
! [N_1] :
( hAPP_nat_nat(div_mod_nat(N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_nat
=> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).
fof(fact_3792_sin__zero__lemma,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> ( hAPP_real_real(sin,X_1) = zero_zero_real
=> ? [N_2] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))
& X_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) ) ) ).
fof(fact_3793_cos__zero__lemma,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> ( hAPP_real_real(cos,X_1) = zero_zero_real
=> ? [N_2] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))
& X_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) ) ) ).
fof(fact_3794_odd__square,axiom,
! [N_1] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
=> ? [X] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X)),one_one_nat) ) ).
fof(fact_3795_sin__coeff__def,axiom,
! [X] :
( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
=> hAPP_nat_real(sin_coeff,X) = zero_zero_real )
& ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
=> hAPP_nat_real(sin_coeff,X) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X),hAPP_nat_nat(suc,zero_zero_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,X))) ) ) ).
fof(fact_3796_even__square,axiom,
! [N_1] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
=> ? [X] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X) ) ).
fof(fact_3797_cos__coeff__def,axiom,
! [X] :
( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
=> hAPP_nat_real(cos_coeff,X) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,X))) )
& ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
=> hAPP_nat_real(cos_coeff,X) = zero_zero_real ) ) ).
fof(fact_3798_fact__add__num__eq__if2__nat,axiom,
! [N_1,M] :
( ( M = zero_zero_nat
=> hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)) = hAPP_nat_nat(fact_fact_nat,N_1) )
& ( M != zero_zero_nat
=> hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N_1))) ) ) ).
fof(fact_3799_fact__add__num__eq__if__nat,axiom,
! [M,N_1] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1) = zero_zero_nat
=> hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)) = one_one_nat )
& ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1) != zero_zero_nat
=> hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)),one_one_nat))) ) ) ).
fof(fact_3800_fact__nonzero__nat,axiom,
! [N_1] : hAPP_nat_nat(fact_fact_nat,N_1) != zero_zero_nat ).
fof(fact_3801_fact__mono__nat,axiom,
! [M,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,hAPP_nat_nat(fact_fact_nat,M)),hAPP_nat_nat(fact_fact_nat,N_1))) ) ).
fof(fact_3802_fact__1__nat,axiom,
hAPP_nat_nat(fact_fact_nat,one_one_nat) = one_one_nat ).
fof(fact_3803_Fact_Ofact__0__nat,axiom,
hAPP_nat_nat(fact_fact_nat,zero_zero_nat) = hAPP_nat_nat(suc,zero_zero_nat) ).
fof(fact_3804_fact__Suc__0__nat,axiom,
hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(suc,zero_zero_nat)) = hAPP_nat_nat(suc,zero_zero_nat) ).
fof(fact_3805_fact__less__mono__nat,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(fact_fact_nat,M)),hAPP_nat_nat(fact_fact_nat,N_1))) ) ) ).
fof(fact_3806_fact__gt__zero__nat,axiom,
! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(fact_fact_nat,N_1))) ).
fof(fact_3807_Fact_Ofact__Suc,axiom,
! [X_1] : hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(suc,X_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,X_1)),hAPP_nat_nat(fact_fact_nat,X_1)) ).
fof(fact_3808_fact__dvd,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(fact_fact_nat,N_1)),hAPP_nat_nat(fact_fact_nat,M))) ) ).
fof(fact_3809_fact__ge__one__nat,axiom,
! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(fact_fact_nat,N_1))) ).
fof(fact_3810_fact__ge__Suc__0__nat,axiom,
! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(fact_fact_nat,N_1))) ).
fof(fact_3811_dvd__fact__nat,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),hAPP_nat_nat(fact_fact_nat,N_1))) ) ) ).
fof(fact_3812_fact__plus__one__nat,axiom,
! [N_1] : hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),hAPP_nat_nat(fact_fact_nat,N_1)) ).
fof(fact_3813_fact__mod,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(fact_fact_nat,N_1)),hAPP_nat_nat(fact_fact_nat,M)) = zero_zero_nat ) ).
fof(fact_3814_fact__num__eq__if__nat,axiom,
! [M] :
( ( M = zero_zero_nat
=> hAPP_nat_nat(fact_fact_nat,M) = one_one_nat )
& ( M != zero_zero_nat
=> hAPP_nat_nat(fact_fact_nat,M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ).
fof(fact_3815_fact__reduce__nat,axiom,
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hAPP_nat_nat(fact_fact_nat,N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat))) ) ).
fof(fact_3816_Ln_Oaux1,axiom,
! [N_1,X_1] :
( 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,X_1),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N_1)))) ) ) ).
fof(fact_3817_aux4,axiom,
! [X_1] :
( 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,X_1),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1))) ) ) ).
fof(fact_3818_exp__less__mono,axiom,
! [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,hAPP_real_real(exp_real,X_1)),hAPP_real_real(exp_real,Y_1))) ) ).
fof(fact_3819_ln__unique,axiom,
! [Y_1,X_1] :
( hAPP_real_real(exp_real,Y_1) = X_1
=> hAPP_real_real(ln,X_1) = Y_1 ) ).
fof(fact_3820_exp__ln__eq,axiom,
! [U_1,X_1] :
( hAPP_real_real(exp_real,U_1) = X_1
=> hAPP_real_real(ln,X_1) = U_1 ) ).
fof(fact_3821_ln__exp,axiom,
! [X_1] : hAPP_real_real(ln,hAPP_real_real(exp_real,X_1)) = X_1 ).
fof(fact_3822_exp__le__cancel__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X_2)),hAPP_real_real(exp_real,Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)) ) ).
fof(fact_3823_exp__less__cancel,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X_1)),hAPP_real_real(exp_real,Y_1)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1)) ) ).
fof(fact_3824_exp__less__cancel__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X_2)),hAPP_real_real(exp_real,Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) ) ).
fof(fact_3825_INVERSE__ZERO,axiom,
hAPP_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).
fof(fact_3826_exp__inj__iff,axiom,
! [X_2,Y_2] :
( hAPP_real_real(exp_real,X_2) = hAPP_real_real(exp_real,Y_2)
<=> X_2 = Y_2 ) ).
fof(fact_3827_abs__exp__cancel,axiom,
! [X_1] : hAPP_real_real(abs_abs_real,hAPP_real_real(exp_real,X_1)) = hAPP_real_real(exp_real,X_1) ).
fof(fact_3828_real__sqrt__inverse,axiom,
! [X_1] : hAPP_real_real(sqrt,hAPP_real_real(inverse_inverse_real,X_1)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_1)) ).
fof(fact_3829_real__divide__def,axiom,
! [R,S_4] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,R),S_4) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R),hAPP_real_real(inverse_inverse_real,S_4)) ).
fof(fact_3830_divide__real__def,axiom,
! [X_1,Y_1] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),Y_1) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),hAPP_real_real(inverse_inverse_real,Y_1)) ).
fof(fact_3831_not__exp__le__zero,axiom,
! [X_1] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X_1)),zero_zero_real)) ).
fof(fact_3832_exp__ge__zero,axiom,
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(exp_real,X_1))) ).
fof(fact_3833_not__exp__less__zero,axiom,
! [X_1] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X_1)),zero_zero_real)) ).
fof(fact_3834_exp__gt__zero,axiom,
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(exp_real,X_1))) ).
fof(fact_3835_exp__eq__one__iff,axiom,
! [X_2] :
( hAPP_real_real(exp_real,X_2) = one_one_real
<=> X_2 = zero_zero_real ) ).
fof(fact_3836_real__mult__inverse__cancel,axiom,
! [Y_1,U_1,X1_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X1_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X1_1),Y_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),U_1)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X_1)),Y_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X1_1)),U_1))) ) ) ) ).
fof(fact_3837_real__mult__inverse__cancel2,axiom,
! [Y_1,U_1,X1_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X1_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X1_1),Y_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),U_1)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_1),hAPP_real_real(inverse_inverse_real,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_1),hAPP_real_real(inverse_inverse_real,X1_1)))) ) ) ) ).
fof(fact_3838_real__mult__inverse__left,axiom,
! [X_1] :
( X_1 != zero_zero_real
=> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X_1)),X_1) = one_one_real ) ).
fof(fact_3839_one__le__exp__iff,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(exp_real,X_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2)) ) ).
fof(fact_3840_exp__le__one__iff,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X_2)),one_one_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),zero_zero_real)) ) ).
fof(fact_3841_one__less__exp__iff,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(exp_real,X_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) ) ).
fof(fact_3842_exp__less__one__iff,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X_2)),one_one_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),zero_zero_real)) ) ).
fof(fact_3843_exp__gt__one,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(exp_real,X_1))) ) ).
fof(fact_3844_exp__ge__add__one__self,axiom,
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1)),hAPP_real_real(exp_real,X_1))) ).
fof(fact_3845_exp__real__of__nat__mult,axiom,
! [N_1,X_1] : hAPP_real_real(exp_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),X_1)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(exp_real,X_1)),N_1) ).
fof(fact_3846_exp__ln,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hAPP_real_real(exp_real,hAPP_real_real(ln,X_1)) = X_1 ) ).
fof(fact_3847_exp__ln__iff,axiom,
! [X_2] :
( hAPP_real_real(exp_real,hAPP_real_real(ln,X_2)) = X_2
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) ) ).
fof(fact_3848_inv__real__of__nat__fact__ge__zero,axiom,
! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1))))) ).
fof(fact_3849_sqrt__divide__self__eq,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,X_1)),X_1) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_1)) ) ).
fof(fact_3850_inv__real__of__nat__fact__gt__zero,axiom,
! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1))))) ).
fof(fact_3851_ln__inverse,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hAPP_real_real(ln,hAPP_real_real(inverse_inverse_real,X_1)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(ln,X_1)) ) ).
fof(fact_3852_exp__ge__add__one__self__aux,axiom,
! [X_1] :
( 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,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1)),hAPP_real_real(exp_real,X_1))) ) ).
fof(fact_3853_ln__x__over__x__mono,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,one_one_real)),X_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,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,Y_1)),Y_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,X_1)),X_1))) ) ) ).
fof(fact_3854_real__inv__sqrt__pow2,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_1))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(inverse_inverse_real,X_1) ) ).
fof(fact_3855_exp__bound,axiom,
! [X_1] :
( 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,X_1),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ) ).
fof(fact_3856_tan__sec,axiom,
! [X_1] :
( hAPP_real_real(cos,X_1) != zero_zero_real
=> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(tan,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(cos,X_1))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) ).
fof(fact_3857_lemma__exp__total,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Y_1))
=> ? [X] :
( 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,X),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_1),one_one_real)))
& hAPP_real_real(exp_real,X) = Y_1 ) ) ).
fof(fact_3858_reals__Archimedean,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ? [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_2)))),X_1)) ) ).
fof(fact_3859_log__base__10__eq1,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hAPP_real_real(log(hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_1) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,hAPP_real_real(exp_real,one_one_real))),hAPP_real_real(ln,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_real_real(ln,X_1)) ) ).
fof(fact_3860_DERIV__arccos,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),one_one_real))
=> hBOOL(hAPP_real_bool(deriv_real(arccos,X_2),hAPP_real_real(inverse_inverse_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) ) ) ).
fof(fact_3861_DERIV__sin,axiom,
! [X_2] : hBOOL(hAPP_real_bool(deriv_real(sin,X_2),hAPP_real_real(cos,X_2))) ).
fof(fact_3862_log__one,axiom,
! [A_3] : hAPP_real_real(log(A_3),one_one_real) = zero_zero_real ).
fof(fact_3863_Log_Olog__def,axiom,
! [A_3,X_1] : hAPP_real_real(log(A_3),X_1) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,X_1)),hAPP_real_real(ln,A_3)) ).
fof(fact_3864_DERIV__cos,axiom,
! [X_2] : hBOOL(hAPP_real_bool(deriv_real(cos,X_2),hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_2)))) ).
fof(fact_3865_DERIV__ln,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> hBOOL(hAPP_real_bool(deriv_real(ln,X_2),hAPP_real_real(inverse_inverse_real,X_2))) ) ).
fof(fact_3866_log__less__cancel__iff,axiom,
! [Y_2,X_2,A] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(log(A),X_2)),hAPP_real_real(log(A),Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) ) ) ) ) ).
fof(fact_3867_log__eq__one,axiom,
! [A_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
=> ( A_3 != one_one_real
=> hAPP_real_real(log(A_3),A_3) = one_one_real ) ) ).
fof(fact_3868_log__ln,axiom,
! [X_1] : hAPP_real_real(ln,X_1) = hAPP_real_real(log(hAPP_real_real(exp_real,one_one_real)),X_1) ).
fof(fact_3869_DERIV__ln__divide,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> hBOOL(hAPP_real_bool(deriv_real(ln,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),X_2))) ) ).
fof(fact_3870_log__le__cancel__iff,axiom,
! [Y_2,X_2,A] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(log(A),X_2)),hAPP_real_real(log(A),Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)) ) ) ) ) ).
fof(fact_3871_log__nat__power,axiom,
! [B_2,N_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hAPP_real_real(log(B_2),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_real_real(log(B_2),X_1)) ) ).
fof(fact_3872_log__mult,axiom,
! [Y_1,X_1,A_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
=> ( A_3 != one_one_real
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1))
=> hAPP_real_real(log(A_3),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(log(A_3),X_1)),hAPP_real_real(log(A_3),Y_1)) ) ) ) ) ).
fof(fact_3873_log__divide,axiom,
! [Y_1,X_1,A_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
=> ( A_3 != one_one_real
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1))
=> hAPP_real_real(log(A_3),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(log(A_3),X_1)),hAPP_real_real(log(A_3),Y_1)) ) ) ) ) ).
fof(fact_3874_log__inverse,axiom,
! [X_1,A_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
=> ( A_3 != one_one_real
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hAPP_real_real(log(A_3),hAPP_real_real(inverse_inverse_real,X_1)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(log(A_3),X_1)) ) ) ) ).
fof(fact_3875_log__eq__div__ln__mult__log,axiom,
! [X_1,B_2,A_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
=> ( A_3 != one_one_real
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_2))
=> ( B_2 != one_one_real
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hAPP_real_real(log(A_3),X_1) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,B_2)),hAPP_real_real(ln,A_3))),hAPP_real_real(log(B_2),X_1)) ) ) ) ) ) ).
fof(fact_3876_DERIV__real__sqrt,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> hBOOL(hAPP_real_bool(deriv_real(sqrt,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_2))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).
fof(fact_3877_DERIV__tan,axiom,
! [X_2] :
( hAPP_real_real(cos,X_2) != zero_zero_real
=> hBOOL(hAPP_real_bool(deriv_real(tan,X_2),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).
fof(fact_3878_DERIV__arctan,axiom,
! [X_2] : hBOOL(hAPP_real_bool(deriv_real(arctan,X_2),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).
fof(fact_3879_DERIV__real__sqrt__generic,axiom,
! [D_1,X_2] :
( X_2 != zero_zero_real
=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> D_1 = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_2))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) )
=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),zero_zero_real))
=> D_1 = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_2)))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) )
=> hBOOL(hAPP_real_bool(deriv_real(sqrt,X_2),D_1)) ) ) ) ).
fof(fact_3880_DERIV__arcsin,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),one_one_real))
=> hBOOL(hAPP_real_bool(deriv_real(arcsin,X_2),hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) ) ) ).
fof(fact_3881_log__base__10__eq2,axiom,
! [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hAPP_real_real(log(hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_1) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(log(hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(exp_real,one_one_real))),hAPP_real_real(ln,X_1)) ) ).
fof(fact_3882_DERIV__const__average,axiom,
! [V,K_1,A,B] :
( A != B
=> ( ! [X] : hBOOL(hAPP_real_bool(deriv_real(V,X),K_1))
=> hAPP_real_real(V,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),B)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(V,A)),hAPP_real_real(V,B))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) ) ).
fof(fact_3883_DERIV__local__max,axiom,
! [D,F,X_2,L] :
( hBOOL(hAPP_real_bool(deriv_real(F,X_2),L))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
=> ( ! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),Y))),D))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,Y)),hAPP_real_real(F,X_2))) )
=> L = zero_zero_real ) ) ) ).
fof(fact_3884_DERIV__local__min,axiom,
! [D,F,X_2,L] :
( hBOOL(hAPP_real_bool(deriv_real(F,X_2),L))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
=> ( ! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),Y))),D))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,X_2)),hAPP_real_real(F,Y))) )
=> L = zero_zero_real ) ) ) ).
fof(fact_3885_DERIV__local__const,axiom,
! [D,F,X_2,L] :
( hBOOL(hAPP_real_bool(deriv_real(F,X_2),L))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
=> ( ! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),Y))),D))
=> hAPP_real_real(F,X_2) = hAPP_real_real(F,Y) )
=> L = zero_zero_real ) ) ) ).
fof(fact_3886_DERIV__pos__inc__left,axiom,
! [F,X_2,L] :
( hBOOL(hAPP_real_bool(deriv_real(F,X_2),L))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),L))
=> ? [D_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
& ! [H_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),H_1))),hAPP_real_real(F,X_2))) ) ) ) ) ) ).
fof(fact_3887_DERIV__neg__dec__left,axiom,
! [F,X_2,L] :
( hBOOL(hAPP_real_bool(deriv_real(F,X_2),L))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,L),zero_zero_real))
=> ? [D_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
& ! [H_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_2)),hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),H_1)))) ) ) ) ) ) ).
fof(fact_3888_DERIV__pos__inc__right,axiom,
! [F,X_2,L] :
( hBOOL(hAPP_real_bool(deriv_real(F,X_2),L))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),L))
=> ? [D_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
& ! [H_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_2)),hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),H_1)))) ) ) ) ) ) ).
fof(fact_3889_DERIV__neg__dec__right,axiom,
! [F,X_2,L] :
( hBOOL(hAPP_real_bool(deriv_real(F,X_2),L))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,L),zero_zero_real))
=> ? [D_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
& ! [H_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),H_1))),hAPP_real_real(F,X_2))) ) ) ) ) ) ).
fof(fact_3890_DERIV__const__ratio__const2,axiom,
! [F,K_1,A,B] :
( A != B
=> ( ! [X] : hBOOL(hAPP_real_bool(deriv_real(F,X),K_1))
=> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)) = K_1 ) ) ).
fof(fact_3891_DERIV__const__ratio__const,axiom,
! [F,K_1,A,B] :
( A != B
=> ( ! [X] : hBOOL(hAPP_real_bool(deriv_real(F,X),K_1))
=> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),K_1) ) ) ).
fof(fact_3892_DERIV__real__root__generic,axiom,
! [D_1,X_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( X_2 != zero_zero_real
=> ( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> D_1 = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))))) ) )
=> ( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),zero_zero_real))
=> D_1 = hAPP_real_real(uminus_uminus_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat)))))) ) )
=> ( ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
=> D_1 = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))))) )
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N),X_2),D_1)) ) ) ) ) ) ).
fof(fact_3893_DERIV__even__real__root,axiom,
! [X_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),zero_zero_real))
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N),X_2),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_nat_real(real_nat,N))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))))))) ) ) ) ).
fof(fact_3894_real__root__zero,axiom,
! [N_1] : hAPP_real_real(hAPP_n546711566l_real(root,N_1),zero_zero_real) = zero_zero_real ).
fof(fact_3895_real__root__Suc__0,axiom,
! [X_1] : hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(suc,zero_zero_nat)),X_1) = X_1 ).
fof(fact_3896_real__root__commute,axiom,
! [X_1,N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hAPP_real_real(hAPP_n546711566l_real(root,M),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)) = hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(hAPP_n546711566l_real(root,M),X_1)) ) ) ).
fof(fact_3897_real__root__eq__iff,axiom,
! [X_2,Y_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hAPP_real_real(hAPP_n546711566l_real(root,N),X_2) = hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2)
<=> X_2 = Y_2 ) ) ).
fof(fact_3898_odd__real__root__power__cancel,axiom,
! [X_1,N_1] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
=> hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1)) = X_1 ) ).
fof(fact_3899_odd__real__root__pow,axiom,
! [X_1,N_1] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),N_1) = X_1 ) ).
fof(fact_3900_odd__real__root__unique,axiom,
! [Y_1,X_1,N_1] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
=> ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),N_1) = X_1
=> hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1) = Y_1 ) ) ).
fof(fact_3901_real__root__le__mono,axiom,
! [X_1,Y_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_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,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_1))) ) ) ).
fof(fact_3902_real__root__le__iff,axiom,
! [X_2,Y_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)) ) ) ).
fof(fact_3903_real__root__eq__0__iff,axiom,
! [X_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hAPP_real_real(hAPP_n546711566l_real(root,N),X_2) = zero_zero_real
<=> X_2 = zero_zero_real ) ) ).
fof(fact_3904_real__root__less__mono,axiom,
! [X_1,Y_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_1))) ) ) ).
fof(fact_3905_real__root__less__iff,axiom,
! [X_2,Y_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) ) ) ).
fof(fact_3906_real__root__mult,axiom,
! [X_1,Y_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_1)) ) ).
fof(fact_3907_real__root__mult__exp,axiom,
! [X_1,N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)),X_1) = hAPP_real_real(hAPP_n546711566l_real(root,M),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)) ) ) ).
fof(fact_3908_real__root__power,axiom,
! [X_1,K_2,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),K_2)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),K_2) ) ).
fof(fact_3909_real__root__eq__1__iff,axiom,
! [X_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hAPP_real_real(hAPP_n546711566l_real(root,N),X_2) = one_one_real
<=> X_2 = one_one_real ) ) ).
fof(fact_3910_real__root__one,axiom,
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hAPP_real_real(hAPP_n546711566l_real(root,N_1),one_one_real) = one_one_real ) ).
fof(fact_3911_real__root__divide,axiom,
! [X_1,Y_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_1)) ) ).
fof(fact_3912_real__root__minus,axiom,
! [X_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(uminus_uminus_real,X_1)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)) ) ).
fof(fact_3913_real__root__abs,axiom,
! [X_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(abs_abs_real,X_1)) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)) ) ).
fof(fact_3914_real__root__inverse,axiom,
! [X_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(inverse_inverse_real,X_1)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)) ) ).
fof(fact_3915_real__root__ge__zero,axiom,
! [X_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( 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),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1))) ) ) ).
fof(fact_3916_real__root__le__0__iff,axiom,
! [X_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),zero_zero_real)) ) ) ).
fof(fact_3917_real__root__ge__0__iff,axiom,
! [Y_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2)) ) ) ).
fof(fact_3918_real__root__gt__zero,axiom,
! [X_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1))) ) ) ).
fof(fact_3919_real__root__lt__0__iff,axiom,
! [X_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),zero_zero_real)) ) ) ).
fof(fact_3920_real__root__gt__0__iff,axiom,
! [Y_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2)) ) ) ).
fof(fact_3921_real__root__decreasing,axiom,
! [X_1,N_5,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_5))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N_5),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1))) ) ) ) ).
fof(fact_3922_real__root__le__1__iff,axiom,
! [X_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),one_one_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),one_one_real)) ) ) ).
fof(fact_3923_real__root__ge__1__iff,axiom,
! [Y_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Y_2)) ) ) ).
fof(fact_3924_real__root__strict__decreasing,axiom,
! [X_1,N_5,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_5))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_5),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1))) ) ) ) ).
fof(fact_3925_real__root__lt__1__iff,axiom,
! [X_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),one_one_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),one_one_real)) ) ) ).
fof(fact_3926_real__root__gt__1__iff,axiom,
! [Y_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Y_2)) ) ) ).
fof(fact_3927_real__root__pos,axiom,
! [N_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(suc,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_nat_nat(suc,N_1))) = X_1 ) ).
fof(fact_3928_real__root__less__mono__lemma,axiom,
! [Y_1,X_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( 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_real,X_1),Y_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_1))) ) ) ) ).
fof(fact_3929_real__root__pos__pos,axiom,
! [X_4,N_6] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_6))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_4))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N_6),X_4))) ) ) ).
fof(fact_3930_real__root__mult__lemma,axiom,
! [Y_1,X_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( 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(hAPP_n546711566l_real(root,N_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_1)) ) ) ) ).
fof(fact_3931_real__root__pos__mult__exp,axiom,
! [X_1,N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)),X_1) = hAPP_real_real(hAPP_n546711566l_real(root,M),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)) ) ) ) ).
fof(fact_3932_real__root__pos__unique,axiom,
! [X_1,Y_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1))
=> ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),N_1) = X_1
=> hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1) = Y_1 ) ) ) ).
fof(fact_3933_real__root__pow__pos2,axiom,
! [X_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),N_1) = X_1 ) ) ).
fof(fact_3934_real__root__pos2,axiom,
! [X_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1)) = X_1 ) ) ).
fof(fact_3935_real__root__pow__pos,axiom,
! [X_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),N_1) = X_1 ) ) ).
fof(fact_3936_real__root__increasing,axiom,
! [X_1,N_5,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_5))
=> ( 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,X_1),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_5),X_1))) ) ) ) ) ).
fof(fact_3937_real__root__strict__increasing,axiom,
! [X_1,N_5,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_5))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_5),X_1))) ) ) ) ) ).
fof(fact_3938_sqrt__def,axiom,
sqrt = hAPP_n546711566l_real(root,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_3939_real__root__inverse__lemma,axiom,
! [X_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(inverse_inverse_real,X_1)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)) ) ) ).
fof(fact_3940_DERIV__odd__real__root,axiom,
! [X_2,N] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
=> ( X_2 != zero_zero_real
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N),X_2),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))))))) ) ) ).
fof(fact_3941_DERIV__real__root,axiom,
! [X_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N),X_2),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))))))) ) ) ).
fof(fact_3942_exp__total,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1))
=> ? [X] : hAPP_real_real(exp_real,X) = Y_1 ) ).
fof(fact_3943_pdivmod__posDivAlg,axiom,
! [K_2,L_2] :
( is_int(L_2)
=> ( ( L_2 = zero_zero_int
=> pdivmod(K_2,L_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),hAPP_int_int(abs_abs_int,K_2)) )
& ( L_2 != zero_zero_int
=> pdivmod(K_2,L_2) = posDivAlg(hAPP_int_int(abs_abs_int,K_2),hAPP_int_int(abs_abs_int,L_2)) ) ) ) ).
fof(fact_3944_MVT2,axiom,
! [F,F_1,A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B))
=> ( ! [X] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B)) )
=> hBOOL(hAPP_real_bool(deriv_real(F,X),hAPP_real_real(F_1,X))) )
=> ? [Z] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),Z))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),B))
& hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),hAPP_real_real(F_1,Z)) ) ) ) ).
fof(fact_3945_pdivmod__def,axiom,
! [K_2,L_2] : pdivmod(K_2,L_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(abs_abs_int,L_2))),hAPP_int_int(div_mod_int(hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(abs_abs_int,L_2))) ).
fof(fact_3946_polar__ex2,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),zero_zero_real))
=> ? [R_1,A_2] :
( X_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_2))
& Y_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_2)) ) ) ).
fof(fact_3947_polar__ex1,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1))
=> ? [R_1,A_2] :
( X_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_2))
& Y_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_2)) ) ) ).
fof(fact_3948_z3mod__def,axiom,
! [K_2,L_2] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2))
=> hAPP_int_int(z3mod(K_2),L_2) = hAPP_int_int(div_mod_int(K_2),L_2) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2))
=> hAPP_int_int(z3mod(K_2),L_2) = hAPP_int_int(div_mod_int(K_2),hAPP_int_int(uminus_uminus_int,L_2)) ) ) ).
fof(fact_3949_z3div__def,axiom,
! [K_2,L_2] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2))
=> hAPP_int_int(z3div(K_2),L_2) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),L_2) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2))
=> hAPP_int_int(z3div(K_2),L_2) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),hAPP_int_int(uminus_uminus_int,L_2))) ) ) ).
fof(fact_3950_DERIV__isconst__all,axiom,
! [X_2,Y_2,F] :
( ! [X] : hBOOL(hAPP_real_bool(deriv_real(F,X),zero_zero_real))
=> hAPP_real_real(F,X_2) = hAPP_real_real(F,Y_2) ) ).
fof(fact_3951_reals__Archimedean2,axiom,
! [X_1] :
? [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_nat_real(real_nat,N_2))) ).
fof(fact_3952_le__Suc__ex,axiom,
! [K_2,L_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L_2))
=> ? [N_2] : L_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N_2) ) ).
fof(fact_3953_complex__norm,axiom,
! [X_1,Y_1] : hAPP_complex_real(norm_norm_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,X_1),Y_1)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_3954_exists__least__lemma,axiom,
! [P_1] :
( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
=> ( ? [X1] : hBOOL(hAPP_nat_bool(P_1,X1))
=> ? [N_2] :
( ~ hBOOL(hAPP_nat_bool(P_1,N_2))
& hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2))) ) ) ) ).
fof(fact_3955_complex_Oinject,axiom,
! [Real1_2,Real2_2,Real1_1,Real2_1] :
( hAPP_real_complex(hAPP_r265291036omplex(complex,Real1_2),Real2_2) = hAPP_real_complex(hAPP_r265291036omplex(complex,Real1_1),Real2_1)
<=> ( Real1_2 = Real1_1
& Real2_2 = Real2_1 ) ) ).
fof(fact_3956_Complex__eq__0,axiom,
! [A,B] :
( hAPP_real_complex(hAPP_r265291036omplex(complex,A),B) = zero_zero_complex
<=> ( A = zero_zero_real
& B = zero_zero_real ) ) ).
fof(fact_3957_complex__zero__def,axiom,
zero_zero_complex = hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),zero_zero_real) ).
fof(fact_3958_Complex__eq__number__of,axiom,
! [A,B,W_1] :
( hAPP_real_complex(hAPP_r265291036omplex(complex,A),B) = hAPP_int_complex(number528085621omplex,W_1)
<=> ( A = hAPP_int_real(number267125858f_real,W_1)
& B = zero_zero_real ) ) ).
fof(fact_3959_Complex__eq__1,axiom,
! [A,B] :
( hAPP_real_complex(hAPP_r265291036omplex(complex,A),B) = one_one_complex
<=> ( A = one_one_real
& B = zero_zero_real ) ) ).
fof(fact_3960_complex__one__def,axiom,
one_one_complex = hAPP_real_complex(hAPP_r265291036omplex(complex,one_one_real),zero_zero_real) ).
fof(fact_3961_complex__add,axiom,
! [A_3,B_2,C_1,D_3] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)),hAPP_real_complex(hAPP_r265291036omplex(complex,C_1),D_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),C_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_2),D_3)) ).
fof(fact_3962_complex__minus,axiom,
! [A_3,B_2] : hAPP_complex_complex(uminus473333897omplex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(uminus_uminus_real,A_3)),hAPP_real_real(uminus_uminus_real,B_2)) ).
fof(fact_3963_complex__diff,axiom,
! [A_3,B_2,C_1,D_3] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)),hAPP_real_complex(hAPP_r265291036omplex(complex,C_1),D_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),C_1)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_2),D_3)) ).
fof(fact_3964_complex__mult,axiom,
! [A_3,B_2,C_1,D_3] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)),hAPP_real_complex(hAPP_r265291036omplex(complex,C_1),D_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),C_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_2),D_3))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),D_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_2),C_1))) ).
fof(fact_3965_cmod__unit__one,axiom,
! [A_3] : hAPP_complex_real(norm_norm_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(cos,A_3)),hAPP_real_real(sin,A_3))) = one_one_real ).
fof(fact_3966_complex__inverse,axiom,
! [A_3,B_2] : hAPP_complex_complex(invers1449016382omplex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,A_3),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,B_2)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).
fof(fact_3967_complex_Osize_I1_J,axiom,
! [Real1,Real2] : complex_size(hAPP_real_complex(hAPP_r265291036omplex(complex,Real1),Real2)) = zero_zero_nat ).
fof(fact_3968_cos__arg__i__mult__zero__neg,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),zero_zero_real))
=> hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),Y_1))) = zero_zero_real ) ).
fof(fact_3969_cos__arg__i__mult__zero__pos,axiom,
! [Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1))
=> hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),Y_1))) = zero_zero_real ) ).
fof(fact_3970_complex_Osize_I2_J,axiom,
! [Real1,Real2] : hAPP_complex_nat(size_size_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,Real1),Real2)) = zero_zero_nat ).
fof(fact_3971_complex__divide__def,axiom,
! [X_1,Y_1] : hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,X_1),Y_1) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_1),hAPP_complex_complex(invers1449016382omplex,Y_1)) ).
fof(fact_3972_cos__arg__i__mult__zero,axiom,
! [Y_1] :
( Y_1 != zero_zero_real
=> hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),Y_1))) = zero_zero_real ) ).
fof(fact_3973_power2__i,axiom,
hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,ii),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_complex(number528085621omplex,min) ).
fof(fact_3974_complex__i__not__zero,axiom,
ii != zero_zero_complex ).
fof(fact_3975_complex__i__not__one,axiom,
ii != one_one_complex ).
fof(fact_3976_complex__i__not__number__of,axiom,
! [W] : ii != hAPP_int_complex(number528085621omplex,W) ).
fof(fact_3977_complex__i__mult__minus,axiom,
! [X_1] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),X_1)) = hAPP_complex_complex(uminus473333897omplex,X_1) ).
fof(fact_3978_inverse__i,axiom,
hAPP_complex_complex(invers1449016382omplex,ii) = hAPP_complex_complex(uminus473333897omplex,ii) ).
fof(fact_3979_Complex__eq__i,axiom,
! [X_2,Y_2] :
( hAPP_real_complex(hAPP_r265291036omplex(complex,X_2),Y_2) = ii
<=> ( X_2 = zero_zero_real
& Y_2 = one_one_real ) ) ).
fof(fact_3980_i__def,axiom,
ii = hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),one_one_real) ).
fof(fact_3981_Complex__mult__i,axiom,
! [A_3,B_2] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)),ii) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(uminus_uminus_real,B_2)),A_3) ).
fof(fact_3982_i__mult__Complex,axiom,
! [A_3,B_2] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(uminus_uminus_real,B_2)),A_3) ).
fof(fact_3983_i__squared,axiom,
hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),ii) = hAPP_int_complex(number528085621omplex,min) ).
fof(fact_3984_i__mult__eq2,axiom,
hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),ii) = hAPP_complex_complex(uminus473333897omplex,one_one_complex) ).
fof(fact_3985_complex__inverse__complex__split,axiom,
! [X_1,Y_1] : hAPP_complex_complex(invers1449016382omplex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,of_real_complex(X_1)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(Y_1)))) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Y_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) ).
fof(fact_3986_complex__eq__cancel__iff2,axiom,
! [X_2,Y_2,Xa_1] :
( hAPP_real_complex(hAPP_r265291036omplex(complex,X_2),Y_2) = of_real_complex(Xa_1)
<=> ( X_2 = Xa_1
& Y_2 = zero_zero_real ) ) ).
fof(fact_3987_complex__of__real__def,axiom,
! [R_3] : of_real_complex(R_3) = hAPP_real_complex(hAPP_r265291036omplex(complex,R_3),zero_zero_real) ).
fof(fact_3988_sgn__eq,axiom,
! [Z_1] : hAPP_complex_complex(sgn_sgn_complex,Z_1) = hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,Z_1),of_real_complex(hAPP_complex_real(norm_norm_complex,Z_1))) ).
fof(fact_3989_Complex__mult__complex__of__real,axiom,
! [X_1,Y_1,R_3] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,X_1),Y_1)),of_real_complex(R_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),R_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_1),R_3)) ).
fof(fact_3990_complex__of__real__mult__Complex,axiom,
! [R_3,X_1,Y_1] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),hAPP_real_complex(hAPP_r265291036omplex(complex,X_1),Y_1)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),Y_1)) ).
fof(fact_3991_complex__of__real__add__Complex,axiom,
! [R_3,X_1,Y_1] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,of_real_complex(R_3)),hAPP_real_complex(hAPP_r265291036omplex(complex,X_1),Y_1)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_3),X_1)),Y_1) ).
fof(fact_3992_Complex__add__complex__of__real,axiom,
! [X_1,Y_1,R_3] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,X_1),Y_1)),of_real_complex(R_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),R_3)),Y_1) ).
fof(fact_3993_i__mult__eq,axiom,
hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),ii) = of_real_complex(hAPP_int_real(number267125858f_real,min)) ).
fof(fact_3994_complex__of__real__i,axiom,
! [R_3] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),ii) = hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),R_3) ).
fof(fact_3995_i__complex__of__real,axiom,
! [R_3] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(R_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),R_3) ).
fof(fact_3996_complex__of__real__minus__one,axiom,
of_real_complex(hAPP_real_real(uminus_uminus_real,one_one_real)) = hAPP_complex_complex(uminus473333897omplex,one_one_complex) ).
fof(fact_3997_cmod__complex__polar,axiom,
! [R_3,A_3] : hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(cos,A_3)),hAPP_real_real(sin,A_3)))) = hAPP_real_real(abs_abs_real,R_3) ).
fof(fact_3998_expi__two__pi__i,axiom,
expi(hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),of_real_complex(pi))),ii)) = one_one_complex ).
fof(fact_3999_expi__add,axiom,
! [A_3,B_2] : expi(hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_3),B_2)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,expi(A_3)),expi(B_2)) ).
fof(fact_4000_expi__zero,axiom,
expi(zero_zero_complex) = one_one_complex ).
fof(fact_4001_complex__split__polar,axiom,
! [Z_1] :
? [R_1,A_2] : Z_1 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_1)),hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(cos,A_2)),hAPP_real_real(sin,A_2))) ).
fof(fact_4002_DERIV__pos__imp__increasing,axiom,
! [F,A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B))
=> ( ! [X] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B)) )
=> ? [Y] :
( hBOOL(hAPP_real_bool(deriv_real(F,X),Y))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,A)),hAPP_real_real(F,B))) ) ) ).
fof(fact_4003_DERIV__neg__imp__decreasing,axiom,
! [F,A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B))
=> ( ! [X] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B)) )
=> ? [Y] :
( hBOOL(hAPP_real_bool(deriv_real(F,X),Y))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),zero_zero_real)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,B)),hAPP_real_real(F,A))) ) ) ).
fof(fact_4004_int__induct,axiom,
! [I_1,P_1,K_1] :
( hBOOL(hAPP_int_bool(P_1,K_1))
=> ( ! [I_2] :
( is_int(I_2)
=> ( 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(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))) ) ) )
=> ( ! [I_2] :
( is_int(I_2)
=> ( 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(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))) ) ) )
=> hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).
fof(fact_4005_plusinfinity,axiom,
! [P_1,P_2,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X,K] :
( ( is_int(X)
& is_int(K) )
=> ( hBOOL(hAPP_int_bool(P_2,X))
<=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D)))) ) )
=> ( ? [Z] :
! [X] :
( is_int(X)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X))
=> ( hBOOL(hAPP_int_bool(P_1,X))
<=> hBOOL(hAPP_int_bool(P_2,X)) ) ) )
=> ( ? [X1] : hBOOL(hAPP_int_bool(P_2,X1))
=> ? [X1] :
( is_int(X1)
& hBOOL(hAPP_int_bool(P_1,X1)) ) ) ) ) ) ).
fof(fact_4006_minusinfinity,axiom,
! [P_1,P1,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X,K] :
( ( is_int(X)
& is_int(K) )
=> ( hBOOL(hAPP_int_bool(P1,X))
<=> hBOOL(hAPP_int_bool(P1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D)))) ) )
=> ( ? [Z] :
! [X] :
( is_int(X)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Z))
=> ( hBOOL(hAPP_int_bool(P_1,X))
<=> hBOOL(hAPP_int_bool(P1,X)) ) ) )
=> ( ? [X1] : hBOOL(hAPP_int_bool(P1,X1))
=> ? [X1] :
( is_int(X1)
& hBOOL(hAPP_int_bool(P_1,X1)) ) ) ) ) ) ).
fof(fact_4007_coprime__sos,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,X_1),Y_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Y_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).
fof(fact_4008_coprime__mul__eq,axiom,
! [D,A,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),B)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),A))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),B)) ) ) ).
fof(fact_4009_coprime__mul,axiom,
! [B_2,D_3,A_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),B_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))) ) ) ).
fof(fact_4010_coprime__rmul2,axiom,
! [D_3,A_3,B_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_3)) ) ).
fof(fact_4011_coprime__lmul2,axiom,
! [D_3,A_3,B_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),B_2)) ) ).
fof(fact_4012_coprime__refl,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,N),N))
<=> N = one_one_nat ) ).
fof(fact_4013_coprime__1,axiom,
! [A_3] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),one_one_nat)) ).
fof(fact_4014_coprime__1_H,axiom,
! [A_3] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,one_one_nat),A_3)) ).
fof(fact_4015_coprime__commute,axiom,
! [A,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,B),A)) ) ).
fof(fact_4016_coprime__divisors,axiom,
! [E_1,B_2,D_3,A_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),A_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E_1),B_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),E_1)) ) ) ) ).
fof(fact_4017_coprime__exp,axiom,
! [N_1,D_3,A_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N_1))) ) ).
fof(fact_4018_coprime__exp__imp,axiom,
! [N_1,A_3,B_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),N_1))) ) ).
fof(fact_4019_floor__real__of__nat,axiom,
! [N_1] : archim1246769320r_real(hAPP_nat_real(real_nat,N_1)) = hAPP_nat_int(semiri1621563631at_int,N_1) ).
fof(fact_4020_natfloor__def,axiom,
! [X_1] : natfloor(X_1) = hAPP_int_nat(nat,archim1246769320r_real(X_1)) ).
fof(fact_4021_coprime__Suc0_H,axiom,
! [A_3] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(suc,zero_zero_nat)),A_3)) ).
fof(fact_4022_coprime__Suc0,axiom,
! [A_3] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),hAPP_nat_nat(suc,zero_zero_nat))) ).
fof(fact_4023_coprime__0_H,axiom,
! [D] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,zero_zero_nat),D))
<=> D = one_one_nat ) ).
fof(fact_4024_coprime__0,axiom,
! [D] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),zero_zero_nat))
<=> D = one_one_nat ) ).
fof(fact_4025_divides__mul,axiom,
! [N_1,M,R_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),R_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),R_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)),R_3)) ) ) ) ).
fof(fact_4026_coprime__divprod,axiom,
! [D_3,A_3,B_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),B_2)) ) ) ).
fof(fact_4027_coprime,axiom,
! [A,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B))
<=> ! [D_2] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),A))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),B)) )
<=> D_2 = one_one_nat ) ) ).
fof(fact_4028_coprime__plus1,axiom,
! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),N_1)) ).
fof(fact_4029_coprime__exp2,axiom,
! [A,N,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),hAPP_nat_nat(suc,N))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),hAPP_nat_nat(suc,N))))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B)) ) ).
fof(fact_4030_coprime__minus1,axiom,
! [N_1] :
( N_1 != zero_zero_nat
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)),N_1)) ) ).
fof(fact_4031_floor__eq3,axiom,
! [N_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1))))
=> hAPP_int_nat(nat,archim1246769320r_real(X_1)) = N_1 ) ) ).
fof(fact_4032_floor__minus__real__of__nat,axiom,
! [N_1] : archim1246769320r_real(hAPP_real_real(uminus_uminus_real,hAPP_nat_real(real_nat,N_1))) = hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N_1)) ).
fof(fact_4033_le__mult__floor,axiom,
! [B_2,A_3] :
( 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,zero_zero_real),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,archim1246769320r_real(A_3)),archim1246769320r_real(B_2))),archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),B_2)))) ) ) ).
fof(fact_4034_floor__eq4,axiom,
! [N_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_1)),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1))))
=> hAPP_int_nat(nat,archim1246769320r_real(X_1)) = N_1 ) ) ).
fof(fact_4035_coprime__bezout,axiom,
! [A,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B))
<=> ? [X,Y] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),X)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),Y)) = one_one_nat
| hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),X)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),Y)) = one_one_nat ) ) ).
fof(fact_4036_coprime__bezout__strong,axiom,
! [A_3,B_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2))
=> ( B_2 != one_one_nat
=> ? [X,Y] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),X) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),Y)),one_one_nat) ) ) ).
fof(fact_4037_nat__number__of__diff__1,axiom,
! [V_2] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_2),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat) = zero_zero_nat )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_2),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(pred,V_2)) ) ) ).
fof(fact_4038_succ__pred,axiom,
! [X_1] :
( is_int(X_1)
=> hAPP_int_int(succ,hAPP_int_int(pred,X_1)) = X_1 ) ).
fof(fact_4039_pred__Bit1,axiom,
! [K_2] : hAPP_int_int(pred,hAPP_int_int(bit1,K_2)) = hAPP_int_int(bit0,K_2) ).
fof(fact_4040_pred__Bit0,axiom,
! [K_2] : hAPP_int_int(pred,hAPP_int_int(bit0,K_2)) = hAPP_int_int(bit1,hAPP_int_int(pred,K_2)) ).
fof(fact_4041_le__iff__pred__less,axiom,
! [K_1,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(pred,K_1)),L)) ) ).
fof(fact_4042_pred__Pls,axiom,
hAPP_int_int(pred,pls) = min ).
fof(fact_4043_pred__Min,axiom,
hAPP_int_int(pred,min) = hAPP_int_int(bit0,min) ).
fof(fact_4044_pred__def,axiom,
! [K_2] : hAPP_int_int(pred,K_2) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),one_one_int) ).
fof(fact_4045_minus__Bit1,axiom,
! [K_2] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(bit1,K_2)) = hAPP_int_int(bit1,hAPP_int_int(pred,hAPP_int_int(uminus_uminus_int,K_2))) ).
fof(fact_4046_add__Min__right,axiom,
! [K_2] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),min) = hAPP_int_int(pred,K_2) ).
fof(fact_4047_add__Min,axiom,
! [K_2] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,min),K_2) = hAPP_int_int(pred,K_2) ).
fof(fact_4048_diff__bin__simps_I8_J,axiom,
! [K_2,L_2] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(pred,K_2)),L_2)) ).
fof(fact_4049_neg__number__of__pred__iff__0,axiom,
! [V] :
( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
<=> hAPP_int_nat(number_number_of_nat,V) = zero_zero_nat ) ).
fof(fact_4050_Suc__diff__number__of,axiom,
! [M,V_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_2))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(pred,V_2))) ) ).
fof(fact_4051_complex__expi__Ex,axiom,
! [Z_1] :
? [A_2,R_1] : Z_1 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_1)),expi(A_2)) ).
fof(fact_4052_ResSet__finite,axiom,
! [X_3,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( hBOOL(hAPP_f448129468l_bool(resSet(Ma),X_3))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,X_3)) ) ) ).
fof(fact_4053_ResSet__def,axiom,
! [Ma,X_3] :
( hBOOL(hAPP_f448129468l_bool(resSet(Ma),X_3))
<=> ! [Y1,Y2] :
( ( is_int(Y1)
& is_int(Y2) )
=> ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y1),X_3))
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y2),X_3))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Y1),Y2),Ma)) )
=> Y1 = Y2 ) ) ) ).
fof(fact_4054_ResSet__SRStar__prop,axiom,
! [P_5] : hBOOL(hAPP_f448129468l_bool(resSet(P_5),hAPP_i1948725293t_bool(sRStar,P_5))) ).
fof(fact_4055_nat_Osize_I2_J,axiom,
! [Nat] : hAPP_nat_nat(nat_size,hAPP_nat_nat(suc,Nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(nat_size,Nat)),hAPP_nat_nat(suc,zero_zero_nat)) ).
fof(fact_4056_nat_Osize_I4_J,axiom,
! [Nat] : hAPP_nat_nat(size_size_nat,hAPP_nat_nat(suc,Nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(size_size_nat,Nat)),hAPP_nat_nat(suc,zero_zero_nat)) ).
fof(fact_4057_code__numeral_Osize_I1_J,axiom,
code_c271388182l_size(zero_z126310315umeral) = zero_zero_nat ).
fof(fact_4058_nat__size,axiom,
! [N_1] : hAPP_nat_nat(size_size_nat,N_1) = N_1 ).
fof(fact_4059_nat_Osize_I3_J,axiom,
hAPP_nat_nat(size_size_nat,zero_zero_nat) = zero_zero_nat ).
fof(fact_4060_nat_Osize_I1_J,axiom,
hAPP_nat_nat(nat_size,zero_zero_nat) = zero_zero_nat ).
fof(fact_4061_code__numeral_Osize_I2_J,axiom,
! [Code_numeral_1] : code_c271388182l_size(code_S1047413653umeral(Code_numeral_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,code_c271388182l_size(Code_numeral_1)),hAPP_nat_nat(suc,zero_zero_nat)) ).
fof(fact_4062_complex__mod__mult__cnj,axiom,
! [Z_1] : hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(norm_norm_complex,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_4063_complex__cnj__inverse,axiom,
! [X_1] : hAPP_complex_complex(cnj,hAPP_complex_complex(invers1449016382omplex,X_1)) = hAPP_complex_complex(invers1449016382omplex,hAPP_complex_complex(cnj,X_1)) ).
fof(fact_4064_complex__cnj__zero__iff,axiom,
! [Z_2] :
( hAPP_complex_complex(cnj,Z_2) = zero_zero_complex
<=> Z_2 = zero_zero_complex ) ).
fof(fact_4065_cnj_Ozero,axiom,
hAPP_complex_complex(cnj,zero_zero_complex) = zero_zero_complex ).
fof(fact_4066_complex__cnj__number__of,axiom,
! [W] : hAPP_complex_complex(cnj,hAPP_int_complex(number528085621omplex,W)) = hAPP_int_complex(number528085621omplex,W) ).
fof(fact_4067_complex__cnj__mult,axiom,
! [X_1,Y_1] : hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_1),Y_1)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(cnj,X_1)),hAPP_complex_complex(cnj,Y_1)) ).
fof(fact_4068_complex__cnj__one,axiom,
hAPP_complex_complex(cnj,one_one_complex) = one_one_complex ).
fof(fact_4069_cnj_Ominus,axiom,
! [X_1] : hAPP_complex_complex(cnj,hAPP_complex_complex(uminus473333897omplex,X_1)) = hAPP_complex_complex(uminus473333897omplex,hAPP_complex_complex(cnj,X_1)) ).
fof(fact_4070_complex__cnj__cancel__iff,axiom,
! [X_2,Y_2] :
( hAPP_complex_complex(cnj,X_2) = hAPP_complex_complex(cnj,Y_2)
<=> X_2 = Y_2 ) ).
fof(fact_4071_complex__cnj__cnj,axiom,
! [Z_1] : hAPP_complex_complex(cnj,hAPP_complex_complex(cnj,Z_1)) = Z_1 ).
fof(fact_4072_code__numeral_Oinject,axiom,
! [Code_numeral_5,Code_numeral_4] :
( code_S1047413653umeral(Code_numeral_5) = code_S1047413653umeral(Code_numeral_4)
<=> Code_numeral_5 = Code_numeral_4 ) ).
fof(fact_4073_complex__cnj__of__nat,axiom,
! [N_1] : hAPP_complex_complex(cnj,hAPP_nat_complex(semiri2020571505omplex,N_1)) = hAPP_nat_complex(semiri2020571505omplex,N_1) ).
fof(fact_4074_complex__mod__cnj,axiom,
! [Z_1] : hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,Z_1)) = hAPP_complex_real(norm_norm_complex,Z_1) ).
fof(fact_4075_cnj_Oadd,axiom,
! [X_1,Y_1] : hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_1),Y_1)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(cnj,X_1)),hAPP_complex_complex(cnj,Y_1)) ).
fof(fact_4076_cnj_Odiff,axiom,
! [X_1,Y_1] : hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_1),Y_1)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(cnj,X_1)),hAPP_complex_complex(cnj,Y_1)) ).
fof(fact_4077_complex__cnj__power,axiom,
! [X_1,N_1] : hAPP_complex_complex(cnj,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_1),N_1)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(cnj,X_1)),N_1) ).
fof(fact_4078_complex__cnj__divide,axiom,
! [X_1,Y_1] : hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,X_1),Y_1)) = hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,hAPP_complex_complex(cnj,X_1)),hAPP_complex_complex(cnj,Y_1)) ).
fof(fact_4079_complex__cnj__complex__of__real,axiom,
! [X_1] : hAPP_complex_complex(cnj,of_real_complex(X_1)) = of_real_complex(X_1) ).
fof(fact_4080_code__numeral_Osimps_I3_J,axiom,
! [Code_numeral_3] : code_S1047413653umeral(Code_numeral_3) != zero_z126310315umeral ).
fof(fact_4081_code__numeral_Osimps_I2_J,axiom,
! [Code_numeral_2] : zero_z126310315umeral != code_S1047413653umeral(Code_numeral_2) ).
fof(fact_4082_complex__cnj,axiom,
! [A_3,B_2] : hAPP_complex_complex(cnj,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),hAPP_real_real(uminus_uminus_real,B_2)) ).
fof(fact_4083_complex__cnj__i,axiom,
hAPP_complex_complex(cnj,ii) = hAPP_complex_complex(uminus473333897omplex,ii) ).
fof(fact_4084_Suc__code__numeral__minus__one,axiom,
! [N_1] : hAPP_C498520661umeral(hAPP_C1594335432umeral(minus_1690775515umeral,code_S1047413653umeral(N_1)),one_on1645066479umeral) = N_1 ).
fof(fact_4085_cnj_Opos__bounded,axiom,
? [K_4] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),K_4))
& ! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X)),K_4))) ) ).
fof(fact_4086_cnj_Ononneg__bounded,axiom,
? [K_4] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),K_4))
& ! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X)),K_4))) ) ).
fof(fact_4087_cnj_Obounded,axiom,
? [K_4] :
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X)),K_4))) ).
fof(fact_4088_code__numeral_Osize_I4_J,axiom,
! [Code_numeral_1] : hAPP_C2129356693al_nat(size_s945831648umeral,code_S1047413653umeral(Code_numeral_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_C2129356693al_nat(size_s945831648umeral,Code_numeral_1)),hAPP_nat_nat(suc,zero_zero_nat)) ).
fof(fact_4089_complex__diff__cnj,axiom,
! [Z_1] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Z_1),hAPP_complex_complex(cnj,Z_1)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_complex_real(im,Z_1)))),ii) ).
fof(fact_4090_code__numeral_Oexhaust,axiom,
! [Y_1] :
( Y_1 != zero_z126310315umeral
=> ~ ! [Code_numeral] : Y_1 != code_S1047413653umeral(Code_numeral) ) ).
fof(fact_4091_Im,axiom,
! [X_1,Y_1] : hAPP_complex_real(im,hAPP_real_complex(hAPP_r265291036omplex(complex,X_1),Y_1)) = Y_1 ).
fof(fact_4092_complex__Im__cnj,axiom,
! [X_1] : hAPP_complex_real(im,hAPP_complex_complex(cnj,X_1)) = hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X_1)) ).
fof(fact_4093_Im__complex__of__real,axiom,
! [Z_1] : hAPP_complex_real(im,of_real_complex(Z_1)) = zero_zero_real ).
fof(fact_4094_complex__Im__i,axiom,
hAPP_complex_real(im,ii) = one_one_real ).
fof(fact_4095_Im_Ozero,axiom,
hAPP_complex_real(im,zero_zero_complex) = zero_zero_real ).
fof(fact_4096_complex__Im__one,axiom,
hAPP_complex_real(im,one_one_complex) = zero_zero_real ).
fof(fact_4097_Im_Oadd,axiom,
! [X_1,Y_1] : hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(im,X_1)),hAPP_complex_real(im,Y_1)) ).
fof(fact_4098_complex__Im__number__of,axiom,
! [V_2] : hAPP_complex_real(im,hAPP_int_complex(number528085621omplex,V_2)) = zero_zero_real ).
fof(fact_4099_Im_Ominus,axiom,
! [X_1] : hAPP_complex_real(im,hAPP_complex_complex(uminus473333897omplex,X_1)) = hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X_1)) ).
fof(fact_4100_Im_Odiff,axiom,
! [X_1,Y_1] : hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(im,X_1)),hAPP_complex_real(im,Y_1)) ).
fof(fact_4101_complex__Im__of__nat,axiom,
! [N_1] : hAPP_complex_real(im,hAPP_nat_complex(semiri2020571505omplex,N_1)) = zero_zero_real ).
fof(fact_4102_abs__Im__le__cmod,axiom,
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_complex_real(im,X_1))),hAPP_complex_real(norm_norm_complex,X_1))) ).
fof(fact_4103_complex__In__mult__cnj__zero,axiom,
! [Z_1] : hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1))) = zero_zero_real ).
fof(fact_4104_Im__sgn,axiom,
! [Z_1] : hAPP_complex_real(im,hAPP_complex_complex(sgn_sgn_complex,Z_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(im,Z_1)),hAPP_complex_real(norm_norm_complex,Z_1)) ).
fof(fact_4105_code__numeral_Osize_I3_J,axiom,
hAPP_C2129356693al_nat(size_s945831648umeral,zero_z126310315umeral) = zero_zero_nat ).
fof(fact_4106_Im_Opos__bounded,axiom,
? [K_4] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),K_4))
& ! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X)),K_4))) ) ).
fof(fact_4107_Im_Ononneg__bounded,axiom,
? [K_4] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),K_4))
& ! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X)),K_4))) ) ).
fof(fact_4108_Im_Obounded,axiom,
? [K_4] :
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X)),K_4))) ).
fof(fact_4109_complex__inverse__def,axiom,
! [X_1] : hAPP_complex_complex(invers1449016382omplex,X_1) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,X_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X_1))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).
fof(fact_4110_complex__mult__cnj,axiom,
! [Z_1] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1)) = of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_4111_complex__Im__inverse,axiom,
! [X_1] : hAPP_complex_real(im,hAPP_complex_complex(invers1449016382omplex,X_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X_1))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_4112_Re,axiom,
! [X_1,Y_1] : hAPP_complex_real(re,hAPP_real_complex(hAPP_r265291036omplex(complex,X_1),Y_1)) = X_1 ).
fof(fact_4113_Re__complex__of__real,axiom,
! [Z_1] : hAPP_complex_real(re,of_real_complex(Z_1)) = Z_1 ).
fof(fact_4114_complex__Re__cnj,axiom,
! [X_1] : hAPP_complex_real(re,hAPP_complex_complex(cnj,X_1)) = hAPP_complex_real(re,X_1) ).
fof(fact_4115_complex__eq__iff,axiom,
! [X_2,Y_2] :
( X_2 = Y_2
<=> ( hAPP_complex_real(re,X_2) = hAPP_complex_real(re,Y_2)
& hAPP_complex_real(im,X_2) = hAPP_complex_real(im,Y_2) ) ) ).
fof(fact_4116_complex__eqI,axiom,
! [X_1,Y_1] :
( hAPP_complex_real(re,X_1) = hAPP_complex_real(re,Y_1)
=> ( hAPP_complex_real(im,X_1) = hAPP_complex_real(im,Y_1)
=> X_1 = Y_1 ) ) ).
fof(fact_4117_complex__Re__le__cmod,axiom,
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(re,X_1)),hAPP_complex_real(norm_norm_complex,X_1))) ).
fof(fact_4118_complex__Re__i,axiom,
hAPP_complex_real(re,ii) = zero_zero_real ).
fof(fact_4119_complex__surj,axiom,
! [Z_1] : hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_complex_real(re,Z_1)),hAPP_complex_real(im,Z_1)) = Z_1 ).
fof(fact_4120_Re_Ozero,axiom,
hAPP_complex_real(re,zero_zero_complex) = zero_zero_real ).
fof(fact_4121_Re_Oadd,axiom,
! [X_1,Y_1] : hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(re,X_1)),hAPP_complex_real(re,Y_1)) ).
fof(fact_4122_complex__Re__one,axiom,
hAPP_complex_real(re,one_one_complex) = one_one_real ).
fof(fact_4123_Re_Ominus,axiom,
! [X_1] : hAPP_complex_real(re,hAPP_complex_complex(uminus473333897omplex,X_1)) = hAPP_real_real(uminus_uminus_real,hAPP_complex_real(re,X_1)) ).
fof(fact_4124_complex__Re__number__of,axiom,
! [V_2] : hAPP_complex_real(re,hAPP_int_complex(number528085621omplex,V_2)) = hAPP_int_real(number267125858f_real,V_2) ).
fof(fact_4125_Re_Odiff,axiom,
! [X_1,Y_1] : hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(re,X_1)),hAPP_complex_real(re,Y_1)) ).
fof(fact_4126_complex__Re__of__nat,axiom,
! [N_1] : hAPP_complex_real(re,hAPP_nat_complex(semiri2020571505omplex,N_1)) = hAPP_nat_real(semiri132038758t_real,N_1) ).
fof(fact_4127_abs__Re__le__cmod,axiom,
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_complex_real(re,X_1))),hAPP_complex_real(norm_norm_complex,X_1))) ).
fof(fact_4128_Re__sgn,axiom,
! [Z_1] : hAPP_complex_real(re,hAPP_complex_complex(sgn_sgn_complex,Z_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,Z_1)),hAPP_complex_real(norm_norm_complex,Z_1)) ).
fof(fact_4129_complex__Im__mult,axiom,
! [X_1,Y_1] : hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X_1)),hAPP_complex_real(im,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X_1)),hAPP_complex_real(re,Y_1))) ).
fof(fact_4130_complex__Re__mult,axiom,
! [X_1,Y_1] : hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X_1)),hAPP_complex_real(re,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X_1)),hAPP_complex_real(im,Y_1))) ).
fof(fact_4131_cnj__def,axiom,
! [Z_1] : hAPP_complex_complex(cnj,Z_1) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_complex_real(re,Z_1)),hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,Z_1))) ).
fof(fact_4132_complex__mod__sqrt__Re__mult__cnj,axiom,
! [Z_1] : hAPP_complex_real(norm_norm_complex,Z_1) = hAPP_real_real(sqrt,hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1)))) ).
fof(fact_4133_complex__add__def,axiom,
! [X_1,Y_1] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_1),Y_1) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(re,X_1)),hAPP_complex_real(re,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(im,X_1)),hAPP_complex_real(im,Y_1))) ).
fof(fact_4134_complex__minus__def,axiom,
! [X_1] : hAPP_complex_complex(uminus473333897omplex,X_1) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(re,X_1))),hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X_1))) ).
fof(fact_4135_complex__mult__def,axiom,
! [X_1,Y_1] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_1),Y_1) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X_1)),hAPP_complex_real(re,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X_1)),hAPP_complex_real(im,Y_1)))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X_1)),hAPP_complex_real(im,Y_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X_1)),hAPP_complex_real(re,Y_1)))) ).
fof(fact_4136_complex__add__cnj,axiom,
! [Z_1] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_1),hAPP_complex_complex(cnj,Z_1)) = of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_complex_real(re,Z_1))) ).
fof(fact_4137_cmod__def,axiom,
! [Z_1] : hAPP_complex_real(norm_norm_complex,Z_1) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_4138_complex__Re__inverse,axiom,
! [X_1] : hAPP_complex_real(re,hAPP_complex_complex(invers1449016382omplex,X_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,X_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_4139_Re_Opos__bounded,axiom,
? [K_4] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),K_4))
& ! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(re,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X)),K_4))) ) ).
fof(fact_4140_Re_Ononneg__bounded,axiom,
? [K_4] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),K_4))
& ! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(re,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X)),K_4))) ) ).
fof(fact_4141_Re_Obounded,axiom,
? [K_4] :
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(re,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X)),K_4))) ).
fof(fact_4142_expi__def,axiom,
! [Z_1] : expi(Z_1) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(hAPP_real_real(exp_real,hAPP_complex_real(re,Z_1)))),cis(hAPP_complex_real(im,Z_1))) ).
fof(fact_4143_sin__n__Im__cis__pow__n,axiom,
! [N_1,A_3] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),A_3)) = hAPP_complex_real(im,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,cis(A_3)),N_1)) ).
fof(fact_4144_cos__n__Re__cis__pow__n,axiom,
! [N_1,A_3] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),A_3)) = hAPP_complex_real(re,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,cis(A_3)),N_1)) ).
fof(fact_4145_Re__cis,axiom,
! [A_3] : hAPP_complex_real(re,cis(A_3)) = hAPP_real_real(cos,A_3) ).
fof(fact_4146_Im__cis,axiom,
! [A_3] : hAPP_complex_real(im,cis(A_3)) = hAPP_real_real(sin,A_3) ).
fof(fact_4147_cis__mult,axiom,
! [A_3,B_2] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,cis(A_3)),cis(B_2)) = cis(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2)) ).
fof(fact_4148_cis__zero,axiom,
cis(zero_zero_real) = one_one_complex ).
fof(fact_4149_cis__inverse,axiom,
! [A_3] : hAPP_complex_complex(invers1449016382omplex,cis(A_3)) = cis(hAPP_real_real(uminus_uminus_real,A_3)) ).
fof(fact_4150_cis__divide,axiom,
! [A_3,B_2] : hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,cis(A_3)),cis(B_2)) = cis(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),B_2)) ).
fof(fact_4151_cis__def,axiom,
! [A_3] : cis(A_3) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(cos,A_3)),hAPP_real_real(sin,A_3)) ).
fof(fact_4152_DeMoivre,axiom,
! [A_3,N_1] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,cis(A_3)),N_1) = cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),A_3)) ).
fof(fact_4153_cis__real__of__nat__Suc__mult,axiom,
! [N_1,A_3] : cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1))),A_3)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,cis(A_3)),cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),A_3))) ).
fof(fact_4154_bezw__0,axiom,
! [X_1] : bezw(X_1,zero_zero_nat) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ).
fof(fact_4155_termination__basic__simps_I4_J,axiom,
! [Y_1,X_1,Z_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Z_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_1),Z_1))) ) ).
fof(fact_4156_termination__basic__simps_I1_J,axiom,
! [Z_1,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,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_1),Z_1))) ) ).
fof(fact_4157_termination__basic__simps_I2_J,axiom,
! [Y_1,X_1,Z_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Z_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_1),Z_1))) ) ).
fof(fact_4158_termination__basic__simps_I5_J,axiom,
! [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_eq_nat,X_1),Y_1)) ) ).
fof(fact_4159_termination__basic__simps_I3_J,axiom,
! [Z_1,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,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_1),Z_1))) ) ).
fof(fact_4160_prime__dvd__power__two,axiom,
! [M,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),M)) ) ) ).
fof(fact_4161_divmod__int__def,axiom,
! [B_2,A_3] :
( is_int(A_3)
=> ( ( 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,zero_zero_int),B_2))
=> divmod_int(A_3,B_2) = posDivAlg(A_3,B_2) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2))
=> ( ( A_3 = zero_zero_int
=> divmod_int(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int) )
& ( A_3 != zero_zero_int
=> divmod_int(A_3,B_2) = hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2))) ) ) ) ) )
& ( ~ 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_int,zero_zero_int),B_2))
=> divmod_int(A_3,B_2) = negDivAlg(A_3,B_2) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> divmod_int(A_3,B_2) = hAPP_P1975530577nt_int(negateSnd,posDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2))) ) ) ) ) ) ).
fof(fact_4162_prime__0,axiom,
~ hBOOL(hAPP_nat_bool(prime,zero_zero_nat)) ).
fof(fact_4163_prime__1,axiom,
~ hBOOL(hAPP_nat_bool(prime,one_one_nat)) ).
fof(fact_4164_primes__eq,axiom,
! [Q_2,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( hBOOL(hAPP_nat_bool(prime,Q_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),Q_2))
=> P_3 = Q_2 ) ) ) ).
fof(fact_4165_distinct__prime__coprime,axiom,
! [Q_2,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( hBOOL(hAPP_nat_bool(prime,Q_2))
=> ( P_3 != Q_2
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),Q_2)) ) ) ) ).
fof(fact_4166_prime__Suc0,axiom,
~ hBOOL(hAPP_nat_bool(prime,hAPP_nat_nat(suc,zero_zero_nat))) ).
fof(fact_4167_prime__g__zero,axiom,
! [P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),P_3)) ) ).
fof(fact_4168_prime__divprod__eq,axiom,
! [A,B,P_5] :
( hBOOL(hAPP_nat_bool(prime,P_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),B)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_5),A))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_5),B)) ) ) ) ).
fof(fact_4169_prime__divprod,axiom,
! [A_3,B_2,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_3))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),B_2)) ) ) ) ).
fof(fact_4170_prime__dvd__mult,axiom,
! [M,N_1,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),M))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),N_1)) ) ) ) ).
fof(fact_4171_prime__product,axiom,
! [P_3,Q_2] :
( hBOOL(hAPP_nat_bool(prime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),Q_2)))
=> ( P_3 = one_one_nat
| Q_2 = one_one_nat ) ) ).
fof(fact_4172_prime__divexp__n,axiom,
! [X_1,N_1,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1))) ) ) ).
fof(fact_4173_prime__divexp,axiom,
! [X_1,N_1,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),X_1)) ) ) ).
fof(fact_4174_prime__dvd__power,axiom,
! [A_3,N_1,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_3)) ) ) ).
fof(fact_4175_prime__exp,axiom,
! [P_5,N] :
( hBOOL(hAPP_nat_bool(prime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_5),N)))
<=> ( hBOOL(hAPP_nat_bool(prime,P_5))
& N = one_one_nat ) ) ).
fof(fact_4176_coprime__prime,axiom,
! [P_3,A_3,B_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2))
=> ~ ( hBOOL(hAPP_nat_bool(prime,P_3))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_3))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),B_2)) ) ) ).
fof(fact_4177_prime__coprime__strong,axiom,
! [N_1,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),N_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),N_1)) ) ) ).
fof(fact_4178_coprime__prime__eq,axiom,
! [A,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B))
<=> ! [P_4] :
~ ( hBOOL(hAPP_nat_bool(prime,P_4))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),A))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),B)) ) ) ).
fof(fact_4179_prime__impl__zprime__int,axiom,
! [A_3] :
( hBOOL(hAPP_nat_bool(prime,A_3))
=> hBOOL(hAPP_int_bool(zprime,hAPP_nat_int(semiri1621563631at_int,A_3))) ) ).
fof(fact_4180_prime__g__one,axiom,
! [P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),P_3)) ) ).
fof(fact_4181_prime__nd__one,axiom,
! [P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(suc,zero_zero_nat))) ) ).
fof(fact_4182_prime__factor__lt,axiom,
! [M,N_1,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( N_1 != zero_zero_nat
=> ( N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),M)
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ) ) ).
fof(fact_4183_prime__def,axiom,
! [P_5] :
( hBOOL(hAPP_nat_bool(prime,P_5))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),P_5))
& ! [M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M_1),P_5))
=> ( M_1 = one_one_nat
| M_1 = P_5 ) ) ) ) ).
fof(fact_4184_divides__primepow,axiom,
! [D,K_1,P_5] :
( hBOOL(hAPP_nat_bool(prime,P_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_5),K_1)))
<=> ? [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),K_1))
& D = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_5),I_2) ) ) ) ).
fof(fact_4185_prime__power__dvd__cancel__right,axiom,
! [N_1,A_3,B_2,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),B_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N_1)),A_3)) ) ) ) ).
fof(fact_4186_prime__coprime__lt,axiom,
! [X_1,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),P_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,X_1),P_3)) ) ) ) ).
fof(fact_4187_prime__coprime,axiom,
! [N_1,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( N_1 = one_one_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),N_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),N_1)) ) ) ).
fof(fact_4188_negateSnd__eq,axiom,
! [Q_2,R_3] : hAPP_P1975530577nt_int(negateSnd,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),hAPP_int_int(uminus_uminus_int,R_3)) ).
fof(fact_4189_divmod__int__rel__neg,axiom,
! [A_3,B_2,Qr_1] :
( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)),Qr_1))
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_P1975530577nt_int(negateSnd,Qr_1))) ) ).
fof(fact_4190_two__is__prime,axiom,
hBOOL(hAPP_nat_bool(prime,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_4191_divmod__int__correct,axiom,
! [A_3,B_2] :
( is_int(B_2)
=> ( B_2 != zero_zero_int
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),divmod_int(A_3,B_2))) ) ) ).
fof(fact_4192_prime__dvd__square,axiom,
! [M,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),M)) ) ) ).
fof(fact_4193_prime__divprod__pow,axiom,
! [N_1,A_3,B_2,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N_1)),A_3))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N_1)),B_2)) ) ) ) ) ).
fof(fact_4194_prime__ge__2,axiom,
! [P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) ) ).
fof(fact_4195_prime__odd,axiom,
! [P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( P_3 = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))
| ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,P_3)) ) ) ).
fof(fact_4196_divmod__int__mod__div,axiom,
! [P_3,Q_2] : divmod_int(P_3,Q_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,P_3),Q_2)),hAPP_int_int(div_mod_int(P_3),Q_2)) ).
fof(fact_4197_div__mod__code__numeral__def,axiom,
! [N_1,M] : code_d418564891umeral(N_1,M) = hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,N_1),M)),hAPP_C498520661umeral(div_mo1740067990umeral(N_1),M)) ).
fof(fact_4198_xzgcd__linear,axiom,
! [M,R_3,S_1,T_1,N_1] :
( is_int(R_3)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
=> ( xzgcd(M,N_1) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,R_3),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_1),T_1))
=> R_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_1),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_1),N_1)) ) ) ) ).
fof(fact_4199_xzgcda_Osimps,axiom,
! [M,N_1,R_5,S_3,S_1,T_2,T_1,R_3] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int))
=> xzgcda(M,N_1,R_5,R_3,S_3,S_1,T_2,T_1) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,R_5),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_3),T_2)) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int))
=> xzgcda(M,N_1,R_5,R_3,S_3,S_1,T_2,T_1) = xzgcda(M,N_1,R_3,hAPP_int_int(div_mod_int(R_5),R_3),S_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,S_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),S_1)),T_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,T_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),T_1))) ) ) ).
fof(fact_4200_xzgcd__def,axiom,
! [M,N_1] : xzgcd(M,N_1) = xzgcda(M,N_1,M,N_1,one_one_int,zero_zero_int,zero_zero_int,one_one_int) ).
fof(fact_4201_xzgcda__linear,axiom,
! [M,N_1,R_5,S_3,S_1,T_2,T_1,Rn,Sn_1,Tn_1,R_3] :
( is_int(Rn)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_3))
=> ( xzgcda(M,N_1,R_5,R_3,S_3,S_1,T_2,T_1) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,Rn),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Sn_1),Tn_1))
=> ( R_5 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_3),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_2),N_1))
=> ( R_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_1),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_1),N_1))
=> Rn = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Sn_1),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Tn_1),N_1)) ) ) ) ) ) ).
fof(fact_4202_coprime__prime__dvd__ex,axiom,
! [X_1,Y_1] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,X_1),Y_1))
=> ? [P_4] :
( hBOOL(hAPP_nat_bool(prime,P_4))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),X_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),Y_1)) ) ) ).
fof(fact_4203_prime__power__exp,axiom,
! [X_1,K_2,N_1,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( N_1 != zero_zero_nat
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),K_2)
=> ? [I_2] : X_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),I_2) ) ) ) ).
fof(fact_4204_prime__factor,axiom,
! [N_1] :
( N_1 != one_one_nat
=> ? [P_4] :
( hBOOL(hAPP_nat_bool(prime,P_4))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),N_1)) ) ) ).
fof(fact_4205_divmod__nat__step,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> divmod_nat(M,N_1) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1)) ) ) ).
fof(fact_4206_bezout__prime,axiom,
! [A_3,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_3))
=> ? [X,Y] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),X) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),Y)),one_one_nat) ) ) ).
fof(fact_4207_not__prime__ex__mk,axiom,
! [N_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1))
& ~ hBOOL(hAPP_nat_bool(prime,N_1)) )
=> ? [M_1,K] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),K))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),N_1))
& N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),K) ) ) ).
fof(fact_4208_divmod__nat__zero,axiom,
! [M] : divmod_nat(M,zero_zero_nat) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),M) ).
fof(fact_4209_divmod__nat__base,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> divmod_nat(M,N_1) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),M) ) ).
fof(fact_4210_divmod__nat__div__mod,axiom,
! [M,N_1] : divmod_nat(M,N_1) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1)),hAPP_nat_nat(div_mod_nat(M),N_1)) ).
fof(fact_4211_divmod__nat__rel__mult2__eq,axiom,
! [C_1,A_3,B_2,Q_2,R_3] :
( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(A_3,B_2),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_2),R_3)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1))
=> hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(A_3,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Q_2),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),hAPP_nat_nat(div_mod_nat(Q_2),C_1))),R_3)))) ) ) ) ).
fof(fact_4212_divmod__nat__rel__mult1__eq,axiom,
! [A_3,B_2,C_1,Q_2,R_3] :
( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(B_2,C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_2),R_3)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1))
=> hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2),C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),Q_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),R_3)),C_1))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),R_3)),C_1)))) ) ) ).
fof(fact_4213_divmod__nat__rel__add1__eq,axiom,
! [B_2,Bq,Br,A_3,C_1,Aq,Ar] :
( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(A_3,C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Aq),Ar)))
=> ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(B_2,C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Bq),Br)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1))
=> hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_3),B_2),C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Aq),Bq)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ar),Br)),C_1))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ar),Br)),C_1)))) ) ) ) ).
fof(fact_4214_divmod__nat__rel__unique,axiom,
! [Qr_2,M,N_1,Qr_1] :
( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N_1),Qr_1))
=> ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N_1),Qr_2))
=> Qr_1 = Qr_2 ) ) ).
fof(fact_4215_divmod__nat__eq,axiom,
! [M,N_1,Qr_1] :
( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N_1),Qr_1))
=> divmod_nat(M,N_1) = Qr_1 ) ).
fof(fact_4216_divmod__nat__rel__divmod__nat,axiom,
! [M,N_1] : hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N_1),divmod_nat(M,N_1))) ).
fof(fact_4217_div__eq,axiom,
! [M,N_1,Q_2,R_3] :
( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_2),R_3)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1) = Q_2 ) ).
fof(fact_4218_mod__eq,axiom,
! [M,N_1,Q_2,R_3] :
( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_2),R_3)))
=> hAPP_nat_nat(div_mod_nat(M),N_1) = R_3 ) ).
fof(fact_4219_divmod__nat__rel,axiom,
! [M,N_1] : hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1)),hAPP_nat_nat(div_mod_nat(M),N_1)))) ).
fof(fact_4220_divmod__nat__rel__ex,axiom,
! [M,N_1] :
~ ! [Q_4,R_1] : ~ hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_4),R_1))) ).
fof(fact_4221_euclid,axiom,
! [N_1] :
? [P_4] :
( hBOOL(hAPP_nat_bool(prime,P_4))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),P_4)) ) ).
fof(fact_4222_div__pos__neg__1__number__of,axiom,
! [W] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),zero_zero_int))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,one_one_int),hAPP_int_int(number_number_of_int,W)) = hAPP_P1175774780nt_int(product_fst_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,one_one_int),hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W))))) ) ).
fof(fact_4223_div__int__def,axiom,
! [A_3,B_2] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_P1175774780nt_int(product_fst_int_int,divmod_int(A_3,B_2)) ).
fof(fact_4224_div__neg__pos,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_P1175774780nt_int(product_fst_int_int,negDivAlg(A_3,B_2)) ) ) ).
fof(fact_4225_div__pos__pos,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_P1175774780nt_int(product_fst_int_int,posDivAlg(A_3,B_2)) ) ) ).
fof(fact_4226_div__pos__pos__1__number__of,axiom,
! [W] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,one_one_int),hAPP_int_int(number_number_of_int,W)) = hAPP_P1175774780nt_int(product_fst_int_int,posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W))) ) ).
fof(fact_4227_div__pos__neg,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_P1175774780nt_int(product_fst_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)))) ) ) ).
fof(fact_4228_div__neg__neg,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_P1175774780nt_int(product_fst_int_int,hAPP_P1975530577nt_int(negateSnd,posDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)))) ) ) ).
fof(fact_4229_bezw_Osimps,axiom,
! [X_1,Y_1] :
( ( Y_1 = zero_zero_nat
=> bezw(X_1,Y_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),zero_zero_int) )
& ( Y_1 != zero_zero_nat
=> bezw(X_1,Y_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y_1,hAPP_nat_nat(div_mod_nat(X_1),Y_1)))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_P1175774780nt_int(product_fst_int_int,bezw(Y_1,hAPP_nat_nat(div_mod_nat(X_1),Y_1)))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y_1,hAPP_nat_nat(div_mod_nat(X_1),Y_1)))),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_1),Y_1))))) ) ) ).
fof(fact_4230_bezw__non__0,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Y_1))
=> bezw(X_1,Y_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y_1,hAPP_nat_nat(div_mod_nat(X_1),Y_1)))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_P1175774780nt_int(product_fst_int_int,bezw(Y_1,hAPP_nat_nat(div_mod_nat(X_1),Y_1)))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y_1,hAPP_nat_nat(div_mod_nat(X_1),Y_1)))),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_1),Y_1))))) ) ).
fof(fact_4231_mod__pos__neg__1__number__of,axiom,
! [W] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),zero_zero_int))
=> hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,W)) = hAPP_P1175774780nt_int(product_snd_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,one_one_int),hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W))))) ) ).
fof(fact_4232_mod__int__def,axiom,
! [A_3,B_2] : hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_P1175774780nt_int(product_snd_int_int,divmod_int(A_3,B_2)) ).
fof(fact_4233_mod__neg__pos,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_P1175774780nt_int(product_snd_int_int,negDivAlg(A_3,B_2)) ) ) ).
fof(fact_4234_mod__pos__pos,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2))
=> hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_P1175774780nt_int(product_snd_int_int,posDivAlg(A_3,B_2)) ) ) ).
fof(fact_4235_mod__pos__pos__1__number__of,axiom,
! [W] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
=> hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,W)) = hAPP_P1175774780nt_int(product_snd_int_int,posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W))) ) ).
fof(fact_4236_mod__pos__neg,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int))
=> hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_P1175774780nt_int(product_snd_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)))) ) ) ).
fof(fact_4237_mod__neg__neg,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int))
=> hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_P1175774780nt_int(product_snd_int_int,hAPP_P1975530577nt_int(negateSnd,posDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)))) ) ) ).
fof(fact_4238_div__nat__def,axiom,
! [M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N_1) = hAPP_P1654798560at_nat(product_fst_nat_nat,divmod_nat(M,N_1)) ).
fof(fact_4239_mod__nat__def,axiom,
! [M,N_1] : hAPP_nat_nat(div_mod_nat(M),N_1) = hAPP_P1654798560at_nat(product_snd_nat_nat,divmod_nat(M,N_1)) ).
fof(fact_4240_divmod__nat__rel__def,axiom,
! [Ma,N,Qr] :
( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(Ma,N),Qr))
<=> ( Ma = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_P1654798560at_nat(product_fst_nat_nat,Qr)),N)),hAPP_P1654798560at_nat(product_snd_nat_nat,Qr))
& ( N = zero_zero_nat
=> hAPP_P1654798560at_nat(product_fst_nat_nat,Qr) = zero_zero_nat )
& ( N != zero_zero_nat
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)),N)) ) )
& ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)),zero_zero_nat)) ) ) ) ) ) ) ).
fof(fact_4241_Bolzano__bisect__diff,axiom,
! [P_1,N,A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
=> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N))),hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)) ) ).
fof(fact_4242_Bolzano__bisect__le,axiom,
! [P_1,A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
=> ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2)))) ) ).
fof(fact_4243_Bolzano__bisect_Osimps_I1_J,axiom,
! [P_1,A,B] : hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),zero_zero_nat) = hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A),B) ).
fof(fact_4244_Bolzano__bisect__fst__le__Suc,axiom,
! [P_1,A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
=> ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))),hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),hAPP_nat_nat(suc,N_2))))) ) ).
fof(fact_4245_Bolzano__bisect__Suc__le__snd,axiom,
! [P_1,A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
=> ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),hAPP_nat_nat(suc,N_2)))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2)))) ) ).
fof(fact_4246_not__P__Bolzano__bisect,axiom,
! [N,A,B,P_1] :
( ! [A_2,B_4,C_3] :
( hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4)))
=> ( hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_4),C_3)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3))
=> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),C_3))) ) ) ) )
=> ( ~ hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A),B)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
=> ~ hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N))))) ) ) ) ).
fof(fact_4247_not__P__Bolzano__bisect_H,axiom,
! [A,B,P_1] :
( ! [A_2,B_4,C_3] :
( ( hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4)))
& hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_4),C_3)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3)) )
=> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),C_3))) )
=> ( ~ hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A),B)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
=> ! [N_2] : ~ hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))))) ) ) ) ).
fof(fact_4248_one__code__int__code,axiom,
one_on1684967323de_int = hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit1,pls)) ).
fof(fact_4249_Quickcheck__Narrowing_Oint__of__code,axiom,
! [K_2] :
( ( K_2 = zero_z891286103de_int
=> quickcheck_int_of(K_2) = zero_zero_int )
& ( K_2 != zero_z891286103de_int
=> ( ( hAPP_Q1762011733de_int(div_mo231679042de_int(K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_z891286103de_int
=> quickcheck_int_of(K_2) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),quickcheck_int_of(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) )
& ( hAPP_Q1762011733de_int(div_mo231679042de_int(K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_z891286103de_int
=> quickcheck_int_of(K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),quickcheck_int_of(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),one_one_int) ) ) ) ) ).
fof(fact_4250_DeMoivre2,axiom,
! [R_3,A_3,N_1] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,rcis(R_3,A_3)),N_1) = rcis(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_3),N_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),A_3)) ).
fof(fact_4251_less__eq__code__int__def,axiom,
! [N,Ma] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,N),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,quickcheck_int_of(N)),quickcheck_int_of(Ma))) ) ).
fof(fact_4252_less__code__int__def,axiom,
! [N,Ma] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,N),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,quickcheck_int_of(N)),quickcheck_int_of(Ma))) ) ).
fof(fact_4253_code__int_Oint__of__inject,axiom,
! [X_2,Y_2] :
( quickcheck_int_of(X_2) = quickcheck_int_of(Y_2)
<=> X_2 = Y_2 ) ).
fof(fact_4254_Quickcheck__Narrowing_Oint__of__inject,axiom,
! [K_1,L] :
( quickcheck_int_of(K_1) = quickcheck_int_of(L)
<=> K_1 = L ) ).
fof(fact_4255_int__of__number,axiom,
! [K_2] : quickcheck_int_of(hAPP_i1732201573de_int(number1226105091de_int,K_2)) = hAPP_int_int(number_number_of_int,K_2) ).
fof(fact_4256_zero__code__int__code,axiom,
zero_z891286103de_int = hAPP_i1732201573de_int(number1226105091de_int,pls) ).
fof(fact_4257_rcis__zero__arg,axiom,
! [R_3] : rcis(R_3,zero_zero_real) = of_real_complex(R_3) ).
fof(fact_4258_complex__mod__rcis,axiom,
! [R_3,A_3] : hAPP_complex_real(norm_norm_complex,rcis(R_3,A_3)) = hAPP_real_real(abs_abs_real,R_3) ).
fof(fact_4259_cis__rcis__eq,axiom,
! [A_3] : cis(A_3) = rcis(one_one_real,A_3) ).
fof(fact_4260_rcis__zero__mod,axiom,
! [A_3] : rcis(zero_zero_real,A_3) = zero_zero_complex ).
fof(fact_4261_Re__rcis,axiom,
! [R_3,A_3] : hAPP_complex_real(re,rcis(R_3,A_3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(cos,A_3)) ).
fof(fact_4262_Im__rcis,axiom,
! [R_3,A_3] : hAPP_complex_real(im,rcis(R_3,A_3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(sin,A_3)) ).
fof(fact_4263_rcis__mult,axiom,
! [R1,A_3,R2,B_2] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,rcis(R1,A_3)),rcis(R2,B_2)) = rcis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R1),R2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2)) ).
fof(fact_4264_rcis__divide,axiom,
! [R1,A_3,R2,B_2] : hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,rcis(R1,A_3)),rcis(R2,B_2)) = rcis(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,R1),R2),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),B_2)) ).
fof(fact_4265_rcis__def,axiom,
! [R_3,A_3] : rcis(R_3,A_3) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),cis(A_3)) ).
fof(fact_4266_rcis__inverse,axiom,
! [R_3,A_3] : hAPP_complex_complex(invers1449016382omplex,rcis(R_3,A_3)) = rcis(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),R_3),hAPP_real_real(uminus_uminus_real,A_3)) ).
fof(fact_4267_div__mod__code__int__def,axiom,
! [N_1,M] : quickc495462417de_int(N_1,M) = hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,N_1),M)),hAPP_Q1762011733de_int(div_mo231679042de_int(N_1),M)) ).
fof(fact_4268_nat__of__def,axiom,
! [I] : quickcheck_nat_of(I) = hAPP_int_nat(nat,quickcheck_int_of(I)) ).
fof(fact_4269_around__zero_Opinduct,axiom,
! [P_1,A0] :
( hBOOL(hAPP_int_bool(accp_int(quickc1265749348ro_rel),A0))
=> ( ! [I_2] :
( is_int(I_2)
=> ( hBOOL(hAPP_int_bool(accp_int(quickc1265749348ro_rel),I_2))
=> ( ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),zero_zero_int))
=> ( I_2 != zero_zero_int
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_1,I_2)) ) ) )
=> hBOOL(hAPP_int_bool(P_1,A0)) ) ) ).
fof(fact_4270_fact__int__def,axiom,
! [X_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hAPP_int_int(fact_fact_int,X_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(fact_fact_nat,hAPP_int_nat(nat,X_1))) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hAPP_int_int(fact_fact_int,X_1) = zero_zero_int ) ) ).
fof(fact_4271_fact__mono__int,axiom,
! [M,N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(fact_fact_int,M)),hAPP_int_int(fact_fact_int,N_1))) ) ).
fof(fact_4272_fact__1__int,axiom,
hAPP_int_int(fact_fact_int,one_one_int) = one_one_int ).
fof(fact_4273_fact__less__mono__int,axiom,
! [N_1,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(fact_fact_int,M)),hAPP_int_int(fact_fact_int,N_1))) ) ) ).
fof(fact_4274_fact__neg__int,axiom,
! [M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M),zero_zero_int))
=> hAPP_int_int(fact_fact_int,M) = zero_zero_int ) ).
fof(fact_4275_fact__0__int,axiom,
hAPP_int_int(fact_fact_int,zero_zero_int) = one_one_int ).
fof(fact_4276_fact__ge__zero__int,axiom,
! [M] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(fact_fact_int,M))) ).
fof(fact_4277_fact__nonzero__int,axiom,
! [N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1))
=> hAPP_int_int(fact_fact_int,N_1) != zero_zero_int ) ).
fof(fact_4278_transfer__nat__int__factorial__closure,axiom,
! [X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(fact_fact_int,X_1))) ) ).
fof(fact_4279_transfer__int__nat__factorial,axiom,
! [X_1] : hAPP_int_int(fact_fact_int,hAPP_nat_int(semiri1621563631at_int,X_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(fact_fact_nat,X_1)) ).
fof(fact_4280_fact__gt__zero__int,axiom,
! [N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(fact_fact_int,N_1))) ) ).
fof(fact_4281_fact__ge__one__int,axiom,
! [N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_int_int(fact_fact_int,N_1))) ) ).
fof(fact_4282_fact__mono__int__aux,axiom,
! [M,K_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(fact_fact_int,M)),hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),K_2)))) ) ).
fof(fact_4283_dvd__fact__int,axiom,
! [N_1,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),hAPP_int_int(fact_fact_int,N_1))) ) ) ).
fof(fact_4284_transfer__nat__int__factorial,axiom,
! [X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hAPP_nat_nat(fact_fact_nat,hAPP_int_nat(nat,X_1)) = hAPP_int_nat(nat,hAPP_int_int(fact_fact_int,X_1)) ) ).
fof(fact_4285_fact__less__mono__int__aux,axiom,
! [M,K_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(fact_fact_int,M)),hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),one_one_int)),K_2)))) ) ) ).
fof(fact_4286_fact__plus__one__int,axiom,
! [N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1))
=> hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),one_one_int)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),one_one_int)),hAPP_int_int(fact_fact_int,N_1)) ) ).
fof(fact_4287_fact__reduce__int,axiom,
! [N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
=> hAPP_int_int(fact_fact_int,N_1) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_1),hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N_1),one_one_int))) ) ).
fof(fact_4288_length__around__zero,axiom,
! [I] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),I))
=> hAPP_list_int_nat(size_size_list_int,quickc666637781d_zero(I)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_nat(nat,I))),one_one_nat) ) ).
fof(fact_4289_Bnor__prime,axiom,
! [A,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),P_5))
=> hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(bnorRset(A),P_5)) = hAPP_int_nat(nat,A) ) ) ).
fof(fact_4290_lemma__BOLZANO2,axiom,
! [P_1] :
( ( ! [A_2,B_4,C_3] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3))
& hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4)))
& hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_4),C_3))) )
=> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),C_3))) )
& ! [X] :
? [D_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
& ! [A_2,B_4] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B_4))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_4),A_2)),D_2)) )
=> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))) ) ) )
=> ! [A_2,B_4] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4))
=> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))) ) ) ).
fof(fact_4291_Bnor__mem__zle__swap,axiom,
! [Ma,A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B))
=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(bnorRset(A),Ma))) ) ).
fof(fact_4292_Bnor__mem__zle,axiom,
! [B,A,Ma] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(bnorRset(A),Ma)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) ) ).
fof(fact_4293_Bnor__fin,axiom,
! [A,Ma] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(bnorRset(A),Ma))) ).
fof(fact_4294_Bnor__mem__zg,axiom,
! [B,A,Ma] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(bnorRset(A),Ma)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) ) ).
fof(fact_4295_lemma__BOLZANO,axiom,
! [A,B,P_1] :
( ! [A_2,B_4,C_3] :
( ( hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4)))
& hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_4),C_3)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3)) )
=> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),C_3))) )
=> ( ! [X] :
? [D_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
& ! [A_2,B_4] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B_4))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_4),A_2)),D_2)) )
=> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))) ) )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
=> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A),B))) ) ) ) ).
fof(fact_4296_bezw__aux,axiom,
! [X_1,Y_1] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,bezw(X_1,Y_1))),hAPP_nat_int(semiri1621563631at_int,X_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(X_1,Y_1))),hAPP_nat_int(semiri1621563631at_int,Y_1))) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1)) ).
fof(fact_4297_gcd__dvd2__nat,axiom,
! [M,N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1)),N_1)) ).
fof(fact_4298_gcd__dvd1__nat,axiom,
! [M,N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1)),M)) ).
fof(fact_4299_gcd__dvd__prod__nat,axiom,
! [M,N_1,K_2] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1))) ).
fof(fact_4300_coprime__exp2__nat,axiom,
! [N_1,M,A_3,B_2] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) = one_one_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),M)) = one_one_nat ) ).
fof(fact_4301_gcd__exp__nat,axiom,
! [A_3,N_1,B_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),N_1) ).
fof(fact_4302_gcd__semilattice__nat_Oinf_Oidem,axiom,
! [A_3] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),A_3) = A_3 ).
fof(fact_4303_gcd__idem__nat,axiom,
! [X_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),X_1) = X_1 ).
fof(fact_4304_gcd__commute__nat,axiom,
! [A_3,B_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_2),A_3) ).
fof(fact_4305_gcd__semilattice__nat_Oinf__commute,axiom,
! [X_1,Y_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_1),X_1) ).
fof(fact_4306_gcd__semilattice__nat_Oinf_Oleft__idem,axiom,
! [A_3,B_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) ).
fof(fact_4307_gcd__semilattice__nat_Oinf__left__idem,axiom,
! [X_1,Y_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1) ).
fof(fact_4308_gcd__left__commute__nat,axiom,
! [B_2,A_3,C_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_2),C_1)) ).
fof(fact_4309_gcd__semilattice__nat_Oinf__left__commute,axiom,
! [X_1,Y_1,Z_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_1),Z_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Z_1)) ).
fof(fact_4310_gcd__assoc__nat,axiom,
! [A_3,B_2,C_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),C_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_2),C_1)) ).
fof(fact_4311_gcd__semilattice__nat_Oinf__assoc,axiom,
! [X_1,Y_1,Z_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1)),Z_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_1),Z_1)) ).
fof(fact_4312_gcd__semilattice__nat_Oless__infI2,axiom,
! [A_3,B_2,X_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),X_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),B_2)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),X_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))) ) ) ).
fof(fact_4313_gcd__semilattice__nat_Oless__infI1,axiom,
! [B_2,A_3,X_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),X_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),A_3)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),X_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))) ) ) ).
fof(fact_4314_gcd__semilattice__nat_Ole__infE,axiom,
! [X_1,A_3,B_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)))
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),A_3))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),B_2)) ) ) ).
fof(fact_4315_dvd__gcd__D2__nat,axiom,
! [K_2,M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N_1)) ) ).
fof(fact_4316_dvd__gcd__D1__nat,axiom,
! [K_2,M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M)) ) ).
fof(fact_4317_gcd__semilattice__nat_Oinf__mono,axiom,
! [B_2,D_3,A_3,C_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),D_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,C_1),D_3))) ) ) ).
fof(fact_4318_gcd__semilattice__nat_Oinf__greatest,axiom,
! [Z_1,X_1,Y_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_1),Z_1))) ) ) ).
fof(fact_4319_gcd__semilattice__nat_Ole__infI,axiom,
! [B_2,X_1,A_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),A_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),B_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))) ) ) ).
fof(fact_4320_gcd__greatest__nat,axiom,
! [N_1,K_2,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1))) ) ) ).
fof(fact_4321_gcd__proj2__if__dvd__nat,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1) = Y_1 ) ).
fof(fact_4322_gcd__proj1__if__dvd__nat,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1) = X_1 ) ).
fof(fact_4323_gcd__semilattice__nat_Ole__infI2,axiom,
! [A_3,B_2,X_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),X_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),X_1)) ) ).
fof(fact_4324_gcd__semilattice__nat_Ole__infI1,axiom,
! [B_2,A_3,X_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),X_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),X_1)) ) ).
fof(fact_4325_gcd__unique__nat,axiom,
! [B,D,A] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),A))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),B))
& ! [E] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),A))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),B)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),D)) ) )
<=> D = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ).
fof(fact_4326_gcd__semilattice__nat_Ole__inf__iff,axiom,
! [X_2,Y_2,Z_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_2),Z_2)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Z_2)) ) ) ).
fof(fact_4327_gcd__greatest__iff__nat,axiom,
! [K_1,Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ma),N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),Ma))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N)) ) ) ).
fof(fact_4328_gcd__semilattice__nat_Ole__iff__inf,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))
<=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_2),Y_2) = X_2 ) ).
fof(fact_4329_gcd__semilattice__nat_Oinf__le2,axiom,
! [X_1,Y_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1)),Y_1)) ).
fof(fact_4330_gcd__semilattice__nat_Oinf__le1,axiom,
! [X_1,Y_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1)),X_1)) ).
fof(fact_4331_gcd__add2__nat,axiom,
! [M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1) ).
fof(fact_4332_gcd__add1__nat,axiom,
! [M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1) ).
fof(fact_4333_gcd__mult__distrib__nat,axiom,
! [K_2,M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)) ).
fof(fact_4334_gcd__lcm__complete__lattice__nat_Oinf__bot__left,axiom,
! [X_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,one_one_nat),X_1) = one_one_nat ).
fof(fact_4335_gcd__1__nat,axiom,
! [M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),one_one_nat) = one_one_nat ).
fof(fact_4336_gcd__lcm__complete__lattice__nat_Oinf__bot__right,axiom,
! [X_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),one_one_nat) = one_one_nat ).
fof(fact_4337_gcd__0__left__nat,axiom,
! [X_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),X_1) = X_1 ).
fof(fact_4338_gcd__0__nat,axiom,
! [X_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),zero_zero_nat) = X_1 ).
fof(fact_4339_gcd__zero__nat,axiom,
! [Ma,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ma),N) = zero_zero_nat
<=> ( Ma = zero_zero_nat
& N = zero_zero_nat ) ) ).
fof(fact_4340_gcd__lcm__complete__lattice__nat_Oinf__eq__top__iff,axiom,
! [X_2,Y_2] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_2),Y_2) = zero_zero_nat
<=> ( X_2 = zero_zero_nat
& Y_2 = zero_zero_nat ) ) ).
fof(fact_4341_gcd__red__nat,axiom,
! [X_1,Y_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_1),hAPP_nat_nat(div_mod_nat(X_1),Y_1)) ).
fof(fact_4342_gcd__Suc__0,axiom,
! [M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),hAPP_nat_nat(suc,zero_zero_nat)) = hAPP_nat_nat(suc,zero_zero_nat) ).
fof(fact_4343_gcd__pos__nat,axiom,
! [Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ma),N)))
<=> ( Ma != zero_zero_nat
| N != zero_zero_nat ) ) ).
fof(fact_4344_gcd__le2__nat,axiom,
! [A_3,B_2] :
( B_2 != zero_zero_nat
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),B_2)) ) ).
fof(fact_4345_gcd__le1__nat,axiom,
! [B_2,A_3] :
( A_3 != zero_zero_nat
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),A_3)) ) ).
fof(fact_4346_coprime__Suc__nat,axiom,
! [N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(suc,N_1)),N_1) = one_one_nat ).
fof(fact_4347_gcd__add__mult__nat,axiom,
! [M,K_2,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1) ).
fof(fact_4348_coprime__common__divisor__nat,axiom,
! [X_1,A_3,B_2] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) = one_one_nat
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),A_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),B_2))
=> X_1 = one_one_nat ) ) ) ).
fof(fact_4349_coprime__nat,axiom,
! [A,B] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A),B) = one_one_nat
<=> ! [D_2] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),A))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),B)) )
<=> D_2 = one_one_nat ) ) ).
fof(fact_4350_coprime__plus__one__nat,axiom,
! [N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),N_1) = one_one_nat ).
fof(fact_4351_coprime__rmult__nat,axiom,
! [D_3,A_3,B_2] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)) = one_one_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),B_2) = one_one_nat ) ).
fof(fact_4352_coprime__lmult__nat,axiom,
! [D_3,A_3,B_2] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)) = one_one_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),A_3) = one_one_nat ) ).
fof(fact_4353_coprime__crossproduct__nat,axiom,
! [B,C,A,D] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A),D) = one_one_nat
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B),C) = one_one_nat
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),C) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),D)
<=> ( A = B
& C = D ) ) ) ) ).
fof(fact_4354_coprime__mult__nat,axiom,
! [B_2,D_3,A_3] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),A_3) = one_one_nat
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),B_2) = one_one_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)) = one_one_nat ) ) ).
fof(fact_4355_gcd__mult__cancel__nat,axiom,
! [M,K_2,N_1] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K_2),N_1) = one_one_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1) ) ).
fof(fact_4356_coprime__mul__eq__nat,axiom,
! [D,A,B] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),B)) = one_one_nat
<=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D),A) = one_one_nat
& hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D),B) = one_one_nat ) ) ).
fof(fact_4357_gcd__diff2__nat,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),M)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1) ) ).
fof(fact_4358_gcd__diff1__nat,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1) ) ).
fof(fact_4359_gcd__nat_Osimps,axiom,
! [X_1,Y_1] :
( ( Y_1 = zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1) = X_1 )
& ( Y_1 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_1),hAPP_nat_nat(div_mod_nat(X_1),Y_1)) ) ) ).
fof(fact_4360_gcd__non__0__nat,axiom,
! [X_1,Y_1] :
( Y_1 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_1),hAPP_nat_nat(div_mod_nat(X_1),Y_1)) ) ).
fof(fact_4361_coprime__exp__nat,axiom,
! [N_1,D_3,A_3] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),A_3) = one_one_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N_1)) = one_one_nat ) ).
fof(fact_4362_ceiling__real__of__nat,axiom,
! [N_1] : archim856651990g_real(hAPP_nat_real(real_nat,N_1)) = hAPP_nat_int(semiri1621563631at_int,N_1) ).
fof(fact_4363_natceiling__def,axiom,
! [X_1] : natceiling(X_1) = hAPP_int_nat(nat,archim856651990g_real(X_1)) ).
fof(fact_4364_gcd__coprime__nat,axiom,
! [B_5,A_5,A_3,B_2] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) != zero_zero_nat
=> ( A_3 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))
=> ( B_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_5),B_5) = one_one_nat ) ) ) ).
fof(fact_4365_coprime__minus__one__nat,axiom,
! [N_1] :
( N_1 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)),N_1) = one_one_nat ) ).
fof(fact_4366_divides__mult__nat,axiom,
! [N_1,M,R_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),R_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),R_3))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N_1) = one_one_nat
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)),R_3)) ) ) ) ).
fof(fact_4367_coprime__dvd__mult__iff__nat,axiom,
! [Ma,K_1,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K_1),N) = one_one_nat
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),Ma)) ) ) ).
fof(fact_4368_coprime__dvd__mult__nat,axiom,
! [M,K_2,N_1] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K_2),N_1) = one_one_nat
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M)) ) ) ).
fof(fact_4369_div__gcd__coprime__nat,axiom,
! [B_2,A_3] :
( ( A_3 != zero_zero_nat
| B_2 != zero_zero_nat )
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))) = one_one_nat ) ).
fof(fact_4370_invertible__coprime__nat,axiom,
! [X_1,Y_1,M] :
( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Y_1)),M) = one_one_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),M) = one_one_nat ) ).
fof(fact_4371_coprime__Suc__0__nat,axiom,
! [A,B] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A),B) = one_one_nat
<=> ! [D_2] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),A))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),B)) )
<=> D_2 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).
fof(fact_4372_gcd__coprime__exists__nat,axiom,
! [A_3,B_2] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) != zero_zero_nat
=> ? [A_4,B_3] :
( A_3 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))
& B_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))
& hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_4),B_3) = one_one_nat ) ) ).
fof(fact_4373_bezout__nat,axiom,
! [B_2,A_3] :
( A_3 != zero_zero_nat
=> ? [X,Y] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),X) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),Y)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) ) ).
fof(fact_4374_Cauchy__iff2,axiom,
! [X_3] :
( hBOOL(cauchy_real(X_3))
<=> ! [J_1] :
? [M_3] :
! [M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),M_1))
=> ! [N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),N_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(X_3,M_1)),hAPP_nat_real(X_3,N_2)))),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,J_1))))) ) ) ) ).
fof(fact_4375_ceiling__eq2,axiom,
! [N_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N_1)),one_one_real)))
=> archim856651990g_real(X_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),one_one_int) ) ) ).
fof(fact_4376_ln__powr__bound,axiom,
! [A_3,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),A_3))) ) ) ).
fof(fact_4377_powr__not__zero,axiom,
! [X_1,A_3] : hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3) != zero_zero_real ).
fof(fact_4378_powr__one__eq__one,axiom,
! [A_3] : hAPP_real_real(hAPP_r1250527377l_real(powr,one_one_real),A_3) = one_one_real ).
fof(fact_4379_floor__real__of__int,axiom,
! [N_1] :
( is_int(N_1)
=> archim1246769320r_real(hAPP_int_real(real_int,N_1)) = N_1 ) ).
fof(fact_4380_real__of__int__floor__cancel,axiom,
! [X_2] :
( hAPP_int_real(real_int,archim1246769320r_real(X_2)) = X_2
<=> ? [N_2] :
( is_int(N_2)
& X_2 = hAPP_int_real(real_int,N_2) ) ) ).
fof(fact_4381_powr__powr,axiom,
! [X_1,A_3,B_2] : hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),B_2) = hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),B_2)) ).
fof(fact_4382_powr__powr__swap,axiom,
! [X_1,A_3,B_2] : hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),B_2) = hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),B_2)),A_3) ).
fof(fact_4383_real__of__int__inject,axiom,
! [X_2,Y_2] :
( ( is_int(X_2)
& is_int(Y_2) )
=> ( hAPP_int_real(real_int,X_2) = hAPP_int_real(real_int,Y_2)
<=> X_2 = Y_2 ) ) ).
fof(fact_4384_real__of__int__zero,axiom,
hAPP_int_real(real_int,zero_zero_int) = zero_zero_real ).
fof(fact_4385_real__of__int__zero__cancel,axiom,
! [X_2] :
( is_int(X_2)
=> ( hAPP_int_real(real_int,X_2) = zero_zero_real
<=> X_2 = zero_zero_int ) ) ).
fof(fact_4386_real__of__int__le__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,X_2)),hAPP_int_real(real_int,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_2)) ) ).
fof(fact_4387_real__of__int__less__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,X_2)),hAPP_int_real(real_int,Y_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Y_2)) ) ).
fof(fact_4388_real__of__int__diff,axiom,
! [X_1,Y_1] : hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,X_1)),hAPP_int_real(real_int,Y_1)) ).
fof(fact_4389_power__real__of__int,axiom,
! [X_1,N_1] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(real_int,X_1)),N_1) = hAPP_int_real(real_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N_1)) ).
fof(fact_4390_real__of__int__power,axiom,
! [X_1,N_1] : hAPP_int_real(real_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N_1)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(real_int,X_1)),N_1) ).
fof(fact_4391_real__of__int__of__nat__eq,axiom,
! [N_1] : hAPP_int_real(real_int,hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_real(real_nat,N_1) ).
fof(fact_4392_real__of__int__add,axiom,
! [X_1,Y_1] : hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,X_1)),hAPP_int_real(real_int,Y_1)) ).
fof(fact_4393_real__number__of,axiom,
! [V_2] : hAPP_int_real(real_int,hAPP_int_int(number_number_of_int,V_2)) = hAPP_int_real(number267125858f_real,V_2) ).
fof(fact_4394_real__of__one,axiom,
hAPP_int_real(real_int,one_one_int) = one_one_real ).
fof(fact_4395_real__of__int__mult,axiom,
! [X_1,Y_1] : hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(real_int,X_1)),hAPP_int_real(real_int,Y_1)) ).
fof(fact_4396_ceiling__real__of__int,axiom,
! [N_1] :
( is_int(N_1)
=> archim856651990g_real(hAPP_int_real(real_int,N_1)) = N_1 ) ).
fof(fact_4397_real__of__int__ceiling__cancel,axiom,
! [X_2] :
( hAPP_int_real(real_int,archim856651990g_real(X_2)) = X_2
<=> ? [N_2] :
( is_int(N_2)
& X_2 = hAPP_int_real(real_int,N_2) ) ) ).
fof(fact_4398_real__of__int__minus,axiom,
! [X_1] : hAPP_int_real(real_int,hAPP_int_int(uminus_uminus_int,X_1)) = hAPP_real_real(uminus_uminus_real,hAPP_int_real(real_int,X_1)) ).
fof(fact_4399_real__of__int__abs,axiom,
! [X_1] : hAPP_int_real(real_int,hAPP_int_int(abs_abs_int,X_1)) = hAPP_real_real(abs_abs_real,hAPP_int_real(real_int,X_1)) ).
fof(fact_4400_real__of__int__floor__le,axiom,
! [R_3] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,archim1246769320r_real(R_3))),R_3)) ).
fof(fact_4401_real__of__int__ceiling__ge,axiom,
! [R_3] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R_3),hAPP_int_real(real_int,archim856651990g_real(R_3)))) ).
fof(fact_4402_powr__ge__pzero,axiom,
! [X_1,Y_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),Y_1))) ).
fof(fact_4403_powr__less__mono2__neg,axiom,
! [Y_1,X_1,A_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,Y_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3))) ) ) ) ).
fof(fact_4404_powr__less__mono2,axiom,
! [Y_1,X_1,A_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y_1),A_3))) ) ) ) ).
fof(fact_4405_powr__gt__zero,axiom,
! [X_1,A_3] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3))) ).
fof(fact_4406_powr__mono,axiom,
! [X_1,A_3,B_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_3),B_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),B_2))) ) ) ).
fof(fact_4407_powr__zero__eq__one,axiom,
! [X_1] : hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),zero_zero_real) = one_one_real ).
fof(fact_4408_powr__less__cancel,axiom,
! [X_1,A_3,B_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),B_2)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),B_2)) ) ) ).
fof(fact_4409_powr__less__mono,axiom,
! [X_1,A_3,B_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),B_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),B_2))) ) ) ).
fof(fact_4410_powr__less__cancel__iff,axiom,
! [A,B,X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_2),A)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_2),B)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) ) ) ).
fof(fact_4411_powr__add,axiom,
! [X_1,A_3,B_2] : hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),B_2)) ).
fof(fact_4412_powr__divide2,axiom,
! [X_1,A_3,B_2] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),B_2)) = hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),B_2)) ).
fof(fact_4413_powr__minus,axiom,
! [X_1,A_3] : hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),hAPP_real_real(uminus_uminus_real,A_3)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)) ).
fof(fact_4414_real__of__int__div4,axiom,
! [N_1,X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N_1),X_1))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N_1)),hAPP_int_real(real_int,X_1)))) ).
fof(fact_4415_floor__less__eq,axiom,
! [X_2,A] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim1246769320r_real(X_2)),A))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_int_real(real_int,A))) ) ).
fof(fact_4416_le__floor__eq,axiom,
! [A,X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),archim1246769320r_real(X_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A)),X_2)) ) ).
fof(fact_4417_real__le__floor,axiom,
! [A_3,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),archim1246769320r_real(X_1)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A_3)),X_1)) ) ).
fof(fact_4418_le__floor,axiom,
! [A_3,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A_3)),X_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),archim1246769320r_real(X_1))) ) ).
fof(fact_4419_floor__add,axiom,
! [X_1,A_3] : archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_int_real(real_int,A_3))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim1246769320r_real(X_1)),A_3) ).
fof(fact_4420_less__ceiling__eq,axiom,
! [A,X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),archim856651990g_real(X_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,A)),X_2)) ) ).
fof(fact_4421_ceiling__le__real,axiom,
! [X_1,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X_1)),A_3))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_int_real(real_int,A_3))) ) ).
fof(fact_4422_ceiling__le,axiom,
! [X_1,A_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_int_real(real_int,A_3)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X_1)),A_3)) ) ).
fof(fact_4423_ceiling__le__eq,axiom,
! [X_2,A] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X_2)),A))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),hAPP_int_real(real_int,A))) ) ).
fof(fact_4424_floor__power,axiom,
! [N_1,X_1] :
( X_1 = hAPP_int_real(real_int,archim1246769320r_real(X_1))
=> archim1246769320r_real(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,archim1246769320r_real(X_1)),N_1) ) ).
fof(fact_4425_floor__subtract,axiom,
! [X_1,A_3] : archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),hAPP_int_real(real_int,A_3))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim1246769320r_real(X_1)),A_3) ).
fof(fact_4426_ceiling__add,axiom,
! [X_1,A_3] : archim856651990g_real(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_int_real(real_int,A_3))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim856651990g_real(X_1)),A_3) ).
fof(fact_4427_floor__minus__real__of__int,axiom,
! [N_1] : archim1246769320r_real(hAPP_real_real(uminus_uminus_real,hAPP_int_real(real_int,N_1))) = hAPP_int_int(uminus_uminus_int,N_1) ).
fof(fact_4428_ceiling__subtract,axiom,
! [X_1,A_3] : archim856651990g_real(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),hAPP_int_real(real_int,A_3))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim856651990g_real(X_1)),A_3) ).
fof(fact_4429_powr__mono2,axiom,
! [Y_1,X_1,A_3] :
( 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_real,zero_zero_real),X_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,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y_1),A_3))) ) ) ) ).
fof(fact_4430_powr__mult,axiom,
! [A_3,Y_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1))
=> hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_1)),A_3) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y_1),A_3)) ) ) ).
fof(fact_4431_ge__one__powr__ge__zero,axiom,
! [A_3,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1))
=> ( 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,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3))) ) ) ).
fof(fact_4432_powr__le__cancel__iff,axiom,
! [A,B,X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_2),A)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_2),B)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) ) ) ).
fof(fact_4433_powr__one__gt__zero__iff,axiom,
! [X_2] :
( hAPP_real_real(hAPP_r1250527377l_real(powr,X_2),one_one_real) = X_2
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) ) ).
fof(fact_4434_powr__divide,axiom,
! [A_3,Y_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1))
=> hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),Y_1)),A_3) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y_1),A_3)) ) ) ).
fof(fact_4435_powr__minus__divide,axiom,
! [X_1,A_3] : hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),hAPP_real_real(uminus_uminus_real,A_3)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3)) ).
fof(fact_4436_powr__def,axiom,
! [X_1,A_3] : hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A_3) = hAPP_real_real(exp_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),hAPP_real_real(ln,X_1))) ).
fof(fact_4437_real__of__int__lt__zero__cancel__iff,axiom,
! [N] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N),zero_zero_int)) ) ).
fof(fact_4438_real__of__int__gt__zero__cancel__iff,axiom,
! [N] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_int_real(real_int,N)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) ) ).
fof(fact_4439_real__of__int__le__zero__cancel__iff,axiom,
! [N] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),zero_zero_int)) ) ).
fof(fact_4440_real__of__int__ge__zero__cancel__iff,axiom,
! [N] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_int_real(real_int,N)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N)) ) ).
fof(fact_4441_number__of__less__real__of__int__iff2,axiom,
! [Ma,N] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,Ma)),hAPP_int_real(number267125858f_real,N)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ma),hAPP_int_int(number_number_of_int,N))) ) ).
fof(fact_4442_number__of__less__real__of__int__iff,axiom,
! [N,Ma] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,N)),hAPP_int_real(real_int,Ma)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,N)),Ma)) ) ).
fof(fact_4443_number__of__le__real__of__int__iff2,axiom,
! [Ma,N] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,Ma)),hAPP_int_real(number267125858f_real,N)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ma),hAPP_int_int(number_number_of_int,N))) ) ).
fof(fact_4444_number__of__le__real__of__int__iff,axiom,
! [N,Ma] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,N)),hAPP_int_real(real_int,Ma)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,N)),Ma)) ) ).
fof(fact_4445_int__less__real__le,axiom,
! [N,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N),Ma))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real)),hAPP_int_real(real_int,Ma))) ) ).
fof(fact_4446_lemma__floor2,axiom,
! [N_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,X_1)),one_one_real)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),X_1)) ) ).
fof(fact_4447_int__le__real__less,axiom,
! [N,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),Ma))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,Ma)),one_one_real))) ) ).
fof(fact_4448_real__nat__eq__real,axiom,
! [X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hAPP_nat_real(real_nat,hAPP_int_nat(nat,X_1)) = hAPP_int_real(real_int,X_1) ) ).
fof(fact_4449_real__of__int__div,axiom,
! [N_1,D_3] :
( is_int(D_3)
=> ( D_3 != zero_zero_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),N_1))
=> hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N_1),D_3)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N_1)),hAPP_int_real(real_int,D_3)) ) ) ) ).
fof(fact_4450_real__of__int__floor__add__one__ge,axiom,
! [R_3] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R_3),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,archim1246769320r_real(R_3))),one_one_real))) ).
fof(fact_4451_floor__eq,axiom,
! [N_1,X_1] :
( is_int(N_1)
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N_1)),one_one_real)))
=> archim1246769320r_real(X_1) = N_1 ) ) ) ).
fof(fact_4452_real__of__int__floor__add__one__gt,axiom,
! [R_3] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,archim1246769320r_real(R_3))),one_one_real))) ).
fof(fact_4453_tan__periodic__int,axiom,
! [X_1,I] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(real_int,I)),pi))) = hAPP_real_real(tan,X_1) ).
fof(fact_4454_real__of__int__floor__ge__diff__one,axiom,
! [R_3] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_3),one_one_real)),hAPP_int_real(real_int,archim1246769320r_real(R_3)))) ).
fof(fact_4455_real__of__int__floor__gt__diff__one,axiom,
! [R_3] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_3),one_one_real)),hAPP_int_real(real_int,archim1246769320r_real(R_3)))) ).
fof(fact_4456_real__of__int__ceiling__le__add__one,axiom,
! [R_3] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,archim856651990g_real(R_3))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_3),one_one_real))) ).
fof(fact_4457_real__of__int__ceiling__diff__one__le,axiom,
! [R_3] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,archim856651990g_real(R_3))),one_one_real)),R_3)) ).
fof(fact_4458_powr__realpow,axiom,
! [N_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),hAPP_nat_real(real_nat,N_1)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1) ) ).
fof(fact_4459_ln__powr,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1))
=> hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_1),hAPP_real_real(ln,X_1)) ) ) ).
fof(fact_4460_log__powr__cancel,axiom,
! [Y_1,A_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
=> ( A_3 != one_one_real
=> hAPP_real_real(log(A_3),hAPP_real_real(hAPP_r1250527377l_real(powr,A_3),Y_1)) = Y_1 ) ) ).
fof(fact_4461_powr__log__cancel,axiom,
! [X_1,A_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
=> ( A_3 != one_one_real
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
=> hAPP_real_real(hAPP_r1250527377l_real(powr,A_3),hAPP_real_real(log(A_3),X_1)) = X_1 ) ) ) ).
fof(fact_4462_lemma__floor,axiom,
! [N_1,M,R_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,M)),R_3))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_real_real(hAPP_r1250527377l_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)) ) ) ).
fof(fact_4463_real__of__int__div2,axiom,
! [N_1,X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N_1)),hAPP_int_real(real_int,X_1))),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N_1),X_1))))) ).
fof(fact_4464_real__of__int__div3,axiom,
! [N_1,X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N_1)),hAPP_int_real(real_int,X_1))),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N_1),X_1)))),one_one_real)) ).
fof(fact_4465_real__of__int__div__aux,axiom,
! [X_1,D_3] :
( is_int(D_3)
=> ( D_3 != zero_zero_int
=> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,X_1)),hAPP_int_real(real_int,D_3)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),D_3))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,hAPP_int_int(div_mod_int(X_1),D_3))),hAPP_int_real(real_int,D_3))) ) ) ).
fof(fact_4466_floor__eq2,axiom,
! [N_1,X_1] :
( is_int(N_1)
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_1)),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N_1)),one_one_real)))
=> archim1246769320r_real(X_1) = N_1 ) ) ) ).
fof(fact_4467_less__floor__eq,axiom,
! [A,X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),archim1246769320r_real(X_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,A)),one_one_real)),X_2)) ) ).
fof(fact_4468_floor__le__eq,axiom,
! [X_2,A] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim1246769320r_real(X_2)),A))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,A)),one_one_real))) ) ).
fof(fact_4469_ceiling__eq3,axiom,
! [N_1,X_1] :
( is_int(N_1)
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,N_1)),one_one_real)),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_int_real(real_int,N_1)))
=> archim856651990g_real(X_1) = N_1 ) ) ) ).
fof(fact_4470_ceiling__less__eq,axiom,
! [X_2,A] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim856651990g_real(X_2)),A))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,A)),one_one_real))) ) ).
fof(fact_4471_le__ceiling__eq,axiom,
! [A,X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),archim856651990g_real(X_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,A)),one_one_real)),X_2)) ) ).
fof(fact_4472_log__powr,axiom,
! [B_2,Y_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_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(log(B_2),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),Y_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_1),hAPP_real_real(log(B_2),X_1)) ) ) ).
fof(fact_4473_ceiling__eq,axiom,
! [N_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N_1)),one_one_real)))
=> archim856651990g_real(X_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),one_one_int) ) ) ).
fof(fact_4474_powr__realpow2,axiom,
! [N_1,X_1] :
( 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_nat,zero_zero_nat),N_1))
=> ( ( X_1 = zero_zero_real
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1) = zero_zero_real )
& ( X_1 != zero_zero_real
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1) = hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),hAPP_nat_real(real_nat,N_1)) ) ) ) ) ).
fof(fact_4475_ln__powr__bound2,axiom,
! [A_3,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(ln,X_1)),A_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,A_3),A_3)),X_1))) ) ) ).
fof(fact_4476_reals__Archimedean__6c__int,axiom,
! [R_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),zero_zero_real))
=> ? [N_2] :
( is_int(N_2)
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_2)),R_3))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_2),one_one_int)))) ) ) ).
fof(fact_4477_reals__Archimedean__6b__int,axiom,
! [R_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_3))
=> ? [N_2] :
( is_int(N_2)
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_2)),R_3))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_2),one_one_int)))) ) ) ).
fof(fact_4478_real__lb__ub__int,axiom,
! [R_3] :
? [N_2] :
( is_int(N_2)
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_2)),R_3))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_2),one_one_int)))) ) ).
fof(fact_4479_coprime__pow,axiom,
! [C_1,N_1,A_3,B_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,C_1),N_1)
=> ? [R_1,S_2] :
( A_3 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,R_1),N_1)
& B_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,S_2),N_1) ) ) ) ).
fof(fact_4480_prime__power__mult,axiom,
! [X_1,Y_1,K_2,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Y_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),K_2)
=> ? [I_2,J_1] :
( X_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),I_2)
& Y_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),J_1) ) ) ) ).
fof(fact_4481_Euler__Fermat,axiom,
! [X_1,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
=> ( hAPP_int_int(legacy_zgcd(X_1),M) = one_one_int
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(phi,M))),one_one_int),M)) ) ) ).
fof(fact_4482_zgcd__zdvd1,axiom,
! [I,J_2] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(I),J_2)),I)) ).
fof(fact_4483_zgcd__zdvd2,axiom,
! [I,J_2] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(I),J_2)),J_2)) ).
fof(fact_4484_zgcd__eq,axiom,
! [M,N_1] : hAPP_int_int(legacy_zgcd(M),N_1) = hAPP_int_int(legacy_zgcd(N_1),hAPP_int_int(div_mod_int(M),N_1)) ).
fof(fact_4485_zgcd__1,axiom,
! [M] : hAPP_int_int(legacy_zgcd(M),one_one_int) = one_one_int ).
fof(fact_4486_zgcd__1__left,axiom,
! [M] : hAPP_int_int(legacy_zgcd(one_one_int),M) = one_one_int ).
fof(fact_4487_zgcd0,axiom,
! [I_1,J] :
( ( is_int(I_1)
& is_int(J) )
=> ( hAPP_int_int(legacy_zgcd(I_1),J) = zero_zero_int
<=> ( I_1 = zero_zero_int
& J = zero_zero_int ) ) ) ).
fof(fact_4488_zgcd__greatest__iff,axiom,
! [K_1,Ma,N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(legacy_zgcd(Ma),N)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),Ma))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N)) ) ) ).
fof(fact_4489_zgcd__greatest,axiom,
! [N_1,K_2,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(legacy_zgcd(M),N_1))) ) ) ).
fof(fact_4490_zgcd__assoc,axiom,
! [K_2,M,N_1] : hAPP_int_int(legacy_zgcd(hAPP_int_int(legacy_zgcd(K_2),M)),N_1) = hAPP_int_int(legacy_zgcd(K_2),hAPP_int_int(legacy_zgcd(M),N_1)) ).
fof(fact_4491_zgcd__left__commute,axiom,
! [K_2,M,N_1] : hAPP_int_int(legacy_zgcd(K_2),hAPP_int_int(legacy_zgcd(M),N_1)) = hAPP_int_int(legacy_zgcd(M),hAPP_int_int(legacy_zgcd(K_2),N_1)) ).
fof(fact_4492_zgcd__commute,axiom,
! [I,J_2] : hAPP_int_int(legacy_zgcd(I),J_2) = hAPP_int_int(legacy_zgcd(J_2),I) ).
fof(fact_4493_zgcd__power__distrib,axiom,
! [A_3,B_2,N_1] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(legacy_zgcd(A_3),B_2)),N_1) = hAPP_int_int(legacy_zgcd(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N_1)) ).
fof(fact_4494_zgcd__zminus,axiom,
! [I,J_2] : hAPP_int_int(legacy_zgcd(hAPP_int_int(uminus_uminus_int,I)),J_2) = hAPP_int_int(legacy_zgcd(I),J_2) ).
fof(fact_4495_zgcd__zminus2,axiom,
! [I,J_2] : hAPP_int_int(legacy_zgcd(I),hAPP_int_int(uminus_uminus_int,J_2)) = hAPP_int_int(legacy_zgcd(I),J_2) ).
fof(fact_4496_zgcd__pos,axiom,
! [I,J_2] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(legacy_zgcd(I),J_2))) ).
fof(fact_4497_zgcd__self,axiom,
! [M] :
( is_int(M)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M))
=> hAPP_int_int(legacy_zgcd(M),M) = M ) ) ).
fof(fact_4498_zgcd__geq__zero,axiom,
! [X_1,Y_1] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(legacy_zgcd(X_1),Y_1))) ).
fof(fact_4499_zgcd__zadd__zmult,axiom,
! [M,N_1,K_2] : hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_1),K_2))),N_1) = hAPP_int_int(legacy_zgcd(M),N_1) ).
fof(fact_4500_zgcd__zmult__cancel,axiom,
! [M,K_2,N_1] :
( hAPP_int_int(legacy_zgcd(K_2),N_1) = one_one_int
=> hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N_1) = hAPP_int_int(legacy_zgcd(M),N_1) ) ).
fof(fact_4501_zgcd__zgcd__zmult,axiom,
! [N_1,K_2,M] :
( hAPP_int_int(legacy_zgcd(K_2),M) = one_one_int
=> ( hAPP_int_int(legacy_zgcd(N_1),M) = one_one_int
=> hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1)),M) = one_one_int ) ) ).
fof(fact_4502_zgcd__zmult__distrib2__abs,axiom,
! [K_2,M,N_1] : hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(legacy_zgcd(M),N_1)) ).
fof(fact_4503_zgcd__0__left,axiom,
! [M] : hAPP_int_int(legacy_zgcd(zero_zero_int),M) = hAPP_int_int(abs_abs_int,M) ).
fof(fact_4504_zgcd__0,axiom,
! [M] : hAPP_int_int(legacy_zgcd(M),zero_zero_int) = hAPP_int_int(abs_abs_int,M) ).
fof(fact_4505_zgcd__1__power__left__distrib,axiom,
! [N_1,A_3,B_2] :
( hAPP_int_int(legacy_zgcd(A_3),B_2) = one_one_int
=> hAPP_int_int(legacy_zgcd(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N_1)),B_2) = one_one_int ) ).
fof(fact_4506_zgcd__1__power__distrib,axiom,
! [N_1,A_3,B_2] :
( hAPP_int_int(legacy_zgcd(A_3),B_2) = one_one_int
=> hAPP_int_int(legacy_zgcd(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N_1)) = one_one_int ) ).
fof(fact_4507_zgcd__zdvd__zgcd__zmult,axiom,
! [M,N_1,K_2] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(M),N_1)),hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N_1))) ).
fof(fact_4508_zgcd__zmult__eq__self,axiom,
! [N_1,K_2] :
( is_int(K_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
=> hAPP_int_int(legacy_zgcd(K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1)) = K_2 ) ) ).
fof(fact_4509_zgcd__zmult__distrib2,axiom,
! [M,N_1,K_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),hAPP_int_int(legacy_zgcd(M),N_1)) = hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1)) ) ).
fof(fact_4510_zgcd__zmult__eq__self2,axiom,
! [N_1,K_2] :
( is_int(K_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
=> hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1)),K_2) = K_2 ) ) ).
fof(fact_4511_zdvd__iff__zgcd,axiom,
! [N,Ma] :
( is_int(Ma)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),N))
<=> hAPP_int_int(legacy_zgcd(N),Ma) = Ma ) ) ) ).
fof(fact_4512_div__zgcd__relprime,axiom,
! [B_2,A_3] :
( ( is_int(B_2)
& is_int(A_3) )
=> ( ( A_3 != zero_zero_int
| B_2 != zero_zero_int )
=> hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(legacy_zgcd(A_3),B_2))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),hAPP_int_int(legacy_zgcd(A_3),B_2))) = one_one_int ) ) ).
fof(fact_4513_zgcd__non__0,axiom,
! [M,N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
=> hAPP_int_int(legacy_zgcd(M),N_1) = hAPP_int_int(legacy_zgcd(N_1),hAPP_int_int(div_mod_int(M),N_1)) ) ).
fof(fact_4514_zcong__zgcd__zmult__zmod,axiom,
! [N_1,A_3,B_2,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),N_1))
=> ( hAPP_int_int(legacy_zgcd(M),N_1) = one_one_int
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N_1))) ) ) ) ).
fof(fact_4515_zrelprime__zdvd__zmult,axiom,
! [M,N_1,K_2] :
( hAPP_int_int(legacy_zgcd(N_1),K_2) = one_one_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) ) ) ).
fof(fact_4516_zgcd__zmult__zdvd__zgcd,axiom,
! [M,K_2,N_1] :
( hAPP_int_int(legacy_zgcd(K_2),N_1) = one_one_int
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N_1)),hAPP_int_int(legacy_zgcd(M),N_1))) ) ).
fof(fact_4517_zrelprime__dvd__mult,axiom,
! [K_2,I,J_2] :
( hAPP_int_int(legacy_zgcd(I),J_2) = one_one_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),J_2)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),K_2)) ) ) ).
fof(fact_4518_zgcd__0__1__iff,axiom,
! [Ma] :
( hAPP_int_int(legacy_zgcd(zero_zero_int),Ma) = one_one_int
<=> hAPP_int_int(abs_abs_int,Ma) = one_one_int ) ).
fof(fact_4519_zgcd__code,axiom,
! [K_1,L] : hAPP_int_int(legacy_zgcd(K_1),L) = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,L),zero_zero_int)),K_1),hAPP_int_int(legacy_zgcd(L),hAPP_int_int(div_mod_int(hAPP_int_int(abs_abs_int,K_1)),hAPP_int_int(abs_abs_int,L))))) ).
fof(fact_4520_zprime__imp__zrelprime,axiom,
! [N_1,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),N_1))
=> hAPP_int_int(legacy_zgcd(N_1),P_3) = one_one_int ) ) ).
fof(fact_4521_zgcd1__iff__no__common__primedivisor,axiom,
! [A,B] :
( hAPP_int_int(legacy_zgcd(A),B) = one_one_int
<=> ~ ? [P_4] :
( is_int(P_4)
& hBOOL(hAPP_int_bool(zprime,P_4))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),A))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),B)) ) ) ).
fof(fact_4522_zgcd__zcong__zgcd,axiom,
! [B_2,A_3,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
=> ( hAPP_int_int(legacy_zgcd(A_3),M) = one_one_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
=> hAPP_int_int(legacy_zgcd(B_2),M) = one_one_int ) ) ) ).
fof(fact_4523_zless__zprime__imp__zrelprime,axiom,
! [N_1,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N_1),P_3))
=> hAPP_int_int(legacy_zgcd(N_1),P_3) = one_one_int ) ) ) ).
fof(fact_4524_zcong__cancel,axiom,
! [A,B,K_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ma))
=> ( hAPP_int_int(legacy_zgcd(K_1),Ma) = one_one_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),K_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),K_1)),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)) ) ) ) ).
fof(fact_4525_zcong__cancel2,axiom,
! [A,B,K_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ma))
=> ( hAPP_int_int(legacy_zgcd(K_1),Ma) = one_one_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),B)),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)) ) ) ) ).
fof(fact_4526_zrelprime__zdvd__zmult__aux,axiom,
! [M,N_1,K_2] :
( hAPP_int_int(legacy_zgcd(N_1),K_2) = one_one_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) ) ) ) ).
fof(fact_4527_Bnor__mem__if,axiom,
! [A,B,Ma] :
( hAPP_int_int(legacy_zgcd(B),Ma) = one_one_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(bnorRset(A),Ma))) ) ) ) ).
fof(fact_4528_int__relprime__odd__power__divisors,axiom,
! [A,B,C,X_2] :
( is_int(A)
=> ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,C),hAPP_int_nat(nat,X_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_int_nat(nat,X_2)))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd))
=> ( hAPP_int_int(legacy_zgcd(A),B) = one_one_int
=> ? [K] :
( is_int(K)
& A = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),hAPP_int_nat(nat,X_2)) ) ) ) ) ) ) ).
fof(fact_4529_xzgcd__correct,axiom,
! [Ma,K_1,N] :
( is_int(K_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
=> ( hAPP_int_int(legacy_zgcd(Ma),N) = K_1
<=> ? [S_2,T] :
( is_int(S_2)
& is_int(T)
& xzgcd(Ma,N) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,K_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_2),T)) ) ) ) ) ).
fof(fact_4530_int__relprime__power__divisors,axiom,
! [A_3,B_2,C_1,N_1] :
( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,C_1),N_1)
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_1))
=> ( hAPP_int_int(legacy_zgcd(A_3),B_2) = one_one_int
=> ? [K] :
( is_int(K)
& hAPP_int_int(abs_abs_int,A_3) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),N_1) ) ) ) ) ).
fof(fact_4531_xzgcd__correct__aux2,axiom,
! [M,N_1,R_5,R_3,S_3,S_1,T_2,T_1,K_2] :
( is_int(K_2)
=> ( ? [Sn,Tn] : xzgcda(M,N_1,R_5,R_3,S_3,S_1,T_2,T_1) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,K_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Sn),Tn))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_3))
=> hAPP_int_int(legacy_zgcd(R_5),R_3) = K_2 ) ) ) ).
fof(fact_4532_zcong__lineq__unique,axiom,
! [B_2,A_3,N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
=> ( hAPP_int_int(legacy_zgcd(A_3),N_1) = one_one_int
=> ? [X] :
( is_int(X)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),N_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),X)),B_2),N_1))
& ! [Y] :
( is_int(Y)
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),N_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Y)),B_2),N_1)) )
=> Y = X ) ) ) ) ) ).
fof(fact_4533_zcong__lineq__ex,axiom,
! [A_3,N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
=> ( hAPP_int_int(legacy_zgcd(A_3),N_1) = one_one_int
=> ? [X] :
( is_int(X)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),X)),one_one_int),N_1)) ) ) ) ).
fof(fact_4534_xzgcd__correct__aux1,axiom,
! [M,N_1,S_3,S_1,T_2,T_1,R_5,R_3,K_2] :
( hAPP_int_int(legacy_zgcd(R_5),R_3) = K_2
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_3))
=> ? [Sn,Tn] :
( is_int(Sn)
& is_int(Tn)
& xzgcda(M,N_1,R_5,R_3,S_3,S_1,T_2,T_1) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,K_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Sn),Tn)) ) ) ) ).
fof(fact_4535_int__triple__relprime__odd__power__divisors,axiom,
! [A,B,C,D,X_2] :
( ( is_int(A)
& is_int(B)
& is_int(C) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B)),C) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,D),hAPP_int_nat(nat,X_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_int_nat(nat,X_2)))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),zOdd))
=> ( hAPP_int_int(legacy_zgcd(A),B) = one_one_int
=> ( hAPP_int_int(legacy_zgcd(B),C) = one_one_int
=> ( hAPP_int_int(legacy_zgcd(C),A) = one_one_int
=> ? [K,L_1,M_1] :
( is_int(K)
& is_int(L_1)
& is_int(M_1)
& A = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),hAPP_int_nat(nat,X_2))
& B = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,L_1),hAPP_int_nat(nat,X_2))
& C = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_1),hAPP_int_nat(nat,X_2)) ) ) ) ) ) ) ) ) ).
fof(fact_4536_zgcd__ex__linear,axiom,
! [M,K_2,N_1] :
( is_int(K_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
=> ( hAPP_int_int(legacy_zgcd(M),N_1) = K_2
=> ? [S_2,T] :
( is_int(S_2)
& is_int(T)
& K_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T),N_1)) ) ) ) ) ).
fof(fact_4537_int__triple__relprime__power__divisors,axiom,
! [A_3,B_2,C_1,D_3,N_1] :
( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),C_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,D_3),N_1)
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_1))
=> ( hAPP_int_int(legacy_zgcd(A_3),B_2) = one_one_int
=> ( hAPP_int_int(legacy_zgcd(B_2),C_1) = one_one_int
=> ( hAPP_int_int(legacy_zgcd(C_1),A_3) = one_one_int
=> ? [K,L_1,M_1] :
( is_int(K)
& is_int(L_1)
& is_int(M_1)
& hAPP_int_int(abs_abs_int,A_3) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),N_1)
& hAPP_int_int(abs_abs_int,B_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,L_1),N_1)
& hAPP_int_int(abs_abs_int,C_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_1),N_1) ) ) ) ) ) ) ).
fof(fact_4538_make__zrelprime,axiom,
! [B_2,A_3] :
( ( is_int(B_2)
& is_int(A_3) )
=> ( ( A_3 != zero_zero_int
| B_2 != zero_zero_int )
=> ? [C_3,D_2] :
( is_int(C_3)
& is_int(D_2)
& A_3 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(legacy_zgcd(A_3),B_2)),C_3)
& B_2 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(legacy_zgcd(A_3),B_2)),D_2)
& hAPP_int_int(legacy_zgcd(C_3),D_2) = one_one_int ) ) ) ).
fof(fact_4539_pair__lessI2,axiom,
! [Sa,Ta,A,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Sa),Ta))
=> hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A),Sa)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B),Ta))),pair_less)) ) ) ).
fof(fact_4540_pair__lessI1,axiom,
! [Sa,Ta,A,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B))
=> hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A),Sa)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B),Ta))),pair_less)) ) ).
fof(fact_4541_pair__leqI2,axiom,
! [Sa,Ta,A,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Sa),Ta))
=> hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A),Sa)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B),Ta))),pair_leq)) ) ) ).
fof(fact_4542_pair__leqI1,axiom,
! [Sa,Ta,A,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B))
=> hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A),Sa)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B),Ta))),pair_leq)) ) ).
fof(fact_4543_polar__Ex,axiom,
! [Y_1,X_1] :
? [R_1,A_2] :
( X_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_2))
& Y_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_2)) ) ).
fof(fact_4544_dist__real__def,axiom,
! [X_1,Y_1] : hAPP_real_real(dist_dist_real(X_1),Y_1) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1)) ).
fof(fact_4545_dist__complex__def,axiom,
! [X_1,Y_1] : hAPP_complex_real(dist_dist_complex(X_1),Y_1) = hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_1),Y_1)) ).
fof(fact_4546_int__cases,axiom,
! [Z_1] :
( is_int(Z_1)
=> ( ! [N_2] : Z_1 != hAPP_nat_int(semiri1621563631at_int,N_2)
=> ~ ! [N_2] : Z_1 != hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2))) ) ) ).
fof(fact_4547_int__of__nat__induct,axiom,
! [Z_2,P_1] :
( ! [N_2] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,N_2)))
=> ( ! [N_2] : hBOOL(hAPP_int_bool(P_1,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2)))))
=> hBOOL(hAPP_int_bool(P_1,Z_2)) ) ) ).
fof(fact_4548_DERIV__nonpos__imp__nonincreasing,axiom,
! [F,A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
=> ( ! [X] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B)) )
=> ? [Y] :
( hBOOL(hAPP_real_bool(deriv_real(F,X),Y))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),zero_zero_real)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,B)),hAPP_real_real(F,A))) ) ) ).
fof(fact_4549_DERIV__nonneg__imp__nonincreasing,axiom,
! [F,A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
=> ( ! [X] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B)) )
=> ? [Y] :
( hBOOL(hAPP_real_bool(deriv_real(F,X),Y))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,A)),hAPP_real_real(F,B))) ) ) ).
fof(fact_4550_gcd__nat__induct,axiom,
! [Ma,N,P_1] :
( ! [M_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,M_1),zero_zero_nat))
=> ( ! [M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_2),hAPP_nat_nat(div_mod_nat(M_1),N_2)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,M_1),N_2)) ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Ma),N)) ) ) ).
fof(fact_4551_divides__fact,axiom,
! [N_1,P_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),P_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_3),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(fact,N_1))) ) ) ).
fof(fact_4552_mod__code__int__def,axiom,
! [N_1,M] : hAPP_Q1762011733de_int(div_mo231679042de_int(N_1),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(div_mod_int(quickcheck_int_of(N_1)),quickcheck_int_of(M))) ).
fof(fact_4553_Quickcheck__Narrowing_Oof__int__inject,axiom,
! [N,Ma] :
( ( is_int(N)
& is_int(Ma) )
=> ( hAPP_i1732201573de_int(quickcheck_of_int,N) = hAPP_i1732201573de_int(quickcheck_of_int,Ma)
<=> N = Ma ) ) ).
fof(fact_4554_int__of__inverse,axiom,
! [X_1] : hAPP_i1732201573de_int(quickcheck_of_int,quickcheck_int_of(X_1)) = X_1 ).
fof(fact_4555_of__int__int__of,axiom,
! [K_2] : hAPP_i1732201573de_int(quickcheck_of_int,quickcheck_int_of(K_2)) = K_2 ).
fof(fact_4556_int__of__of__int,axiom,
! [N_1] :
( is_int(N_1)
=> quickcheck_int_of(hAPP_i1732201573de_int(quickcheck_of_int,N_1)) = N_1 ) ).
fof(fact_4557_number__of__code__int__def,axiom,
number1226105091de_int = quickcheck_of_int ).
fof(fact_4558_fact__mono,axiom,
! [M,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,hAPP_nat_nat(fact,M)),hAPP_nat_nat(fact,N_1))) ) ).
fof(fact_4559_times__code__int__def,axiom,
! [N_1,M] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,N_1),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,quickcheck_int_of(N_1)),quickcheck_int_of(M))) ).
fof(fact_4560_plus__code__int__def,axiom,
! [N_1,M] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,N_1),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,quickcheck_int_of(N_1)),quickcheck_int_of(M))) ).
fof(fact_4561_minus__code__int__def,axiom,
! [N_1,M] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(minus_534354567de_int,N_1),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,quickcheck_int_of(N_1)),quickcheck_int_of(M))) ).
fof(fact_4562_zero__code__int__def,axiom,
zero_z891286103de_int = hAPP_i1732201573de_int(quickcheck_of_int,zero_zero_int) ).
fof(fact_4563_fact__lt,axiom,
! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(fact,N_1))) ).
fof(fact_4564_fact_Osimps_I1_J,axiom,
hAPP_nat_nat(fact,zero_zero_nat) = one_one_nat ).
fof(fact_4565_fact_Osimps_I2_J,axiom,
! [N_1] : hAPP_nat_nat(fact,hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,N_1)),hAPP_nat_nat(fact,N_1)) ).
fof(fact_4566_fact__le,axiom,
! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(fact,N_1))) ).
fof(fact_4567_one__code__int__def,axiom,
one_on1684967323de_int = hAPP_i1732201573de_int(quickcheck_of_int,one_one_int) ).
fof(fact_4568_div__code__int__def,axiom,
! [N_1,M] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,N_1),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,quickcheck_int_of(N_1)),quickcheck_int_of(M))) ).
fof(fact_4569_euclid__bound,axiom,
! [N_1] :
? [P_4] :
( hBOOL(hAPP_nat_bool(prime,P_4))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),P_4))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_4),hAPP_nat_nat(suc,hAPP_nat_nat(fact,N_1)))) ) ).
fof(fact_4570_weak__decr__stable,axiom,
! [F] :
( ! [I_2] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,I_2))),hAPP_nat_nat(F,I_2)))
=> ? [I_2] :
! [K] : hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),K)) = hAPP_nat_nat(F,I_2) ) ).
fof(fact_4571_norm__frac_Opinduct,axiom,
! [P_1,A0,A1] :
( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(norm_frac_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1)))
=> ( ! [A_2,B_4] :
( ( is_int(A_2)
& is_int(B_4) )
=> ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(norm_frac_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_4)))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_4),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(uminus_uminus_int,A_2)),hAPP_int_int(uminus_uminus_int,B_4))) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),B_4)) ) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1)) ) ) ).
fof(fact_4572_upto_Opinduct,axiom,
! [P_1,A0,A1] :
( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(upto_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1)))
=> ( ! [I_2,J_1] :
( ( is_int(I_2)
& is_int(J_1) )
=> ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(upto_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,I_2),J_1)))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),J_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int)),J_1)) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1)) ) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1)) ) ) ).
fof(fact_4573_small__lazy_H_Opinduct,axiom,
! [P_1,A0,A1] :
( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(lazy_small_lazy_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1)))
=> ( ! [D_2,I_2] :
( ( is_int(D_2)
& is_int(I_2) )
=> ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(lazy_small_lazy_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,D_2),I_2)))
=> ( ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,D_2),I_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,D_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,D_2),I_2)) ) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1)) ) ) ).
fof(fact_4574_norRRset__def,axiom,
! [Ma] : hAPP_i1948725293t_bool(norRRset,Ma) = hAPP_i1948725293t_bool(bnorRset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ma),one_one_int)),Ma) ).
fof(fact_4575_real__eq__of__int,axiom,
real_int = ring_1_of_int_real ).
fof(fact_4576_real__of__int__def,axiom,
real_int = ring_1_of_int_real ).
fof(fact_4577_complex__Re__of__int,axiom,
! [Z_1] : hAPP_complex_real(re,hAPP_int_complex(ring_11397209091omplex,Z_1)) = hAPP_int_real(ring_1_of_int_real,Z_1) ).
fof(fact_4578_complex__cnj__of__int,axiom,
! [Z_1] : hAPP_complex_complex(cnj,hAPP_int_complex(ring_11397209091omplex,Z_1)) = hAPP_int_complex(ring_11397209091omplex,Z_1) ).
fof(fact_4579_of__int__int__eq,axiom,
! [N_1] : hAPP_int_int(ring_1_of_int_int,hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,N_1) ).
fof(fact_4580_number__of__real__def,axiom,
! [X_1] : hAPP_int_real(number267125858f_real,X_1) = hAPP_int_real(ring_1_of_int_real,X_1) ).
fof(fact_4581_int__number__of__def,axiom,
! [W] : hAPP_int_int(number_number_of_int,W) = hAPP_int_int(ring_1_of_int_int,W) ).
fof(fact_4582_complex__number__of__def,axiom,
! [W] : hAPP_int_complex(number528085621omplex,W) = hAPP_int_complex(ring_11397209091omplex,W) ).
fof(fact_4583_complex__Im__of__int,axiom,
! [Z_1] : hAPP_complex_real(im,hAPP_int_complex(ring_11397209091omplex,Z_1)) = zero_zero_real ).
fof(fact_4584_phi__def,axiom,
! [Ma] : hAPP_int_nat(phi,Ma) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(norRRset,Ma)) ).
fof(fact_4585_norR__mem__unique,axiom,
! [A,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
=> ( hAPP_int_int(legacy_zgcd(A),Ma) = one_one_int
=> ? [X] :
( is_int(X)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),X),Ma))
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X),hAPP_i1948725293t_bool(norRRset,Ma)))
& ! [Y] :
( is_int(Y)
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),Y),Ma))
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y),hAPP_i1948725293t_bool(norRRset,Ma))) )
=> Y = X ) ) ) ) ) ).
fof(fact_4586_card__nor__eq__noX,axiom,
! [X_2,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( hAPP_int_int(legacy_zgcd(X_2),Ma) = one_one_int
=> hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(noXRRset(Ma),X_2)) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(norRRset,Ma)) ) ) ).
fof(fact_4587_noX__is__RRset,axiom,
! [X_2,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( hAPP_int_int(legacy_zgcd(X_2),Ma) = one_one_int
=> hBOOL(hAPP_int_bool(is_RRset(hAPP_i1948725293t_bool(noXRRset(Ma),X_2)),Ma)) ) ) ).
fof(fact_4588_RRset__gcd,axiom,
! [A,A_1,Ma] :
( hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1))
=> hAPP_int_int(legacy_zgcd(A),Ma) = one_one_int ) ) ).
fof(fact_4589_RRset__zcong__eq,axiom,
! [A,B,A_1,Ma] :
( ( is_int(A)
& is_int(B) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
=> ( hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),A_1))
=> A = B ) ) ) ) ) ) ).
fof(fact_4590_RRset2norRR__correct,axiom,
! [A,A_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
=> ( hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),hAPP_int_int(rRset2norRR(A_1,Ma),A)),Ma))
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(rRset2norRR(A_1,Ma),A)),hAPP_i1948725293t_bool(norRRset,Ma))) ) ) ) ) ).
fof(fact_4591_RRset2norRR__correct2,axiom,
! [A,A_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
=> ( hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(rRset2norRR(A_1,Ma),A)),hAPP_i1948725293t_bool(norRRset,Ma))) ) ) ) ).
fof(fact_4592_RRset2norRR__correct1,axiom,
! [A,A_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
=> ( hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),hAPP_int_int(rRset2norRR(A_1,Ma),A)),Ma)) ) ) ) ).
fof(fact_4593_is__RRset__def,axiom,
! [A_1,Ma] :
( hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
<=> ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A_1),rsetR(Ma)))
& hAPP_f957591787ol_nat(finite_card_int,A_1) = hAPP_int_nat(phi,Ma) ) ) ).
fof(fact_4594_RsetR__fin,axiom,
! [A_1,Ma] :
( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A_1),rsetR(Ma)))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_1)) ) ).
fof(fact_4595_Bnor__in__RsetR,axiom,
! [A,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),Ma))
=> hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,hAPP_i1948725293t_bool(bnorRset(A),Ma)),rsetR(Ma))) ) ).
fof(fact_4596_RRset2norRR__eq__norR,axiom,
! [A_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
=> ( hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
=> image_int_int(rRset2norRR(A_1,Ma),A_1) = hAPP_i1948725293t_bool(norRRset,Ma) ) ) ).
fof(fact_4597_Nitpick_Oint__lcm__def,axiom,
! [X_1,Y_1] : hAPP_int_int(int_lcm(X_1),Y_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(nat_lcm(hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,X_1))),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Y_1)))) ).
fof(fact_4598_ResSet__image,axiom,
! [F,A_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( hBOOL(hAPP_f448129468l_bool(resSet(Ma),A_1))
=> ( ! [X] :
( is_int(X)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X),A_1))
=> ! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(F,X)),hAPP_int_int(F,Xa)),Ma))
=> X = Xa ) ) ) ) )
=> hBOOL(hAPP_f448129468l_bool(resSet(Ma),image_int_int(F,A_1))) ) ) ) ).
fof(fact_4599_transfer__nat__int__set__relations_I5_J,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_1))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(semiri1621563631at_int,A_1)),image_nat_int(semiri1621563631at_int,B_1))) ) ).
fof(fact_4600_transfer__nat__int__set__relations_I4_J,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),B_1))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,image_nat_int(semiri1621563631at_int,A_1)),image_nat_int(semiri1621563631at_int,B_1))) ) ).
fof(fact_4601_transfer__nat__int__set__relations_I3_J,axiom,
! [A_1,B_1] :
( A_1 = B_1
<=> image_nat_int(semiri1621563631at_int,A_1) = image_nat_int(semiri1621563631at_int,B_1) ) ).
fof(fact_4602_transfer__nat__int__set__relations_I2_J,axiom,
! [X_2,A_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_1))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(semiri1621563631at_int,X_2)),image_nat_int(semiri1621563631at_int,A_1))) ) ).
fof(fact_4603_transfer__int__nat__set__return__embed,axiom,
! [A_1] : image_int_nat(nat,image_nat_int(semiri1621563631at_int,A_1)) = A_1 ).
fof(fact_4604_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J,axiom,
! [A_1] : hAPP_f22106695ol_nat(finite_card_nat,A_1) = hAPP_f957591787ol_nat(finite_card_int,image_nat_int(semiri1621563631at_int,A_1)) ).
fof(fact_4605_transfer__nat__int__set__relations_I1_J,axiom,
! [A_1] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
<=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(semiri1621563631at_int,A_1))) ) ).
fof(fact_4606_SetS__def,axiom,
! [A,P_5] : setS(A,P_5) = image_275383677t_bool(multInvPair(A,P_5),hAPP_i1948725293t_bool(sRStar,P_5)) ).
fof(fact_4607_infinite__nat__iff__unbounded__le,axiom,
! [S] :
( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S))
<=> ! [M_1] :
? [N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2))
& hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,N_2),S)) ) ) ).
fof(fact_4608_infinite__nat__iff__unbounded,axiom,
! [S] :
( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S))
<=> ! [M_1] :
? [N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
& hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,N_2),S)) ) ) ).
fof(fact_4609_Nitpick_Onat__lcm__def,axiom,
! [X_1,Y_1] : hAPP_nat_nat(nat_lcm(X_1),Y_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Y_1)),hAPP_nat_nat(nat_gcd(X_1),Y_1)) ).
fof(fact_4610_transfer__int__nat__set__relations_I4_J,axiom,
! [B_1,A_1] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
=> ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,B_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_1),B_1))
<=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,image_int_nat(nat,A_1)),image_int_nat(nat,B_1))) ) ) ) ).
fof(fact_4611_gcd__eq__nitpick__gcd,axiom,
! [X_1,Y_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1) = hAPP_nat_nat(nat_gcd(X_1),Y_1) ).
fof(fact_4612_nat__set__def,axiom,
! [S] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,S))
<=> ! [X] :
( is_int(X)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X),S))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) ) ) ) ).
fof(fact_4613_Nat__Transfer_Otransfer__nat__int__set__function__closures_I6_J,axiom,
! [X_2,A_1] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) ) ) ).
fof(fact_4614_Nat__Transfer_Otransfer__int__nat__set__function__closures_I5_J,axiom,
! [C_2] : hBOOL(hAPP_f448129468l_bool(nat_nat_set,image_nat_int(semiri1621563631at_int,C_2))) ).
fof(fact_4615_transfer__int__nat__set__relations_I3_J,axiom,
! [B_1,A_1] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
=> ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,B_1))
=> ( A_1 = B_1
<=> image_int_nat(nat,A_1) = image_int_nat(nat,B_1) ) ) ) ).
fof(fact_4616_nat__gcd_Osimps,axiom,
! [X_1,Y_1] :
( ( Y_1 = zero_zero_nat
=> hAPP_nat_nat(nat_gcd(X_1),Y_1) = X_1 )
& ( Y_1 != zero_zero_nat
=> hAPP_nat_nat(nat_gcd(X_1),Y_1) = hAPP_nat_nat(nat_gcd(Y_1),hAPP_nat_nat(div_mod_nat(X_1),Y_1)) ) ) ).
fof(fact_4617_transfer__nat__int__set__return__embed,axiom,
! [A_1] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
=> image_nat_int(semiri1621563631at_int,image_int_nat(nat,A_1)) = A_1 ) ).
fof(fact_4618_transfer__int__nat__set__relations_I1_J,axiom,
! [A_1] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_1))
<=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_int_nat(nat,A_1))) ) ) ).
fof(fact_4619_Nat__Transfer_Otransfer__int__nat__set__functions_I1_J,axiom,
! [A_1] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
=> hAPP_f957591787ol_nat(finite_card_int,A_1) = hAPP_f22106695ol_nat(finite_card_nat,image_int_nat(nat,A_1)) ) ).
fof(fact_4620_transfer__int__nat__set__relations_I5_J,axiom,
! [B_1,A_1] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
=> ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,B_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_1),B_1))
<=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,image_int_nat(nat,A_1)),image_int_nat(nat,B_1))) ) ) ) ).
fof(fact_4621_nat__gcd_Opsimps,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_P1555980039t_bool(accp_P490777396at_nat(nat_gcd_rel),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,X_2),Y_2)))
=> ( ( Y_2 = zero_zero_nat
=> hAPP_nat_nat(nat_gcd(X_2),Y_2) = X_2 )
& ( Y_2 != zero_zero_nat
=> hAPP_nat_nat(nat_gcd(X_2),Y_2) = hAPP_nat_nat(nat_gcd(Y_2),hAPP_nat_nat(div_mod_nat(X_2),Y_2)) ) ) ) ).
fof(fact_4622_Nitpick_Oint__gcd__def,axiom,
! [X_1,Y_1] : hAPP_int_int(hAPP_int_fun_int_int(int_gcd,X_1),Y_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(nat_gcd(hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,X_1))),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Y_1)))) ).
fof(fact_4623_nat__gcd_Opinduct,axiom,
! [P_1,A0,A1] :
( hBOOL(hAPP_P1555980039t_bool(accp_P490777396at_nat(nat_gcd_rel),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A0),A1)))
=> ( ! [X,Y] :
( hBOOL(hAPP_P1555980039t_bool(accp_P490777396at_nat(nat_gcd_rel),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,X),Y)))
=> ( ( Y != zero_zero_nat
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Y),hAPP_nat_nat(div_mod_nat(X),Y))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,X),Y)) ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,A0),A1)) ) ) ).
fof(fact_4624_finite__nat__set__iff__bounded__le,axiom,
! [N_4] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_4))
<=> ? [M_1] :
! [X] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),N_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),M_1)) ) ) ).
fof(fact_4625_finite__nat__set__iff__bounded,axiom,
! [N_4] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_4))
<=> ? [M_1] :
! [X] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),N_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),M_1)) ) ) ).
fof(fact_4626_bounded__nat__set__is__finite,axiom,
! [N,N_4] :
( ! [X] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),N_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),N)) )
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_4)) ) ).
fof(fact_4627_card__greaterThanLessThan__int,axiom,
! [L,U] : hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_gr1297742076an_int(L),U)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,L),one_one_int))) ).
fof(fact_4628_finite__greaterThanLessThan__int,axiom,
! [L,U] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_gr1297742076an_int(L),U))) ).
fof(fact_4629_Rats__abs__nat__div__natE,axiom,
! [X_2] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),field_1210416355s_real))
=> ~ ! [M_1,N_2] :
( N_2 != zero_zero_nat
=> ( hAPP_real_real(abs_abs_real,X_2) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,M_1)),hAPP_nat_real(real_nat,N_2))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_2) != one_one_nat ) ) ) ).
fof(fact_4630_finite__greaterThanLessThan,axiom,
! [L,U] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_gr660468384an_nat(L),U))) ).
fof(fact_4631_Rats__real__nat,axiom,
! [N] : hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_nat_real(real_nat,N)),field_1210416355s_real)) ).
fof(fact_4632_card__greaterThanLessThan,axiom,
! [L,U] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_gr660468384an_nat(L),U)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,U),hAPP_nat_nat(suc,L)) ).
fof(fact_4633_Rats__dense__in__nn__real,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2))
=> ? [X] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X),field_1210416355s_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y_2)) ) ) ) ).
fof(fact_4634_DERIV__isconst3,axiom,
! [F,Y_2,X_2,A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B))
=> ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),hAPP_r1134773055l_bool(ord_gr788844697n_real(A),B)))
=> ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Y_2),hAPP_r1134773055l_bool(ord_gr788844697n_real(A),B)))
=> ( ! [X] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X),hAPP_r1134773055l_bool(ord_gr788844697n_real(A),B)))
=> hBOOL(hAPP_real_bool(deriv_real(F,X),zero_zero_real)) )
=> hAPP_real_real(F,X_2) = hAPP_real_real(F,Y_2) ) ) ) ) ).
fof(fact_4635_iszero__rat,axiom,
! [K_1] :
( hBOOL(hAPP_rat_bool(iszero_rat,hAPP_int_rat(number_number_of_rat,K_1)))
<=> hBOOL(hAPP_int_bool(iszero_int,hAPP_int_int(number_number_of_int,K_1))) ) ).
fof(fact_4636_Rats__dense__in__real,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2))
=> ? [X] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X),field_1210416355s_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y_2)) ) ) ).
fof(fact_4637_Sup__greaterThanLessThan,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_2))
=> comple124823625p_real(hAPP_r1134773055l_bool(ord_gr788844697n_real(Y_2),X_2)) = X_2 ) ).
fof(fact_4638_Sup__upper__EX,axiom,
! [X_2,X_3] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),X_3))
=> ( ? [Z] :
! [X] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X),X_3))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Z)) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),comple124823625p_real(X_3))) ) ) ).
fof(fact_4639_SupInf_OSup__upper,axiom,
! [Z_2,X_2,X_3] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),X_3))
=> ( ! [X] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X),X_3))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Z_2)) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),comple124823625p_real(X_3))) ) ) ).
fof(fact_4640_Frct__code__post_I4_J,axiom,
! [K_2] : frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,K_2)),one_one_int)) = hAPP_int_rat(number_number_of_rat,K_2) ).
fof(fact_4641_Frct__code__post_I1_J,axiom,
! [K_2] : frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K_2)) = zero_zero_rat ).
fof(fact_4642_Frct__code__post_I2_J,axiom,
! [K_2] : frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,K_2),zero_zero_int)) = zero_zero_rat ).
fof(fact_4643_Frct__code__post_I3_J,axiom,
frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),one_one_int)) = one_one_rat ).
fof(fact_4644_Frct__code__post_I6_J,axiom,
! [K_2,L_2] : frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,K_2)),hAPP_int_int(number_number_of_int,L_2))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(number_number_of_rat,K_2)),hAPP_int_rat(number_number_of_rat,L_2)) ).
fof(fact_4645_Frct__code__post_I5_J,axiom,
! [K_2] : frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),hAPP_int_int(number_number_of_int,K_2))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,K_2)) ).
fof(fact_4646_Sup__eq__maximum,axiom,
! [Z_2,X_3] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Z_2),X_3))
=> ( ! [X] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X),X_3))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Z_2)) )
=> comple124823625p_real(X_3) = Z_2 ) ) ).
fof(fact_4647_SupInf_OSup__upper2,axiom,
! [Z_2,Y_2,X_2,X_3] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),X_3))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),X_2))
=> ( ! [X] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X),X_3))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Z_2)) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),comple124823625p_real(X_3))) ) ) ) ).
fof(fact_4648_abs__rat__def,axiom,
! [Q_2] :
( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Q_2),zero_zero_rat))
=> hAPP_rat_rat(abs_abs_rat,Q_2) = hAPP_rat_rat(uminus_uminus_rat,Q_2) )
& ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Q_2),zero_zero_rat))
=> hAPP_rat_rat(abs_abs_rat,Q_2) = Q_2 ) ) ).
fof(fact_4649_sgn__rat__def,axiom,
! [Q_2] :
( ( Q_2 = zero_zero_rat
=> hAPP_rat_rat(sgn_sgn_rat,Q_2) = zero_zero_rat )
& ( Q_2 != zero_zero_rat
=> ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Q_2))
=> hAPP_rat_rat(sgn_sgn_rat,Q_2) = one_one_rat )
& ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Q_2))
=> hAPP_rat_rat(sgn_sgn_rat,Q_2) = hAPP_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ) ).
fof(fact_4650_divide__rat__def,axiom,
! [Q_2,R_3] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,Q_2),R_3) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Q_2),hAPP_rat_rat(inverse_inverse_rat,R_3)) ).
fof(fact_4651_normalize__negative,axiom,
! [P_3,Q_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Q_2),zero_zero_int))
=> normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2)) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(uminus_uminus_int,P_3)),hAPP_int_int(uminus_uminus_int,Q_2))) ) ).
fof(fact_4652_rat__sgn__code,axiom,
! [P_3] : quotient_of(hAPP_rat_rat(sgn_sgn_rat,P_3)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(sgn_sgn_int,hAPP_P1175774780nt_int(product_fst_int_int,quotient_of(P_3)))),one_one_int) ).
fof(fact_4653_quotient__of__inject,axiom,
! [A_3,B_2] :
( quotient_of(A_3) = quotient_of(B_2)
=> A_3 = B_2 ) ).
fof(fact_4654_quotient__of__inject__eq,axiom,
! [A,B] :
( quotient_of(A) = quotient_of(B)
<=> A = B ) ).
fof(fact_4655_diff__rat__def,axiom,
! [Q_2,R_3] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Q_2),R_3) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Q_2),hAPP_rat_rat(uminus_uminus_rat,R_3)) ).
fof(fact_4656_less__rat__def,axiom,
! [Z_2,W_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_2),W_1))
<=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Z_2),W_1))
& Z_2 != W_1 ) ) ).
fof(fact_4657_quotient__of__denom__pos,axiom,
! [R_3,P_3,Q_2] :
( quotient_of(R_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2)
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_2)) ) ).
fof(fact_4658_rat__one__code,axiom,
quotient_of(one_one_rat) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),one_one_int) ).
fof(fact_4659_normalize__denom__pos,axiom,
! [R_3,P_3,Q_2] :
( normalize(R_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2)
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_2)) ) ).
fof(fact_4660_normalize__denom__zero,axiom,
! [P_3] : normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),zero_zero_int)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).
fof(fact_4661_normalize__crossproduct,axiom,
! [P_3,R_3,S_1,Q_2] :
( ( is_int(S_1)
& is_int(Q_2) )
=> ( Q_2 != zero_zero_int
=> ( S_1 != zero_zero_int
=> ( normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2)) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,R_3),S_1))
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,P_3),S_1) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),Q_2) ) ) ) ) ).
fof(fact_4662_rat__zero__code,axiom,
quotient_of(zero_zero_rat) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).
fof(fact_4663_quotient__of__number_I3_J,axiom,
! [K_2] : quotient_of(hAPP_int_rat(number_number_of_rat,K_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,K_2)),one_one_int) ).
fof(fact_4664_obtain__pos__sum,axiom,
! [R_3] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_3))
=> ~ ! [S_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),S_2))
=> ! [T] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),T))
=> R_3 != hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,S_2),T) ) ) ) ).
fof(fact_4665_ratrel__iff,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_f526364711l_bool(hAPP_P1062890741l_bool(member2143287562nt_int,hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,X_2),Y_2)),ratrel))
<=> ( hAPP_P1175774780nt_int(product_snd_int_int,X_2) != zero_zero_int
& hAPP_P1175774780nt_int(product_snd_int_int,Y_2) != zero_zero_int
& hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,X_2)),hAPP_P1175774780nt_int(product_snd_int_int,Y_2)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,Y_2)),hAPP_P1175774780nt_int(product_snd_int_int,X_2)) ) ) ).
fof(fact_4666_Fract__1__number__of,axiom,
! [K_2] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,one_one_int),hAPP_int_int(number_number_of_int,K_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,K_2)) ).
fof(fact_4667_Ratreal__number__of__quotient2,axiom,
! [R_3,S_1] : ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(number_number_of_rat,R_3)),hAPP_int_rat(number_number_of_rat,S_1))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(number267125858f_real,R_3)),hAPP_int_real(number267125858f_real,S_1)) ).
fof(fact_4668_quotient__of__eq,axiom,
! [A_3,B_2,P_3,Q_2] :
( quotient_of(hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2)
=> hAPP_int_rat(hAPP_int_fun_int_rat(fract,P_3),Q_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2) ) ).
fof(fact_4669_eq__rat_I2_J,axiom,
! [A_3] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),zero_zero_int) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),one_one_int) ).
fof(fact_4670_one__rat,axiom,
one_one_rat = hAPP_int_rat(hAPP_int_fun_int_rat(fract,one_one_int),one_one_int) ).
fof(fact_4671_rat__number__collapse_I2_J,axiom,
hAPP_int_rat(hAPP_int_fun_int_rat(fract,one_one_int),one_one_int) = one_one_rat ).
fof(fact_4672_rat__number__collapse_I4_J,axiom,
! [K_2] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),zero_zero_int) = zero_zero_rat ).
fof(fact_4673_rat__number__collapse_I1_J,axiom,
! [K_2] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),K_2) = zero_zero_rat ).
fof(fact_4674_eq__rat_I3_J,axiom,
! [A_3,C_1] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),A_3) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),C_1) ).
fof(fact_4675_of__nat__rat,axiom,
! [K_2] : hAPP_nat_rat(semiri151668891at_rat,K_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_nat_int(semiri1621563631at_int,K_2)),one_one_int) ).
fof(fact_4676_Fract__of__nat__eq,axiom,
! [K_2] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_nat_int(semiri1621563631at_int,K_2)),one_one_int) = hAPP_nat_rat(semiri151668891at_rat,K_2) ).
fof(fact_4677_of__int__rat,axiom,
! [K_2] : hAPP_int_rat(ring_1_of_int_rat,K_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),one_one_int) ).
fof(fact_4678_Fract__of__int__eq,axiom,
! [K_2] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),one_one_int) = hAPP_int_rat(ring_1_of_int_rat,K_2) ).
fof(fact_4679_Fract__of__int__quotient,axiom,
! [K_2,L_2] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),L_2) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(ring_1_of_int_rat,K_2)),hAPP_int_rat(ring_1_of_int_rat,L_2)) ).
fof(fact_4680_rat__number__of__def,axiom,
! [W] : hAPP_int_rat(number_number_of_rat,W) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,W),one_one_int) ).
fof(fact_4681_real__floor__code,axiom,
! [X_1] : archim1246769320r_real(ratreal(X_1)) = archim791455193or_rat(X_1) ).
fof(fact_4682_floor__Fract,axiom,
! [A_3,B_2] : archim791455193or_rat(hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) ).
fof(fact_4683_inverse__rat,axiom,
! [A_3,B_2] : hAPP_rat_rat(inverse_inverse_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,B_2),A_3) ).
fof(fact_4684_minus__rat__cancel,axiom,
! [A_3,B_2] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(uminus_uminus_int,A_3)),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2) ).
fof(fact_4685_mult__rat__cancel,axiom,
! [A_3,B_2,C_1] :
( is_int(C_1)
=> ( C_1 != zero_zero_int
=> hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),A_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),B_2)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2) ) ) ).
fof(fact_4686_eq__rat_I1_J,axiom,
! [A,C,D,B] :
( ( is_int(D)
& is_int(B) )
=> ( B != zero_zero_int
=> ( D != zero_zero_int
=> ( hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D)
<=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B) ) ) ) ) ).
fof(fact_4687_divide__rat,axiom,
! [A_3,B_2,C_1,D_3] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),D_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)) ).
fof(fact_4688_minus__rat,axiom,
! [A_3,B_2] : hAPP_rat_rat(uminus_uminus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(uminus_uminus_int,A_3)),B_2) ).
fof(fact_4689_mult__rat,axiom,
! [A_3,B_2,C_1,D_3] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),D_3)) ).
fof(fact_4690_abs__rat,axiom,
! [A_3,B_2] : hAPP_rat_rat(abs_abs_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(abs_abs_int,A_3)),hAPP_int_int(abs_abs_int,B_2)) ).
fof(fact_4691_normalize__eq,axiom,
! [A_3,B_2,P_3,Q_2] :
( normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_3),B_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2)
=> hAPP_int_rat(hAPP_int_fun_int_rat(fract,P_3),Q_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2) ) ).
fof(fact_4692_real__less__code,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,ratreal(X_2)),ratreal(Y_2)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_2),Y_2)) ) ).
fof(fact_4693_Ratreal__number__collapse_I1_J,axiom,
ratreal(zero_zero_rat) = zero_zero_real ).
fof(fact_4694_zero__real__code,axiom,
zero_zero_real = ratreal(zero_zero_rat) ).
fof(fact_4695_number__of__real__code,axiom,
! [K_2] : hAPP_int_real(number267125858f_real,K_2) = ratreal(hAPP_int_rat(number_number_of_rat,K_2)) ).
fof(fact_4696_Ratreal__number__collapse_I3_J,axiom,
! [K_2] : ratreal(hAPP_int_rat(number_number_of_rat,K_2)) = hAPP_int_real(number267125858f_real,K_2) ).
fof(fact_4697_Ratreal__number__collapse_I2_J,axiom,
ratreal(one_one_rat) = one_one_real ).
fof(fact_4698_one__real__code,axiom,
one_one_real = ratreal(one_one_rat) ).
fof(fact_4699_real__less__eq__code,axiom,
! [X_2,Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,ratreal(X_2)),ratreal(Y_2)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_2),Y_2)) ) ).
fof(fact_4700_real__plus__code,axiom,
! [X_1,Y_1] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,ratreal(X_1)),ratreal(Y_1)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_1),Y_1)) ).
fof(fact_4701_real__minus__code,axiom,
! [X_1,Y_1] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,ratreal(X_1)),ratreal(Y_1)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_1),Y_1)) ).
fof(fact_4702_real__uminus__code,axiom,
! [X_1] : hAPP_real_real(uminus_uminus_real,ratreal(X_1)) = ratreal(hAPP_rat_rat(uminus_uminus_rat,X_1)) ).
fof(fact_4703_real__divide__code,axiom,
! [X_1,Y_1] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,ratreal(X_1)),ratreal(Y_1)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,X_1),Y_1)) ).
fof(fact_4704_real__times__code,axiom,
! [X_1,Y_1] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,ratreal(X_1)),ratreal(Y_1)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_1),Y_1)) ).
fof(fact_4705_real__inverse__code,axiom,
! [X_1] : hAPP_real_real(inverse_inverse_real,ratreal(X_1)) = ratreal(hAPP_rat_rat(inverse_inverse_rat,X_1)) ).
fof(fact_4706_zero__rat,axiom,
zero_zero_rat = hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),one_one_int) ).
fof(fact_4707_rat__number__collapse_I3_J,axiom,
! [K_2] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(number_number_of_int,K_2)),one_one_int) = hAPP_int_rat(number_number_of_rat,K_2) ).
fof(fact_4708_rat__number__expand_I3_J,axiom,
! [K_2] : hAPP_int_rat(number_number_of_rat,K_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(number_number_of_int,K_2)),one_one_int) ).
fof(fact_4709_Fract__number__of__quotient,axiom,
! [K_2,L_2] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(number_number_of_int,K_2)),hAPP_int_int(number_number_of_int,L_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(number_number_of_rat,K_2)),hAPP_int_rat(number_number_of_rat,L_2)) ).
fof(fact_4710_quotient__of__Fract,axiom,
! [A_3,B_2] : quotient_of(hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_3),B_2)) ).
fof(fact_4711_Frct__def,axiom,
! [P_3] : frct(P_3) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3)) ).
fof(fact_4712_less__rat,axiom,
! [A,C,D,B] :
( ( is_int(D)
& is_int(B) )
=> ( B != zero_zero_int
=> ( D != zero_zero_int
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D)))) ) ) ) ) ).
fof(fact_4713_Ratreal__number__of__quotient,axiom,
! [R_3,S_1] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,ratreal(hAPP_int_rat(number_number_of_rat,R_3))),ratreal(hAPP_int_rat(number_number_of_rat,S_1))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(number267125858f_real,R_3)),hAPP_int_real(number267125858f_real,S_1)) ).
fof(fact_4714_sgn__rat,axiom,
! [A_3,B_2] : hAPP_rat_rat(sgn_sgn_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) = hAPP_int_rat(ring_1_of_int_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(sgn_sgn_int,A_3)),hAPP_int_int(sgn_sgn_int,B_2))) ).
fof(fact_4715_Fract__less__zero__iff,axiom,
! [A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),zero_zero_rat))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),zero_zero_int)) ) ) ).
fof(fact_4716_zero__less__Fract__iff,axiom,
! [A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) ) ) ).
fof(fact_4717_le__rat,axiom,
! [A,C,D,B] :
( ( is_int(D)
& is_int(B) )
=> ( B != zero_zero_int
=> ( D != zero_zero_int
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D)))) ) ) ) ) ).
fof(fact_4718_add__rat,axiom,
! [A_3,C_1,D_3,B_2] :
( ( is_int(D_3)
& is_int(B_2) )
=> ( B_2 != zero_zero_int
=> ( D_3 != zero_zero_int
=> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),D_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),D_3)) ) ) ) ).
fof(fact_4719_diff__rat,axiom,
! [A_3,C_1,D_3,B_2] :
( ( is_int(D_3)
& is_int(B_2) )
=> ( B_2 != zero_zero_int
=> ( D_3 != zero_zero_int
=> hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),D_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),D_3)) ) ) ) ).
fof(fact_4720_Fract__less__one__iff,axiom,
! [A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),one_one_rat))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) ) ) ).
fof(fact_4721_one__less__Fract__iff,axiom,
! [A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A)) ) ) ).
fof(fact_4722_Fract__add__one,axiom,
! [M,N_1] :
( is_int(N_1)
=> ( N_1 != zero_zero_int
=> hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),N_1)),N_1) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,M),N_1)),one_one_rat) ) ) ).
fof(fact_4723_Fract__le__zero__iff,axiom,
! [A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),zero_zero_rat))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int)) ) ) ).
fof(fact_4724_zero__le__Fract__iff,axiom,
! [A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).
fof(fact_4725_rat__floor__lemma,axiom,
! [A_3,B_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(ring_1_of_int_rat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(ring_1_of_int_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),one_one_int)))) ) ).
fof(fact_4726_one__le__Fract__iff,axiom,
! [A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) ) ) ).
fof(fact_4727_Fract__le__one__iff,axiom,
! [A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),one_one_rat))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B)) ) ) ).
fof(fact_4728_Rat__induct__pos,axiom,
! [Q_1,P_1] :
( ! [A_2,B_4] :
( ( is_int(A_2)
& is_int(B_4) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4))
=> hBOOL(hAPP_rat_bool(P_1,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_4))) ) )
=> hBOOL(hAPP_rat_bool(P_1,Q_1)) ) ).
fof(fact_4729_adjust__eq,axiom,
! [B,Q_1,R_2] : hAPP_P1975530577nt_int(adjust(B),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_2)) = hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,R_2),B))),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Q_1)),one_one_int)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,R_2),B))),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Q_1)),R_2)) ).
fof(fact_4730_finite__Collect__less__nat,axiom,
! [K_1] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),K_1)))) ).
fof(fact_4731_finite__Collect__le__nat,axiom,
! [K_1] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),K_1)))) ).
fof(fact_4732_transfer__morphism__int__nat,axiom,
hBOOL(hAPP_f54304608l_bool(nat_tr160667106at_int(semiri1621563631at_int),cOMBK_bool_nat(fTrue))) ).
fof(fact_4733_Nat__Transfer_Otransfer__int__nat__set__functions_I5_J,axiom,
! [P_1] : collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_1)) = image_nat_int(semiri1621563631at_int,collect_nat(hAPP_f1154658180t_bool(hAPP_f1361476881t_bool(cOMBB_int_bool_nat,P_1),semiri1621563631at_int))) ).
fof(fact_4734_int__card__bdd__int__set__l__le,axiom,
! [N] :
( is_int(N)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N))))) = N ) ) ).
fof(fact_4735_bdd__int__set__l__le__finite,axiom,
! [N] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N))))) ).
fof(fact_4736_bdd__int__set__l__finite,axiom,
! [N] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N))))) ).
fof(fact_4737_card__bdd__int__set__l__le,axiom,
! [N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
=> hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N)))) = hAPP_int_nat(nat,N) ) ).
fof(fact_4738_card__bdd__int__set__l,axiom,
! [N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
=> hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N)))) = hAPP_int_nat(nat,N) ) ).
fof(fact_4739_DERIV__cos__add,axiom,
! [K_1,Xa_1] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),K_1)),Xa_1),hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa_1),K_1))))) ).
fof(fact_4740_less__eq__Suc__le__raw,axiom,
! [X] : hAPP_n1699378549t_bool(ord_less_nat,X) = hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,X)) ).
fof(fact_4741_DERIV__sin__sin__mult,axiom,
! [X_2] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),sin),X_2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_2)),hAPP_real_real(sin,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_2)),hAPP_real_real(sin,X_2))))) ).
fof(fact_4742_DERIV__sin__add,axiom,
! [K_1,Xa_1] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),K_1)),Xa_1),hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa_1),K_1)))) ).
fof(fact_4743_bdd__nat__set__le__finite,axiom,
! [X_2] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),X_2)))) ).
fof(fact_4744_card__Collect__le__nat,axiom,
! [N] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),N))) = hAPP_nat_nat(suc,N) ).
fof(fact_4745_card__bdd__nat__set__le,axiom,
! [X_2] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),X_2))) = hAPP_nat_nat(suc,X_2) ).
fof(fact_4746_nat__number__of__Bit0,axiom,
! [W] : hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,W)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,W)),hAPP_int_nat(number_number_of_nat,W)) ).
fof(fact_4747_Nat__Transfer_Otransfer__int__nat__set__function__closures_I4_J,axiom,
! [P_1] : hBOOL(hAPP_f448129468l_bool(nat_nat_set,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_1)))) ).
fof(fact_4748_Nat__Transfer_Otransfer__nat__int__set__functions_I5_J,axiom,
! [P_1] : collect_nat(P_1) = image_int_nat(nat,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_f1448486952t_bool(hAPP_f696844121t_bool(cOMBB_nat_bool_int,P_1),nat)))) ).
fof(fact_4749_Bseq__realpow,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),one_one_real))
=> hBOOL(bseq_real(hAPP_r474017924t_real(power_power_real,X_2))) ) ) ).
fof(fact_4750_bdd__int__set__le__finite,axiom,
! [N] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N))))) ).
fof(fact_4751_diff__nat__eq__if,axiom,
! [Z_2,Z_3] :
( ( hBOOL(hAPP_int_bool(nat_neg,Z_3))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,Z_2)),hAPP_int_nat(nat,Z_3)) = hAPP_int_nat(nat,Z_2) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,Z_3))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,Z_2)),hAPP_int_nat(nat,Z_3)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),Z_3))),zero_zero_nat),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),Z_3))) ) ) ).
fof(fact_4752_finite__divisors__int,axiom,
! [I_1] :
( is_int(I_1)
=> ( I_1 != zero_zero_int
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,dvd_dvd_int),I_1)))) ) ) ).
fof(fact_4753_finite__divisors__nat,axiom,
! [Ma] :
( Ma != zero_zero_nat
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,dvd_dvd_nat),Ma)))) ) ).
fof(fact_4754_primes__infinite,axiom,
~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(prime))) ).
fof(fact_4755_cnj_OCauchy,axiom,
! [X_3] :
( hBOOL(cauchy_complex(X_3))
=> hBOOL(cauchy_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3))) ) ).
fof(fact_4756_Im_OCauchy,axiom,
! [X_3] :
( hBOOL(cauchy_complex(X_3))
=> hBOOL(cauchy_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3))) ) ).
fof(fact_4757_Re_OCauchy,axiom,
! [X_3] :
( hBOOL(cauchy_complex(X_3))
=> hBOOL(cauchy_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3))) ) ).
fof(fact_4758_Ints__real__of__nat,axiom,
! [N] : hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_nat_real(real_nat,N)),ring_1_Ints_real)) ).
fof(fact_4759_Ints__real__of__int,axiom,
! [X_2] : hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_int_real(real_int,X_2)),ring_1_Ints_real)) ).
fof(fact_4760_DERIV__fun__cos,axiom,
! [G,X_2,Ma] :
( hBOOL(hAPP_real_bool(deriv_real(G,X_2),Ma))
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),G),X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,hAPP_real_real(G,X_2)))),Ma))) ) ).
fof(fact_4761_DERIV__fun__sin,axiom,
! [G,X_2,Ma] :
( hBOOL(hAPP_real_bool(deriv_real(G,X_2),Ma))
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),G),X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,hAPP_real_real(G,X_2))),Ma))) ) ).
fof(fact_4762_bdd__nat__set__l__finite,axiom,
! [X_2] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),X_2)))) ).
fof(fact_4763_card__Collect__less__nat,axiom,
! [N] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),N))) = N ).
fof(fact_4764_card__bdd__nat__set__l,axiom,
! [X_2] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),X_2))) = X_2 ).
fof(fact_4765_finite__M__bounded__by__nat,axiom,
! [P_1,I_1] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),P_1)),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),I_1))))) ).
fof(fact_4766_zpower__number__of__even,axiom,
! [Z_1,W] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W))) ).
fof(fact_4767_card__less__Suc2,axiom,
! [I_1,M_2] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_2))
=> hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,hAPP_f66927821l_bool(hAPP_f2037783381l_bool(cOMBB_1146692694ol_nat,member_nat),suc)),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),I_1)))) = hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),hAPP_nat_nat(suc,I_1))))) ) ).
fof(fact_4768_card__less,axiom,
! [I_1,M_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_2))
=> hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),hAPP_nat_nat(suc,I_1))))) != zero_zero_nat ) ).
fof(fact_4769_card__less__Suc,axiom,
! [I_1,M_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_2))
=> hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,hAPP_f66927821l_bool(hAPP_f2037783381l_bool(cOMBB_1146692694ol_nat,member_nat),suc)),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),I_1))))) = hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),hAPP_nat_nat(suc,I_1))))) ) ).
fof(fact_4770_bdd__int__set__l__l__finite,axiom,
! [N] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N))))) ).
fof(fact_4771_noXRRset__def,axiom,
! [Ma,X_2] : hAPP_i1948725293t_bool(noXRRset(Ma),X_2) = image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,times_times_int),X_2),hAPP_i1948725293t_bool(norRRset,Ma)) ).
fof(fact_4772_DERIV__fun__pow,axiom,
! [N,G,X_2,Ma] :
( hBOOL(hAPP_real_bool(deriv_real(G,X_2),Ma))
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),G)),N),X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(G,X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))),Ma))) ) ).
fof(fact_4773_card__bdd__int__set__le,axiom,
! [N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
=> hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,N)),one_one_nat) ) ).
fof(fact_4774_int__card__bdd__int__set__l__l,axiom,
! [N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N))))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N),one_one_int) ) ).
fof(fact_4775_card__bdd__int__set__l__l,axiom,
! [N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
=> hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,N)),one_one_nat) ) ).
fof(fact_4776_DERIV__cos__cos__mult,axiom,
! [X_2] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),cos),X_2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_2))),hAPP_real_real(cos,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_2))),hAPP_real_real(cos,X_2))))) ).
fof(fact_4777_Suc__eq__number__of,axiom,
! [N,V] :
( hAPP_nat_nat(suc,N) = hAPP_int_nat(number_number_of_nat,V)
<=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
=> hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))) = N ) ) ) ).
fof(fact_4778_eq__number__of__Suc,axiom,
! [V,N] :
( hAPP_int_nat(number_number_of_nat,V) = hAPP_nat_nat(suc,N)
<=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
=> hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))) = N ) ) ) ).
fof(fact_4779_nat__number__of__Bit1,axiom,
! [W] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W)))
=> hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W)) = zero_zero_nat )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W)))
=> hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,W)),hAPP_int_nat(number_number_of_nat,W))) ) ) ).
fof(fact_4780_DERIV__pow,axiom,
! [N,X_2] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),N),X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat)))))) ).
fof(fact_4781_DERIV__cos__cos__mult2,axiom,
! [X_2] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),cos),X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,min))),hAPP_real_real(cos,X_2))),hAPP_real_real(sin,X_2)))) ).
fof(fact_4782_less__number__of__Suc,axiom,
! [V,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_nat_nat(suc,N)))
<=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),N)) ) ) ).
fof(fact_4783_less__Suc__number__of,axiom,
! [N,V] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,N)),hAPP_int_nat(number_number_of_nat,V)))
<=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))) ) ) ) ).
fof(fact_4784_le__Suc__number__of,axiom,
! [N,V] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),hAPP_int_nat(number_number_of_nat,V)))
<=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))) ) ) ) ).
fof(fact_4785_le__number__of__Suc,axiom,
! [V,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_nat_nat(suc,N)))
<=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),N)) ) ) ).
fof(fact_4786_SR__pos,axiom,
! [X_3,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_int_int(standardRes(Ma),X_3)),collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),Ma))))) ) ).
fof(fact_4787_RsetR__zmult__mono,axiom,
! [X_2,A_1,Ma] :
( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A_1),rsetR(Ma)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( hAPP_int_int(legacy_zgcd(X_2),Ma) = one_one_int
=> hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,times_times_int),X_2),A_1)),rsetR(Ma))) ) ) ) ).
fof(fact_4788_zpower__number__of__odd,axiom,
! [Z_1,W] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W))) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W))) = one_one_int ) ) ).
fof(fact_4789_DERIV__sin__sin__mult2,axiom,
! [X_2] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),sin),X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_2))),hAPP_real_real(sin,X_2)))) ).
fof(fact_4790_DERIV__log,axiom,
! [B,X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> hBOOL(hAPP_real_bool(deriv_real(log(B),X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(ln,B)),X_2)))) ) ).
fof(fact_4791_DERIV__cos__cos__mult3,axiom,
! [X_2] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),cos),X_2),hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_2))),hAPP_real_real(sin,X_2))))) ).
fof(fact_4792_DERIV__sin__circle__all__zero,axiom,
! [X] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X),zero_zero_real)) ).
fof(fact_4793_DERIV__sin__realpow2a,axiom,
! [X_2] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_2))),hAPP_real_real(sin,X_2)))) ).
fof(fact_4794_DERIV__sin__realpow2,axiom,
! [X_2] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_2)),hAPP_real_real(sin,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_2)),hAPP_real_real(sin,X_2))))) ).
fof(fact_4795_diff__nat__number__of,axiom,
! [V,V_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1)) = hAPP_int_nat(number_number_of_nat,V) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V),hAPP_int_int(uminus_uminus_int,V_1))))),zero_zero_nat),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V),hAPP_int_int(uminus_uminus_int,V_1))))) ) ) ).
fof(fact_4796_DERIV__cos__realpow2a,axiom,
! [X_2] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,min))),hAPP_real_real(cos,X_2))),hAPP_real_real(sin,X_2)))) ).
fof(fact_4797_DERIV__cos__realpow2b,axiom,
! [X_2] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2),hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_2))),hAPP_real_real(sin,X_2))))) ).
fof(fact_4798_DERIV__cos__realpow2,axiom,
! [X_2] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_2))),hAPP_real_real(cos,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_2))),hAPP_real_real(cos,X_2))))) ).
fof(fact_4799_lemma__DERIV__tan,axiom,
! [X_2] :
( hAPP_real_real(cos,X_2) != zero_zero_real
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real),sin)),cos),X_2),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).
fof(fact_4800_lemma__DERIV__sin__cos__add,axiom,
! [Y_2,X] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),Y_2)))),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),hAPP_real_real(cos,Y_2)))),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),hAPP_real_real(sin,Y_2)))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),Y_2)))),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),hAPP_real_real(cos,Y_2)))),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),hAPP_real_real(sin,Y_2)))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X),zero_zero_real)) ).
fof(fact_4801_DERIV__sin__circle__all,axiom,
! [X] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X))),hAPP_real_real(sin,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X))),hAPP_real_real(sin,X))))) ).
fof(fact_4802_lemma__DERIV__sin__cos__minus,axiom,
! [X] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),uminus_uminus_real))),sin))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),uminus_uminus_real))),cos))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X),zero_zero_real)) ).
fof(fact_4803_norm__frac_Opsimps,axiom,
! [A,B] :
( ( is_int(A)
& is_int(B) )
=> ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(norm_frac_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A),B)))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int))
=> norm_frac(A,B) = norm_frac(hAPP_int_int(uminus_uminus_int,A),hAPP_int_int(uminus_uminus_int,B)) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int))
=> ( ( ( A = zero_zero_int
| B = zero_zero_int )
=> norm_frac(A,B) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) )
& ( ~ ( A = zero_zero_int
| B = zero_zero_int )
=> norm_frac(A,B) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A),B))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A),B))) ) ) ) ) ) ) ).
fof(fact_4804_monoseq__arctan__series,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_2)),one_one_real))
=> hBOOL(monoseq_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))) ) ).
fof(fact_4805_monoseq__realpow,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),one_one_real))
=> hBOOL(monoseq_real(hAPP_r474017924t_real(power_power_real,X_2))) ) ) ).
fof(fact_4806_norm__frac_Osimps,axiom,
! [A_3,B_2] :
( ( is_int(A_3)
& is_int(B_2) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int))
=> norm_frac(A_3,B_2) = norm_frac(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int))
=> ( ( ( A_3 = zero_zero_int
| B_2 = zero_zero_int )
=> norm_frac(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) )
& ( ~ ( A_3 = zero_zero_int
| B_2 = zero_zero_int )
=> norm_frac(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A_3),B_2))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A_3),B_2))) ) ) ) ) ) ).
fof(fact_4807_arg__def,axiom,
! [Z_2] : arg(Z_2) = hilbert_Eps_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_f1402056515l_bool(hAPP_f824790735l_bool(cOMBB_real_bool_real,hAPP_r1134773055l_bool(fequal_real,hAPP_complex_real(re,hAPP_complex_complex(sgn_sgn_complex,Z_2)))),cos))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_f1402056515l_bool(hAPP_f824790735l_bool(cOMBB_real_bool_real,hAPP_r1134773055l_bool(fequal_real,hAPP_complex_real(im,hAPP_complex_complex(sgn_sgn_complex,Z_2)))),sin))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,pi)))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),pi))))) ).
fof(fact_4808_Union__SetS__setprod__prop1,axiom,
! [A,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,comple219730294t_bool(setS(A,P_5)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_5)) ) ) ) ) ).
fof(fact_4809_d22set__prod__zfact,axiom,
! [A] : big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(d22set,A)) = hAPP_int_int(zfact,A) ).
fof(fact_4810_Bnor__prod__zgcd,axiom,
! [A,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),Ma))
=> hAPP_int_int(legacy_zgcd(big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(bnorRset(A),Ma))),Ma) = one_one_int ) ).
fof(fact_4811_Bnor__prod__power,axiom,
! [A,Ma,X_2] :
( is_int(X_2)
=> ( X_2 != zero_zero_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),Ma))
=> big_co1548731110nt_int(cOMBI_int,image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,times_times_int),X_2),hAPP_i1948725293t_bool(bnorRset(A),Ma))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(bnorRset(A),Ma))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_2),hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(bnorRset(A),Ma)))) ) ) ) ).
fof(fact_4812_SetS__setprod__prop,axiom,
! [X_2,A,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A))
=> ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_2),setS(A,P_5)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,X_2)),A),P_5)) ) ) ) ) ) ).
fof(fact_4813_wset__zcong__prod__1,axiom,
! [A,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(wset(A),P_5))),one_one_int),P_5)) ) ) ) ).
fof(fact_4814_Union__SetS__setprod__prop2,axiom,
! [A,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5))
=> big_co1548731110nt_int(cOMBI_int,comple219730294t_bool(setS(A,P_5))) = hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)) ) ) ) ).
fof(fact_4815_transfer__nat__int__sum__prod__closure_I2_J,axiom,
! [F,A_1] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
=> ( ! [X] :
( is_int(X)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(F,X))) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),big_co1548731110nt_int(F,A_1))) ) ) ).
fof(fact_4816_arctan__series,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_2)),one_one_real))
=> hAPP_real_real(arctan,X_2) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))) ) ).
fof(fact_4817_DERIV__arctan__series,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X_2)),one_one_real))
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1122023986l_real(hAPP_f1786887945l_real(cOMBB_1896127834l_real,suminf_real),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))))),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))))),hAPP_f1501741374t_real(hAPP_f1135648319t_real(cOMBC_1329014627t_real,hAPP_f1163894827t_real(hAPP_f1998755145t_real(cOMBB_421464275l_real,cOMBB_nat_real_nat),power_power_real)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),X_2),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) ) ).
fof(fact_4818_sin__def,axiom,
! [X_2] : hAPP_real_real(sin,X_2) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_2))) ).
fof(fact_4819_lemma__sin__ext,axiom,
! [X] : hAPP_real_real(sin,X) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X))) ).
fof(fact_4820_cos__def,axiom,
! [X_2] : hAPP_real_real(cos,X_2) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X_2))) ).
fof(fact_4821_lemma__cos__ext,axiom,
! [X] : hAPP_real_real(cos,X) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X))) ).
fof(fact_4822_lemma__sin__minus,axiom,
! [X_2] : hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_2)) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,uminus_uminus_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_2)))) ).
fof(fact_4823_exp__first__two__terms,axiom,
! [X_2] : hAPP_real_real(exp_real,X_2) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_2)),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).
fof(fact_4824_pi__series,axiom,
hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),one_one_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))) ).
fof(fact_4825_ln__series,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
=> hAPP_real_real(ln,X_2) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),one_one_nat)))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),one_one_real))),suc))) ) ) ).
fof(fact_4826_summable__arctan__series,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_2)),one_one_real))
=> hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))))) ) ).
fof(fact_4827_summable__rabs__cancel,axiom,
! [F] :
( hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F)))
=> hBOOL(summable_real(F)) ) ).
fof(fact_4828_summable__rabs__comparison__test,axiom,
! [F,G] :
( ? [N_3] :
! [N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_nat_real(F,N_2))),hAPP_nat_real(G,N_2))) )
=> ( hBOOL(summable_real(G))
=> hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F))) ) ) ).
fof(fact_4829_summable__rabs,axiom,
! [F] :
( hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_f352196356l_real(suminf_real,F))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F)))) ) ).
fof(fact_4830_summable__sin,axiom,
! [X_2] : hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_2)))) ).
fof(fact_4831_summable__cos,axiom,
! [X_2] : hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X_2)))) ).
fof(fact_4832_rat__floor__code,axiom,
! [P_5] : archim791455193or_rat(P_5) = hAPP_P1175774780nt_int(produc1298267108nt_int(div_div_int),quotient_of(P_5)) ).
fof(fact_4833_rat__uminus__code,axiom,
! [P_5] : quotient_of(hAPP_rat_rat(uminus_uminus_rat,P_5)) = hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),uminus_uminus_int)),quotient_of(P_5)) ).
fof(fact_4834_summable__exp,axiom,
! [X_2] : hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_2)))) ).
fof(fact_4835_rat__abs__code,axiom,
! [P_5] : quotient_of(hAPP_rat_rat(abs_abs_rat,P_5)) = hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),abs_abs_int)),quotient_of(P_5)) ).
fof(fact_4836_divmod__int__rel__def,axiom,
! [A,B] : divmod_int_rel(A,B) = produc93441119t_bool(hAPP_f1048215610t_bool(hAPP_f472159229t_bool(cOMBC_1683390479t_bool,hAPP_f1545556668t_bool(hAPP_f85237525t_bool(cOMBB_389152643ol_int,cOMBS_int_bool_bool),hAPP_f1344929775l_bool(hAPP_f1830214065l_bool(cOMBB_793166024ol_int,hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj)),hAPP_f720654810t_bool(hAPP_f908327869t_bool(cOMBB_1290694363ol_int,hAPP_f1512942609t_bool(cOMBB_int_bool_int,hAPP_i1948725293t_bool(fequal_int,A))),hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,plus_plus_int),hAPP_int_fun_int_int(times_times_int,B)))))),hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,fimplies(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))),hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),B))))),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,fimplies(hAPP_bool_bool(fNot,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)))),hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,B))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),zero_zero_int)))))) ).
fof(fact_4837_sum2sq__def,axiom,
twoSqu1907779896sum2sq = produc1298267108nt_int(hAPP_f1760145644nt_int(hAPP_f808647893nt_int(cOMBC_736024425nt_int,hAPP_f836602773nt_int(hAPP_f888249301nt_int(cOMBB_1875755114nt_int,cOMBB_int_int_int),hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,plus_plus_int),hAPP_nat_fun_int_int(hAPP_f2105620693nt_int(cOMBC_int_nat_int,power_power_int),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_nat_fun_int_int(hAPP_f2105620693nt_int(cOMBC_int_nat_int,power_power_int),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_4838_rat__inverse__code,axiom,
! [P_5] : quotient_of(hAPP_rat_rat(inverse_inverse_rat,P_5)) = hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f400200275nt_int(hAPP_f1731308757nt_int(cOMBB_735549356nt_int,cOMBB_1389251822nt_int),hAPP_P621635040nt_int(hAPP_f1582586579nt_int(cOMBC_707974837nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),zero_zero_int))),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int)))),hAPP_f1574055885nt_int(hAPP_f434654626nt_int(cOMBS_2011279539nt_int,hAPP_f2094497995nt_int(hAPP_f534361363nt_int(cOMBB_1657320178nt_int,cOMBC_1964283556nt_int),hAPP_f2004554147nt_int(hAPP_f1228585167nt_int(cOMBB_1102189010nt_int,hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int)),hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,times_times_int),sgn_sgn_int)))),abs_abs_int))),quotient_of(P_5)) ).
fof(fact_4839_exp__tail__after__first__two__terms__summable,axiom,
! [X_2] : hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).
fof(fact_4840_adjust__def,axiom,
! [B] : adjust(B) = produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,cOMBS_1891347093nt_int),hAPP_f55007289nt_int(hAPP_f747475781nt_int(cOMBB_222539774nt_int,hAPP_f1311564564nt_int(cOMBS_1640193733nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_f429935748t_bool(hAPP_f1512942609t_bool(cOMBB_int_bool_int,hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int)),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,minus_minus_int),B))))),hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,cOMBB_1846624359nt_int),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,plus_plus_int),hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_int)))),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,minus_minus_int),B))))),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).
fof(fact_4841_DERIV__power__series_H,axiom,
! [X0,F,R_4] :
( ! [X] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X),hAPP_r1134773055l_bool(ord_gr788844697n_real(hAPP_real_real(uminus_uminus_real,R_4)),R_4)))
=> hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc)))),hAPP_r474017924t_real(power_power_real,X)))) )
=> ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X0),hAPP_r1134773055l_bool(ord_gr788844697n_real(hAPP_real_real(uminus_uminus_real,R_4)),R_4)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_4))
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1122023986l_real(hAPP_f1786887945l_real(cOMBB_1896127834l_real,suminf_real),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F))),hAPP_f1501741374t_real(hAPP_f1135648319t_real(cOMBC_1329014627t_real,hAPP_f1163894827t_real(hAPP_f1998755145t_real(cOMBB_421464275l_real,cOMBB_nat_real_nat),power_power_real)),suc))),X0),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc)))),hAPP_r474017924t_real(power_power_real,X0))))) ) ) ) ).
fof(fact_4842_Nitpick_OFrac__def,axiom,
frac = produc93441119t_bool(hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int)))),hAPP_i345030060t_bool(hAPP_f791698359t_bool(cOMBC_241247961t_bool,hAPP_f791698359t_bool(hAPP_f847817363t_bool(cOMBB_562416518ol_int,cOMBC_int_int_bool),hAPP_f725770969t_bool(hAPP_f983569339t_bool(cOMBB_550562460ol_int,hAPP_f1399575567t_bool(cOMBB_118231410ol_int,fequal_int)),int_gcd))),one_one_int))) ).
fof(fact_4843_cnj_Osuminf,axiom,
! [X_3] :
( hBOOL(summable_complex(X_3))
=> hAPP_complex_complex(cnj,hAPP_f213450978omplex(suminf_complex,X_3)) = hAPP_f213450978omplex(suminf_complex,hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3)) ) ).
fof(fact_4844_cnj_Osummable,axiom,
! [X_3] :
( hBOOL(summable_complex(X_3))
=> hBOOL(summable_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3))) ) ).
fof(fact_4845_Re_Osummable,axiom,
! [X_3] :
( hBOOL(summable_complex(X_3))
=> hBOOL(summable_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3))) ) ).
fof(fact_4846_Im_Osummable,axiom,
! [X_3] :
( hBOOL(summable_complex(X_3))
=> hBOOL(summable_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3))) ) ).
fof(fact_4847_Re_Osuminf,axiom,
! [X_3] :
( hBOOL(summable_complex(X_3))
=> hAPP_complex_real(re,hAPP_f213450978omplex(suminf_complex,X_3)) = hAPP_f352196356l_real(suminf_real,hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3)) ) ).
fof(fact_4848_Im_Osuminf,axiom,
! [X_3] :
( hBOOL(summable_complex(X_3))
=> hAPP_complex_real(im,hAPP_f213450978omplex(suminf_complex,X_3)) = hAPP_f352196356l_real(suminf_real,hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3)) ) ).
fof(fact_4849_ratrel__def,axiom,
ratrel = collec50511176nt_int(produc973645403t_bool(hAPP_f1781059817t_bool(hAPP_f125837377t_bool(cOMBS_1610675060t_bool,hAPP_f1965248563t_bool(hAPP_f1273270095t_bool(cOMBB_595786433nt_int,cOMBB_1344650379nt_int),hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,fconj),hAPP_f237669397t_bool(hAPP_f513299955t_bool(cOMBB_1344650379nt_int,fNot),hAPP_i119638232t_bool(hAPP_f2018027565t_bool(cOMBC_33652895t_bool,hAPP_f291087797t_bool(hAPP_f214467655t_bool(cOMBB_182850971nt_int,fequal_int),product_snd_int_int)),zero_zero_int))))),hAPP_f1781059817t_bool(hAPP_f326728547t_bool(cOMBB_2086486755nt_int,hAPP_f461489489t_bool(cOMBS_400904860l_bool,hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,fconj),hAPP_f237669397t_bool(hAPP_f513299955t_bool(cOMBB_1344650379nt_int,fNot),hAPP_i119638232t_bool(hAPP_f2018027565t_bool(cOMBC_33652895t_bool,hAPP_f291087797t_bool(hAPP_f214467655t_bool(cOMBB_182850971nt_int,fequal_int),product_snd_int_int)),zero_zero_int))))),hAPP_f1187443546t_bool(hAPP_f11709033t_bool(cOMBS_1676841879t_bool,hAPP_f223623489t_bool(hAPP_f1815301987t_bool(cOMBB_650041899nt_int,cOMBS_480885903t_bool),hAPP_f1198736473t_bool(hAPP_f2042071795t_bool(cOMBB_773384995nt_int,hAPP_f214467655t_bool(cOMBB_182850971nt_int,fequal_int)),hAPP_f167782601nt_int(hAPP_f1769710159nt_int(cOMBC_2037038898nt_int,hAPP_f301246617nt_int(hAPP_f1151807883nt_int(cOMBB_563965291nt_int,cOMBB_47643171nt_int),hAPP_f1855025978nt_int(hAPP_f1038672605nt_int(cOMBB_1506897658nt_int,times_times_int),product_fst_int_int))),product_snd_int_int)))),hAPP_f167782601nt_int(hAPP_f400538607nt_int(cOMBB_866192175nt_int,hAPP_f578362681nt_int(cOMBC_1417754532nt_int,hAPP_f1855025978nt_int(hAPP_f1038672605nt_int(cOMBB_1506897658nt_int,times_times_int),product_fst_int_int))),product_snd_int_int)))))) ).
fof(fact_4850_divmod__nat__if,axiom,
! [Ma,N] :
( ( ( N = zero_zero_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) )
=> divmod_nat(Ma,N) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),Ma) )
& ( ~ ( N = zero_zero_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) )
=> divmod_nat(Ma,N) = hAPP_P1586233937at_nat(produc1391996073at_nat(hAPP_f837737749at_nat(hAPP_f999082675at_nat(cOMBB_348438404at_nat,product_Pair_nat_nat),suc)),divmod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),N),N)) ) ) ).
fof(fact_4851_Bolzano__bisect_Osimps_I2_J,axiom,
! [P_1,A,B,N] : hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),hAPP_nat_nat(suc,N)) = hAPP_P1860904029l_real(produc595218619l_real(hAPP_f2072293513l_real(hAPP_f1763467369l_real(cOMBS_163651580l_real,hAPP_f1663701695l_real(hAPP_f774948957l_real(cOMBB_3698607l_real,cOMBS_902912708l_real),hAPP_f324002831l_real(hAPP_f677014621l_real(cOMBS_107495110l_real,hAPP_f1161935955l_real(hAPP_f328572565l_real(cOMBB_520243667l_real,cOMBS_794971624l_real),hAPP_f517738447l_real(hAPP_f1879982819l_real(cOMBB_731595089l_real,hAPP_f1076489843l_real(cOMBB_415306361l_real,if_Pro313124157l_real)),hAPP_f1884005689l_bool(hAPP_f172451459l_bool(cOMBB_421479067l_real,hAPP_f940061907l_bool(cOMBB_1795223523l_real,P_1)),hAPP_f320596107l_real(hAPP_f850851693l_real(cOMBS_193881978l_real,hAPP_f1119546819l_real(hAPP_f555557763l_real(cOMBB_422423973l_real,cOMBB_2107848501l_real),produc865579683l_real)),hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,cOMBC_real_real_real),hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real)),plus_plus_real))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1591894897l_real(hAPP_f644105375l_real(cOMBC_467724138l_real,hAPP_f1108539711l_real(hAPP_f1109256153l_real(cOMBB_44084907l_real,cOMBS_511527878l_real),hAPP_f823702159l_real(hAPP_f1040184293l_real(cOMBB_557258983l_real,hAPP_f2114902373l_real(cOMBB_237479429l_real,produc865579683l_real)),hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,cOMBC_real_real_real),hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real)),plus_plus_real))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),cOMBI_real)))),hAPP_f320596107l_real(hAPP_f850851693l_real(cOMBS_193881978l_real,hAPP_f1119546819l_real(hAPP_f555557763l_real(cOMBB_422423973l_real,cOMBB_2107848501l_real),produc865579683l_real)),hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,cOMBC_real_real_real),hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real)),plus_plus_real))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N)) ).
fof(fact_4852_suminf__exist__split,axiom,
! [A,R_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_2))
=> ( hBOOL(summable_real(A))
=> ? [N_3] :
! [N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_f352196356l_real(suminf_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,A),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),N_2))))),R_2)) ) ) ) ).
fof(fact_4853_divmod__int__pdivmod,axiom,
! [L,K_1] :
( ( is_int(L)
& is_int(K_1) )
=> ( ( K_1 = zero_zero_int
=> divmod_int(K_1,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int) )
& ( K_1 != zero_zero_int
=> ( ( L = zero_zero_int
=> divmod_int(K_1,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K_1) )
& ( L != zero_zero_int
=> divmod_int(K_1,L) = hAPP_P1975530577nt_int(produc713050258nt_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(sgn_sgn_int,L))),hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fdisj(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),L)),hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1)),hAPP_bool_bool(hAPP_b589554111l_bool(fconj,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,L),zero_zero_int)),hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int)))),pdivmod(K_1,L)),hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,cOMBS_1891347093nt_int),hAPP_f1148171384nt_int(hAPP_f334321603nt_int(cOMBB_1484610687nt_int,hAPP_f1582586579nt_int(cOMBC_707974837nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),zero_zero_int)))),hAPP_i1584592887nt_int(hAPP_f465821005nt_int(cOMBC_1964283556nt_int,hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),uminus_uminus_int)),zero_zero_int)))),hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,cOMBB_1846624359nt_int),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,minus_minus_int),uminus_uminus_int)),one_one_int)))),hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(abs_abs_int,L))))),pdivmod(K_1,L)))) ) ) ) ) ) ).
fof(fact_4854_negateSnd__def,axiom,
negateSnd = produc713050258nt_int(uminus_uminus_int) ).
fof(fact_4855_divmod__int__code,axiom,
! [L,K_1] :
( ( is_int(L)
& is_int(K_1) )
=> ( ( K_1 = zero_zero_int
=> divmod_int(K_1,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int) )
& ( K_1 != zero_zero_int
=> ( ( L = zero_zero_int
=> divmod_int(K_1,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K_1) )
& ( L != zero_zero_int
=> divmod_int(K_1,L) = hAPP_P1975530577nt_int(produc713050258nt_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(sgn_sgn_int,L))),hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,hAPP_int_int(sgn_sgn_int,K_1)),hAPP_int_int(sgn_sgn_int,L))),pdivmod(K_1,L)),hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,cOMBS_1891347093nt_int),hAPP_f1148171384nt_int(hAPP_f334321603nt_int(cOMBB_1484610687nt_int,hAPP_f1582586579nt_int(cOMBC_707974837nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),zero_zero_int)))),hAPP_i1584592887nt_int(hAPP_f465821005nt_int(cOMBC_1964283556nt_int,hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),uminus_uminus_int)),zero_zero_int)))),hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,cOMBB_1846624359nt_int),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,minus_minus_int),uminus_uminus_int)),one_one_int)))),hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(abs_abs_int,L))))),pdivmod(K_1,L)))) ) ) ) ) ) ).
fof(fact_4856_summable__le2,axiom,
! [F,G] :
( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_nat_real(F,N_2))),hAPP_nat_real(G,N_2)))
=> ( hBOOL(summable_real(G))
=> ( hBOOL(summable_real(F))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,F)),hAPP_f352196356l_real(suminf_real,G))) ) ) ) ).
fof(fact_4857_suminf__gt__zero,axiom,
! [F] :
( hBOOL(summable_real(F))
=> ( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(F,N_2)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_f352196356l_real(suminf_real,F))) ) ) ).
fof(fact_4858_suminf__ge__zero,axiom,
! [F] :
( hBOOL(summable_real(F))
=> ( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_2)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_f352196356l_real(suminf_real,F))) ) ) ).
fof(fact_4859_suminf__0__le,axiom,
! [F] :
( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_2)))
=> ( hBOOL(summable_real(F))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_f352196356l_real(suminf_real,F))) ) ) ).
fof(fact_4860_int__ge__less__than2__def,axiom,
! [D] : int_ge_less_than2(D) = collec1347809874nt_int(produc93441119t_bool(hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,D)))),ord_less_int))) ).
fof(fact_4861_int__ge__less__than__def,axiom,
! [D] : int_ge_less_than(D) = collec1347809874nt_int(produc93441119t_bool(hAPP_f428220345t_bool(hAPP_f1878066172t_bool(cOMBS_1720984575t_bool,hAPP_f1315737299t_bool(hAPP_f538565589t_bool(cOMBB_1211207212ol_int,cOMBB_bool_bool_int),hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,D)))),ord_less_int))) ).
fof(fact_4862_sin__paired,axiom,
! [X_2] : hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))),hAPP_real_real(sin,X_2))) ).
fof(fact_4863_sin__converges,axiom,
! [X_2] : hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_real_real(sin,X_2))) ).
fof(fact_4864_cos__converges,axiom,
! [X_2] : hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_real_real(cos,X_2))) ).
fof(fact_4865_power__half__series,axiom,
hBOOL(hAPP_real_bool(sums_real(hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),suc)),one_one_real)) ).
fof(fact_4866_Ln_Oaux2,axiom,
! [X_2] : hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_4867_sums__if_H,axiom,
! [G,X_2] :
( hBOOL(hAPP_real_bool(sums_real(G),X_2))
=> hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,G),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),one_one_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),X_2)) ) ).
fof(fact_4868_sums__if,axiom,
! [F,Y_2,G,X_2] :
( hBOOL(hAPP_real_bool(sums_real(G),X_2))
=> ( hBOOL(hAPP_real_bool(sums_real(F),Y_2))
=> hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,G),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),one_one_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Y_2))) ) ) ).
fof(fact_4869_cos__paired,axiom,
! [X_2] : hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(cos,X_2))) ).
fof(fact_4870_cnj_Osums,axiom,
! [X_3,A] :
( hBOOL(hAPP_complex_bool(sums_complex(X_3),A))
=> hBOOL(hAPP_complex_bool(sums_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3)),hAPP_complex_complex(cnj,A))) ) ).
fof(fact_4871_Re_Osums,axiom,
! [X_3,A] :
( hBOOL(hAPP_complex_bool(sums_complex(X_3),A))
=> hBOOL(hAPP_real_bool(sums_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3)),hAPP_complex_real(re,A))) ) ).
fof(fact_4872_Im_Osums,axiom,
! [X_3,A] :
( hBOOL(hAPP_complex_bool(sums_complex(X_3),A))
=> hBOOL(hAPP_real_bool(sums_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3)),hAPP_complex_real(im,A))) ) ).
fof(fact_4873_sin__fdiffs2,axiom,
! [N] : hAPP_nat_real(diffs_real(sin_coeff),N) = hAPP_nat_real(cos_coeff,N) ).
fof(fact_4874_sin__fdiffs,axiom,
diffs_real(sin_coeff) = cos_coeff ).
fof(fact_4875_exp__fdiffs,axiom,
! [X] : hAPP_nat_real(diffs_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))),X) = hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,X))) ).
fof(fact_4876_less__eq__nat_Osimps_I2_J,axiom,
! [Ma,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Ma)),N))
<=> hBOOL(hAPP_nat_bool(nat_case_bool(fFalse,hAPP_n1699378549t_bool(ord_less_eq_nat,Ma)),N)) ) ).
fof(fact_4877_cos__fdiffs2,axiom,
! [N] : hAPP_nat_real(diffs_real(cos_coeff),N) = hAPP_real_real(uminus_uminus_real,hAPP_nat_real(sin_coeff,N)) ).
fof(fact_4878_cos__fdiffs,axiom,
! [X] : hAPP_nat_real(diffs_real(cos_coeff),X) = hAPP_real_real(uminus_uminus_real,hAPP_nat_real(sin_coeff,X)) ).
fof(fact_4879_diff__Suc,axiom,
! [Ma,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(nat_case_nat(zero_zero_nat,cOMBI_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),N)) ).
fof(fact_4880_summable__le,axiom,
! [F,G] :
( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),hAPP_nat_real(G,N_2)))
=> ( hBOOL(summable_real(F))
=> ( hBOOL(summable_real(G))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,F)),hAPP_f352196356l_real(suminf_real,G))) ) ) ) ).
fof(fact_4881_arctan__def,axiom,
! [Y_2] : hAPP_real_real(arctan,Y_2) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),tan)),Y_2)))) ).
fof(fact_4882_arcsin__def,axiom,
! [Y_2] : hAPP_real_real(arcsin,Y_2) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),sin)),Y_2)))) ).
fof(fact_4883_ln__def,axiom,
! [X_2] : hAPP_real_real(ln,X_2) = the_real(hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),exp_real)),X_2)) ).
fof(fact_4884_root__def,axiom,
! [N,X_2] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> hAPP_real_real(hAPP_n546711566l_real(root,N),X_2) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,zero_zero_real))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),N))),X_2))) )
& ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),zero_zero_real))
=> hAPP_real_real(hAPP_n546711566l_real(root,N),X_2) = hAPP_real_real(uminus_uminus_real,the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,zero_zero_real))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),N))),hAPP_real_real(uminus_uminus_real,X_2))))) )
& ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),zero_zero_real))
=> hAPP_real_real(hAPP_n546711566l_real(root,N),X_2) = zero_zero_real ) ) ) ) ).
fof(fact_4885_arccos__def,axiom,
! [Y_2] : hAPP_real_real(arccos,Y_2) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),pi))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),cos)),Y_2)))) ).
fof(fact_4886_pi__def,axiom,
pi = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),cos)),zero_zero_real))))) ).
fof(fact_4887_pi__half,axiom,
hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),cos)),zero_zero_real)))) ).
fof(fact_4888_divmod__nat__def,axiom,
! [Ma,N] : divmod_nat(Ma,N) = the_Pr588456374at_nat(divmod_nat_rel(Ma,N)) ).
fof(fact_4889_floor__real__def,axiom,
! [X_2] : archim1246769320r_real(X_2) = the_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_r481142414t_bool(hAPP_f1845081853t_bool(cOMBC_int_real_bool,hAPP_f801032407l_bool(hAPP_f499530369l_bool(cOMBB_1317819104ol_int,ord_less_eq_real),ring_1_of_int_real)),X_2))),hAPP_f429384717t_bool(hAPP_f1173902867t_bool(cOMBB_real_bool_int,hAPP_r1134773055l_bool(ord_less_real,X_2)),hAPP_f563293398t_real(hAPP_f134192053t_real(cOMBB_int_real_int,ring_1_of_int_real),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,plus_plus_int),one_one_int))))) ).
fof(fact_4890_floor__rat__def,axiom,
! [X_2] : archim791455193or_rat(X_2) = the_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_r1900510169t_bool(hAPP_f1480506641t_bool(cOMBC_int_rat_bool,hAPP_f601898971t_bool(hAPP_f857296063t_bool(cOMBB_214503962ol_int,ord_less_eq_rat),ring_1_of_int_rat)),X_2))),hAPP_f1241350704t_bool(hAPP_f1131424617t_bool(cOMBB_rat_bool_int,hAPP_r2115560837t_bool(ord_less_rat,X_2)),hAPP_f1900950721nt_rat(hAPP_f1270196437nt_rat(cOMBB_int_rat_int,ring_1_of_int_rat),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,plus_plus_int),one_one_int))))) ).
fof(fact_4891_RRset2norRR__def,axiom,
! [A,A_1,Ma] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
& hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) )
=> hAPP_int_int(rRset2norRR(A_1,Ma),A) = hilbert_Eps_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,zcong(A)),Ma))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_i1948725293t_bool(norRRset,Ma)))) )
& ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
& hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) )
=> hAPP_int_int(rRset2norRR(A_1,Ma),A) = zero_zero_int ) ) ).
fof(fact_4892_aux__some,axiom,
! [A,A_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
=> ( hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),hilbert_Eps_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,zcong(A)),Ma))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_i1948725293t_bool(norRRset,Ma))))),Ma))
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hilbert_Eps_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,zcong(A)),Ma))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_i1948725293t_bool(norRRset,Ma))))),hAPP_i1948725293t_bool(norRRset,Ma))) ) ) ) ) ).
fof(fact_4893_Maclaurin__sin__bound,axiom,
! [X_2,N] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(sin,X_2)),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(abs_abs_real,X_2)),N)))) ).
fof(fact_4894_finite__less__ub,axiom,
! [U,F] :
( ! [N_2] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),hAPP_nat_nat(F,N_2)))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_less_eq_nat),F)),U)))) ) ).
fof(fact_4895_finite__atLeastLessThan,axiom,
! [L,U] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_at4362885an_nat(L),U))) ).
fof(fact_4896_sumr__diff__mult__const,axiom,
! [F,N,R_2] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),R_2)) = hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,minus_minus_real),F)),R_2)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N)) ).
fof(fact_4897_sumr__offset2,axiom,
! [K_1,N,F_2] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F_2),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K_1)))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),K_1))) ).
fof(fact_4898_sumr__offset4,axiom,
! [K_1,N_2,F_2] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_2),K_1))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F_2),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_2))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),K_1))) ).
fof(fact_4899_all__nat__less__eq,axiom,
! [P_1,N] :
( ! [M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
=> hBOOL(hAPP_nat_bool(P_1,M_1)) )
<=> ! [X] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N)))
=> hBOOL(hAPP_nat_bool(P_1,X)) ) ) ).
fof(fact_4900_ex__nat__less__eq,axiom,
! [P_1,N] :
( ? [M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
& hBOOL(hAPP_nat_bool(P_1,M_1)) )
<=> ? [X] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N)))
& hBOOL(hAPP_nat_bool(P_1,X)) ) ) ).
fof(fact_4901_card__atLeastLessThan,axiom,
! [L,U] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_at4362885an_nat(L),U)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,U),L) ).
fof(fact_4902_sumr__geometric,axiom,
! [N,X_2] :
( X_2 != one_one_real
=> hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N)),one_one_real)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),one_one_real)) ) ).
fof(fact_4903_sumr__one__lb__realpow__zero,axiom,
! [F,N] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F)),hAPP_r474017924t_real(power_power_real,zero_zero_real))),hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(suc,zero_zero_nat)),N)) = zero_zero_real ).
fof(fact_4904_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L,U] : hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(suc,L)),U) = hAPP_n1699378549t_bool(ord_gr660468384an_nat(L),U) ).
fof(fact_4905_image__add__atLeastLessThan,axiom,
! [K_1,I_1,J] : image_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1),hAPP_n1699378549t_bool(ord_at4362885an_nat(I_1),J)) = hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_1)) ).
fof(fact_4906_image__Suc__atLeastLessThan,axiom,
! [I_1,J] : image_nat_nat(suc,hAPP_n1699378549t_bool(ord_at4362885an_nat(I_1),J)) = hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(suc,I_1)),hAPP_nat_nat(suc,J)) ).
fof(fact_4907_Maclaurin__zero,axiom,
! [Diff,N,X_2] :
( X_2 = zero_zero_real
=> ( N != zero_zero_nat
=> hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N)) = hAPP_real_real(hAPP_n546711566l_real(Diff,zero_zero_nat),zero_zero_real) ) ) ).
fof(fact_4908_subset__card__intvl__is__intvl,axiom,
! [A_1,K_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),hAPP_n1699378549t_bool(ord_at4362885an_nat(K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_f22106695ol_nat(finite_card_nat,A_1)))))
=> A_1 = hAPP_n1699378549t_bool(ord_at4362885an_nat(K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_f22106695ol_nat(finite_card_nat,A_1))) ) ).
fof(fact_4909_sum__split__even__odd,axiom,
! [F,G,N] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),F)),G)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,G),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))) ).
fof(fact_4910_sumr__minus__one__realpow__zero,axiom,
! [N] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),suc)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N))) = zero_zero_real ).
fof(fact_4911_lemma__STAR__cos2,axiom,
! [N] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,zero_zero_real)))),zero_zero_real)),hAPP_n1699378549t_bool(ord_at4362885an_nat(one_one_nat),N)) = zero_zero_real ).
fof(fact_4912_sumr__cos__zero__one,axiom,
! [N] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),zero_zero_real))),hAPP_r474017924t_real(power_power_real,zero_zero_real))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(suc,N))) = one_one_real ).
fof(fact_4913_Maclaurin__sin__expansion2,axiom,
! [N,X_2] :
? [T] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2)))
& hAPP_real_real(sin,X_2) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,T),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(real_nat,N))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N))) ) ).
fof(fact_4914_Maclaurin__cos__expansion,axiom,
! [N,X_2] :
? [T] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2)))
& hAPP_real_real(cos,X_2) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),zero_zero_real))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,T),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(real_nat,N))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N))) ) ).
fof(fact_4915_finite__atLeastLessThan__int,axiom,
! [L,U] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at641636577an_int(L),U))) ).
fof(fact_4916_Sup__atLeastLessThan,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_2))
=> comple124823625p_real(hAPP_r1134773055l_bool(ord_at1496968948n_real(Y_2),X_2)) = X_2 ) ).
fof(fact_4917_finite__atLeastZeroLessThan__int,axiom,
! [U] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U))) ).
fof(fact_4918_card__atLeastZeroLessThan__int,axiom,
! [U] : hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U)) = hAPP_int_nat(nat,U) ).
fof(fact_4919_card__atLeastLessThan__int,axiom,
! [L,U] : hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at641636577an_int(L),U)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),L)) ).
fof(fact_4920_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L,U] : hAPP_i1948725293t_bool(ord_at641636577an_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,L),one_one_int)),U) = hAPP_i1948725293t_bool(ord_gr1297742076an_int(L),U) ).
fof(fact_4921_image__add__int__atLeastLessThan,axiom,
! [L,U] : image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,plus_plus_int),L),hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),L))) = hAPP_i1948725293t_bool(ord_at641636577an_int(L),U) ).
fof(fact_4922_Maclaurin__sin__expansion,axiom,
! [X_2,N] :
? [T] : hAPP_real_real(sin,X_2) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,T),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(real_nat,N))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N))) ).
fof(fact_4923_Maclaurin__lemma2,axiom,
! [B_1,K_1,Diff,H,N] :
( ! [M_1,T] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) )
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))) )
=> ( N = hAPP_nat_nat(suc,K_1)
=> ! [M_1,T] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) )
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_n546711566l_real(Diff,M_1))),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_f2119584768l_real(hAPP_f59594445l_real(cOMBC_2010481033l_real,hAPP_f1523861863l_real(hAPP_f1067681227l_real(cOMBB_1934313333l_real,big_co604158596t_real),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f1904307534l_real(hAPP_f224989205l_real(cOMBB_1419615765al_nat,Diff),hAPP_nat_fun_nat_nat(plus_plus_nat,M_1))),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),power_power_real))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M_1))))),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,hAPP_r1250527377l_real(times_times_real,B_1)),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M_1)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M_1))))))),T),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f1904307534l_real(hAPP_f224989205l_real(cOMBB_1419615765al_nat,Diff),hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M_1)))),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,T))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,M_1))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,T),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,M_1)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,M_1)))))))))) ) ) ) ).
fof(fact_4924_Maclaurin__exp__lt,axiom,
! [N,X_2] :
( X_2 != zero_zero_real
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ? [T] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,T)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2)))
& hAPP_real_real(exp_real,X_2) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(exp_real,T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N))) ) ) ) ).
fof(fact_4925_Maclaurin__exp__le,axiom,
! [N,X_2] :
? [T] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2)))
& hAPP_real_real(exp_real,X_2) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(exp_real,T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N))) ) ).
fof(fact_4926_sumr__pos__lt__pair,axiom,
! [K_1,F] :
( hBOOL(summable_real(F))
=> ( ! [D_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),D_2)))),hAPP_nat_real(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),D_2)),one_one_nat))))))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),K_1))),hAPP_f352196356l_real(suminf_real,F))) ) ) ).
fof(fact_4927_Maclaurin__lemma,axiom,
! [F,J,N,H] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H))
=> ? [B_6] : hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),J)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_6),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N))))) ) ).
fof(fact_4928_series__pos__less,axiom,
! [N,F] :
( hBOOL(summable_real(F))
=> ( ! [M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(F,M_1))) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_f352196356l_real(suminf_real,F))) ) ) ).
fof(fact_4929_series__pos__le,axiom,
! [N,F] :
( hBOOL(summable_real(F))
=> ( ! [M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,M_1))) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_f352196356l_real(suminf_real,F))) ) ) ).
fof(fact_4930_Maclaurin__all__lt,axiom,
! [X_2,N,Diff,F] :
( hAPP_n546711566l_real(Diff,zero_zero_nat) = F
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( X_2 != zero_zero_real
=> ( ! [M_1,X] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X)))
=> ? [T] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,T)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2)))
& hAPP_real_real(F,X_2) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N))) ) ) ) ) ) ).
fof(fact_4931_Maclaurin__all__lt__objl,axiom,
! [N,X_2,Diff,F] :
( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F
& ! [M_1,X] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X)))
& X_2 != zero_zero_real
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) )
=> ? [T] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,T)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2)))
& hAPP_real_real(F,X_2) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N))) ) ) ).
fof(fact_4932_Maclaurin__bi__le,axiom,
! [X_2,N,Diff,F] :
( hAPP_n546711566l_real(Diff,zero_zero_nat) = F
=> ( ! [M_1,T] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2))) )
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))) )
=> ? [T] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2)))
& hAPP_real_real(F,X_2) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N))) ) ) ) ).
fof(fact_4933_Maclaurin__objl,axiom,
! [Diff,F,N,H] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
& hAPP_n546711566l_real(Diff,zero_zero_nat) = F
& ! [M_1,T] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) )
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))) ) )
=> ? [T] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T),H))
& hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N))) ) ) ).
fof(fact_4934_Maclaurin2__objl,axiom,
! [N,Diff,F,H] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H))
& hAPP_n546711566l_real(Diff,zero_zero_nat) = F
& ! [M_1,T] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) )
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))) ) )
=> ? [T] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H))
& hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N))) ) ) ).
fof(fact_4935_Maclaurin__minus,axiom,
! [Diff,F,N,H] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),zero_zero_real))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F
=> ( ! [M_1,T] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,H),T))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),zero_zero_real)) )
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))) )
=> ? [T] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),T))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T),zero_zero_real))
& hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N))) ) ) ) ) ) ).
fof(fact_4936_Maclaurin2,axiom,
! [N,Diff,F,H] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H))
=> ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F
=> ( ! [M_1,T] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) )
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))) )
=> ? [T] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H))
& hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N))) ) ) ) ) ).
fof(fact_4937_Maclaurin__minus__objl,axiom,
! [Diff,F,N,H] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),zero_zero_real))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
& hAPP_n546711566l_real(Diff,zero_zero_nat) = F
& ! [M_1,T] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,H),T))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),zero_zero_real)) )
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))) ) )
=> ? [T] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),T))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T),zero_zero_real))
& hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N))) ) ) ).
fof(fact_4938_Maclaurin,axiom,
! [Diff,F,N,H] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F
=> ( ! [M_1,T] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) )
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))) )
=> ? [T] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T),H))
& hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N))) ) ) ) ) ) ).
fof(fact_4939_Maclaurin__all__le,axiom,
! [N,X_2,Diff,F] :
( hAPP_n546711566l_real(Diff,zero_zero_nat) = F
=> ( ! [M_1,X] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X)))
=> ? [T] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2)))
& hAPP_real_real(F,X_2) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N))) ) ) ) ).
fof(fact_4940_Maclaurin__all__le__objl,axiom,
! [N,X_2,Diff,F] :
( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F
& ! [M_1,X] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X))) )
=> ? [T] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_2)))
& hAPP_real_real(F,X_2) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_2))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N))) ) ) ).
fof(fact_4941_real__setsum__nat__ivl__bounded,axiom,
! [F,K_3,N] :
( ! [P_4] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,P_4),N))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,P_4)),K_3)) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),K_3))) ) ).
fof(fact_4942_DERIV__sumr,axiom,
! [F,X_2,F_1,N,Ma] :
( ! [R_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),R_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N))) )
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(F,R_1),X_2),hAPP_real_real(hAPP_n546711566l_real(F_1,R_1),X_2))) )
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_f2119584768l_real(hAPP_f59594445l_real(cOMBC_2010481033l_real,hAPP_f1523861863l_real(hAPP_f1067681227l_real(cOMBB_1934313333l_real,big_co604158596t_real),hAPP_f42270917t_real(cOMBC_nat_real_real,F))),hAPP_n1699378549t_bool(ord_at4362885an_nat(Ma),N)),X_2),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,F_1),X_2)),hAPP_n1699378549t_bool(ord_at4362885an_nat(Ma),N)))) ) ).
fof(fact_4943_suminf__le,axiom,
! [X_2,F] :
( hBOOL(summable_real(F))
=> ( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_2))),X_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,F)),X_2)) ) ) ).
fof(fact_4944_transfer__nat__int__sum__prod__closure_I1_J,axiom,
! [F,A_1] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
=> ( ! [X] :
( is_int(X)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(F,X))) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),A_1))) ) ) ).
fof(fact_4945_pos__summable,axiom,
! [X_2,F] :
( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_2)))
=> ( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_2))),X_2))
=> hBOOL(summable_real(F)) ) ) ).
fof(fact_4946_arith__series__int,axiom,
! [A,D,N] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,hAPP_f1139079189at_int(hAPP_f1431025877at_int(cOMBB_int_int_nat,hAPP_int_fun_int_int(plus_plus_int,A)),hAPP_int_fun_nat_int(hAPP_f1673103573at_int(cOMBC_nat_int_int,hAPP_f1935805932nt_int(hAPP_f125788821nt_int(cOMBB_859312247nt_nat,times_times_int),semiri1621563631at_int)),D))),hAPP_n1699378549t_bool(ord_lessThan_nat,N))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,N)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),D)))) ).
fof(fact_4947_BseqI2,axiom,
! [K_3,K_1,F] :
( ! [N_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,K_1),hAPP_nat_real(F,N_2)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),K_3)) )
=> hBOOL(bseq_real(F)) ) ).
fof(fact_4948_finite__lessThan,axiom,
! [K_1] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_lessThan_nat,K_1))) ).
fof(fact_4949_finite__nat__iff__bounded,axiom,
! [S] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S))
<=> ? [K] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_lessThan_nat,K))) ) ).
fof(fact_4950_card__lessThan,axiom,
! [U] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_lessThan_nat,U)) = U ).
fof(fact_4951_atLeast0LessThan,axiom,
! [N] : hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N) = hAPP_n1699378549t_bool(ord_lessThan_nat,N) ).
fof(fact_4952_arith__series__nat,axiom,
! [A,D,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,hAPP_nat_fun_nat_nat(plus_plus_nat,A)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),D))),hAPP_n1699378549t_bool(ord_lessThan_nat,N))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),D)))) ).
fof(fact_4953_image__atLeastZeroLessThan__int,axiom,
! [U] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U))
=> hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U) = image_nat_int(semiri1621563631at_int,hAPP_n1699378549t_bool(ord_lessThan_nat,hAPP_int_nat(nat,U))) ) ).
fof(fact_4954_finite__nat__bounded,axiom,
! [S] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S))
=> ? [K] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_lessThan_nat,K))) ) ).
fof(fact_4955_sum__diff__distrib,axiom,
! [N,Q_3,P_1] :
( ! [X] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(Q_3,X)),hAPP_nat_nat(P_1,X)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,P_1),hAPP_n1699378549t_bool(ord_lessThan_nat,N))),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,Q_3),hAPP_n1699378549t_bool(ord_lessThan_nat,N))) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,minus_minus_nat),P_1)),Q_3)),hAPP_n1699378549t_bool(ord_lessThan_nat,N)) ) ).
fof(fact_4956_Sup__lessThan,axiom,
! [X_2] : comple124823625p_real(hAPP_r1134773055l_bool(ord_lessThan_real,X_2)) = X_2 ).
fof(fact_4957_transfer__nat__int__set__cong,axiom,
! [P_2,P_1] :
( ! [X] :
( is_int(X)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(P_1,X))
<=> hBOOL(hAPP_int_bool(P_2,X)) ) ) )
=> collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_1)) = collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_2)) ) ).
fof(fact_4958_normalize__def,axiom,
! [P_3] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,P_3)))
=> normalize(P_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_snd_int_int,P_3)),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3)))) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,P_3)))
=> ( ( hAPP_P1175774780nt_int(product_snd_int_int,P_3) = zero_zero_int
=> normalize(P_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) )
& ( hAPP_P1175774780nt_int(product_snd_int_int,P_3) != zero_zero_int
=> normalize(P_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3))))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_snd_int_int,P_3)),hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3))))) ) ) ) ) ).
fof(fact_4959_gcd__dvd1__int,axiom,
! [X_1,Y_1] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1)),X_1)) ).
fof(fact_4960_gcd__dvd2__int,axiom,
! [X_1,Y_1] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1)),Y_1)) ).
fof(fact_4961_coprime__exp2__int,axiom,
! [N_1,M,A_3,B_2] :
( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) = one_one_int
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),M)) = one_one_int ) ).
fof(fact_4962_invertible__coprime__int,axiom,
! [X_1,Y_1,M] :
( hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1)),M) = one_one_int
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),M) = one_one_int ) ).
fof(fact_4963_divides__mult__int,axiom,
! [N_1,M,R_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),R_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),R_3))
=> ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N_1) = one_one_int
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N_1)),R_3)) ) ) ) ).
fof(fact_4964_coprime__dvd__mult__iff__int,axiom,
! [Ma,K_1,N] :
( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_1),N) = one_one_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),N)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),Ma)) ) ) ).
fof(fact_4965_coprime__dvd__mult__int,axiom,
! [M,K_2,N_1] :
( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_2),N_1) = one_one_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) ) ) ).
fof(fact_4966_div__gcd__coprime__int,axiom,
! [B_2,A_3] :
( ( is_int(B_2)
& is_int(A_3) )
=> ( ( A_3 != zero_zero_int
| B_2 != zero_zero_int )
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2))) = one_one_int ) ) ).
fof(fact_4967_gcd__coprime__int,axiom,
! [B_5,A_5,A_3,B_2] :
( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) != zero_zero_int
=> ( A_3 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_5),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2))
=> ( B_2 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2))
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_5),B_5) = one_one_int ) ) ) ).
fof(fact_4968_transfer__int__nat__gcd_I1_J,axiom,
! [X_1,Y_1] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_1)) ).
fof(fact_4969_gcd__add__mult__int,axiom,
! [M,K_2,N_1] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N_1) ).
fof(fact_4970_gcd__add1__int,axiom,
! [M,N_1] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),N_1)),N_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N_1) ).
fof(fact_4971_gcd__add2__int,axiom,
! [M,N_1] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N_1) ).
fof(fact_4972_coprime__plus__one__int,axiom,
! [N_1] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),one_one_int)),N_1) = one_one_int ).
fof(fact_4973_coprime__minus__one__int,axiom,
! [N_1] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N_1),one_one_int)),N_1) = one_one_int ).
fof(fact_4974_coprime__divisors__nat,axiom,
! [E_1,B_2,D_3,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E_1),B_2))
=> ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) = one_one_int
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),E_1) = one_one_int ) ) ) ).
fof(fact_4975_coprime__exp__int,axiom,
! [N_1,D_3,A_3] :
( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),A_3) = one_one_int
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N_1)) = one_one_int ) ).
fof(fact_4976_gcd__1__int,axiom,
! [M] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),one_one_int) = one_one_int ).
fof(fact_4977_gcd__zero__int,axiom,
! [Ma,N] :
( ( is_int(Ma)
& is_int(N) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Ma),N) = zero_zero_int
<=> ( Ma = zero_zero_int
& N = zero_zero_int ) ) ) ).
fof(fact_4978_gcd__exp__int,axiom,
! [A_3,N_1,B_2] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N_1)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)),N_1) ).
fof(fact_4979_gcd__commute__int,axiom,
! [A_3,B_2] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_2),A_3) ).
fof(fact_4980_gcd__int_Oleft__commute,axiom,
! [B_2,A_3,C_1] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),C_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_2),C_1)) ).
fof(fact_4981_gcd__assoc__int,axiom,
! [A_3,B_2,C_1] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)),C_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_2),C_1)) ).
fof(fact_4982_dvd__gcd__D2__int,axiom,
! [I,M,N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),N_1)) ) ).
fof(fact_4983_dvd__gcd__D1__int,axiom,
! [I,M,N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),M)) ) ).
fof(fact_4984_gcd__greatest__int,axiom,
! [N_1,K_2,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N_1))) ) ) ).
fof(fact_4985_gcd__greatest__iff__int,axiom,
! [K_1,Ma,N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Ma),N)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),Ma))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N)) ) ) ).
fof(fact_4986_gcd__red__int,axiom,
! [X_1,Y_1] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Y_1),hAPP_int_int(div_mod_int(X_1),Y_1)) ).
fof(fact_4987_abs__gcd__int,axiom,
! [X_1,Y_1] : hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1) ).
fof(fact_4988_gcd__idem__int,axiom,
! [X_1] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),X_1) = hAPP_int_int(abs_abs_int,X_1) ).
fof(fact_4989_gcd__abs__int,axiom,
! [X_1,Y_1] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(abs_abs_int,X_1)),hAPP_int_int(abs_abs_int,Y_1)) ).
fof(fact_4990_gcd__abs2__int,axiom,
! [X_1,Y_1] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),hAPP_int_int(abs_abs_int,Y_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1) ).
fof(fact_4991_gcd__abs1__int,axiom,
! [X_1,Y_1] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(abs_abs_int,X_1)),Y_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1) ).
fof(fact_4992_gcd__neg1__int,axiom,
! [X_1,Y_1] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,X_1)),Y_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1) ).
fof(fact_4993_gcd__neg2__int,axiom,
! [X_1,Y_1] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),hAPP_int_int(uminus_uminus_int,Y_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1) ).
fof(fact_4994_gcd__proj1__if__dvd__int,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_1),Y_1))
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1) = hAPP_int_int(abs_abs_int,X_1) ) ).
fof(fact_4995_gcd__proj2__if__dvd__int,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y_1),X_1))
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1) = hAPP_int_int(abs_abs_int,Y_1) ) ).
fof(fact_4996_gcd__mult__distrib__int,axiom,
! [K_2,M,N_1] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1)) ).
fof(fact_4997_gcd__pos__int,axiom,
! [Ma,N] :
( ( is_int(Ma)
& is_int(N) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Ma),N)))
<=> ( Ma != zero_zero_int
| N != zero_zero_int ) ) ) ).
fof(fact_4998_coprime__rmult__int,axiom,
! [D_3,A_3,B_2] :
( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)) = one_one_int
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),B_2) = one_one_int ) ).
fof(fact_4999_coprime__lmult__int,axiom,
! [D_3,A_3,B_2] :
( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)) = one_one_int
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),A_3) = one_one_int ) ).
fof(fact_5000_coprime__mult__int,axiom,
! [B_2,D_3,A_3] :
( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),A_3) = one_one_int
=> ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),B_2) = one_one_int
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)) = one_one_int ) ) ).
fof(fact_5001_gcd__mult__cancel__int,axiom,
! [M,K_2,N_1] :
( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_2),N_1) = one_one_int
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N_1) ) ).
fof(fact_5002_coprime__mul__eq__int,axiom,
! [D,A,B] :
( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B)) = one_one_int
<=> ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D),A) = one_one_int
& hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D),B) = one_one_int ) ) ).
fof(fact_5003_gcd__0__left__int,axiom,
! [X_1] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,zero_zero_int),X_1) = hAPP_int_int(abs_abs_int,X_1) ).
fof(fact_5004_gcd__0__int,axiom,
! [X_1] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),zero_zero_int) = hAPP_int_int(abs_abs_int,X_1) ).
fof(fact_5005_Fract__coprime,axiom,
! [A_3,B_2] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2))) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2) ).
fof(fact_5006_gcd__non__0__int,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y_1))
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Y_1),hAPP_int_int(div_mod_int(X_1),Y_1)) ) ).
fof(fact_5007_transfer__nat__int__gcd__closures_I1_J,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1))) ) ) ).
fof(fact_5008_gcd__ge__0__int,axiom,
! [X_1,Y_1] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1))) ).
fof(fact_5009_gcd__le1__int,axiom,
! [B_2,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)),A_3)) ) ).
fof(fact_5010_gcd__le2__int,axiom,
! [A_3,B_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)),B_2)) ) ).
fof(fact_5011_gcd__unique__int,axiom,
! [B,A,D] :
( is_int(D)
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),D))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),A))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),B))
& ! [E] :
( is_int(E)
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),A))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),B)) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),D)) ) ) )
<=> D = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A),B) ) ) ).
fof(fact_5012_gcd__cases__int,axiom,
! [P_1,Y_2,X_2] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_2),Y_2))) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_2),zero_zero_int))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_2),hAPP_int_int(uminus_uminus_int,Y_2)))) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,X_2)),Y_2))) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_2),zero_zero_int))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,X_2)),hAPP_int_int(uminus_uminus_int,Y_2)))) ) )
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_2),Y_2))) ) ) ) ) ).
fof(fact_5013_coprime__crossproduct__int,axiom,
! [B,C,A,D] :
( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A),D) = one_one_int
=> ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B),C) = one_one_int
=> ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,A)),hAPP_int_int(abs_abs_int,C)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,B)),hAPP_int_int(abs_abs_int,D))
<=> ( hAPP_int_int(abs_abs_int,A) = hAPP_int_int(abs_abs_int,B)
& hAPP_int_int(abs_abs_int,C) = hAPP_int_int(abs_abs_int,D) ) ) ) ) ).
fof(fact_5014_gcd__code__int,axiom,
! [K_1,L] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_1),L) = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,L),zero_zero_int)),K_1),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,L),hAPP_int_int(div_mod_int(hAPP_int_int(abs_abs_int,K_1)),hAPP_int_int(abs_abs_int,L))))) ).
fof(fact_5015_coprime__common__divisor__int,axiom,
! [X_1,A_3,B_2] :
( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) = one_one_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_1),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_1),B_2))
=> hAPP_int_int(abs_abs_int,X_1) = one_one_int ) ) ) ).
fof(fact_5016_quotient__of__coprime,axiom,
! [R_3,P_3,Q_2] :
( quotient_of(R_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2)
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_2) = one_one_int ) ).
fof(fact_5017_normalize__coprime,axiom,
! [R_3,P_3,Q_2] :
( normalize(R_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2)
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_2) = one_one_int ) ).
fof(fact_5018_coprime__int,axiom,
! [A,B] :
( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A),B) = one_one_int
<=> ! [D_2] :
( is_int(D_2)
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),D_2))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_2),A))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_2),B)) )
<=> D_2 = one_one_int ) ) ) ).
fof(fact_5019_transfer__nat__int__gcd_I1_J,axiom,
! [Y_1,X_1] :
( 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_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_int_nat(nat,X_1)),hAPP_int_nat(nat,Y_1)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1)) ) ) ).
fof(fact_5020_gcd__int__def,axiom,
! [X_1,Y_1] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,X_1))),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Y_1)))) ).
fof(fact_5021_normalize__stable,axiom,
! [P_3,Q_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_2))
=> ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_2) = one_one_int
=> normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) ) ) ).
fof(fact_5022_quotient__of__def,axiom,
! [X_2] : quotient_of(X_2) = the_Pr2103884470nt_int(hAPP_f237669397t_bool(hAPP_f461489489t_bool(cOMBS_400904860l_bool,hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,fconj),hAPP_f1084724746t_bool(hAPP_f374502857t_bool(cOMBB_569260360nt_int,hAPP_r2115560837t_bool(fequal_rat,X_2)),hAPP_f1065357839nt_rat(hAPP_f1120137083nt_rat(cOMBS_549334880nt_rat,hAPP_f1547007718nt_rat(hAPP_f382317021nt_rat(cOMBB_170828966nt_int,fract),product_fst_int_int)),product_snd_int_int)))),hAPP_f237669397t_bool(hAPP_f461489489t_bool(cOMBS_400904860l_bool,hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,fconj),hAPP_f892584630t_bool(hAPP_f492612297t_bool(cOMBB_798864284nt_int,hAPP_i1948725293t_bool(ord_less_int,zero_zero_int)),product_snd_int_int))),hAPP_i119638232t_bool(hAPP_f2018027565t_bool(cOMBC_33652895t_bool,hAPP_f291087797t_bool(hAPP_f214467655t_bool(cOMBB_182850971nt_int,fequal_int),hAPP_f521271395nt_int(hAPP_f1914742907nt_int(cOMBS_2019229620nt_int,hAPP_f1855025978nt_int(hAPP_f1038672605nt_int(cOMBB_1506897658nt_int,gcd_gcd_int),product_fst_int_int)),product_snd_int_int))),one_one_int)))) ).
fof(fact_5023_quotient__of__unique,axiom,
! [R_3] :
? [X] :
( R_3 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,X)),hAPP_P1175774780nt_int(product_snd_int_int,X))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,X)))
& hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,X)),hAPP_P1175774780nt_int(product_snd_int_int,X)) = one_one_int
& ! [Y] :
( ( R_3 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,Y)),hAPP_P1175774780nt_int(product_snd_int_int,Y))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,Y)))
& hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,Y)),hAPP_P1175774780nt_int(product_snd_int_int,Y)) = one_one_int )
=> Y = X ) ) ).
fof(fact_5024_Rat__cases__nonzero,axiom,
! [Q_2] :
( ! [A_2,B_4] :
( ( is_int(A_2)
& is_int(B_4) )
=> ( Q_2 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_4)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4))
=> ( A_2 != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_4) != one_one_int ) ) ) )
=> Q_2 = zero_zero_rat ) ).
fof(fact_5025_Rat__cases,axiom,
! [Q_2] :
~ ! [A_2,B_4] :
( ( is_int(A_2)
& is_int(B_4) )
=> ( Q_2 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_4)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4))
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_4) != one_one_int ) ) ) ).
fof(fact_5026_Rat__induct,axiom,
! [Q_1,P_1] :
( ! [A_2,B_4] :
( ( is_int(A_2)
& is_int(B_4) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4))
=> ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_4) = one_one_int
=> hBOOL(hAPP_rat_bool(P_1,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_4))) ) ) )
=> hBOOL(hAPP_rat_bool(P_1,Q_1)) ) ).
fof(fact_5027_card__setsum__aux,axiom,
! [N,S] :
( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,S))
=> ( ! [X] :
( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X),S))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,X)) )
=> ( ! [X] :
( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X),S))
=> hAPP_f957591787ol_nat(finite_card_int,X) = N )
=> hAPP_f521865025ol_nat(hAPP_f93439879ol_nat(big_co1758102091ol_nat,finite_card_int),S) = hAPP_f521865025ol_nat(hAPP_f93439879ol_nat(big_co1758102091ol_nat,cOMBK_1643584600t_bool(N)),S) ) ) ) ).
fof(fact_5028_zOdd__def,axiom,
zOdd = collect_int(hAPP_f2119767738t_bool(hAPP_f423804115t_bool(cOMBB_1418110531ol_int,fEx_int),hAPP_f1791153283t_bool(hAPP_f735247205t_bool(cOMBC_983868720t_bool,hAPP_f1251097615t_bool(hAPP_f1475822739t_bool(cOMBB_2027861294ol_int,cOMBB_int_bool_int),fequal_int)),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,plus_plus_int),hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_int)))) ).
fof(fact_5029_Rats__eq__int__div__int,axiom,
field_1210416355s_real = collect_real(hAPP_f101635158l_bool(hAPP_f1925507189l_bool(cOMBB_1121934354l_real,fEx_int),hAPP_f1625314454t_bool(hAPP_f340152631t_bool(cOMBB_1453147152l_real,hAPP_f423804115t_bool(cOMBB_1418110531ol_int,fEx_int)),hAPP_f1832844496t_bool(hAPP_f1990636069t_bool(cOMBC_412863505t_bool,hAPP_f704188951t_bool(hAPP_f1456711209t_bool(cOMBB_690122963l_real,cOMBC_1683390479t_bool),hAPP_f1063001928t_bool(hAPP_f2050994651t_bool(cOMBB_2138578194l_real,hAPP_f85237525t_bool(cOMBB_389152643ol_int,cOMBS_int_bool_bool)),hAPP_f381769257l_bool(hAPP_f512004221l_bool(cOMBB_188663909l_real,hAPP_f1830214065l_bool(cOMBB_793166024ol_int,hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj))),hAPP_f956464319t_bool(hAPP_f422375781t_bool(cOMBC_243356514t_bool,hAPP_f332806115t_bool(hAPP_f1177545951t_bool(cOMBB_1459832681l_real,cOMBB_95956018ol_int),hAPP_f1920544743t_bool(hAPP_f866672739t_bool(cOMBB_1247074793l_real,cOMBB_real_bool_int),fequal_real))),hAPP_f1323932638t_real(hAPP_f352226557t_real(cOMBC_824100275t_real,hAPP_f74147283t_real(hAPP_f1169087445t_real(cOMBB_2129825580al_int,cOMBB_real_real_int),hAPP_f1352333353l_real(hAPP_f318350629l_real(cOMBB_1946221070al_int,inverse_divide_real),real_int))),real_int)))))),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,fNot),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),zero_zero_int)))))) ).
fof(fact_5030_zEven__def,axiom,
zEven = collect_int(hAPP_f2119767738t_bool(hAPP_f423804115t_bool(cOMBB_1418110531ol_int,fEx_int),hAPP_f1791153283t_bool(hAPP_f735247205t_bool(cOMBC_983868720t_bool,hAPP_f1251097615t_bool(hAPP_f1475822739t_bool(cOMBB_2027861294ol_int,cOMBB_int_bool_int),fequal_int)),hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).
fof(fact_5031_summable__Leibniz_I5_J,axiom,
! [A] :
( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
=> ( hBOOL(monoseq_real(A))
=> hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),sequentially)) ) ) ).
fof(fact_5032_LIMSEQ__realpow__zero,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),one_one_real))
=> hBOOL(tendsto_nat_real(hAPP_r474017924t_real(power_power_real,X_2),zero_zero_real,sequentially)) ) ) ).
fof(fact_5033_LIMSEQ__le__const,axiom,
! [A,X_3,X_2] :
( hBOOL(tendsto_nat_real(X_3,X_2,sequentially))
=> ( ? [N_3] :
! [N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),hAPP_nat_real(X_3,N_2))) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2)) ) ) ).
fof(fact_5034_LIMSEQ__le__const2,axiom,
! [A,X_3,X_2] :
( hBOOL(tendsto_nat_real(X_3,X_2,sequentially))
=> ( ? [N_3] :
! [N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(X_3,N_2)),A)) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),A)) ) ) ).
fof(fact_5035_LIMSEQ__le,axiom,
! [Y_3,Y_2,X_3,X_2] :
( hBOOL(tendsto_nat_real(X_3,X_2,sequentially))
=> ( hBOOL(tendsto_nat_real(Y_3,Y_2,sequentially))
=> ( ? [N_3] :
! [N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(X_3,N_2)),hAPP_nat_real(Y_3,N_2))) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)) ) ) ) ).
fof(fact_5036_monoseq__le,axiom,
! [X_2,A] :
( hBOOL(monoseq_real(A))
=> ( hBOOL(tendsto_nat_real(A,X_2,sequentially))
=> ( ( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,N_2)),X_2))
& ! [M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,M_1)),hAPP_nat_real(A,N_2))) ) )
| ( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),hAPP_nat_real(A,N_2)))
& ! [M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,N_2)),hAPP_nat_real(A,M_1))) ) ) ) ) ) ).
fof(fact_5037_LIMSEQ__rabs__realpow__zero2,axiom,
! [C] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,C)),one_one_real))
=> hBOOL(tendsto_nat_real(hAPP_r474017924t_real(power_power_real,C),zero_zero_real,sequentially)) ) ).
fof(fact_5038_LIMSEQ__rabs__realpow__zero,axiom,
! [C] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,C)),one_one_real))
=> hBOOL(tendsto_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(abs_abs_real,C)),zero_zero_real,sequentially)) ) ).
fof(fact_5039_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R_2] : hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(plus_plus_real,R_2)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc))),R_2,sequentially)) ).
fof(fact_5040_LIMSEQ__imp__rabs,axiom,
! [F,L] :
( hBOOL(tendsto_nat_real(F,L,sequentially))
=> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F),hAPP_real_real(abs_abs_real,L),sequentially)) ) ).
fof(fact_5041_LIMSEQ__rabs__zero,axiom,
! [F] :
( hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F),zero_zero_real,sequentially))
<=> hBOOL(tendsto_nat_real(F,zero_zero_real,sequentially)) ) ).
fof(fact_5042_LIMSEQ__inverse__real__of__nat,axiom,
hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc)),zero_zero_real,sequentially)) ).
fof(fact_5043_LIMSEQ__divide__realpow__zero,axiom,
! [A,X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_2))
=> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,A)),hAPP_r474017924t_real(power_power_real,X_2)),zero_zero_real,sequentially)) ) ).
fof(fact_5044_LIMSEQ__inverse__realpow__zero,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_2))
=> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_r474017924t_real(power_power_real,X_2)),zero_zero_real,sequentially)) ) ).
fof(fact_5045_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R_2] : hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(plus_plus_real,R_2)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,uminus_uminus_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc)))),R_2,sequentially)) ).
fof(fact_5046_LIMSEQ__neg__powr,axiom,
! [Sa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Sa))
=> hBOOL(tendsto_nat_real(hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,powr),real_nat)),hAPP_real_real(uminus_uminus_real,Sa)),zero_zero_real,sequentially)) ) ).
fof(fact_5047_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R_2] : hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(times_times_real,R_2)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(plus_plus_real,one_one_real)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,uminus_uminus_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc))))),R_2,sequentially)) ).
fof(fact_5048_summable__Leibniz_I1_J,axiom,
! [A] :
( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
=> ( hBOOL(monoseq_real(A))
=> hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))) ) ) ).
fof(fact_5049_zeroseq__arctan__series,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_2)),one_one_real))
=> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_2)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),zero_zero_real,sequentially)) ) ).
fof(fact_5050_Rats__eq__int__div__nat,axiom,
field_1210416355s_real = collect_real(hAPP_f101635158l_bool(hAPP_f1925507189l_bool(cOMBB_1121934354l_real,fEx_int),hAPP_f1341793266t_bool(hAPP_f571820015t_bool(cOMBB_1159313332l_real,hAPP_f389116883t_bool(cOMBB_1593174431ol_int,fEx_nat)),hAPP_f1115375056t_bool(hAPP_f271976045t_bool(cOMBC_1801450329t_bool,hAPP_f1927674023t_bool(hAPP_f1261108921t_bool(cOMBB_172850899l_real,cOMBC_678732631t_bool),hAPP_f2084827628t_bool(hAPP_f1121609875t_bool(cOMBB_2125215742l_real,hAPP_f1182713365t_bool(cOMBB_2087181719ol_int,cOMBS_nat_bool_bool)),hAPP_f834119977l_bool(hAPP_f848516149l_bool(cOMBB_270132509l_real,hAPP_f643658017l_bool(cOMBB_2046938128ol_int,hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj))),hAPP_f1459420095t_bool(hAPP_f693283245t_bool(cOMBC_1113900202t_bool,hAPP_f77155443t_bool(hAPP_f403082607t_bool(cOMBB_984071017l_real,cOMBB_1647910634ol_int),hAPP_f1840085295t_bool(hAPP_f1083049571t_bool(cOMBB_2050079537l_real,cOMBB_real_bool_nat),fequal_real))),hAPP_f1613846310t_real(hAPP_f1720068861t_real(cOMBC_295337211t_real,hAPP_f1295083291t_real(hAPP_f212164053t_real(cOMBB_929094628al_int,cOMBB_real_real_nat),hAPP_f1352333353l_real(hAPP_f318350629l_real(cOMBB_1946221070al_int,inverse_divide_real),real_int))),real_nat)))))),hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,fNot),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),zero_zero_nat)))))) ).
fof(fact_5051_summable__Leibniz_I4_J,axiom,
! [A] :
( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
=> ( hBOOL(monoseq_real(A))
=> hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),sequentially)) ) ) ).
fof(fact_5052_summable__Leibniz_H_I5_J,axiom,
! [A] :
( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
=> ( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))
=> ( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))
=> hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),sequentially)) ) ) ) ).
fof(fact_5053_summable__Leibniz_H_I4_J,axiom,
! [N,A] :
( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
=> ( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))
=> ( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)),one_one_nat))))) ) ) ) ).
fof(fact_5054_cnj_OLIMSEQ,axiom,
! [X_3,A] :
( hBOOL(tendsto_nat_complex(X_3,A,sequentially))
=> hBOOL(tendsto_nat_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3),hAPP_complex_complex(cnj,A),sequentially)) ) ).
fof(fact_5055_Re_OLIMSEQ,axiom,
! [X_3,A] :
( hBOOL(tendsto_nat_complex(X_3,A,sequentially))
=> hBOOL(tendsto_nat_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3),hAPP_complex_real(re,A),sequentially)) ) ).
fof(fact_5056_Im_OLIMSEQ,axiom,
! [X_3,A] :
( hBOOL(tendsto_nat_complex(X_3,A,sequentially))
=> hBOOL(tendsto_nat_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3),hAPP_complex_real(im,A),sequentially)) ) ).
fof(fact_5057_LIMSEQ__Complex,axiom,
! [Y_3,B,X_3,A] :
( hBOOL(tendsto_nat_real(X_3,A,sequentially))
=> ( hBOOL(tendsto_nat_real(Y_3,B,sequentially))
=> hBOOL(tendsto_nat_complex(hAPP_f219720582omplex(hAPP_f1052840010omplex(cOMBS_696648398omplex,hAPP_f23851874omplex(hAPP_f406530075omplex(cOMBB_1714920583ex_nat,complex),X_3)),Y_3),hAPP_real_complex(hAPP_r265291036omplex(complex,A),B),sequentially)) ) ) ).
fof(fact_5058_Bolzano__nest__unique,axiom,
! [P_1,A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
=> ( hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,minus_minus_real),hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc1935615926l_real),bolzano_bisect(P_1,A,B)))),hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc556554744l_real),bolzano_bisect(P_1,A,B))),zero_zero_real,sequentially))
=> ? [L_1] :
( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))),L_1))
& hBOOL(tendsto_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc1935615926l_real),bolzano_bisect(P_1,A,B)),L_1,sequentially))
& ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,L_1),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))))
& hBOOL(tendsto_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc556554744l_real),bolzano_bisect(P_1,A,B)),L_1,sequentially)) ) ) ) ) ) ).
fof(fact_5059_summable__Leibniz_H_I2_J,axiom,
! [N,A] :
( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
=> ( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))
=> ( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)))) ) ) ) ).
fof(fact_5060_summable__Leibniz_H_I3_J,axiom,
! [A] :
( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
=> ( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))
=> ( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))
=> hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),sequentially)) ) ) ) ).
fof(fact_5061_summable__Leibniz_I3_J,axiom,
! [A] :
( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
=> ( hBOOL(monoseq_real(A))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(A,zero_zero_nat)),zero_zero_real))
=> ! [N_2] : hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_r1134773055l_bool(ord_at1589558736t_real(hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2)),one_one_nat)))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2)))))) ) ) ) ).
fof(fact_5062_summable__Leibniz_I2_J,axiom,
! [A] :
( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
=> ( hBOOL(monoseq_real(A))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(A,zero_zero_nat)))
=> ! [N_2] : hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_r1134773055l_bool(ord_at1589558736t_real(hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2)))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2)),one_one_nat)))))) ) ) ) ).
fof(fact_5063_Sup__atLeastAtMost,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),X_2))
=> comple124823625p_real(hAPP_r1134773055l_bool(ord_at1589558736t_real(Y_2),X_2)) = X_2 ) ).
fof(fact_5064_summable,axiom,
! [A] :
( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
=> ( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))
=> ( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))
=> hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))) ) ) ) ).
fof(fact_5065_finite__atLeastAtMost,axiom,
! [L,U] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U))) ).
fof(fact_5066_finite__atLeastAtMost__int,axiom,
! [L,U] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at875362053st_int(L),U))) ).
fof(fact_5067_SetInterval_Otransfer__nat__int__set__function__closures,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> hBOOL(hAPP_f448129468l_bool(nat_nat_set,hAPP_i1948725293t_bool(ord_at875362053st_int(X_2),Y_2))) ) ).
fof(fact_5068_ex__nat__less,axiom,
! [P_1,N] :
( ? [M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
& hBOOL(hAPP_nat_bool(P_1,M_1)) )
<=> ? [X] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),N)))
& hBOOL(hAPP_nat_bool(P_1,X)) ) ) ).
fof(fact_5069_all__nat__less,axiom,
! [P_1,N] :
( ! [M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
=> hBOOL(hAPP_nat_bool(P_1,M_1)) )
<=> ! [X] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),N)))
=> hBOOL(hAPP_nat_bool(P_1,X)) ) ) ).
fof(fact_5070_SetInterval_Otransfer__nat__int__set__functions_I2_J,axiom,
! [Ma,N] : hAPP_n1699378549t_bool(ord_at238088361st_nat(Ma),N) = image_int_nat(nat,hAPP_i1948725293t_bool(ord_at875362053st_int(hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N))) ).
fof(fact_5071_image__Suc__atLeastAtMost,axiom,
! [I_1,J] : image_nat_nat(suc,hAPP_n1699378549t_bool(ord_at238088361st_nat(I_1),J)) = hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(suc,I_1)),hAPP_nat_nat(suc,J)) ).
fof(fact_5072_image__add__atLeastAtMost,axiom,
! [K_1,I_1,J] : image_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1),hAPP_n1699378549t_bool(ord_at238088361st_nat(I_1),J)) = hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_1)) ).
fof(fact_5073_fact__altdef__nat,axiom,
! [N] : hAPP_nat_nat(fact_fact_nat,N) = big_co1705425894at_nat(cOMBI_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(one_one_nat),N)) ).
fof(fact_5074_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L,U] : hAPP_n1699378549t_bool(ord_at4362885an_nat(L),hAPP_nat_nat(suc,U)) = hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U) ).
fof(fact_5075_Re_Ocont,axiom,
! [A] : hBOOL(tendsto_complex_real(re,hAPP_complex_real(re,A),at_complex(A))) ).
fof(fact_5076_Im_Ocont,axiom,
! [A] : hBOOL(tendsto_complex_real(im,hAPP_complex_real(im,A),at_complex(A))) ).
fof(fact_5077_cnj_Ocont,axiom,
! [A] : hBOOL(tendst1507391555omplex(cnj,hAPP_complex_complex(cnj,A),at_complex(A))) ).
fof(fact_5078_card__atLeastAtMost,axiom,
! [L,U] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,U)),L) ).
fof(fact_5079_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L,U] : hAPP_i1948725293t_bool(ord_at641636577an_int(L),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U),one_one_int)) = hAPP_i1948725293t_bool(ord_at875362053st_int(L),U) ).
fof(fact_5080_setsum__shift__lb__Suc0__0,axiom,
! [K_1,F] :
( hAPP_nat_nat(F,zero_zero_nat) = zero_zero_nat
=> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(suc,zero_zero_nat)),K_1)) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),K_1)) ) ).
fof(fact_5081_card__atLeastAtMost__int,axiom,
! [L,U] : hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at875362053st_int(L),U)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),L)),one_one_int)) ).
fof(fact_5082_fact__div__fact,axiom,
! [N,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),Ma))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(fact_fact_nat,Ma)),hAPP_nat_nat(fact_fact_nat,N)) = big_co1705425894at_nat(cOMBI_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),Ma)) ) ).
fof(fact_5083_fact__altdef__int,axiom,
! [N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
=> hAPP_int_int(fact_fact_int,N) = big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),N)) ) ).
fof(fact_5084_LIM__cos__div__sin,axiom,
hBOOL(tendsto_real_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real),cos)),sin),zero_zero_real,at_real(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).
fof(fact_5085_LIM__fun__less__zero,axiom,
! [F,L,C] :
( hBOOL(tendsto_real_real(F,L,at_real(C)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,L),zero_zero_real))
=> ? [R_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))
& ! [X] :
( ( X != C
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),X))),R_1)) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X)),zero_zero_real)) ) ) ) ) ).
fof(fact_5086_LIM__fun__gt__zero,axiom,
! [F,L,C] :
( hBOOL(tendsto_real_real(F,L,at_real(C)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),L))
=> ? [R_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))
& ! [X] :
( ( X != C
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),X))),R_1)) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(F,X))) ) ) ) ) ).
fof(fact_5087_LIM__fun__not__zero,axiom,
! [F,L,C] :
( hBOOL(tendsto_real_real(F,L,at_real(C)))
=> ( L != zero_zero_real
=> ? [R_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))
& ! [X] :
( ( X != C
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),X))),R_1)) )
=> hAPP_real_real(F,X) != zero_zero_real ) ) ) ) ).
fof(fact_5088_aset_I8_J,axiom,
! [Ta,A_1,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ! [X] :
( is_int(X)
=> ( ! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1))
=> X != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),X))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D_1))) ) ) ) ) ).
fof(fact_5089_aset_I6_J,axiom,
! [Ta,A_1,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A_1))
=> ! [X] :
( is_int(X)
=> ( ! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1))
=> X != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),Ta))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D_1)),Ta)) ) ) ) ) ) ).
fof(fact_5090_bset_I8_J,axiom,
! [Ta,B_1,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B_1))
=> ! [X] :
( is_int(X)
=> ( ! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1))
=> X != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),X))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D_1))) ) ) ) ) ) ).
fof(fact_5091_bset_I6_J,axiom,
! [Ta,B_1,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ! [X] :
( is_int(X)
=> ( ! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1))
=> X != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),Ta))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D_1)),Ta)) ) ) ) ) ).
fof(fact_5092_bset_I3_J,axiom,
! [Ta,B_1,D_1] :
( is_int(Ta)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B_1))
=> ! [X] :
( is_int(X)
=> ( ! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1))
=> X != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) ) )
=> ( X = Ta
=> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D_1) = Ta ) ) ) ) ) ) ).
fof(fact_5093_aset_I3_J,axiom,
! [Ta,A_1,D_1] :
( is_int(Ta)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A_1))
=> ! [X] :
( is_int(X)
=> ( ! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1))
=> X != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) ) )
=> ( X = Ta
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D_1) = Ta ) ) ) ) ) ) ).
fof(fact_5094_aset_I7_J,axiom,
! [Ta,A_1,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ! [X] :
( is_int(X)
=> ( ! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1))
=> X != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),X))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D_1))) ) ) ) ) ).
fof(fact_5095_bset_I5_J,axiom,
! [Ta,B_1,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ! [X] :
( is_int(X)
=> ( ! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1))
=> X != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Ta))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D_1)),Ta)) ) ) ) ) ).
fof(fact_5096_aset_I5_J,axiom,
! [Ta,A_1,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),A_1))
=> ! [X] :
( is_int(X)
=> ( ! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1))
=> X != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Ta))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D_1)),Ta)) ) ) ) ) ) ).
fof(fact_5097_bset_I7_J,axiom,
! [Ta,B_1,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),B_1))
=> ! [X] :
( is_int(X)
=> ( ! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1))
=> X != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),X))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D_1))) ) ) ) ) ) ).
fof(fact_5098_aset_I4_J,axiom,
! [Ta,A_1,D_1] :
( is_int(Ta)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),A_1))
=> ! [X] :
( is_int(X)
=> ( ! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1))
=> X != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) ) )
=> ( X != Ta
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D_1) != Ta ) ) ) ) ) ) ).
fof(fact_5099_bset_I4_J,axiom,
! [Ta,B_1,D_1] :
( is_int(Ta)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),B_1))
=> ! [X] :
( is_int(X)
=> ( ! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1))
=> X != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) ) )
=> ( X != Ta
=> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D_1) != Ta ) ) ) ) ) ) ).
fof(fact_5100_periodic__finite__ex,axiom,
! [P_1,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X,K] :
( ( is_int(X)
& is_int(K) )
=> ( hBOOL(hAPP_int_bool(P_1,X))
<=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D)))) ) )
=> ( ? [X1] :
( is_int(X1)
& hBOOL(hAPP_int_bool(P_1,X1)) )
<=> ? [X] :
( is_int(X)
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
& hBOOL(hAPP_int_bool(P_1,X)) ) ) ) ) ).
fof(fact_5101_aset_I10_J,axiom,
! [Ta,A_1,D,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1))
=> ! [X] :
( is_int(X)
=> ( ! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1))
=> X != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) ) )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Ta)))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D_1)),Ta))) ) ) ) ) ).
fof(fact_5102_bset_I10_J,axiom,
! [Ta,B_1,D,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1))
=> ! [X] :
( is_int(X)
=> ( ! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1))
=> X != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) ) )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Ta)))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D_1)),Ta))) ) ) ) ) ).
fof(fact_5103_bset_I9_J,axiom,
! [Ta,B_1,D,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1))
=> ! [X] :
( is_int(X)
=> ( ! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1))
=> X != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Ta)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D_1)),Ta))) ) ) ) ) ).
fof(fact_5104_aset_I9_J,axiom,
! [Ta,A_1,D,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1))
=> ! [X] :
( is_int(X)
=> ( ! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1))
=> X != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Ta)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D_1)),Ta))) ) ) ) ) ).
fof(fact_5105_LIMSEQ__inverse__zero,axiom,
! [X_3] :
( ! [R_1] :
? [N_3] :
! [N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_1),hAPP_nat_real(X_3,N_2))) )
=> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),X_3),zero_zero_real,sequentially)) ) ).
fof(fact_5106_real__scaleR__def,axiom,
! [A_3,X_1] : hAPP_real_real(scaleR_scaleR_real(A_3),X_1) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),X_1) ).
fof(fact_5107_complex__Im__scaleR,axiom,
! [R_3,X_1] : hAPP_complex_real(im,hAPP_complex_complex(scaleR1652505878omplex(R_3),X_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(im,X_1)) ).
fof(fact_5108_complex__Re__scaleR,axiom,
! [R_3,X_1] : hAPP_complex_real(re,hAPP_complex_complex(scaleR1652505878omplex(R_3),X_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(re,X_1)) ).
fof(fact_5109_complex__scaleR,axiom,
! [R_3,A_3,B_2] : hAPP_complex_complex(scaleR1652505878omplex(R_3),hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),A_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),B_2)) ).
fof(fact_5110_Re_OscaleR,axiom,
! [R_3,X_1] : hAPP_complex_real(re,hAPP_complex_complex(scaleR1652505878omplex(R_3),X_1)) = hAPP_real_real(scaleR_scaleR_real(R_3),hAPP_complex_real(re,X_1)) ).
fof(fact_5111_Im_OscaleR,axiom,
! [R_3,X_1] : hAPP_complex_real(im,hAPP_complex_complex(scaleR1652505878omplex(R_3),X_1)) = hAPP_real_real(scaleR_scaleR_real(R_3),hAPP_complex_real(im,X_1)) ).
fof(fact_5112_cnj_OscaleR,axiom,
! [R_3,X_1] : hAPP_complex_complex(cnj,hAPP_complex_complex(scaleR1652505878omplex(R_3),X_1)) = hAPP_complex_complex(scaleR1652505878omplex(R_3),hAPP_complex_complex(cnj,X_1)) ).
fof(fact_5113_complex__sgn__def,axiom,
! [X_1] : hAPP_complex_complex(sgn_sgn_complex,X_1) = hAPP_complex_complex(scaleR1652505878omplex(hAPP_real_real(inverse_inverse_real,hAPP_complex_real(norm_norm_complex,X_1))),X_1) ).
fof(fact_5114_complex__scaleR__def,axiom,
! [R_3,X_1] : hAPP_complex_complex(scaleR1652505878omplex(R_3),X_1) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(re,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(im,X_1))) ).
fof(fact_5115_trivial__limit__sequentially,axiom,
~ hBOOL(trivial_limit_nat(sequentially)) ).
fof(fact_5116_pred__nat__def,axiom,
pred_nat = collec1979865426at_nat(produc1922581855t_bool(hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat)),suc))) ).
fof(fact_5117_gcd__coprime__exists__int,axiom,
! [A_3,B_2] :
( ( is_int(A_3)
& is_int(B_2) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) != zero_zero_int
=> ? [A_4,B_3] :
( is_int(A_4)
& is_int(B_3)
& A_3 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_4),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2))
& B_2 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2))
& hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_4),B_3) = one_one_int ) ) ) ).
fof(fact_5118_min__number__of__Suc,axiom,
! [N,V] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,N)),hAPP_int_nat(number_number_of_nat,V)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),zero_zero_nat),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,N),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))))) ).
fof(fact_5119_min__Suc__number__of,axiom,
! [V,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),zero_zero_nat),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),N))) ).
fof(fact_5120_min__diff,axiom,
! [M,I,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),I)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M),N_1)),I) ).
fof(fact_5121_min__0R,axiom,
! [N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,N_1),zero_zero_nat) = zero_zero_nat ).
fof(fact_5122_min__0L,axiom,
! [N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N_1) = zero_zero_nat ).
fof(fact_5123_min__Suc__Suc,axiom,
! [M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M),N_1)) ).
fof(fact_5124_min__Suc2,axiom,
! [Ma,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Ma),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(nat_case_nat(zero_zero_nat,hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,ord_min_nat),N))),Ma) ).
fof(fact_5125_min__Suc1,axiom,
! [N,Ma] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,N)),Ma) = hAPP_nat_nat(nat_case_nat(zero_zero_nat,hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(ord_min_nat,N))),Ma) ).
fof(fact_5126_vanishes__mult__bounded,axiom,
! [Y_3,X_3] :
( ? [A_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_2))
& ! [N_2] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),A_2)) )
=> ( hBOOL(vanishes(Y_3))
=> hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,times_times_rat),X_3)),Y_3))) ) ) ).
fof(fact_5127_SetInterval_Otransfer__int__nat__set__functions,axiom,
! [N,Ma] :
( hBOOL(hAPP_int_bool(nat_is_nat,Ma))
=> ( hBOOL(hAPP_int_bool(nat_is_nat,N))
=> hAPP_i1948725293t_bool(ord_at875362053st_int(Ma),N) = image_nat_int(semiri1621563631at_int,hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_int_nat(nat,Ma)),hAPP_int_nat(nat,N))) ) ) ).
fof(fact_5128_is__nat__def,axiom,
! [X_2] :
( hBOOL(hAPP_int_bool(nat_is_nat,X_2))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) ) ).
fof(fact_5129_vanishes__diff,axiom,
! [Y_3,X_3] :
( hBOOL(vanishes(X_3))
=> ( hBOOL(vanishes(Y_3))
=> hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,minus_minus_rat),X_3)),Y_3))) ) ) ).
fof(fact_5130_vanishes__add,axiom,
! [Y_3,X_3] :
( hBOOL(vanishes(X_3))
=> ( hBOOL(vanishes(Y_3))
=> hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,plus_plus_rat),X_3)),Y_3))) ) ) ).
fof(fact_5131_vanishes__const,axiom,
! [C] :
( hBOOL(vanishes(cOMBK_rat_nat(C)))
<=> C = zero_zero_rat ) ).
fof(fact_5132_vanishes__minus,axiom,
! [X_3] :
( hBOOL(vanishes(X_3))
=> hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1082758869at_rat(cOMBB_rat_rat_nat,uminus_uminus_rat),X_3))) ) ).
fof(fact_5133_Nat__Transfer_Otransfer__int__nat__function__closures_I2_J,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(nat_is_nat,X_1))
=> ( hBOOL(hAPP_int_bool(nat_is_nat,Y_1))
=> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1))) ) ) ).
fof(fact_5134_Nat__Transfer_Otransfer__int__nat__function__closures_I4_J,axiom,
! [N_1,X_1] :
( hBOOL(hAPP_int_bool(nat_is_nat,X_1))
=> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N_1))) ) ).
fof(fact_5135_Divides_Otransfer__int__nat__function__closures_I2_J,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(nat_is_nat,X_1))
=> ( hBOOL(hAPP_int_bool(nat_is_nat,Y_1))
=> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(div_mod_int(X_1),Y_1))) ) ) ).
fof(fact_5136_Divides_Otransfer__int__nat__function__closures_I1_J,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(nat_is_nat,X_1))
=> ( hBOOL(hAPP_int_bool(nat_is_nat,Y_1))
=> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),Y_1))) ) ) ).
fof(fact_5137_Nat__Transfer_Otransfer__int__nat__function__closures_I3_J,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(nat_is_nat,X_1))
=> ( hBOOL(hAPP_int_bool(nat_is_nat,Y_1))
=> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(nat_tsub(X_1),Y_1))) ) ) ).
fof(fact_5138_Nat__Transfer_Otransfer__int__nat__set__function__closures_I6_J,axiom,
! [X_2,A_1] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A_1))
=> hBOOL(hAPP_int_bool(nat_is_nat,X_2)) ) ) ).
fof(fact_5139_Nat__Transfer_Otransfer__int__nat__function__closures_I5_J,axiom,
hBOOL(hAPP_int_bool(nat_is_nat,zero_zero_int)) ).
fof(fact_5140_Nat__Transfer_Otransfer__int__nat__function__closures_I6_J,axiom,
hBOOL(hAPP_int_bool(nat_is_nat,one_one_int)) ).
fof(fact_5141_Nat__Transfer_Otransfer__int__nat__function__closures_I1_J,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(nat_is_nat,X_1))
=> ( hBOOL(hAPP_int_bool(nat_is_nat,Y_1))
=> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_1))) ) ) ).
fof(fact_5142_Nat__Transfer_Otransfer__int__nat__function__closures_I9_J,axiom,
! [Z_1] : hBOOL(hAPP_int_bool(nat_is_nat,hAPP_nat_int(semiri1621563631at_int,Z_1))) ).
fof(fact_5143_SetInterval_Otransfer__int__nat__set__function__closures,axiom,
! [Y_2,X_2] :
( hBOOL(hAPP_int_bool(nat_is_nat,X_2))
=> hBOOL(hAPP_f448129468l_bool(nat_nat_set,hAPP_i1948725293t_bool(ord_at875362053st_int(X_2),Y_2))) ) ).
fof(fact_5144_Nat__Transfer_Otransfer__int__nat__function__closures_I8_J,axiom,
hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) ).
fof(fact_5145_transfer__int__nat__gcd__closures_I1_J,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(nat_is_nat,X_1))
=> ( hBOOL(hAPP_int_bool(nat_is_nat,Y_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_1))) ) ) ).
fof(fact_5146_transfer__int__nat__factorial__closure,axiom,
! [X_1] :
( hBOOL(hAPP_int_bool(nat_is_nat,X_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(fact_fact_int,X_1))) ) ).
fof(fact_5147_transfer__int__nat__set__relations_I2_J,axiom,
! [A_1,X_2] :
( hBOOL(hAPP_int_bool(nat_is_nat,X_2))
=> ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A_1))
<=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_int_nat(nat,X_2)),image_int_nat(nat,A_1))) ) ) ) ).
fof(fact_5148_Nat__Transfer_Otransfer__int__nat__function__closures_I7_J,axiom,
hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_5149_vanishes__def,axiom,
! [X_3] :
( hBOOL(vanishes(X_3))
<=> ! [R_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_1))
=> ? [K] :
! [N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_2))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),R_1)) ) ) ) ).
fof(fact_5150_vanishesD,axiom,
! [R_2,X_3] :
( hBOOL(vanishes(X_3))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_2))
=> ? [K] :
! [N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_2))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),R_2)) ) ) ) ).
fof(fact_5151_vanishesI,axiom,
! [X_3] :
( ! [R_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_1))
=> ? [K] :
! [N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_2))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),R_1)) ) )
=> hBOOL(vanishes(X_3)) ) ).
fof(fact_5152_zcongm__def,axiom,
! [Ma,X,Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcongm(Ma),X),Xa))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Xa),Ma)) ) ).
fof(fact_5153_cpmi,axiom,
! [B_1,P_2,P_1,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ( ? [Z] :
! [X] :
( is_int(X)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Z))
=> ( hBOOL(hAPP_int_bool(P_1,X))
<=> hBOOL(hAPP_int_bool(P_2,X)) ) ) )
=> ( ! [X] :
( is_int(X)
=> ( ! [Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
=> ! [Xb] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1))
=> X != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) )
=> ( hBOOL(hAPP_int_bool(P_1,X))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),D_1))) ) ) )
=> ( ! [X,K] :
( ( is_int(X)
& is_int(K) )
=> ( hBOOL(hAPP_int_bool(P_2,X))
<=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1)))) ) )
=> ( ? [X1] :
( is_int(X1)
& hBOOL(hAPP_int_bool(P_1,X1)) )
<=> ( ? [X] :
( is_int(X)
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
& hBOOL(hAPP_int_bool(P_2,X)) )
| ? [X] :
( is_int(X)
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
& ? [Xa] :
( is_int(Xa)
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),B_1))
& hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xa),X))) ) ) ) ) ) ) ) ) ).
fof(fact_5154_bijzcong__zcong__prod,axiom,
! [A_1,B_1,Ma] :
( hBOOL(hAPP_f1469637905l_bool(hAPP_P1681069257l_bool(member308914164t_bool,hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,A_1),B_1)),bijR_int_int(zcongm(Ma))))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,A_1)),big_co1548731110nt_int(cOMBI_int,B_1)),Ma)) ) ).
fof(fact_5155_cppi,axiom,
! [A_1,P_2,P_1,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ( ? [Z] :
! [X] :
( is_int(X)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X))
=> ( hBOOL(hAPP_int_bool(P_1,X))
<=> hBOOL(hAPP_int_bool(P_2,X)) ) ) )
=> ( ! [X] :
( is_int(X)
=> ( ! [Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
=> ! [Xb] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1))
=> X != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) )
=> ( hBOOL(hAPP_int_bool(P_1,X))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),D_1))) ) ) )
=> ( ! [X,K] :
( ( is_int(X)
& is_int(K) )
=> ( hBOOL(hAPP_int_bool(P_2,X))
<=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1)))) ) )
=> ( ? [X1] :
( is_int(X1)
& hBOOL(hAPP_int_bool(P_1,X1)) )
<=> ( ? [X] :
( is_int(X)
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
& hBOOL(hAPP_int_bool(P_2,X)) )
| ? [X] :
( is_int(X)
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
& ? [Xa] :
( is_int(Xa)
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A_1))
& hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),X))) ) ) ) ) ) ) ) ) ).
fof(fact_5156_increasing__LIMSEQ,axiom,
! [L,F] :
( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),hAPP_nat_real(F,hAPP_nat_nat(suc,N_2))))
=> ( ! [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),L))
=> ( ! [E] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E))
=> ? [N_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,L),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(F,N_2)),E))) )
=> hBOOL(tendsto_nat_real(F,L,sequentially)) ) ) ) ).
fof(fact_5157_isCont__ln,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> hBOOL(hAPP_real_bool(isCont_real_real(ln),X_2)) ) ).
fof(fact_5158_isCont__real__root,axiom,
! [X_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N)),X_2)) ) ).
fof(fact_5159_isCont__tan,axiom,
! [X_2] :
( hAPP_real_real(cos,X_2) != zero_zero_real
=> hBOOL(hAPP_real_bool(isCont_real_real(tan),X_2)) ) ).
fof(fact_5160_isCont__cos,axiom,
! [X_2] : hBOOL(hAPP_real_bool(isCont_real_real(cos),X_2)) ).
fof(fact_5161_isCont__arctan,axiom,
! [X_2] : hBOOL(hAPP_real_bool(isCont_real_real(arctan),X_2)) ).
fof(fact_5162_isCont__real__sqrt,axiom,
! [X_2] : hBOOL(hAPP_real_bool(isCont_real_real(sqrt),X_2)) ).
fof(fact_5163_isCont__sin,axiom,
! [X_2] : hBOOL(hAPP_real_bool(isCont_real_real(sin),X_2)) ).
fof(fact_5164_cnj_OisCont,axiom,
! [A] : hBOOL(hAPP_complex_bool(isCont156215680omplex(cnj),A)) ).
fof(fact_5165_Im_OisCont,axiom,
! [A] : hBOOL(hAPP_complex_bool(isCont_complex_real(im),A)) ).
fof(fact_5166_Re_OisCont,axiom,
! [A] : hBOOL(hAPP_complex_bool(isCont_complex_real(re),A)) ).
fof(fact_5167_isCont__abs,axiom,
! [A] : hBOOL(hAPP_real_bool(isCont_real_real(abs_abs_real),A)) ).
fof(fact_5168_isCont__root__zero,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N)),zero_zero_real)) ) ).
fof(fact_5169_isCont__root__pos,axiom,
! [X_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
=> hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N)),X_2)) ) ) ).
fof(fact_5170_isCont__root__neg,axiom,
! [X_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),zero_zero_real))
=> hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N)),X_2)) ) ) ).
fof(fact_5171_isCont__arccos,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),one_one_real))
=> hBOOL(hAPP_real_bool(isCont_real_real(arccos),X_2)) ) ) ).
fof(fact_5172_isCont__arcsin,axiom,
! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),one_one_real))
=> hBOOL(hAPP_real_bool(isCont_real_real(arcsin),X_2)) ) ) ).
fof(fact_5173_LIM__less__bound,axiom,
! [F,B,X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B),X_2))
=> ( ! [X] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X),hAPP_r1134773055l_bool(ord_gr788844697n_real(B),X_2)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(F,X))) )
=> ( hBOOL(hAPP_real_bool(isCont_real_real(F),X_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(F,X_2))) ) ) ) ).
fof(fact_5174_DERIV__inverse__function,axiom,
! [B,A,F,G,X_2,D_1] :
( hBOOL(hAPP_real_bool(deriv_real(F,hAPP_real_real(G,X_2)),D_1))
=> ( D_1 != zero_zero_real
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),B))
=> ( ! [Y] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),Y))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),B)) )
=> hAPP_real_real(F,hAPP_real_real(G,Y)) = Y )
=> ( hBOOL(hAPP_real_bool(isCont_real_real(G),X_2))
=> hBOOL(hAPP_real_bool(deriv_real(G,X_2),hAPP_real_real(inverse_inverse_real,D_1))) ) ) ) ) ) ) ).
fof(fact_5175_isCont__inv__fun,axiom,
! [G,F,X_2,D] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
=> ( ! [Z] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D))
=> hAPP_real_real(G,hAPP_real_real(F,Z)) = Z )
=> ( ! [Z] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D))
=> hBOOL(hAPP_real_bool(isCont_real_real(F),Z)) )
=> hBOOL(hAPP_real_bool(isCont_real_real(G),hAPP_real_real(F,X_2))) ) ) ) ).
fof(fact_5176_DERIV__isconst2,axiom,
! [X_2,F,A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B))
=> ( ! [X] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B)) )
=> hBOOL(hAPP_real_bool(isCont_real_real(F),X)) )
=> ( ! [X] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),B)) )
=> hBOOL(hAPP_real_bool(deriv_real(F,X),zero_zero_real)) )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B))
=> hAPP_real_real(F,X_2) = hAPP_real_real(F,A) ) ) ) ) ) ).
fof(fact_5177_DERIV__isconst__end,axiom,
! [F,A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B))
=> ( ! [X] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B)) )
=> hBOOL(hAPP_real_bool(isCont_real_real(F),X)) )
=> ( ! [X] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),B)) )
=> hBOOL(hAPP_real_bool(deriv_real(F,X),zero_zero_real)) )
=> hAPP_real_real(F,B) = hAPP_real_real(F,A) ) ) ) ).
fof(fact_5178_DERIV__isconst1,axiom,
! [F,A,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B))
=> ( ! [X] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B)) )
=> hBOOL(hAPP_real_bool(isCont_real_real(F),X)) )
=> ( ! [X] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),B)) )
=> hBOOL(hAPP_real_bool(deriv_real(F,X),zero_zero_real)) )
=> ! [X] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),B)) )
=> hAPP_real_real(F,X) = hAPP_real_real(F,A) ) ) ) ) ).
fof(fact_5179_isCont__inv__fun__inv,axiom,
! [G,F,X_2,D] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
=> ( ! [Z] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D))
=> hAPP_real_real(G,hAPP_real_real(F,Z)) = Z )
=> ( ! [Z] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D))
=> hBOOL(hAPP_real_bool(isCont_real_real(F),Z)) )
=> ? [E] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E))
& ! [Y] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y),hAPP_real_real(F,X_2)))))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y),hAPP_real_real(F,X_2)))),E)) )
=> hAPP_real_real(F,hAPP_real_real(G,Y)) = Y ) ) ) ) ) ).
fof(fact_5180_lemma__isCont__inj2,axiom,
! [G,F,X_2,D] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
=> ( ! [Z] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D))
=> hAPP_real_real(G,hAPP_real_real(F,Z)) = Z )
=> ( ! [Z] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D))
=> hBOOL(hAPP_real_bool(isCont_real_real(F),Z)) )
=> ? [Z] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,Z)),hAPP_real_real(F,X_2))) ) ) ) ) ).
fof(fact_5181_lemma__isCont__inj,axiom,
! [G,F,X_2,D] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
=> ( ! [Z] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D))
=> hAPP_real_real(G,hAPP_real_real(F,Z)) = Z )
=> ( ! [Z] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D))
=> hBOOL(hAPP_real_bool(isCont_real_real(F),Z)) )
=> ? [Z] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_2))),D))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_2)),hAPP_real_real(F,Z))) ) ) ) ) ).
fof(fact_5182_SetInterval_Otransfer__nat__int__set__functions_I1_J,axiom,
! [N] : hAPP_n1699378549t_bool(ord_atMost_nat,N) = image_int_nat(nat,hAPP_i1948725293t_bool(ord_at875362053st_int(zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N))) ).
fof(fact_5183_max__number__of__Suc,axiom,
! [N,V] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,N)),hAPP_int_nat(number_number_of_nat,V)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),hAPP_nat_nat(suc,N)),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))))) ).
fof(fact_5184_finite__atMost,axiom,
! [K_1] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_atMost_nat,K_1))) ).
fof(fact_5185_atLeast0AtMost,axiom,
! [N] : hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),N) = hAPP_n1699378549t_bool(ord_atMost_nat,N) ).
fof(fact_5186_lessThan__Suc__atMost,axiom,
! [K_1] : hAPP_n1699378549t_bool(ord_lessThan_nat,hAPP_nat_nat(suc,K_1)) = hAPP_n1699378549t_bool(ord_atMost_nat,K_1) ).
fof(fact_5187_nat__minus__add__max,axiom,
! [N_1,M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),M)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N_1),M) ).
fof(fact_5188_max__0R,axiom,
! [N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N_1),zero_zero_nat) = N_1 ).
fof(fact_5189_max__0L,axiom,
! [N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N_1) = N_1 ).
fof(fact_5190_max__Suc__Suc,axiom,
! [M,N_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,M),N_1)) ).
fof(fact_5191_card__atMost,axiom,
! [U] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_atMost_nat,U)) = hAPP_nat_nat(suc,U) ).
fof(fact_5192_max__Suc1,axiom,
! [N,Ma] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,N)),Ma) = hAPP_nat_nat(nat_case_nat(hAPP_nat_nat(suc,N),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(ord_max_nat,N))),Ma) ).
fof(fact_5193_max__Suc2,axiom,
! [Ma,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Ma),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(nat_case_nat(hAPP_nat_nat(suc,N),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,ord_max_nat),N))),Ma) ).
fof(fact_5194_finite__nat__iff__bounded__le,axiom,
! [S] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S))
<=> ? [K] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_atMost_nat,K))) ) ).
fof(fact_5195_max__Suc__number__of,axiom,
! [V,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),hAPP_nat_nat(suc,N)),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),N))) ).
%----Helper facts (224)
fof(help_fNot_1_1_U,axiom,
! [P] :
( ~ hBOOL(hAPP_bool_bool(fNot,P))
| ~ hBOOL(P) ) ).
fof(help_fNot_2_1_U,axiom,
! [P] :
( hBOOL(P)
| hBOOL(hAPP_bool_bool(fNot,P)) ) ).
fof(help_fTrue_1_1_U,axiom,
hBOOL(fTrue) ).
fof(help_fTrue_1_1_T,axiom,
! [P] :
( is_bool(P)
=> ( P = fTrue
| P = fFalse ) ) ).
fof(help_fconj_1_1_U,axiom,
! [Q,P] :
( ~ hBOOL(P)
| ~ hBOOL(Q)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) ) ).
fof(help_fconj_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
| hBOOL(P) ) ).
fof(help_fconj_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
| hBOOL(Q) ) ).
fof(help_fdisj_1_1_U,axiom,
! [Q,P] :
( ~ hBOOL(P)
| hBOOL(fdisj(P,Q)) ) ).
fof(help_fdisj_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(Q)
| hBOOL(fdisj(P,Q)) ) ).
fof(help_fdisj_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(fdisj(P,Q))
| hBOOL(P)
| hBOOL(Q) ) ).
fof(help_fFalse_1_1_U,axiom,
~ hBOOL(fFalse) ).
fof(help_fFalse_1_1_T,axiom,
! [P] :
( is_bool(P)
=> ( P = fTrue
| P = fFalse ) ) ).
fof(help_fimplies_1_1_U,axiom,
! [Q,P] :
( hBOOL(P)
| hBOOL(hAPP_bool_bool(fimplies(P),Q)) ) ).
fof(help_fimplies_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(Q)
| hBOOL(hAPP_bool_bool(fimplies(P),Q)) ) ).
fof(help_fimplies_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP_bool_bool(fimplies(P),Q))
| ~ hBOOL(P)
| hBOOL(Q) ) ).
fof(help_If_1_1_If_000tc__Int__Oint_T,axiom,
! [X_1,Y_1] :
( is_int(X_1)
=> hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,fTrue),X_1),Y_1) = X_1 ) ).
fof(help_If_2_1_If_000tc__Int__Oint_T,axiom,
! [X_1,Y_1] :
( is_int(Y_1)
=> hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,fFalse),X_1),Y_1) = Y_1 ) ).
fof(help_If_3_1_If_000tc__Int__Oint_T,axiom,
! [P] :
( is_bool(P)
=> ( P = fTrue
| P = fFalse ) ) ).
fof(help_If_1_1_If_000tc__Nat__Onat_T,axiom,
! [X_1,Y_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,fTrue),X_1),Y_1) = X_1 ).
fof(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
! [X_1,Y_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,fFalse),X_1),Y_1) = Y_1 ).
fof(help_If_3_1_If_000tc__Nat__Onat_T,axiom,
! [P] :
( is_bool(P)
=> ( P = fTrue
| P = fFalse ) ) ).
fof(help_fEx_1_1_fEx_000tc__Int__Oint_U,axiom,
! [P,X_1] :
( ~ hBOOL(hAPP_int_bool(P,X_1))
| hBOOL(hAPP_f448129468l_bool(fEx_int,P)) ) ).
fof(help_fEx_1_1_fEx_000tc__Nat__Onat_U,axiom,
! [P,X_1] :
( ~ hBOOL(hAPP_nat_bool(P,X_1))
| hBOOL(hAPP_f54304608l_bool(fEx_nat,P)) ) ).
fof(help_COMBI_1_1_COMBI_000tc__Int__Oint_U,axiom,
! [P] :
( is_int(P)
=> hAPP_int_int(cOMBI_int,P) = P ) ).
fof(help_COMBI_1_1_COMBI_000tc__Nat__Onat_U,axiom,
! [P] : hAPP_nat_nat(cOMBI_nat,P) = P ).
fof(help_fequal_1_1_fequal_000tc__Int__Oint_T,axiom,
! [X_1,Y_1] :
( ( is_int(X_1)
& is_int(Y_1) )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X_1),Y_1))
| X_1 = Y_1 ) ) ).
fof(help_fequal_2_1_fequal_000tc__Int__Oint_T,axiom,
! [X_1,Y_1] :
( X_1 != Y_1
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X_1),Y_1)) ) ).
fof(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
! [X_1,Y_1] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X_1),Y_1))
| X_1 = Y_1 ) ).
fof(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
! [X_1,Y_1] :
( X_1 != Y_1
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X_1),Y_1)) ) ).
fof(help_fequal_1_1_fequal_000tc__Rat__Orat_T,axiom,
! [X_1,Y_1] :
( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(fequal_rat,X_1),Y_1))
| X_1 = Y_1 ) ).
fof(help_fequal_2_1_fequal_000tc__Rat__Orat_T,axiom,
! [X_1,Y_1] :
( X_1 != Y_1
| hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(fequal_rat,X_1),Y_1)) ) ).
fof(help_If_1_1_If_000tc__RealDef__Oreal_T,axiom,
! [X_1,Y_1] : hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,fTrue),X_1),Y_1) = X_1 ).
fof(help_If_2_1_If_000tc__RealDef__Oreal_T,axiom,
! [X_1,Y_1] : hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,fFalse),X_1),Y_1) = Y_1 ).
fof(help_If_3_1_If_000tc__RealDef__Oreal_T,axiom,
! [P] :
( is_bool(P)
=> ( P = fTrue
| P = fFalse ) ) ).
fof(help_COMBI_1_1_COMBI_000tc__RealDef__Oreal_U,axiom,
! [P] : hAPP_real_real(cOMBI_real,P) = P ).
fof(help_fequal_1_1_fequal_000tc__RealDef__Oreal_T,axiom,
! [X_1,Y_1] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(fequal_real,X_1),Y_1))
| X_1 = Y_1 ) ).
fof(help_fequal_2_1_fequal_000tc__RealDef__Oreal_T,axiom,
! [X_1,Y_1] :
( X_1 != Y_1
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(fequal_real,X_1),Y_1)) ) ).
fof(help_COMBK_1_1_COMBK_000tc__Rat__Orat_000tc__Nat__Onat_U,axiom,
! [P,Q] : hAPP_nat_rat(cOMBK_rat_nat(P),Q) = P ).
fof(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
! [P,Q] :
( is_bool(P)
=> hAPP_nat_bool(cOMBK_bool_nat(P),Q) = P ) ).
fof(help_If_1_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T,axiom,
! [X_1,Y_1] : hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fTrue),X_1),Y_1) = X_1 ).
fof(help_If_2_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T,axiom,
! [X_1,Y_1] : hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fFalse),X_1),Y_1) = Y_1 ).
fof(help_If_3_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T,axiom,
! [P] :
( is_bool(P)
=> ( P = fTrue
| P = fFalse ) ) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__Int__Oint_U,axiom,
! [P,Q,R] : hAPP_int_int(hAPP_f1356864277nt_int(hAPP_f747162069nt_int(cOMBB_int_int_int,P),Q),R) = hAPP_int_int(P,hAPP_int_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__Nat__Onat_U,axiom,
! [P,Q,R] : hAPP_nat_int(hAPP_f1139079189at_int(hAPP_f1431025877at_int(cOMBB_int_int_nat,P),Q),R) = hAPP_int_int(P,hAPP_nat_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Rat__Orat_000tc__Int__Oint_U,axiom,
! [P,Q,R] : hAPP_int_rat(hAPP_f1900950721nt_rat(hAPP_f1270196437nt_rat(cOMBB_int_rat_int,P),Q),R) = hAPP_int_rat(P,hAPP_int_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
! [P,Q,R] : hAPP_nat_nat(hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,P),Q),R) = hAPP_nat_nat(P,hAPP_nat_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__Rat__Orat_000tc__Nat__Onat_U,axiom,
! [P,Q,R] : hAPP_nat_rat(hAPP_f2062659861at_rat(hAPP_f1082758869at_rat(cOMBB_rat_rat_nat,P),Q),R) = hAPP_rat_rat(P,hAPP_nat_rat(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__Int__Oint_U,axiom,
! [P,Q,R] : hAPP_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,P),Q),R) = hAPP_int_int(hAPP_int_fun_int_int(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Nat__Onat_000tc__Int__Oint_U,axiom,
! [P,Q,R] : hAPP_int_int(hAPP_nat_fun_int_int(hAPP_f2105620693nt_int(cOMBC_int_nat_int,P),Q),R) = hAPP_nat_int(hAPP_int_fun_nat_int(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Int__Oint_000tc__Int__Oint_U,axiom,
! [P,Q,R] : hAPP_nat_int(hAPP_int_fun_nat_int(hAPP_f1673103573at_int(cOMBC_nat_int_int,P),Q),R) = hAPP_int_int(hAPP_nat_fun_int_int(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
! [P,Q,R] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,P),Q),R) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(P,R),Q) ).
fof(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
! [P,Q,R] : hAPP_nat_nat(hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,P),Q),R) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(P,R),hAPP_nat_nat(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__Rat__Orat_000tc__Rat__Orat_U,axiom,
! [P,Q,R] : hAPP_nat_rat(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,P),Q),R) = hAPP_rat_rat(hAPP_nat_fun_rat_rat(P,R),hAPP_nat_rat(Q,R)) ).
fof(help_If_1_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T,axiom,
! [X_1,Y_1] : hAPP_P1860904029l_real(hAPP_P1982799375l_real(hAPP_b1482221219l_real(if_Pro313124157l_real,fTrue),X_1),Y_1) = X_1 ).
fof(help_If_2_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T,axiom,
! [X_1,Y_1] : hAPP_P1860904029l_real(hAPP_P1982799375l_real(hAPP_b1482221219l_real(if_Pro313124157l_real,fFalse),X_1),Y_1) = Y_1 ).
fof(help_If_3_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T,axiom,
! [P] :
( is_bool(P)
=> ( P = fTrue
| P = fFalse ) ) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
! [P,Q,R] : hAPP_int_bool(hAPP_f429935748t_bool(hAPP_f1512942609t_bool(cOMBB_int_bool_int,P),Q),R) = hAPP_int_bool(P,hAPP_int_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
! [P,Q,R] : hAPP_nat_bool(hAPP_f1154658180t_bool(hAPP_f1361476881t_bool(cOMBB_int_bool_nat,P),Q),R) = hAPP_int_bool(P,hAPP_nat_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
! [P,Q,R] : hAPP_int_bool(hAPP_f1448486952t_bool(hAPP_f696844121t_bool(cOMBB_nat_bool_int,P),Q),R) = hAPP_nat_bool(P,hAPP_int_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
! [P,Q,R] : hAPP_int_bool(hAPP_f1241350704t_bool(hAPP_f1131424617t_bool(cOMBB_rat_bool_int,P),Q),R) = hAPP_rat_bool(P,hAPP_int_rat(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__HOL__Obool_U,axiom,
! [P,Q,R] : hAPP_int_bool(hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,P),Q),R) = hAPP_int_bool(hAPP_i1948725293t_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Rat__Orat_000tc__HOL__Obool_U,axiom,
! [P,Q,R] : hAPP_int_bool(hAPP_r1900510169t_bool(hAPP_f1480506641t_bool(cOMBC_int_rat_bool,P),Q),R) = hAPP_rat_bool(hAPP_i16292313t_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool_U,axiom,
! [P,Q,R] : hAPP_nat_bool(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,P),Q),R) = hAPP_nat_bool(hAPP_n1699378549t_bool(P,R),Q) ).
fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
! [P,Q,R] : hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,P),Q),R) = hAPP_bool_bool(P,hAPP_int_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
! [P,Q,R] : hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,P),Q),R) = hAPP_bool_bool(P,hAPP_nat_bool(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P,Q,R] : hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_i68813070l_bool(P,R),hAPP_int_bool(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P,Q,R] : hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_n1006566506l_bool(P,R),hAPP_nat_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__RealDef__Oreal_000tc__Int__Oint_U,axiom,
! [P,Q,R] : hAPP_int_real(hAPP_f563293398t_real(hAPP_f134192053t_real(cOMBB_int_real_int,P),Q),R) = hAPP_int_real(P,hAPP_int_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__Nat__Onat_U,axiom,
! [P,Q,R] : hAPP_nat_real(hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,P),Q),R) = hAPP_nat_real(P,hAPP_nat_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
! [P,Q,R] : hAPP_int_bool(hAPP_f429384717t_bool(hAPP_f1173902867t_bool(cOMBB_real_bool_int,P),Q),R) = hAPP_real_bool(P,hAPP_int_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
! [P,Q,R] : hAPP_nat_bool(hAPP_f737665877t_bool(hAPP_f605995867t_bool(cOMBB_real_bool_nat,P),Q),R) = hAPP_real_bool(P,hAPP_nat_real(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__RealDef__Oreal_000tc__HOL__Obool_U,axiom,
! [P,Q,R] : hAPP_int_bool(hAPP_r481142414t_bool(hAPP_f1845081853t_bool(cOMBC_int_real_bool,P),Q),R) = hAPP_real_bool(hAPP_i688218208l_bool(P,R),Q) ).
fof(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P,Q,R] : hAPP_real_bool(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_r1938957037l_bool(P,R),hAPP_real_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__Int__Oin,axiom,
! [P,Q,R] : hAPP_int_real(hAPP_f269654879t_real(hAPP_f766218899t_real(cOMBB_real_real_int,P),Q),R) = hAPP_real_real(P,hAPP_int_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__Nat__Ona,axiom,
! [P,Q,R] : hAPP_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,P),Q),R) = hAPP_real_real(P,hAPP_nat_real(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__RealDef__Orea,axiom,
! [P,Q,R] : hAPP_nat_real(hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,P),Q),R) = hAPP_real_real(hAPP_n546711566l_real(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__Nat__Onat_000tc__RealDef__Orea,axiom,
! [P,Q,R] : hAPP_real_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,P),Q),R) = hAPP_nat_real(hAPP_r474017924t_real(P,R),Q) ).
fof(help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
! [P,Q] : hAPP_f957591787ol_nat(cOMBK_1643584600t_bool(P),Q) = P ).
fof(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__RealDef__Orea,axiom,
! [P,Q,R] : hAPP_nat_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,P),Q),R) = hAPP_real_real(hAPP_n546711566l_real(P,R),hAPP_nat_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__RealDef__Ore,axiom,
! [P,Q,R] : hAPP_real_bool(hAPP_f1402056515l_bool(hAPP_f824790735l_bool(cOMBB_real_bool_real,P),Q),R) = hAPP_real_bool(P,hAPP_real_real(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__HOL__Obo,axiom,
! [P,Q,R] : hAPP_real_bool(hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,P),Q),R) = hAPP_real_bool(hAPP_r1134773055l_bool(P,R),Q) ).
fof(help_COMBB_1_1_COMBB_000tc__Complex__Ocomplex_000tc__RealDef__Oreal_000tc__Nat__,axiom,
! [P,Q,R] : hAPP_nat_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,P),Q),R) = hAPP_complex_real(P,hAPP_nat_complex(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__Complex__Ocom,axiom,
! [P,Q,R] : hAPP_nat_complex(hAPP_f219720582omplex(hAPP_f1052840010omplex(cOMBS_696648398omplex,P),Q),R) = hAPP_real_complex(hAPP_n1983812447omplex(P,R),hAPP_nat_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef_,axiom,
! [P,Q,R] : hAPP_real_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,P),Q),R) = hAPP_real_real(P,hAPP_real_real(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef_,axiom,
! [P,Q,R] : hAPP_real_real(hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,P),Q),R) = hAPP_real_real(hAPP_r1250527377l_real(P,R),Q) ).
fof(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef_,axiom,
! [P,Q,R] : hAPP_real_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,P),Q),R) = hAPP_real_real(hAPP_r1250527377l_real(P,R),hAPP_real_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Complex__Ocomplex_000tc__Complex__Ocomplex_000tc__Na,axiom,
! [P,Q,R] : hAPP_nat_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,P),Q),R) = hAPP_complex_complex(P,hAPP_nat_complex(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J,axiom,
! [P,Q,R] : hAPP_int_fun_int_int(hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,P),Q),R) = hAPP_int_fun_int_int(P,hAPP_int_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_001,axiom,
! [P,Q,R] : hAPP_nat_fun_int_int(hAPP_f1935805932nt_int(hAPP_f125788821nt_int(cOMBB_859312247nt_nat,P),Q),R) = hAPP_int_fun_int_int(P,hAPP_nat_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J,axiom,
! [P,Q,R] : hAPP_nat_fun_nat_nat(hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,P),Q),R) = hAPP_nat_fun_nat_nat(P,hAPP_nat_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__fun_Itc__Rat__Orat_Mtc__Rat__Orat_J,axiom,
! [P,Q,R] : hAPP_nat_fun_rat_rat(hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,P),Q),R) = hAPP_rat_fun_rat_rat(P,hAPP_nat_rat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__prod_Itc__Int__Oin,axiom,
! [P,Q,R] : hAPP_P1175774780nt_int(hAPP_f521271395nt_int(hAPP_f11151005nt_int(cOMBB_47643171nt_int,P),Q),R) = hAPP_int_int(P,hAPP_P1175774780nt_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
! [P,Q,R] : hAPP_i1948725293t_bool(hAPP_f1791153283t_bool(hAPP_f1399575567t_bool(cOMBB_118231410ol_int,P),Q),R) = hAPP_i1948725293t_bool(P,hAPP_int_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_,axiom,
! [P,Q,R] : hAPP_i1524277240nt_int(hAPP_f883232462nt_int(hAPP_f2002277285nt_int(cOMBB_1846624359nt_int,P),Q),R) = hAPP_i1524277240nt_int(P,hAPP_int_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
! [P,Q,R] : hAPP_n1699378549t_bool(hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,P),Q),R) = hAPP_n1699378549t_bool(P,hAPP_nat_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__fun_Itc__Rat__Orat_Mtc__HOL__Obool_,axiom,
! [P,Q,R] : hAPP_i16292313t_bool(hAPP_f601898971t_bool(hAPP_f857296063t_bool(cOMBB_214503962ol_int,P),Q),R) = hAPP_r2115560837t_bool(P,hAPP_int_rat(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__fun_Itc__Int__Oint,axiom,
! [P,Q,R] : hAPP_i1948725293t_bool(hAPP_i345030060t_bool(hAPP_f791698359t_bool(cOMBC_241247961t_bool,P),Q),R) = hAPP_i1948725293t_bool(hAPP_i345030060t_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__prod_Itc__Int__Oin,axiom,
! [P,Q,R] : hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(hAPP_f465821005nt_int(cOMBC_1964283556nt_int,P),Q),R) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin,axiom,
! [P,Q,R] : hAPP_P1175774780nt_int(hAPP_i61245889nt_int(hAPP_f578362681nt_int(cOMBC_1417754532nt_int,P),Q),R) = hAPP_int_int(hAPP_P2110489235nt_int(P,R),Q) ).
fof(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin,axiom,
! [P,Q,R] : hAPP_P1175774780nt_int(hAPP_f521271395nt_int(hAPP_f1914742907nt_int(cOMBS_2019229620nt_int,P),Q),R) = hAPP_int_int(hAPP_P2110489235nt_int(P,R),hAPP_P1175774780nt_int(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin_002,axiom,
! [P,Q,R] : hAPP_P1853363688nt_rat(hAPP_f1065357839nt_rat(hAPP_f1120137083nt_rat(cOMBS_549334880nt_rat,P),Q),R) = hAPP_int_rat(hAPP_P507092031nt_rat(P,R),hAPP_P1175774780nt_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__prod_Itc__Int__Oi,axiom,
! [P,Q,R] : hAPP_P603027463t_bool(hAPP_f892584630t_bool(hAPP_f492612297t_bool(cOMBB_798864284nt_int,P),Q),R) = hAPP_int_bool(P,hAPP_P1175774780nt_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__HOL__Obool_000tc__prod_Itc__Int__Oi,axiom,
! [P,Q,R] : hAPP_P603027463t_bool(hAPP_f1084724746t_bool(hAPP_f374502857t_bool(cOMBB_569260360nt_int,P),Q),R) = hAPP_rat_bool(P,hAPP_P1853363688nt_rat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
! [P,Q,R] : hAPP_int_bool(hAPP_f2119767738t_bool(hAPP_f423804115t_bool(cOMBB_1418110531ol_int,P),Q),R) = hAPP_f448129468l_bool(P,hAPP_i1948725293t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
! [P,Q,R] : hAPP_int_bool(hAPP_f41603166t_bool(hAPP_f389116883t_bool(cOMBB_1593174431ol_int,P),Q),R) = hAPP_f54304608l_bool(P,hAPP_i418383825t_bool(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
! [P,Q,R] : hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,P),Q),R) = hAPP_f448129468l_bool(hAPP_i2112223885l_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
! [P,Q,R] : hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,P),Q),R) = hAPP_f54304608l_bool(hAPP_n215258509l_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin_003,axiom,
! [P,Q,R] : hAPP_P603027463t_bool(hAPP_i119638232t_bool(hAPP_f2018027565t_bool(cOMBC_33652895t_bool,P),Q),R) = hAPP_int_bool(hAPP_P252494982t_bool(P,R),Q) ).
fof(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin_004,axiom,
! [P,Q,R] : hAPP_P603027463t_bool(hAPP_f892584630t_bool(hAPP_f1325804733t_bool(cOMBS_480885903t_bool,P),Q),R) = hAPP_int_bool(hAPP_P252494982t_bool(P,R),hAPP_P1175774780nt_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__prod_Itc__Int__O,axiom,
! [P,Q,R] : hAPP_P603027463t_bool(hAPP_f237669397t_bool(hAPP_f513299955t_bool(cOMBB_1344650379nt_int,P),Q),R) = hAPP_bool_bool(P,hAPP_P603027463t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo,axiom,
! [P,Q,R] : hAPP_i68813070l_bool(hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_int_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_005,axiom,
! [P,Q,R] : hAPP_n1006566506l_bool(hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_nat_bool(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__HOL__Obo,axiom,
! [P,Q,R] : hAPP_P603027463t_bool(hAPP_f237669397t_bool(hAPP_f461489489t_bool(cOMBS_400904860l_bool,P),Q),R) = hAPP_bool_bool(hAPP_P921862901l_bool(P,R),hAPP_P603027463t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__RealDef_,axiom,
! [P,Q,R] : hAPP_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,P),Q),R) = hAPP_f1406902706l_real(P,hAPP_n1699378549t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obo_006,axiom,
! [P,Q,R] : hAPP_real_bool(hAPP_f101635158l_bool(hAPP_f1925507189l_bool(cOMBB_1121934354l_real,P),Q),R) = hAPP_f448129468l_bool(P,hAPP_r481142414t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_007,axiom,
! [P,Q,R] : hAPP_r1938957037l_bool(hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_real_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDe,axiom,
! [P,Q,R] : hAPP_n546711566l_real(hAPP_f1904307534l_real(hAPP_f224989205l_real(cOMBB_1419615765al_nat,P),Q),R) = hAPP_n546711566l_real(P,hAPP_nat_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__H,axiom,
! [P,Q,R] : hAPP_i688218208l_bool(hAPP_f801032407l_bool(hAPP_f499530369l_bool(cOMBB_1317819104ol_int,P),Q),R) = hAPP_r1134773055l_bool(P,hAPP_int_real(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__O,axiom,
! [P,Q,R] : hAPP_real_real(hAPP_f2119584768l_real(hAPP_f59594445l_real(cOMBC_2010481033l_real,P),Q),R) = hAPP_f1406902706l_real(hAPP_r1409264866l_real(P,R),Q) ).
fof(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__RealDe,axiom,
! [P,Q,R] : hAPP_r474017924t_real(hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,P),Q),R) = hAPP_r474017924t_real(P,hAPP_real_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R,axiom,
! [P,Q,R] : hAPP_i371647666l_real(hAPP_f1352333353l_real(hAPP_f318350629l_real(cOMBB_1946221070al_int,P),Q),R) = hAPP_r1250527377l_real(P,hAPP_int_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R_008,axiom,
! [P,Q,R] : hAPP_n546711566l_real(hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,P),Q),R) = hAPP_r1250527377l_real(P,hAPP_nat_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__Real,axiom,
! [P,Q,R] : hAPP_real_real(hAPP_f1122023986l_real(hAPP_f1786887945l_real(cOMBB_1896127834l_real,P),Q),R) = hAPP_f352196356l_real(P,hAPP_r474017924t_real(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__fun_Itc__Real,axiom,
! [P,Q,R] : hAPP_n546711566l_real(hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,P),Q),R) = hAPP_r1250527377l_real(hAPP_n682674106l_real(P,R),Q) ).
fof(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__fun_Itc__Real,axiom,
! [P,Q,R] : hAPP_n546711566l_real(hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,P),Q),R) = hAPP_r1250527377l_real(hAPP_n682674106l_real(P,R),hAPP_nat_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__H_009,axiom,
! [P,Q,R] : hAPP_r1134773055l_bool(hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,P),Q),R) = hAPP_r1134773055l_bool(P,hAPP_real_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc,axiom,
! [P,Q,R] : hAPP_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,P),Q),R) = hAPP_P731461727l_real(P,hAPP_n332733730l_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc_010,axiom,
! [P,Q,R] : hAPP_real_bool(hAPP_f330985925l_bool(hAPP_f940061907l_bool(cOMBB_1795223523l_real,P),Q),R) = hAPP_P1333854989l_bool(P,hAPP_r1195171167l_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__C,axiom,
! [P,Q,R] : hAPP_n1983812447omplex(hAPP_f23851874omplex(hAPP_f406530075omplex(cOMBB_1714920583ex_nat,P),Q),R) = hAPP_r265291036omplex(P,hAPP_nat_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R_011,axiom,
! [P,Q,R] : hAPP_r1250527377l_real(hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,P),Q),R) = hAPP_r1250527377l_real(P,hAPP_real_real(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__fun_Itc_,axiom,
! [P,Q,R] : hAPP_r1250527377l_real(hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,P),Q),R) = hAPP_r1250527377l_real(hAPP_r337325687l_real(P,R),Q) ).
fof(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc__,axiom,
! [P,Q,R] : hAPP_r1195171167l_real(hAPP_f617735571l_real(hAPP_f800361423l_real(cOMBB_2107848501l_real,P),Q),R) = hAPP_r1195171167l_real(P,hAPP_real_real(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__prod_Itc,axiom,
! [P,Q,R] : hAPP_r1195171167l_real(hAPP_f617735571l_real(hAPP_f1730463541l_real(cOMBS_511527878l_real,P),Q),R) = hAPP_r1195171167l_real(hAPP_r2019696015l_real(P,R),hAPP_real_real(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J,axiom,
! [P,Q,R] : hAPP_int_fun_int_int(hAPP_f1760145644nt_int(hAPP_f808647893nt_int(cOMBC_736024425nt_int,P),Q),R) = hAPP_f1356864277nt_int(hAPP_i744251628nt_int(P,R),Q) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_012,axiom,
! [P,Q,R] : hAPP_P2110489235nt_int(hAPP_f1855025978nt_int(hAPP_f1038672605nt_int(cOMBB_1506897658nt_int,P),Q),R) = hAPP_int_fun_int_int(P,hAPP_P1175774780nt_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Rat__Orat_J,axiom,
! [P,Q,R] : hAPP_P507092031nt_rat(hAPP_f1547007718nt_rat(hAPP_f382317021nt_rat(cOMBB_170828966nt_int,P),Q),R) = hAPP_int_fun_int_rat(P,hAPP_P1175774780nt_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__I,axiom,
! [P,Q,R] : hAPP_i1584592887nt_int(hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,P),Q),R) = hAPP_i1584592887nt_int(P,hAPP_int_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__prod_Itc__N,axiom,
! [P,Q,R] : hAPP_n1865633855at_nat(hAPP_f837737749at_nat(hAPP_f999082675at_nat(cOMBB_348438404at_nat,P),Q),R) = hAPP_n1865633855at_nat(P,hAPP_nat_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__,axiom,
! [P,Q,R] : hAPP_i1948725293t_bool(hAPP_f720654810t_bool(hAPP_f908327869t_bool(cOMBB_1290694363ol_int,P),Q),R) = hAPP_f429935748t_bool(P,hAPP_int_fun_int_int(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_013,axiom,
! [P,Q,R] : hAPP_i1948725293t_bool(hAPP_f1791153283t_bool(hAPP_f735247205t_bool(cOMBC_983868720t_bool,P),Q),R) = hAPP_f429935748t_bool(hAPP_i872162435t_bool(P,R),Q) ).
fof(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__Int__Oint_000tc__fun_Itc__Int__Oint,axiom,
! [P,Q,R] : hAPP_i1584592887nt_int(hAPP_f1574055885nt_int(hAPP_f434654626nt_int(cOMBS_2011279539nt_int,P),Q),R) = hAPP_i1584592887nt_int(hAPP_i1778081398nt_int(P,R),hAPP_int_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__014,axiom,
! [P,Q,R] : hAPP_P252494982t_bool(hAPP_f291087797t_bool(hAPP_f214467655t_bool(cOMBB_182850971nt_int,P),Q),R) = hAPP_i1948725293t_bool(P,hAPP_P1175774780nt_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HO,axiom,
! [P,Q,R] : hAPP_n215258509l_bool(hAPP_f66927821l_bool(hAPP_f2037783381l_bool(cOMBB_1146692694ol_nat,P),Q),R) = hAPP_n215258509l_bool(P,hAPP_nat_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
! [P,Q,R] : hAPP_i1948725293t_bool(hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,P),Q),R) = hAPP_f1805168059t_bool(P,hAPP_i1948725293t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod_Itc,axiom,
! [P,Q,R] : hAPP_i1524277240nt_int(hAPP_f1735705551nt_int(hAPP_f151095827nt_int(cOMBB_1389251822nt_int,P),Q),R) = hAPP_P1975530577nt_int(P,hAPP_i1524277240nt_int(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__015,axiom,
! [P,Q,R] : hAPP_i1948725293t_bool(hAPP_f1048215610t_bool(hAPP_f472159229t_bool(cOMBC_1683390479t_bool,P),Q),R) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
! [P,Q,R] : hAPP_i418383825t_bool(hAPP_f289318786t_bool(hAPP_f729588221t_bool(cOMBC_678732631t_bool,P),Q),R) = hAPP_f800510211t_bool(hAPP_i1796122964t_bool(P,R),Q) ).
fof(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
! [P,Q,R] : hAPP_i1948725293t_bool(hAPP_f428220345t_bool(hAPP_f1878066172t_bool(cOMBS_1720984575t_bool,P),Q),R) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(P,R),hAPP_i1948725293t_bool(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_,axiom,
! [P,Q,R] : hAPP_i1524277240nt_int(hAPP_f1735705551nt_int(hAPP_f713949180nt_int(cOMBS_1891347093nt_int,P),Q),R) = hAPP_P1975530577nt_int(hAPP_i1216375074nt_int(P,R),hAPP_i1524277240nt_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_016,axiom,
! [P,Q,R] : hAPP_P921862901l_bool(hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_P603027463t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__RealDef__Oreal_J_000tc__fun_,axiom,
! [P,Q,R] : hAPP_i1948725293t_bool(hAPP_f351634443t_bool(hAPP_f1576199059t_bool(cOMBB_95956018ol_int,P),Q),R) = hAPP_f429384717t_bool(P,hAPP_i627165055t_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun_,axiom,
! [P,Q,R] : hAPP_i418383825t_bool(hAPP_f1222178131t_bool(hAPP_f743790483t_bool(cOMBB_1647910634ol_int,P),Q),R) = hAPP_f737665877t_bool(P,hAPP_i157317923t_real(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__RealDef__Or,axiom,
! [P,Q,R] : hAPP_i627165055t_real(hAPP_f1323932638t_real(hAPP_f352226557t_real(cOMBC_824100275t_real,P),Q),R) = hAPP_f269654879t_real(hAPP_i1671359920t_real(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Or,axiom,
! [P,Q,R] : hAPP_i157317923t_real(hAPP_f1613846310t_real(hAPP_f1720068861t_real(cOMBC_295337211t_real,P),Q),R) = hAPP_f1888375463t_real(hAPP_i447943416t_real(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__Nat__O,axiom,
! [P,Q,R] : hAPP_r474017924t_real(hAPP_f1501741374t_real(hAPP_f1135648319t_real(cOMBC_1329014627t_real,P),Q),R) = hAPP_f1676298106t_real(hAPP_r50993370t_real(P,R),Q) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun__017,axiom,
! [P,Q,R] : hAPP_r474017924t_real(hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,P),Q),R) = hAPP_f1888375463t_real(P,hAPP_r474017924t_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_I,axiom,
! [P,Q,R] : hAPP_n682674106l_real(hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,P),Q),R) = hAPP_b646336293l_real(P,hAPP_nat_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__p,axiom,
! [P,Q,R] : hAPP_r2019696015l_real(hAPP_f1591894897l_real(hAPP_f2114902373l_real(cOMBB_237479429l_real,P),Q),R) = hAPP_r2019696015l_real(P,hAPP_real_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___018,axiom,
! [P,Q,R] : hAPP_i345030060t_bool(hAPP_f725770969t_bool(hAPP_f983569339t_bool(cOMBB_550562460ol_int,P),Q),R) = hAPP_f1791153283t_bool(P,hAPP_int_fun_int_int(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_019,axiom,
! [P,Q,R] : hAPP_i1584592887nt_int(hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,P),Q),R) = hAPP_f883232462nt_int(hAPP_i998473677nt_int(P,R),Q) ).
fof(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc__,axiom,
! [P,Q,R] : hAPP_r1195171167l_real(hAPP_f1736103445l_real(hAPP_f1908301369l_real(cOMBS_902912708l_real,P),Q),R) = hAPP_P1860904029l_real(hAPP_r938550413l_real(P,R),hAPP_r1195171167l_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__I,axiom,
! [P,Q,R] : hAPP_P1668131738nt_int(hAPP_f167782601nt_int(hAPP_f400538607nt_int(cOMBB_866192175nt_int,P),Q),R) = hAPP_i61245889nt_int(P,hAPP_P1175774780nt_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__020,axiom,
! [P,Q,R] : hAPP_i1823918693l_bool(hAPP_f1344929775l_bool(hAPP_f1830214065l_bool(cOMBB_793166024ol_int,P),Q),R) = hAPP_f2144054103l_bool(P,hAPP_i1948725293t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
! [P,Q,R] : hAPP_i614188481l_bool(hAPP_f252743847l_bool(hAPP_f643658017l_bool(cOMBB_2046938128ol_int,P),Q),R) = hAPP_f1146629647l_bool(P,hAPP_i418383825t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL_,axiom,
! [P,Q,R] : hAPP_r481142414t_bool(hAPP_f1625314454t_bool(hAPP_f340152631t_bool(cOMBB_1453147152l_real,P),Q),R) = hAPP_f2119767738t_bool(P,hAPP_r632163725t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Nat__Onat_Mtc__HOL_,axiom,
! [P,Q,R] : hAPP_r481142414t_bool(hAPP_f1341793266t_bool(hAPP_f571820015t_bool(cOMBB_1159313332l_real,P),Q),R) = hAPP_f41603166t_bool(P,hAPP_r373117745t_bool(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Int__Oint_Mtc__HOL__O,axiom,
! [P,Q,R] : hAPP_r632163725t_bool(hAPP_f1832844496t_bool(hAPP_f1990636069t_bool(cOMBC_412863505t_bool,P),Q),R) = hAPP_f1048215610t_bool(hAPP_r1623423338t_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__O_021,axiom,
! [P,Q,R] : hAPP_r373117745t_bool(hAPP_f1115375056t_bool(hAPP_f271976045t_bool(cOMBC_1801450329t_bool,P),Q),R) = hAPP_f289318786t_bool(hAPP_r653725362t_bool(P,R),Q) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun__022,axiom,
! [P,Q,R] : hAPP_r1409264866l_real(hAPP_f1523861863l_real(hAPP_f1067681227l_real(cOMBB_1934313333l_real,P),Q),R) = hAPP_f1509287679l_real(P,hAPP_r474017924t_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___023,axiom,
! [P,Q,R] : hAPP_i744251628nt_int(hAPP_f836602773nt_int(hAPP_f888249301nt_int(cOMBB_1875755114nt_int,P),Q),R) = hAPP_f747162069nt_int(P,hAPP_int_fun_int_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___024,axiom,
! [P,Q,R] : hAPP_i1778081398nt_int(hAPP_f2004554147nt_int(hAPP_f1228585167nt_int(cOMBB_1102189010nt_int,P),Q),R) = hAPP_f1574055885nt_int(P,hAPP_int_fun_int_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__prod_Itc__RealDef__Orea,axiom,
! [P,Q,R] : hAPP_r1134773055l_bool(hAPP_f1884005689l_bool(hAPP_f172451459l_bool(cOMBB_421479067l_real,P),Q),R) = hAPP_f330985925l_bool(P,hAPP_r2019696015l_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__025,axiom,
! [P,Q,R] : hAPP_i872162435t_bool(hAPP_f1251097615t_bool(hAPP_f1475822739t_bool(cOMBB_2027861294ol_int,P),Q),R) = hAPP_f1512942609t_bool(P,hAPP_i1948725293t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__026,axiom,
! [P,Q,R] : hAPP_i345030060t_bool(hAPP_f791698359t_bool(hAPP_f847817363t_bool(cOMBB_562416518ol_int,P),Q),R) = hAPP_f428220345t_bool(P,hAPP_i345030060t_bool(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__I,axiom,
! [P,Q,R] : hAPP_i1584592887nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,P),Q),R) = hAPP_f1735705551nt_int(hAPP_i1836591008nt_int(P,R),hAPP_i1584592887nt_int(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_,axiom,
! [P,Q,R] : hAPP_i1216375074nt_int(hAPP_P621635040nt_int(hAPP_f1582586579nt_int(cOMBC_707974837nt_int,P),Q),R) = hAPP_P1145851913nt_int(hAPP_i2104874254nt_int(P,R),Q) ).
fof(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint__027,axiom,
! [P,Q,R] : hAPP_i1216375074nt_int(hAPP_f1375227553nt_int(hAPP_f1311564564nt_int(cOMBS_1640193733nt_int,P),Q),R) = hAPP_P1145851913nt_int(hAPP_i2104874254nt_int(P,R),hAPP_i1524277240nt_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__fun_Itc,axiom,
! [P,Q,R] : hAPP_i1529485324t_bool(hAPP_f1315737299t_bool(hAPP_f538565589t_bool(cOMBB_1211207212ol_int,P),Q),R) = hAPP_f627970963t_bool(P,hAPP_i68813070l_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc_,axiom,
! [P,Q,R] : hAPP_r337325687l_real(hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,P),Q),R) = hAPP_f556082547l_real(P,hAPP_r1250527377l_real(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R,axiom,
! [P,Q,R] : hAPP_r2019696015l_real(hAPP_f1591894897l_real(hAPP_f644105375l_real(cOMBC_467724138l_real,P),Q),R) = hAPP_f617735571l_real(hAPP_r712953929l_real(P,R),Q) ).
fof(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R,axiom,
! [P,Q,R] : hAPP_r2019696015l_real(hAPP_f320596107l_real(hAPP_f850851693l_real(cOMBS_193881978l_real,P),Q),R) = hAPP_f617735571l_real(hAPP_r712953929l_real(P,R),hAPP_r1250527377l_real(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Int__Oint_Mtc__fun_It,axiom,
! [P,Q,R] : hAPP_r632163725t_bool(hAPP_f956464319t_bool(hAPP_f422375781t_bool(cOMBC_243356514t_bool,P),Q),R) = hAPP_f351634443t_bool(hAPP_r1392521531t_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Int__Oint_Mtc__fun_It_028,axiom,
! [P,Q,R] : hAPP_r373117745t_bool(hAPP_f1459420095t_bool(hAPP_f693283245t_bool(cOMBC_1113900202t_bool,P),Q),R) = hAPP_f1222178131t_bool(hAPP_r2052264067t_bool(P,R),Q) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun__029,axiom,
! [P,Q,R] : hAPP_r50993370t_real(hAPP_f1163894827t_real(hAPP_f1998755145t_real(cOMBB_421464275l_real,P),Q),R) = hAPP_f1566856189t_real(P,hAPP_r474017924t_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__fun,axiom,
! [P,Q,R] : hAPP_r847664445t_bool(hAPP_f1920544743t_bool(hAPP_f866672739t_bool(cOMBB_1247074793l_real,P),Q),R) = hAPP_f1173902867t_bool(P,hAPP_r1134773055l_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__fun_030,axiom,
! [P,Q,R] : hAPP_r279757445t_bool(hAPP_f1840085295t_bool(hAPP_f1083049571t_bool(cOMBB_2050079537l_real,P),Q),R) = hAPP_f605995867t_bool(P,hAPP_r1134773055l_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__031,axiom,
! [P,Q,R] : hAPP_i1671359920t_real(hAPP_f74147283t_real(hAPP_f1169087445t_real(cOMBB_2129825580al_int,P),Q),R) = hAPP_f766218899t_real(P,hAPP_i371647666l_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__032,axiom,
! [P,Q,R] : hAPP_i447943416t_real(hAPP_f1295083291t_real(hAPP_f212164053t_real(cOMBB_929094628al_int,P),Q),R) = hAPP_f1690286043t_real(P,hAPP_i371647666l_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc_,axiom,
! [P,Q,R] : hAPP_i1153444985nt_int(hAPP_f1148171384nt_int(hAPP_f334321603nt_int(cOMBB_1484610687nt_int,P),Q),R) = hAPP_P621635040nt_int(P,hAPP_i1524277240nt_int(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc_,axiom,
! [P,Q,R] : hAPP_P1668131738nt_int(hAPP_f167782601nt_int(hAPP_f1769710159nt_int(cOMBC_2037038898nt_int,P),Q),R) = hAPP_f521271395nt_int(hAPP_P752050971nt_int(P,R),Q) ).
fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__,axiom,
! [P,Q,R] : hAPP_i2104874254nt_int(hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,P),Q),R) = hAPP_b40753821nt_int(P,hAPP_int_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__HOL__Obool_Mtc__HOL,axiom,
! [P,Q,R] : hAPP_i1529485324t_bool(hAPP_f1545556668t_bool(hAPP_f85237525t_bool(cOMBB_389152643ol_int,P),Q),R) = hAPP_f727283836t_bool(P,hAPP_i1823918693l_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__HOL__Obool_Mtc__HOL,axiom,
! [P,Q,R] : hAPP_i1796122964t_bool(hAPP_f1902329768t_bool(hAPP_f1182713365t_bool(cOMBB_2087181719ol_int,P),Q),R) = hAPP_f561022312t_bool(P,hAPP_i614188481l_bool(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc_,axiom,
! [P,Q,R] : hAPP_P989584703t_bool(hAPP_f1187443546t_bool(hAPP_f11709033t_bool(cOMBS_1676841879t_bool,P),Q),R) = hAPP_f892584630t_bool(hAPP_P1400155668t_bool(P,R),hAPP_P1668131738nt_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__H,axiom,
! [P,Q,R] : hAPP_P989584703t_bool(hAPP_f1781059817t_bool(hAPP_f326728547t_bool(cOMBB_2086486755nt_int,P),Q),R) = hAPP_f237669397t_bool(P,hAPP_P989584703t_bool(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__033,axiom,
! [P,Q,R] : hAPP_P989584703t_bool(hAPP_f1781059817t_bool(hAPP_f125837377t_bool(cOMBS_1610675060t_bool,P),Q),R) = hAPP_f237669397t_bool(hAPP_P877703507t_bool(P,R),hAPP_P989584703t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__034,axiom,
! [P,Q,R] : hAPP_r1150855451l_bool(hAPP_f381769257l_bool(hAPP_f512004221l_bool(cOMBB_188663909l_real,P),Q),R) = hAPP_f1344929775l_bool(P,hAPP_r632163725t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Nat__Onat_Mtc__HOL__035,axiom,
! [P,Q,R] : hAPP_r285062775l_bool(hAPP_f834119977l_bool(hAPP_f848516149l_bool(cOMBB_270132509l_real,P),Q),R) = hAPP_f252743847l_bool(P,hAPP_r373117745t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__RealDef__Oreal,axiom,
! [P,Q,R] : hAPP_r337325687l_real(hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,P),Q),R) = hAPP_f203520653l_real(P,hAPP_r337325687l_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__036,axiom,
! [P,Q,R] : hAPP_r579619957l_real(hAPP_f823702159l_real(hAPP_f1040184293l_real(cOMBB_557258983l_real,P),Q),R) = hAPP_f1591894897l_real(P,hAPP_r1250527377l_real(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__p,axiom,
! [P,Q,R] : hAPP_r2019696015l_real(hAPP_f2072293513l_real(hAPP_f1763467369l_real(cOMBS_163651580l_real,P),Q),R) = hAPP_f1736103445l_real(hAPP_r224952523l_real(P,R),hAPP_r2019696015l_real(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc___037,axiom,
! [P,Q,R] : hAPP_r938550413l_real(hAPP_f1293550189l_real(hAPP_f183342685l_real(cOMBS_794971624l_real,P),Q),R) = hAPP_P1982799375l_real(hAPP_r931665397l_real(P,R),hAPP_r1195171167l_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__prod,axiom,
! [P,Q,R] : hAPP_i1778081398nt_int(hAPP_f2094497995nt_int(hAPP_f534361363nt_int(cOMBB_1657320178nt_int,P),Q),R) = hAPP_f465821005nt_int(P,hAPP_i1778081398nt_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int,axiom,
! [P,Q,R] : hAPP_i998473677nt_int(hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,P),Q),R) = hAPP_f2002277285nt_int(P,hAPP_i1584592887nt_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int_038,axiom,
! [P,Q,R] : hAPP_i1153444985nt_int(hAPP_f55007289nt_int(hAPP_f747475781nt_int(cOMBB_222539774nt_int,P),Q),R) = hAPP_f1375227553nt_int(P,hAPP_i1584592887nt_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__I,axiom,
! [P,Q,R] : hAPP_P2053772734t_bool(hAPP_f1198736473t_bool(hAPP_f2042071795t_bool(cOMBB_773384995nt_int,P),Q),R) = hAPP_f291087797t_bool(P,hAPP_P1668131738nt_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__prod_Itc__RealDef__Oreal_,axiom,
! [P,Q,R] : hAPP_r931665397l_real(hAPP_f1607917947l_real(hAPP_f1076489843l_real(cOMBB_415306361l_real,P),Q),R) = hAPP_b1482221219l_real(P,hAPP_real_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___039,axiom,
! [P,Q,R] : hAPP_P752050971nt_int(hAPP_f301246617nt_int(hAPP_f1151807883nt_int(cOMBB_563965291nt_int,P),Q),R) = hAPP_f11151005nt_int(P,hAPP_P2110489235nt_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__fun_Itc_040,axiom,
! [P,Q,R] : hAPP_P877703507t_bool(hAPP_f1965248563t_bool(hAPP_f1273270095t_bool(cOMBB_595786433nt_int,P),Q),R) = hAPP_f513299955t_bool(P,hAPP_P921862901l_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__fun_,axiom,
! [P,Q,R] : hAPP_r1962736578t_bool(hAPP_f1063001928t_bool(hAPP_f2050994651t_bool(cOMBB_2138578194l_real,P),Q),R) = hAPP_f1545556668t_bool(P,hAPP_r1150855451l_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Nat__Onat_Mtc__fun_,axiom,
! [P,Q,R] : hAPP_r1562208522t_bool(hAPP_f2084827628t_bool(hAPP_f1121609875t_bool(cOMBB_2125215742l_real,P),Q),R) = hAPP_f1902329768t_bool(P,hAPP_r285062775l_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__prod_Itc__RealDef__Orea_041,axiom,
! [P,Q,R] : hAPP_r712953929l_real(hAPP_f1119546819l_real(hAPP_f555557763l_real(cOMBB_422423973l_real,P),Q),R) = hAPP_f800361423l_real(P,hAPP_r2019696015l_real(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__p_042,axiom,
! [P,Q,R] : hAPP_r1230691443l_real(hAPP_f324002831l_real(hAPP_f677014621l_real(cOMBS_107495110l_real,P),Q),R) = hAPP_f1293550189l_real(hAPP_r1043989325l_real(P,R),hAPP_r2019696015l_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__p,axiom,
! [P,Q,R] : hAPP_i1836591008nt_int(hAPP_f400200275nt_int(hAPP_f1731308757nt_int(cOMBB_735549356nt_int,P),Q),R) = hAPP_f151095827nt_int(P,hAPP_i1216375074nt_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__fun_Itc__Int__Oint_,axiom,
! [P,Q,R] : hAPP_r1623423338t_bool(hAPP_f704188951t_bool(hAPP_f1456711209t_bool(cOMBB_690122963l_real,P),Q),R) = hAPP_f472159229t_bool(P,hAPP_r1962736578t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__fun_Itc__Nat__Onat_,axiom,
! [P,Q,R] : hAPP_r653725362t_bool(hAPP_f1927674023t_bool(hAPP_f1261108921t_bool(cOMBB_172850899l_real,P),Q),R) = hAPP_f729588221t_bool(P,hAPP_r1562208522t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__RealDef__Oreal_J_Mt,axiom,
! [P,Q,R] : hAPP_r1392521531t_bool(hAPP_f332806115t_bool(hAPP_f1177545951t_bool(cOMBB_1459832681l_real,P),Q),R) = hAPP_f1576199059t_bool(P,hAPP_r847664445t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_Mt,axiom,
! [P,Q,R] : hAPP_r2052264067t_bool(hAPP_f77155443t_bool(hAPP_f403082607t_bool(cOMBB_984071017l_real,P),Q),R) = hAPP_f743790483t_bool(P,hAPP_r279757445t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__fun_043,axiom,
! [P,Q,R] : hAPP_r794344869l_real(hAPP_f517738447l_real(hAPP_f1879982819l_real(cOMBB_731595089l_real,P),Q),R) = hAPP_f1607917947l_real(P,hAPP_r1134773055l_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__RealDef__Oreal_044,axiom,
! [P,Q,R] : hAPP_r712953929l_real(hAPP_f1108539711l_real(hAPP_f1109256153l_real(cOMBB_44084907l_real,P),Q),R) = hAPP_f1730463541l_real(P,hAPP_r579619957l_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__prod_Itc__Int__Oint,axiom,
! [P,Q,R] : hAPP_i1836591008nt_int(hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,P),Q),R) = hAPP_f713949180nt_int(P,hAPP_i1153444985nt_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__f,axiom,
! [P,Q,R] : hAPP_P1400155668t_bool(hAPP_f223623489t_bool(hAPP_f1815301987t_bool(cOMBB_650041899nt_int,P),Q),R) = hAPP_f1325804733t_bool(P,hAPP_P2053772734t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__prod_Itc__Real,axiom,
! [P,Q,R] : hAPP_r224952523l_real(hAPP_f1663701695l_real(hAPP_f774948957l_real(cOMBB_3698607l_real,P),Q),R) = hAPP_f1908301369l_real(P,hAPP_r1230691443l_real(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__prod_Itc__Real_045,axiom,
! [P,Q,R] : hAPP_r1043989325l_real(hAPP_f1161935955l_real(hAPP_f328572565l_real(cOMBB_520243667l_real,P),Q),R) = hAPP_f183342685l_real(P,hAPP_r794344869l_real(Q,R)) ).
%----Conjectures (1)
fof(conj_0,conjecture,
? [X,Y] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int) ).
%------------------------------------------------------------------------------